he ee ed ee ed et er Oe “, ee eee ee? τὰν, Ὁ “σα ΣΥῸΣ Satan ares τ ee I ps he Ἢ : bes ὁ ον ὁρῦ. ee He OB, see τους τε ρον Ὧι + eee ὩΣ cea τ γί χες » 08 oe Sees ee -- > tae Oe ot Wire ae eee a A ey ‘at eae ONE ν τ; πῶ Δ᾽ ὁ ἐν * #29 oe a4 ee +4? © 6 + ὴ ὁ ee oe > “ 7 Ἕ » > = ᾷ P . r - ᾿ - we ᾿ πεν : ae ae Ως -t "8 Pai oto © ew Be ΕΣ » * re “ nC woe ow Vee eve * * a Aus ? Ἐν 2" Nace a+ v,v,F,**. ἐ νυν Hee eet se i pititi ; af εἰ γί δε οι fincdeytysitelersierat ereris ΤΉ ΤΣ ι ἷ opt taped taht yee of f δα ἘΝ Ὶ ἡ Ἢ i ib ΤῊΝ i AE i ci ea ait eh ie mah wy PROCEEDINGS OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES. Vou. XLVIII. FROM MAY 1912, TO MAY 1913. BOS TON: PUBLISHED BY THE ACADEMY 1913 The Cosmos Press EDW. W. WHEELER CAMBRIDGE, MASS. || CONTENTS. PAGE On the Ultra Violet Component in Artificial Light. By Louis BELL : Ἐν Sb On eS Ἐς 1 Alexander Agassiz. By Henry Ρ. Waucotr 31 A Theory of Linear Distance and Angle. By H. B. PxHtnures and C. L. E. Moors Serta aes τ 3) Preliminary Diagnoses of New Species of Chaetomium. By A. H. CHIVERS 81 A Study with the Echelon Spectroscope of Certain Lines in the Spectra of the Zinc Arc and Spark at Atmospheric Pressure. By Norton A. Kent 91 The Impedance of Telephone Receivers as affected by the Motion of their Diaphragms. By A. E. KennEtLyY and G.W. Pierce 111 New or Critical Laboulbeniales from the Argentine. By Routanp THAXTER - 155 Culture Studies of Fungi producing Bulbils and Similar Propaga- tive Bodies. By J. W. Hotson + 225 Thermodynamic Properties of Liquid Water to 80° and 12000 Kgm. By P. W. BripGMan Preliminary Descriptions of New Species of Rickia and Treno- myces. By Rontanp THAXTER The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics. By EH. B. Witson and ἃ. N. Lewis : On the Existence and Properties of the Ether. By D.L. WEBSTER The History, Comparative Anatomy and Evolution of the Arau- carioxylon Type. By E. C. Jerrrey iv CONTENTS. XIV. The Action of Sulphur Trioxide on Silicon Tetrachloride. By C. R. Sancer and E. R. Rirceu ot 3 ἡ εν een OS XV. An Electric Heater and Automatic Thermostat. By A. L.CLuarK 597 XVI. Cretaceous Pityoxyia from Cliffwood, New Jersey. By Rutu FGEDENS em) τα Bole he ices eee ey Ubi ρον mL XVII. On the Scalar Functions of Hyper Complex Numbers. By ἘΠΕ ΝΕ YGe GRABER: 91S oe Wa ed ets PLS Doty ΣΉ 099 XVIII. Preliminary Study of the Salinity ue Sea Water in the Bermudas. ihe ἼἼ | By ΠῚ Marx . Bue eee ail eee Unie ee XIX. On Certain Fragments of the Pre-Socratics: Critical Notes and Hiuctdations. By W.:A.. HEIDEL: τ. (0 4. 40 a) ee ῸΠ XX. The Structure of the Gorgonian Coral Pseudoplexaura crassa Wright and Studer, “By W: M. Ztschr. f. Augenheilk., July 1908, and elsewhere. BELL. —~ ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. 3) injuries ascribed to ultra violet rays, like cataract and retinal degen- eration, are charged to the radiations running even up to the visible spectrum, while the extreme ultra violet, absorbed by the cornea, produces only superficial lesions generally recovered in a few days. From the standpoint of the present investigation it did not seem justifiable to attempt to pass without further investigation on the validity of any of the divergent views here noted, but to deal with the radiations of short wave length as a whole, including in the possibly injurious group all those radiations which have been under serious suspicion on clinical evidence by reputable investigators. The line has therefore been drawn between the ordinary lighting radiations and radiations of short wave length in the extreme violet and ultra violet of the spectrum, where the lighting value of the rays is negli- gible and their actinic value notably high. Separation of the Ultra Violet from the Visible Spectrum. — Having determined on such a separation of the radiations under grave sus- picion of injurious action from the rest of the spectrum, it was next in order to find a suitable screen for making just this division of the spectrum, so that it would be possible to measure the energy in the two portions of the spectrum directly and as a whole, without a resort to the extremely difficult and troublesome measures of the energy in separate spectrum lines, a task of great delicacy when discontinuous have to be compared with continuous spectra. After considerable investigation a suitable medium was found in the so- called Euphos glass. This glass, which has been strongly recom- mended by Schanz and Stockhausen as eliminating completely all the harmful ‘rays and which was prepared under the direction of one of them, cuts off the ultra violet spectrum with remarkable definiteness while showing relatively little absorption of the general luminous rays. Plate 1, b, c, d, shows the nature of this absorption very clearly. Spectrogram ᾧ of this Plate is the spectrum of the mercury quartz arc put on merely for reference, the group at 365 wu being at the right of the figure and the brilliant green line exactly in the centre of the plate. Spectrogram c shows the spectrum of the magnetite are which is very rich in the ultra violet and d shows the same as absorbed by a Euphos glass screen 2 mm. thick. The exposure in each case was one minute with a rather wide slit and a very brilliant grating. The cut off of the shorter wave lengths by the Euphos glass in the ultra violet is very clean and sudden at wave length 390 uu, practically just at the end of the visible spectrum as seen by the average eye. The 4 PROCEEDINGS OF THE AMERICAN ACADEMY. absorption continues slightly on into the violet, gradually fading away until the transmission becomes nearly complete for the bright blue mercury line (4385 pu). In examining b, c and d of Plate 1 it must be remembered that the second order ultra violet overlaps the first order so that the group near 365 wu appears in the first order at the extreme right of the figure and in the second order at the extreme left. In d of this Plate the arc spectrum fades off on the left, not from absorption but from the weakening of the photographic action. The Euphos glass is ex- tremely transparent to the radiations throughout all except the ex- treme violet of the visible spectrum, and well into the infra red, as will hereafter be seen. The results here obtained for its absorption of the ultra violet are altogether parallel with those shown in the paper by Schanz and Stockhausen ® and also by Hallauer.?. The Euphos glass thus enables a particularly clean partition of the visible spectrum from the ultra violet and extreme violet to be made. If it were possible to obtain an equally good absorbent for separat- ing the infra red from the visible spectrum radiometric measurements of efficiency would be greatly facilitated. It should here be noted that Euphos glass appears in various shades and some imitations of it are now upon the market, so that a sample of such glass should be tested in the spectrograph before use for such a purpose as the present, inasmuch as in some of the shades the cut-off of the ultra violet is much less sharp and complete. The sample here used was the original No. 1, 2 mm. thick. Method of Investigation. —'The method taken for the evaluation was the familiar one of measuring the radiation directly by means of a thermopile connected with a sensitive galvanometer in a manner familiar in recent experiments on the efficiency of illuminants in the visible spectrum, 6. g., Lux,® Féry.2 The thermopile was chosen as the radiometric instrument merely as a matter of convenience. The instrument actually used was a Rubens linear thermopile, having 20 constantin-iron couples with a total resistance of 4.6 ohms. It was mounted as shown in Figure 1, in a vacuum tube with a quartz window immediately in front of the couples. The inner body of the instru- ment, containing the couples, was taken out of its original mounting and set up in a tube about 37 mm. in diameter, through the upper end of which was sealed a pair of leading-in wires. 6 Zts. f. Augenheilk., May 1910, Table VII, figure 3. 7 Archiv. of Ophthal., Jan. 1910, Plate I, figure 3. 8. Zts. f. Beleuchtungswesen, Heft 16, 1 p. 36, 1907. ® Bull. Soc. Franc. de Physique, p. 148, 1908. BELL. — ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. 5 These were firmly clamped in the binding posts of the instrument by working through the side tube attached for the reception of the quartz window. The thermopile was then pushed up exactly opposite the side tube and wedged in place with cork and cotton wool attached with shellac. The end of the side tube was flanged out and ground flat for the fitting of the quartz window and after the shellac had dried out thoroughly the window was fastened in place and the lower end of the tube drawn out for the attachment of the pump. The tube was pumped to the high vacuum usual in an X-ray tube, and was then sealed. It was mounted as shown in a block of wood to which was secured the disconnecting terminal, reached by a long handled plug, Figure 1. Vacuum thermopile. Figure 2. Quartz cell. and the whole was then surrounded by a pasteboard case having a hole just opposite the quartz window, and packed full with loose cotton wool. The galvanometer was of the D’Arsonval type, having a sensibility of 210° ampere per mm. scale deflection. Its period for the attainment of a complete deflection, was, under the ordinary conditions of its use, 1 minute. The galvanometer deflections were read by a scale and telescope, the scale being a special one bent to 1.5 meters radius. The thermo- pile indications were calibrated in absolute measure by observations 6 PROCEEDINGS OF THE AMERICAN ACADEMY. on the radiation of a standard incandescent lamp supplied by the Bureau of Standards. After applying the proper correction for stray thermal losses and spherical reduction factor and reducing the read- ings as taken to the standard distance of 50 cm. employed throughout this investigation, the constant of the thermopile galvanometer system was found to be 1 mm. = 1 scale division = 35.3 ergs per second per square em. By this constant the observed deviations were reduced to absolute dynamical measure. As a matter of convenience and to establish an approximate ratio between the ultra violet radiation from the various sources studied and the radiation in the visible spectrum, an absorption cell which 8 Transmission i 500 700 900 1100 1300 1500 μμ Figure 3. Absorption curve of water. eliminated nearly all the infra red was kept in front of the thermopile window. This cell, Figure 2, was of glass, ground flat and exactly 1 em. thick, 44 mm. external diameter and 35 mm. internal diameter. The glass ring was provided with a hole for filling and was closed by two quartz plates cut across the axis, each 2.25 mm. thick and 44 mm. diameter. These were fastened with hard shellac to the glass cell, and the cell in use was filled with distilled water. The absorption of a layer of distilled water of this thickness is shown in Figure 3 taken from Nichols’s experiments.’ Quartz has no material absorption in the part of the infra red spectrum transmitted and neither quartz nor 10 Nichols, Physical Review, Vol. 1, p. 1. BELL. — ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. 7 distilled water in this thickness has any material absorption in even the extreme ultra violet up to the limit investigated. The use of this cell therefore could produce no sensible effect on the accuracy of the ultra violet measurements, while it did serve the extremely useful purpose of limiting the total amount of energy to be measured and of eliminating any difficulties that might arise owing to absorption in the further part of the infra red, all the absorbing media incidentally used being, as compared with water, practically entirely transparent to all the radiations that got through the water cell. It would have been convenient if some substance cutting off the infra red sharply at 750 uu or 800 μμ had been available. Unfortu- nately, there is no such substance, so far as has yet been discovered, the very few substances less transparent than water in the region 800 to 1300 μμ being useless for the purpose of this investigation on account of opacity in the ultra violet and generally in the visible spectrum as well. Iron ammonium alum used by Lux (loc. cit.) and the copper salts used by Féry (loc. cit.) are open to this objection and the same is true of all the otherwise useful and promising sub- stances discussed in the very thorough and valuable researches of Coblentz." In some of the experiments a second similar quartz cell was used, particularly in work on are lamps. In this case the Euphos glass used to cut out the ultra violet portion of the spectrum was perma- nently affixed to one of these cells and either the plain quartz cell or the Euphos-quartz cell was thrust into the beam so as quickly to get differential readings. In order to avoid the somewhat large correc- tion due to reflection of energy which would have been produced by the introduction of a plain slip of Euphos glass to cut out the ultra violet the following expedient was adopted. The Euphos glass was attached to the surface of the quartz cell by spring clips with the addition of a thin capillary film of pure glycerine between the quartz and glass surface. Glycerine is immensely trans- parent to all radiations, including the extreme ultra violet, to which Canada balsam and gelatine are quite opaque. Its index of refrac- tion, 1.47 for D, is sufficiently near that for quartz and the various glasses to reduce the loss of light at the surfaces to an entirely negligi- ble amount. As the Euphos has a slightly less index of refraction than quartz, there was a minute residual gain in the total transmis- sion of the system when the Euphos glass was added, in the right direction to compensate for the minute losses by absorption in the glycerine film. 1 Bull. Bureau of Standards, Vol. 2, p. 619. 8 PROCEEDINGS OF THE AMERICAN ACADEMY. As a check on the possible magnitude of this virtual absorption by the glycerine film readings were taken on a tungsten lamp through the quartz cell alone, and through the quartz cell plus a disc of optical crown glass 2 mm. thick secured with glycerine in the ordinary man- ner. The absorption of this crown glass is shown in Plate 1, e, f, g, in which e is the spectrogram of the quartz arc taken with a wide slit and 2 minutes exposure, f the spectrogram through the crown glass in question, and g through the Euphos glass. In spite of the fact that there is a slight absorption by the crown glass in the region near 300 up, the addition of the crown glass and glycerine film reduced the galvanometer deflection by barely 0.5 %, an amount scarcely out- side the errors of observation. ‘The energy cut off from the spectrum of a tungsten lamp by the crown glass would be of course very _ small, but perhaps not negligible, since as Schanz and Stockhausen have shown (loc. cit. table VIII, figure 6) the tungsten lamp spectrum goes quite down to 300 μμ in sufficient strength to give a clear photo- graphic effect. At all events it is evident that the use of the glycerine film involves no material errors. In the ordinary experimentation in using steady sources, sets of readings were taken alternately with and without the Euphos glass, the glass being either added to the clear cell with the glycerine film, or removed and the film quickly washed away with distilled water. With sources which give trouble from unsteadiness the second quartz cell was brought into play as previously mentioned. Aside from a slight drifting of the zero point, which is generally observable in measurements with a thermopile, the method adopted worked very smoothly. The drift, however, was usually small and slow and satis- factorily taken care of by a time correction. With proper attention to this, the readings, although necessarily slow, were nearly as consis- tent as would be found in ordinary photometric measurements. The following string of deflections forming a single group of 5 readings is typical of those obtained under ordinary conditions. Scale readings from bare quartz lamp through quartz cell only. cm. 36.17 36.10 36 .27 36 .36 36 .16 Av. = 36.21 BELL. ——~ ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. 9 The mean departure of a single reading from the average here given is slightly less than 4%, so that the errors of observation, of which this is a fair sample, showed that the thermopile observations are about as reliable as those with a photometer. Some preliminary experiments made on Euphos and other glasses showed that the transmission of the Euphoa glass aside from its absorption in the violet and ultra violet was exceptionally high for such rays as got through the layer of distilled water. In fact the total transmission of energy with Euphos glass was greater than with the ordinary samples of clear glass and was only exceeded by a single sample of optical crown which showed extraordinary transparency to all these radiations, so great that the losses were practically only those charge- able to actual reflection at the surfaces. Measurements on various Illwminants. — With these preliminaries the apparatus was set up permanently and work begun on commercial illuminants. Readings of current and voltage on the electric lamps were taken by Weston instruments freshly calibrated, and the quantity readings on the gas lamps tested were obtained from a newly adjusted standard meter. 100 Watt Tungsten The first source of light investigated was an ordinary 100 watt tungsten lamp, taking actually .951 amperes at 113 volts, i. e. 103.38 watts, and giving 79.4 ο. p. in the direction of the thermopile. With this lamp the mean difference of deflection due to energy cut off by the Euphos glass was 1.9 em. The ultra violet energy cut off, including such losses in the extreme violet as are indi- cated by Plate 1,d, was 6% of the total energy transmitted by the quartz cell. 100 Watt Gem. — The second source studied was anordinary 100 watt Gem lamp, taking 100 watts at 114 volts and giving in the marked direction 39.25 c. p. This lamp of course gave a spectrum relatively weak in the ultra violet, but as will be seen from its spectrogram in Plate 2,6, the ultra violet region down to wave length 330 μμ is by no means negligible. The total differential deflection due to the ultra violet was in this case only 0.61 em., 2.6% of the total deflection. These readings confirm the extraordinarily small absorption of Euphos glass throughout the longer wave lengths, since the transmission ob- served with the known cut off of a very perceptible amount in the ultra violet, leaves no room for any material selective or general absorption elsewhere. It should here be noted that while quartz transmits with extraordi- nary freedom, so far as absorption is concerned, all rays which are 10 PROCEEDINGS OF THE AMERICAN ACADEMY. allowed to pass by a cm. thickness of distilled water, it still exercises a slight selective action by reflection. The index of refraction of quartz for the longer wave lengths of the visible spectrum is 1.54, while for rays in the further ultra violet this figure rises to about 1.6, hence in accordance with Fresnel’s formula (2>;) 2 there is a small amount of selective stopping of the ultra violet rays by reflection. This occurs both at the quartz water cell and at the quartz window in front of the thermopile so that the total selective effect is proportional to the fourth power of the difference due to the change in the index of re- fraction for a single surface of transmission. This difference amounts to approximately 2% as between the red rays and the further part of the ultra violet. The result is to cause a slight under estimation of the ultra violet. No account has been taken in any of these experi- ments of this very small and troublesome correction, which amounts in ordinary cases to only a small fraction of 1% of the total ultra violet. The existence of the effect should, however, be noted as it has a tendency toward causing a slight under estimate rather than an over estimate of the ultra violet component. Cooper Hewitt Tube.—'The next source investigated was the Cooper Hewitt tube. One of the ordinary commercial 22 inch tubes was used, the particular tube having previously been used in another research and very carefully photometered. A section of this tube, giving 100 c. p., was screened off so that the length might be so re- duced that the energy from the whole,section taken could fall freely upon the thermopile without causing a material angular error or forcing one to depart widely from the standard distance of 0.5 meter. The horizontal radiation normal to the tube was of course taken, the reflector being removed. The corrected deflection due to the ultra violet amounted to 1.64 em. which corresponded to 41.7 % of the total energy passing through the quartz cell. The lamp was singularly steady and easy to work with, with the exception of producing an inconveniently small total deflection. The result, however, can be regarded as fairly precise in spite of the small magnitude, the mean deviation of a single reading amounting to barely over .5% in the total deflection. In this lamp the ultra violet energy is nearly all between 365 μμ and the visible spectrum, the extreme ultra violet being entirely cut off by the glass of the tube and the few lines of wave length between 365 and 300 μμ being reduced by the absorption to very feeble intensity. The total deflection produced by this lamp, of which the portion exposed radiated 100 ec. p., was only 17 % of the deflection given by the Gem lamp of the previous experiment, which gave less than 40 ec. p. BELL. — ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. ΠῚ Quartz Mercury Lamp. — Following the examination of the ordi- nary glass Copper Hewitt tube, the next source investigated was the quartz mercury lamp. Two tubes were available, each of the ordi- nary commercial 220 volt type rated at 3.5 amperes. One of these tubes, which is here referred to as the old mercury lamp, was made by the French Cooper-Hewitt Company and_had been already used for experimental purposes for about a year and had seen rather hard service, having often been worked above its rated amperage. The second lamp was entirely new, made in the Cooper-Hewitt factory in this country and was not at any time worked above its rating. The spectrum of the quartz lamp is extremely rich in certain portions of the ultra violet, particularly in rays of wave length less than 300 μμ. It is well shown in Spectrum e of Plate 1. The brilliant lines in this spectrum, counting from the violet, have wave lengths as follows: — 4077 .84 2967 .27 4046 .55 2925 .38 3983 .96 2893 .60 3906 .47 2752 .80 3663 .27 2698 .88 3662.88 | 2655 .14 | 3654 .83 2653 .70 } 3650 .14 2652 .07 | 3341 .48 2536 .52 3131 84] 2483 87 3131.56 } 2482 76 3125 .67 2482.07 3027 .49 2399 .81 3025.61 | 2399 .43 3023 .43 2378 .39 3021 .50 2302 .65 The wave lengths here are taken at the value assigned by Stiles ” in A. u. It willbe observed that a number of the lines are associated in close groups which with small dispersion mass into heavy lines. The relative intensity of the lines, as is well known, shifts consid- erably with the degree of excitation of the tube, so that the relative intensities given by Stiles do not agree with the spectrograms taken from the quartz arc for the same reason that Stiles’ arc and spark intensities do not agree. The quartz arc spectrum resembles Stiles’ are spectrum much more closely than it does the spark spectrum. 12 Astrophysical Journ., Vol. XXX, p. 48. 12 PROCEEDINGS OF THE AMERICAN ACADEMY. In particular the quartz are spectrum displays a very striking gap between wave length 334.14 μμ and the double line at wave length 313.1 up. Save for the very faint haze of continuous spectrum that characterizes the radiation from the quartz tube this part of the spectrum is blank. Indeed the line 334.14 uy itself is far from strong relatively to those in the further part of the ultra violet and there is. very little effect of radiation between wave length 313.1 μμ and 365.2 μμ. This gap is of some significance in interpreting the results: of bactericidal experiments, since any failure of bactericidal action in the region between wave length 350 wu and wave length 313 μμ observed in working with the quartz lamp may be due to the absence of any strong radiation in this region as well as to lack of specific bactericidal power in rays of this particular wave length if they existed. In the radiometric investigations on the old quartz lamp it was run at 3.7 amperes and about 80 volts, an average of about 260 watts, without an external globe. Under these circumstances the corrected deflection due to the total ultra violet was 16.7 cm. The deflections: were not quite so steady as in the case of the ordinary Cooper Hewitt tube, but still the average departure of a single reading was within 1%. After the deflection due to the total ultra violet was determined another set of readings was taken with the bare lamp and quartz cell and then with the Euphos glass replaced by the crown glass: of which the absorption spectrum is shown at f, Plate 1. This glass in effect cuts off substantially the whole of the extreme ultra violet spectrum, letting pass in practically undiminished strength only the lines of greater wave length than 300 wu. This separation is of some importance with respect to the bactericidal power of the lamp in water purification and similar work. The result was to show that the transmission of the crown glass was 54.7 % of the transmission found for the Euphos glass. In other words, nearly one half of the total ultra violet energy in this lamp was of wave length below 300 yu. Of the remaining half the spectrum shows, as just indicated, that by all odds the larger part lies between 365 μμ and the visible spectrum. The new quartz lamp without its globe was then tested, the input in this case being 350 watts. The ultra violet output was greater than in the old tube, the total deflection reduced to the standard distance rising to 32.1 cm. In this case 65.1 % of the energy trans- mitted by the quartz water cell was cut off by the Euphos glass. Following up the radiometric measurement further, the Euphos glass was replaced by the light crown glass as before with the result of showing that substantially one half, 49.9 %, of the total ultra violet BELL. — ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. i153 was cut off by the crown glass and hence substantially this proportion was of wave length less than 300 wu. In running quartz lamps without their globes, as was done in these experiments, the energy output is considerably diminished by the cooling of the tube and the light-giving properties of the lamp are very much reduced. Both the old and the new quartz lamps herein noted were photometered. The lamps were compared against a tungsten secondary standard by means of a Simmance-Abady flicker photo- meter. Thee. p. normal to the length of the tube and in a horizontal direction, was for the old quartz lamp 415, for the new quartz lamp 348, in each case without any enclosing globe. Both lamps were very steady and easy to work with, both on the photometer bar and with the thermopile. Finally the new quartz lamp was fitted with its regular diffusing globe and tested with the thermopile. In working with the globe the tube operated at a higher temperature and far more intensively, the wattage rising to 400. With the Euphos glass in, the total change in deflection amounted to only 3.7 em. although the lamp tested on the photometer as in the previous case reached 820 c. p. in the hori- zontal direction. In percentage the amount of energy cut off by the Euphos glass was 42.5. These figures plainly indicate that the globe absorbed the further ultra violet very strongly, more strongly than the crown glass already referred to. In fact the deflection due to the ultra violet energy which passed through the globe of the lamp was extraordinarily small with respect to the ec. p. of the source, very much smaller than in the case of any other illuminant investigated. With- out the globe the quartz are is a very powerful source of radiation in the extreme ultra violet, below wave length 300 uu. With its ordinary globe on, all this energy in the extreme ultra violet is cut off and the small remaining amount, mostly in that part of the ultra violet nearest the visible spectrum, becomes quite insignificant. The Welsbach Mantle— At this point study of the radiation from the Welsbach light was taken up. The particular form used was a Graetzin street lamp with a single large inverted mantle fitted with a clear glass globe, which obviously eliminated whatever of extreme ultra violet might be present. This burner took 6.4 feet of gas per hour at 3 inches pressure and gave 75 c. p. in the horizontal direction. Its total deflection was slightly greater than that produced by the quartz lamp with its globe tested immediately before. The addition of the Euphos glass cut down the deflection by .924 c. m., an amount equiva- lent to the absorption of 8.4 % of the total radiation recorded. The 14 PROCEEDINGS OF THE AMERICAN ACADEMY. lamp proved fairly easy to work with in point of steadiness and the average variation of a single deflection from the mean was still less than 1%. Acetylene Flame. — Following the trial of the Graetzin lamp a series of measurements was made on an acetylene flame fed from a Prestolite tank. This flame gave on the photometer in the direction of measurement 27.35 c. p. and its change in deflection on interposi- tion of the Euphos was .524 cm., corresponding to a cut off of 4.5% of the total energy. It proved very amenable to measurements and was quite as steady and easy to work with as the mantle burner pre- viously used. The spectrum of the acetylene flame reaches well down into the ultra violet as shown by Schanz and Stockhausen." It reaches, in fact, approximately wave length 310 wu, but the further portion of the spectrum is comparatively weak. The spectrum of the Welsbach mantle with a clear globe, given by the same authorities (loc. cit.), is cut off at about wave length 320 μμ, but is notably bright in the part of the ultra violet toward the visible spectrum. These results are fully checked by the spectrograms taken of the particular burners here indicated. The Carbon Electric Arc. — Next in order the various are lamps were taken up for investigation, beginning with the are between carbon electrodes. On account of the relative instability of the ares the method of experimentation was modified. A second quartz cell similar to the one already in use was constructed and filled with distilled water. The ratio of the absorption between this new cell and the old cell was then determined. From a slight difference in thickness or in polish of the quartz plates the new cell was found to give about 1% more absorption than the original quartz cell and a correction for this difference was introduced in the subsequent meas- urements. The two quartz cells were mounted in recesses in a sliding screen so that either could be brought quickly in front of the thermo- pile window. The Euphos glass screen was then mounted with a glycerine film on one of the quartz cells so that the cells with and without the Euphos could be rapidly interchanged in the beam from the lamp under test and the absorption thus determined without having to depend on the constancy of the lamp for any considerable time. The times of observation were regulated by means of a stop watch so that a time correction for shift of zero could be readily made, and 13 Zts. f. Augenheilk., V. XX XIII, plate 8. BELL. — ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. 15 by taking several preliminary swings, so as to give the thermopile chance to settle into a steady state, the rate of shift of zero was kept pretty steadily and the corrections were easily applied. It was also necessary to photometer the ares in the actual condition in which they were under test. To this end the apparatus was set up as shown in Figure 4. Here A is the are lamp, B the thermopile, C the galva- nometer, D the telescope and scale, EK an adjustable rotating sector dise just in front of the are, F the quartz cells in their sliding screen in front of the thermopile window, G a silvered plate glass mirror which could be quickly interposed in the beam between the arc and © waz A” D \ ve Tl pores es H ΘΟ ΖΜ Φ oe eat Figure 4. Arrangement of radiometric apparatus. the thermopile so as to deflect the rays into the portable photometer H, set up on the other side of the photometer room. The coefficient of reflection of the mirror had previously been many times determined as the mirror had been in use for photometric work. The photometer was ready for use at any time simply by closing the switch on the standard lamp. When in course of a series of thermopile measure- ments it was desired to test the ec. p. of the lamp the disc was started, the mirror swung into place and readings were then taken on the portable photometer. 16 PROCEEDINGS OF THE AMERICAN ACADEMY. The carbon are was first attacked and it proved to be a difficult subject for investigation. The particular lamp used was of the en- closed type, having the globe fitted with a short side tube and a quartz window so as to keep the arc as steady as possible without losing the ultra violet. To the same end it was found desirable to adjust a magnet behind the are so as to keep it burning on the side of the carbons next the thermopile instead of wandering round and round the carbons in the usual manner. The are thus operated gave a prodigious amount of ultra violet radiation, showing a continuous spectrum far down into the ultra violet and the three enormously intensive carbon bands usually ascribed to cyanogen, one of them in the extreme violet and the other two near wave lengths 380 yu and 360 μμ respectively. Reduced to the standard distance the deflection due to the ultra violet cut off by the Euphos glass amounted to 74 em., being 30 % of the whole energy which passed through the quartz cell. It has, of course, been long known that the naked electric are gives off very powerful ultra violet radiations and its effect in the production of ophthalmia electrica has been known for more than half a century, but in this case the extent of the ultra violet activity was somewhat unexpected. It was undoubtedly considerably enhanced by the intensive cyano- gen bands as regards that portion of the radiation lying near the visible spectrum, but on the other hand the extreme ultra violet, wave length 300 uu and less, is unquestionably stronger in the case of an open are than in the enclosed are on account of the very intense continuous spectrum emitted from the crater, which is much lessened when the are is enclosed. No separation between these parts of the ultra violet was attempted with the lamp under consideration since its unsteadiness was a constant source of annoyance and the ordinary variations of independent readings from the mean amounted to 5 or 6%. It was sufficiently evident, however, that a powerful en- closed are in a globe which permits all the radiations to pass is an enormously powerful source of ultra violet light. The carbon arc, however, is rapidly passing out of general use so that attention was next directed to the luminous are. Magnetite Arcs. — The magnetite are is one of the commonest and most generally useful outdoor illuminants. It gives a very intense nearly white light due almost wholly to the arc stream itself. The spectrum of this, the active electrode being composed almost wholly of the oxides of iron and titanium, is immensely complicated, contain- ing thousands of bright lines so closely packed as almost to obtain BELL..— ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. 17 the effect of a continuous spectrum. The actual character of the spectrum photographed with a fairly wide slit, is shown in Plate 2, d. Here, with the quartz are spectrum for reference at a is shown the radiation from the magnetite are through a quartz window and below it the spectrum of the same are taken through its ordinary globe. A quartz window was used merely to insure steadiness of the light, which would have been lost by taking off the globe. A glance shows that this spectrum is exceedingly rich in powerful lines all through the ultra violet clear down to wave length 230 wy. The glass globe cuts off the spectrum quite sharply near wave length 300 up, as in Plate 2, ὁ, but from this region to the visible spectrum lies an almost continuous mass of strong lines, very intense in the region where the quartz mer- cury are is conspicuously weak, say from the group at wave length 313 yu to the group near wave length 365 μμ. For radiometric measurements the magnetite are, which was oper- ated at 6.6 amperes and about 80 volts, proved much more steady than the carbon are, showing more small and quick fluctuations, but fewer of the large and relatively slow variations which interfered most with the readings. As a consequence the deflections obtained agreed more closely, the average variations of a single setting running be- tween 3 and 4%. For the magnetite arc through the quartz window the cut-off of Euphos glass amounted to 29 em., 28% of the total deflection. Through the ordinary glass globe the deflection was reduced to 22.4 cm., 22.5% of the total deflection. The difference between these results shows that while there is a large amount of energy of short wave length produced by the magnetite arc, most of the ultra violet energy is of wave length greater than 300 uu. As compared with the quartz mercury are used without its globe the magnetite are gave relatively about 60% less energy of wave length below 300 uu and about 40% more energy in the wave lengths above 300 wu. The candle power in the horizontal direction as measured by the method just described amounted to 760 in the run with the quartz window, and 700 in the run with the ordinary globe. The Nernst Lamp. — Finally a series of readings was taken on the Nernst lamp. The lamp investigated was of the single glower type for 220 volts, taking 91 watts and giving a downward ec. p. of 68. As the spectrum of this source runs to less than wave length 300 μμ and reaches that vicinity with somewhat material strength an attempt was at first made to run the Nernst glower without a globe. 11 proved so difficult to get steady deflections under these conditions, on account of the effect of air currents, that this measurement was 18 PROCEEDINGS OF THE AMERICAN ACADEMY. abandoned and the readings taken with the globe on, which proved reasonably easy, the precision being comparable with that obtained with the ordinary incandescent lamps. But even then the lamp proved very sensitive to small changes of voltage and only by very careful regulation of the current could consecutive series of readings be held in reasonably close agreement. In the average the deflection due to the ultra violet in the Nernst lamp with its globe was 1.81 em. and the percentage of energy thus cut off was 5.2. This completed the radiometric investigation of ordinary illuminants. Two others which it seemed desirable to investigate, that is the ordinary flame arc, and the are between iron electrodes as used by Finsen were studied on the spectrograph, since their fluctuations were of a character to make their study by means of a galvanometer of so long period as that used in this investigation quite impracticable. The peculiarities of these sources will be referred to in discussion of the general results. Sun Light. — Finally it seemed advisable to take some comparative readings on sunlight as a source of ultra violet radiations, particularly with reference to the amount of ultra violet energy with respect to the intensity of the light. Of course the solar radiation in absolute amount has been investigated with great thoroughness, but the ultra violet has received less attention than the rest of the spectrum. In general the sun radiates energy substantially like an incandescent black body at about 6000 degrees C. except in so far as its energy, particularly in the ultra violet, is cut off by the absorption of its own and the _ terrestrial atmosphere. It behaves then, like an enormously hot incandescent body shining through a medium that cuts off all the ultra violet of less wave length than about 295 wu and greatly dimin- ishes the shorter radiations even into the violet of the visible spec- trum. One would expect therefore to find relatively little total ultra violet per unit of illumination so far as the direct light of the sun is concerned. On the other hand as Schuster ' and others have shown, much of this cutting off of the ultra violet is due to scattering of the short waves by the molecules of the atmosphere and. small bodies suspended in it. In other words, the violet and ultra violet are not wholly lost, but appear in radiation from the blue sky. Ἷ Of the energy thus radiated from the sky the maximum lies almost in the edge of the ultra violet. The arrangement of the apparatus for experiments on sunlight is shown in Figure 5. Through the “Nature, XXXI, p. 97. ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. 19 BELL. courtesy of the Director, this part of the work was done in the Rogers Laboratory of Physics where the conditions for getting natural light were good. In Figure 5, A is a porte lumiére receiving the light from the sun and forming by means of the iris diaphragm B, stopped to 3 mm. diameter, an image of the sun on the thermopile front at C, before which was placed the usual quartz cell D. The thermopile was connected with the galvanometer F, read by the telescope and scale G. By the use of the diaphragm, forming a species of “pin hole” image on the face of the thermopile, at a distance of 3 meters, Figure 5. Apparatus for solar radiation. the light and energy were cut down so as to be readable with compara- tive ease. To measure the intensity of the illumination a Simmance-Abady flicker photometer H was set up close alongside the thermopile so that the solar image could be quickly moved so as to fall squarely on the photometer disc. On the other side of the photometer at I was an 80 watt tantalum lamp which was previously calibrated, in terms of the current flowing through it, against a standardized Gem lamp. From the source of supply the current was taken to this lamp through an adjustable rheostat J and a mil-amperemeter K. In measuring the light-intensity of the beam which was allowed to fall on the thermopile, it was simply shifted from the face of the thermo- pile to the face of the photometer and by means of the rheostat J 20 PROCEEDINGS OF THE AMERICAN ACADEMY. a flicker balance was established. The current read on K and re- ferred to the standardization curve at once gave the ec. p. of I, so that the illumination could be computed. The mirror at A was an electrolytic nickel surface highly polished, inasmuch as nickel gives a considerably higher coefficient of reflection near the end of the solar spectrum than does silver, which is particu- larly weak at this point. To separate the extreme violet and ultra violet as before and on exactly the same basis, the solar readings were taken with simply the quartz cell and then with the Euphos glass and a glycerine film. The cut off of violet and ultra violet pro- duced by the Euphos glass in the first day’s readings was 16.2 % and in a second day’s reading 17.9%, both days being brilliantly clear and cold in late December at noon. The average energy therefore cut off was substantially 17% uncorrected for the coefficient of re- flection of the nickel mirror, or approximately 21 % after the correc- tion for the variation in reflection as between the ultra violet and the visible spectrum. This figure is somewhat large as compared with the data ordinarily quoted for the ultra violet component of the solar spectrum, but it should be noted that this comparison is not with the spectrum as a whole but with that portion of it transmitted by a quartz cell filled with distilled water which cuts off a large part of the infra red. Also the absorption of the Euphos glass extends into the violet as has been previously noted, and finally the observations were taken in cold winter weather when the aqueous vapor, which is important in the absorption of the atmosphere, is pretty well frozen out. The observed difference of deflection in these experiments on the sun due to the cut off of the ultra violet was 2.28 em. and the observed intensity of the illumination was equivalent to 101 foot candles. These readings show precisely what the general theory indicates, that the solar light must be regarded as received from an enormously hot and hence very efficient radiator which has been robbed by atmos- pheric scattering and absorption of a considerable part of its shorter wave lengths. : REcoRD OF GENERAL RESULTS. In these experiments the following artificial sources of light were investigated with respect to the ultra violet component of each as separated from the rest of the spectrum by a disc 2 mm. thick of Euphos glass ¥1:—G. E. M. lamp; tungsten lamp; Cooper Hewitt BELL. — ULTRAVIOLET COMPONENT IN ARTIFICIAL LIGHT. il: tube; quartz lamp of the French Cooper Hewitt Company without globe; quartz lamp, American, without globe; quartz lamp, American, with globe; Graetzin mantle burner; acetylene flame; carbon electric are through quartz window; magnetite are through quartz window; magnetite are with ordinary globe; Nernst glower. In addition, a study was made of sunlight with the thermopile for comparative purposes and spectrographiec studies were also made of the ordinary yellow flame are and of the are between iron terminals such as is used for therapeutic purposes. The Euphos glass was chosen as the medium for the partition of the ultra violet from the rest of the spec- trum for the reason that it cuts out and was intended to cut out by its designers all the rays of any illuminant which are under indictment as having specific harmful action on the eyes. Broadly, the accusations of short wave lengths as injurious to the eye involve the entire ultra violet from the furthest point reached by natural or artificial illuminants up to and into the chemically active rays of the violet. Τῇ on the one hand it is the rays in the extreme ultra violet, wave length 300 μμ and less, which are absorbed by the cornea, that are held responsible for the ordinary phenomena of ophthalmia electrica, it is the rays of ultra violet of greater wave length than this, extending clear into the violet, that have been regarded by some recent investigators as producing perhaps serious lesions of the retina and of the lens. Note Schanz and Stockhausen.” The former class of injuries which have to do with the radiations absorbed by the cornea are wholly superficial and, according to Van Lint 15 the prognosis is generally good and the recovery rapid. Injuries to the retina and the lens, in-so-far as they take place, involvea far greater danger of permanent injury. Glass-blowers cataract is one of the typical injuries which has been ascribed to ultra violet radiations lying adjacent to the visible spectrum by Schanz and Stockhausen, Birch- Hirschfeld and others. Obviously, the temperature of melting glass (1400° C) is too low to give rise to any material amount of energy in the extreme ultra violet. The present investigation, therefore, took account of the whole body of radiations of short wave length. So far as possible injury from the ultra violet component in any artificial light source is con- cerned it is obviously dependent on the amount of actual energy delivered by the source in the ultra violet region and not upon the Ztschr. f. Augenheilk., Mai, 1910. 16 Accidents oculaires provoqué’s par |’électricité, p. 100. 22 PROCEEDINGS OF THE AMERICAN ACADEMY. percentage relation of this energy to the whole input. It is quite clear that in order to do any injury to the eye a certain amount of energy must be spent upon it and must be delivered at a rate in excess of the power of the eye to repair damages. One receives injury from excessive exposure to ultra violet rays just as he receives it by exces- sive exposure to heat rays. In either case the delivery of energy at a very high rate for a considerable time does damage. TABLE I. Source Input Totalu.v. w.v. per watt 100 Watt G. E. M. 100 PNG: 2 15 ΧΟ ΠΩΣ Glass Mercury Lamp (3 length taken) 96 77, 6.02 x 107 Nernst (with globe) 91 640 1203.<0g 100 Watt Tungsten 103 670 6.50) o% 5B.) the summation being for every combination of n—gq A’s in the first factor with the remaining p + q— _n in the second factor so arranged that the two groups in the order written constitute a positive permuta- tion of A; to A,. The form of this expression is evident since the B’s cannot occur in the same factor with a’s and β᾽5 (the other factor then containing a row of zeros). The sign of the term written is positive since it is obtained as a product of principal minors given by moving 4 rows of B’s past n — p rows of a’s and p-+ qg—n rows of A’s, interchanging first x and last n columns and changing n — q minus signs. The result should therefore have a sign (— 1) το» +p +a—n) +n? +n—¢ ΞΞῚ The signs of the other terms then follow, since any positive rearrange- ment of A’s should not change the sign of the term. Now in the expression of A each minor formed of n — p rows of a’s and ἢ —q rows of 6’s is equal to its coefficient. Furthermore A, contains all of the terms inAgiven by such minors taken from the first n columns. Therefore in A, each minor of the matrix [αρε1- - «αι Byii---B»] is equal to its coefficient. These coefficients constitute the matrix > (4, te Aner B, εν .B,) ea nite! fe val PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. δ σι which is therefore equal to the former. We can write this result in the form 7 leper +O Garin Br = [Ay Ay. nA By By. 2 BJ = = (C Bin B,) D (1) C being the product of any combination of n —q of A’s and D the product of the others such that [A, Ap....A,] = [CD] If p+q n, we obtain the expression in the form WA Absa τις B, Bee do = DCA Jaloye «Ale DYE. . (3) where D is any combination of n—p letters B and C the remaining ones so arranged that [B, By...B,] = [0 Ὁ]. 7 Grassmann, Gesamelte Werke, Vol. I, p. 83; Whitehead, Universal Alge- bra, p. 188; H. B. Philips, Proceedings of ‘the American Academy of Arts and Sciences, Vol. 46, p. 909. In this last article the formula obtz tined may havea different sign from the one here given. 56 PROCEEDINGS OF THE AMERICAN ACADEMY. When p+ q < ἢ this determinant gives [apt εὐ τας Bost ες Brl =2 (γ 5: : 5.) ὃ ᾿ἢ . (4) where Ὑ is any combination of qg a’s and ὃ the remaining ones so arranged that [α 81 oan Ay] = Ly δ]. Symbolic notation. 8. The determinants in the matrix representing a space S are the coordinates 5; of S. Of σ is a space complimentary to S, we consider it as represented by a matrix of the same kind as S. It has then a like number of coordniates 7; (algebraic compliments of 5; in the determinant |so7|). Then (S σὴ == δὲ Fj. This is a linear function of the coordinates s; and by a proper choice of σ (perhaps complex) can be made any linear function of those coordinates. To obtain a bilinear function of the coordinates r;, s; of two spaces R and S we take matrices Pp and © complimentary to Rand Ὁ. then (Rp) So) = 2p, 7; 5. : . (ὃ) In order to obtain the most general bilinear function Σ Ay, Ty δὲ we consider the above as a symbolic representation in which p; σι, is to be replaced by az. Thus (Rp) (So) represents symbolically any bilinear function of the coordinates 7;, δ. Any lmear relation connecting the symbolic quantities (R p) (S ©) will be satisfied by the bilinear functions Σ᾽ a, 7; s,. This is the symbolic representa- tion so much used by Clebsch. We can consider (Rp) (So) as resulting from an expression p 7 by operating on the first factor with R and on the second with S. This product p © is the dyadic of Gibbs.8 It may be considered as a distributive product of p and σ. It is called the indeterminate ϑ product. In it the order of factors must be preserved. In fact there is no general functional relation between Pp 7 and @ p. The dyadic 8 Vector Analysis, Gibbs-Wilson, page 265. Ci ΕΠ Bs Phillips: locsent: PHILLIFS AND MOORE.— LINEAR DISTANCE AND ANGLE. ou po represents a transformation which changes a space complimentary to p into a space (Rp) σ which is given by the locus of S in (Rp) (So) = 0. Linear Distance and Angle in Three Dimensions. 9. Linear distance between two points. We define the dis- tance between two points A, B as such a function A B of their coordinates that (1) if one is fixed the other lies in a plane, and (2) for points A, B, C on a line AB + ΒΟ- CA=0. ; : d (6) From the first condition the distance must be of the form Ap alas) (7) Fy (A, B) where F, and F, are bilinear functions of A and B. Putting A, B and C equal in the second condition we get AA=0. Hence F, (A, A) = 0 . . δ : (8) In this last equation replacing A by A + B and cancelling the terms F, (A, A) and F, (B, B) we have F, (4, B)+ F, (BSA e— 0 . δ . (9) The numerator of AB must then change sign when we interchange Aand B. In (6) putting C = B we have AB+BA=0. This shows that F, (A, B) = F, (B, A) . . δ . (10) or the denominator of AB is symmetricin A and B. Let C= A+B. Then (6) becomes Fy τ FP, (B, A) =f Fy (B, B) ae FP, (A, A) -Ξ Fy (B, A) = Ἐς (A, B) F, (B, A) + F,(B,B) " F(A, A) + F,(B, A) © Making use of (8), (9) and (10) this becomes Fy (A, B) [F2 (A, A) Fo (B, B) — F; (A, B)?] = 0. 58 PROCEEDINGS OF THE AMERICAN ACADEMY. Then either ΤΠ ΞΘ or Py» (A, Be ἘΞ FP, (A, A) Fs (B, B). This last equation shows that F, (A, B) factors into a function of A times a function of B. Calling this function (¢ A), and writing F, (A, B) = 3 [F (4, B) — F, (B, 4)] = (2 A) (8 B) — (α Β) (8 A), we have aq . (2 4) (BB) — (@B) (8 A) ( A) ( B) Using the identity (3) this takes the form 10 (@8-AB) _ (qg-AB) (φ A) (6B) (@ A) (φ BY’ where we put gq in place of the two rowed matrix (α A]. 10. Angle between two planes. We define the angle between two planes as such a function af of their coordinates that if the angle is given and one of the planes fixed, the other passes through a point and for three planes of a linear pencil ab+pyt+tya=0. By the same argument as for the clstcaunes between two points we obtain for the angle AB= ae (p-a B) (F a) (FB) where p is a fixed complex and F a fixed point. Distance is a relative invariant under the group of collineations that leave the complex q and the plane ¢ fixed. Similarly angle is a relative invariant under the group leaving p and F fixed. In order that fixed relations may exist between distances and angles we wish, if possible, these groups to be the same. We assume that the complex q does not degenerate into a line. Then the only complex and point determined by g and ¢ is the complex ῳ itself and the polar point of φ with respect to it. Hence we have p=4% F= we PI. 10 We consider ὯΝ 1 Β) as a regressive produel (a.B.A B), in which we expand the product (B.A B) and then multiply by a. PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 59 We choose the unit angle such that ἃ = 1. Then F = [¢ pl. [F pl] = [φΦ».»] -- (yp. ρ)φ = φ if we choose the magnitude of p such that also 11 (p p) = 2. The relations between ¢ and F are then symmetrical. Our formulae are now —._ _—(pAB) - B= 6 4) (¢ B) oe — ἀΑἀ(φ-αβ[) ; ap = (F a) (FB) a) (FB) : ‘ : : (12) with the condition that F = [¢ p], 6 = [F p] and (pp) = 2. The ratio of two distances or of two angles, also the product of a distance and angle are invariant under the seven parameter group of collineations leaving the complex p and the plane ¢ fixed. If one of these transformations leaves a distance or angle unchanged it leaves all distances and angles unchanged. Those quantities then are invariant under a six parameter group. Any tetrahedron can therefore be transformed into an equal tetrahedron (one having equal length of sides) by a collineation leaving distance and angle invariant. From the formula for the distance between two points, it is seen that distances along a line of the complex p are zero provided neither of the points lies on ¢. The distance along a line of p to ¢ is inde- terminate but along any other line it is infinite. Similarly the angle between two planes intersecting in a line of p but neither passing through F is zero. If one of the planes passes through F, the angle is indeterminate or infinite according as the other plane does or does not cut it in a line of p. 11. The locus of points y at a distance from the point αἱ is a plane 11 The formula [ p+ p] = 3 (p p) Φ can be proved as follows. Let p=aB+ γδ. Then (pp) -- 2(αβ γδ) and [p-p] = [φία β΄ γδ)- (@B + γ8)} = (pa B8) y— (ba By) ὃ + (φ α ὃ β)α -- (pyda)B Ξ Φ(αβ γδ) -- ὦ (pp) >. 60 PROCEEDINGS OF THE AMERICAN ACADEMY. E intersecting ¢ on the polar plane of x with respect to p. The corre- spondence between x and & is a correlation. From the equation τ See ee cama On) ian or λ (φ “) (Py) — (pry) = 0, it is seen that the locus of y is E= (ox) φ -- [pa] oe ee ome Cs) Similarly the locus of planes ἢ making a given angle ἃ with the plane & is a point such that the line connecting it to F passes through the polar point of € with respect to p. The locus of planes making with ἕξ an angle — 2 is z= —A(F&) F — [pi]. Substituting in this the value of € from (13) we get z= (F px) F —2 (φ 4) [pg] + [p-p2] since (Ff ¢) = 0, F being a point of ¢. Using the conditions [Fp] = 4, [dp] =F, and [p-pa]=3(pp)zr=e we get Z=2 Hence the correlations determined by a distance » and by an angle — are inverse. Now the correlation set up by an angle — ἃ is inverse to that determined by an angle X\. Hence the equations Δ Ξε δ, Ξ where x and ἕ are given, y and 7 variable, set up the same correlation. Through a correlation = x 1 Sc to 2; and x» correspond the planes λ (φ αἱ) 6 — pay, d (φ x2) 6 — p Xo. The angle between these planes is (p [A ( a1) 6 — pa] [A (Φ a2) — p x) ΤᾺ (φ a) 6 — pal} (F [A (φ a2) 6 — pal} PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 61 Since [F ¢] and [φ φ] are zero this gives —) [(h a1) (ρ- bp x2) + (p a2) (p-p αι" Φ)ὴ + (p:p ay: p x)] (F pai) (F p 22) hes d {(G 21) (φ 29) + (φ 29) (x1 φ)} + (p αἱ 19) (F pai) (F p x2) ΕΞ (p xy 19) (φ x1) (φ ao) Hence the correlation changes 21, 22 into two planes &, & such that £1 & = x1 ὅλ. In particular if ἃ = 0, the correlation between x and é is the null system determined by the complex p. The distance between any two points is therefore equal to the angle between their polar planes with respect to the complex p. 12. Angle between twolines. We define the angle between two lines r, s as such a function rs of their coordinates that, one of them being fixed and the angle constant, the other satisfies a linear relation (ἡ. ὁ. belongs to a linear complex) and for lines r, 8, t of a plane pencil rst+tst+tr=0. By the same argument as for distance between two points we find fi (r, s) fs (r, Oh fi (r, 8) r= -- ἢ (s, r) and [5 (r, 5) factors into a linear function of r times the same linear function of s. Hence rs= (ar) (ὦ 5) — (as) (Ὁ 7) (cr) (ὁ 5) : where a, b, ὁ are matrices of two rows and ab a dyadic setting up a correspondence between lines or complexes. The numerator of rs can be written in a different form. In fact (A Br) (CDs) —(CDr) (ABs) = [486 -τ- Ὁ 5] — [ABD -r-Cs] + [BCD-r- As] — [ACD-r-Bs]}, as is seen by expanding the right hand member. The expression 15 ΞΞ 62 PROCEEDINGS OF THE AMERICAN ACADEMY. in the parentheses may be regarded as gotten by operating on the collineation (dyadic) 1{[A BC] D—[ABD|C+ [BCD] A —[ACD]B} with 7, s. For this collineation the linear invariant Z{(ABCD) —(ABDC)+ (BCD A) — (ACD B)} ac) Such a collineation has sometimes been called normal. By summing we get (ar) (ὖ 5) — @s) (Ὁ Ὁ) = (ar-As) where a A is a collineation such that (aA) — 0: Conversely if a A is any normal collineation ία ἢ 1 ἢ) ΞΞ ἴα ar 0 r being any line or complex. Replacing r by r + s we have (ar-4s)+ (as-Ar) =0, showing that (a r-A 5) changes sign with interchange of r and s and is hence of the type (7) (05) (07 aa (ar-As) (cr) (ὁ 5) It is to be noticed that this formula determines an angle between two complexes as well as between two lines. In particular the angle is zero if the complexes coincide. The system of lines s making a zero angle with a line r = [Ὁ D] may be constructed as follows. Let the correspondents of C and D through the collineation a A be ἘΞ ΟῚ Di Din) eAe We therefore have (14) Then s is determined by an equation (ar-As) = (a-CD-As) = (aD) CAs) — (aC) DAs) = CDEC Naa (Cl IDS) = PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 63 In particular any line of the congruence cutting D’ C and C’ D has the required property. We may use instead of C, D any two points of the line. If then CD and C’ D’ do not intersect this gives us an infinite number of congruences generating the complex to which s belongs. 13. For a general collineation a A these lines r, s making with each other zero angles have an interesting geometrical interpretation. It is well known that a general collineation whose linear invariant (a A) vanishes has a system of tetrahedra A, B, C, D such that each point is carried by the collineation into a point of the opposite face. Two opposite edges A B and C D of such a tetrahedron determine a zero angle. For in this case since C’, D’ are in the planes A B D and A BC, the lines C’ D and C D’ cut A B. Conversely if A B and C D are two non-intersecting lines making with each other a zero angle and those lines are not left entirely invariant by the collineation we construct a tetrahedron upon them as follows. Join A’ and C’, the correspondents of A, C through the collineation a A, to CD and A B respectively and let these planes determine on A B and C D respectively the points B and D. Then B will pass into a point B’ such that A B’ cuts C D (i. 6. a point of ACD). Similarly for D. Thus, with the possible exception of fixed lines, the entire system of non-intersecting lines making with each other a zero angle consists of the opposite edges of these particular tetrahedra associated with the normal collineation a A. If P, Q, R, S are any four points it is seen on expanding the right side that (ab) ΤΠ =] ah O-ARS —«¢PRAO See SAO RI Hence if x is any point and é any plane (a2) (A &) is expressible as a sum of terms of the form ar-As. Under any collineation leaving all angles invariant this last expression must be covariant. Hence the form (a x) (A δ) must also be covariant. Collineations leaving angle invariant must then leave the complex ὁ invariant and the collineation aA fixed. We wish these angles to be invariant under the group of transformations that leave distance fixed. In that case c must coincide with p. There is a transforma- tion of this group changing any distance x y into any equal distance xy’. Since to a there can correspond through a A only one point y, this point must be fixed under all the collineations. Therefore to each point x corresponds the point F. Hence aA=fF 64 PROCEEDINGS OF THE AMERICAN ACADEMY. where 2 is aplane and Fapoint. Dual considerations show that ? is fixed under all the collineations, i. e. coincides with ¢. Hence by a proper choice of units we have Pie ae (or-F 5) = (φ 5-Ε r) (pr) (ps) (pr) (ps) 5), The angle between two lines is zero if they cut a line through F in the plane ¢. The angle is infinite if one of them belongs to the complex p and they are not cut by a line of the plane ¢ passing through the point F. 14. We have seen that xy=R sets up acorrelation. To x and y correspond planes Ao x) φ — pa, Moy) > — py. To x y corresponds the intersection which can be written A[p-xy-F] + (pry) p — ἃ (pp) [ry] Hence to lines r and s correspond lines A\lor-F] + @r) --τ, Alo s:F] + (ps) -- 5. The angle between these lines is | (φ {Alo s:F] + (ps) p—st Fid[or Fl] + (pr) -- τῇ) (pith [o S F] + (ps) p—s}) @idlor F] + (Wr) Ὁ τ) (ὦ. 7) τ (pr) Hence the angle between two lines is equal to that between the lines corresponding to them through the correlation. : a aN In particular when ἃ = Ὁ we see that the angle between two lines is equal to that between their polar lines with respect to the complex p. 15. Distance from point to plane. We wish to determine a function Aa of the coordinates of a point and plane such that if either is fixed the other satisfies a linear relation and such that = ῳ ee PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 65 is a necessary and sufficient condition that A, a be transformable into A’, a’ by a motion leaving distance invariant. Such a function is ΣΕ os See ae ΠΩΣ Let a be a plane BCD. Then ria (ABC Dy 2 (A BCD) (pp) A ©" (¢4) (FBCD) (¢ 4) (@p-BCD) This expression can be written (pA B) (pCD)+ PAC) (pDB)+ (DAD) DBC) (6 A){(o B) (pC D) + (@C) (PD B)+ @D) (pBC)} _ AB‘ CD+AC-DB+ AD: B.C ἦ 0 eC ΠΕ (17) That A a is invariant under the transformations leaving distance unchanged is shown by the last form. Conversely if 4« -- 4’ α' we take in a a triangle BCD and in a’ a corresponding triangle B’C’D’ such that HOG IR, Ξ GS TO GN LO SD SOT Od Then the above equation shows that BD’ = BD. The two tetrahedra have all their edges equal and hence the one is transformable into the other. This quantity Aa we call the distance from the point A to the plane a. It has many of the properties of euclidean distance from point to plane. Thus if the point lies in the plane (point not in @ and plane not through F) the distance is zero. If the plane is held fixed and the distance kept constant the point lies in a plane cutting aon @. Ifthe point is held fixed the locus of the plane is a point on the line joining the given point to F. If aN 66 PROCEEDINGS OF THE AMERICAN ACADEMY. the point corresponding to A (enveloped by a) is B = λ (φ A) lie A. The distance from A to B is ———— edie 3s A? 4B WA Ge) — (φ 4) Thus 4 B= Aa. This shows that 4 a is the distance, measured along A F, from A to the point of intersection of a with A F. 16. Distance from point to line. We define the distance from a point A to a line r as such a function of their coordinates that one of the quantities being fixed and the distance held constant, the other satisfies a linear relation and such that this distance is invariant under the transformations leaving distance between two points un- changed. Such a function is λ. (18) If r joins two points, B, C this can be written 5. _ (ABC-op) _ (Ad) (BOD) + (BS) (CAp) + C9) (ABD) (A φ) (BC p) (A 6) PBC) Dividing numerator and denominator by (4 ¢) (B@¢) (C4), this becomes a Bb CO=- CA AB Ar = Bie (19) This expression shows that Avr is invariant under the distance transformations. Conversely of A A oa there is a transformation changing 4r into A’r’. For let B, C be two points of r. Take on r’ two points δ΄, C’ such that ANB ΞΞΟ a AC=A'C’, then piers ΞΡ and a transformation of the kind desired can be obtained. PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 67 The distance from a point to a line is zero if the point lies on the line or if the plane of the line and the point pass through F (assuming that the point does not lie in ¢ and that the line does not belong to p). Since the order of A and r is immaterial in the formula for A r we write Arr rns 17. Angle between line and plane. Dual considerations give for the angle between a line r and a plane a, the expression Se ee " (F a) (pr) Let a be the plane at distance ἃ from A and r the line all points of which are at distance ἃ from s. Then a=)(¢A)o— pA, ; — Nous | = (ps) 5. Hence τὴς φ A) ¢ -- pA} ἰλ|φ ΕἸ] + (ps) p -- 5}} (F {\(¢A)6—pA}) (p ἵλ Φ5Ε + (ps) p—s}) _ (—[pAli@s) F—s4}) (05) @ 4) + @A)-@ 8) | (φ A) (ps) (¢ A) (ps) But [p A-s φ] = [p {(4 6) s— As φ]], and {p[A-¢ = (A-s-¢ p) = (As F). Ξ πον UPAS) - Hence oa ΠΕΡΙ Σ Ton ΞΞΙ Ὁ: Therefore the angle between a line and plane is equal to the distance between the line and point corresponding to them through the dis- tance correlation. In particular for \ = 0, we see that the distance between point and line is equal to the angle between their polar plane and point with respect to the complex p. 18. Line Area of a triangle. We define the area of a triangle ABC as a function A BC of three points such that if the vertex is fixed and the base moved along its line, the area is proportional to the base. Hence if A is the vertex of the triangle and s the line on which the base BC lies 68 PROCEEDINGS OF THE AMERICAN ACADEMY. where kis a function of A and s. This gives on applying formulae (11) and (18) after replacing s by [B C] ---- (F ABC) nC = Ξ ες sie (φ A) (φ B) @ C) The areas of two triangles having the same vertex and base line are then proportional to the quantities CAR Oe (φ A) ( (φ B) (φ C) By a series of operations consisting of moving one side of the triangle along its line and keeping the opposite vertex fixed we can move the triangle into coincidence with any other having the same area. Under each of these operations the area is proportional to the above quantity. Hence any two areas are to each other as those quantities. Then by a proper choice of unit we have Ἔν = eS OS ene (φ A) (¢ B) (φ C) Writing F = (¢ p) we have ἜΣ. (pp-ABC) _ (ΦΑ) (pBC) + (B) (pCA) + (¢C) (pAB) ~ (pA) (@B) (¢C) ($A) (6 B) (6 ΟἹ = B CAS CANA B= ποι τ Ste eee on) ς : o . : Thus the line area of a triangle is equal to its perimeter. Dually we can find as the trihedral angle between three planes Ὁ: 9. Ys τ ἐξ (φ αβ γ) ' (F a) (F θ) (Fy) Sy SG ey easy Ge, ee et 19. Volume of a tetrahedron. Similarly we define the volume of a tetrahedron A BC D as such a function A BCD of the four points that given the vertex and plane of the base, the volume is proportional to the area of the base. From the definition we have ABCD=kBCD-Aa = k(FBCD)-(Aa) A peels a B CD) ts (φ B) (6C) (6D) (φ A) (Fa) (φ A) (φ B) (φ C) (¢ DY’ PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 69 where a represents the plane BC D, in which the base lies and i is afunction of A anda. By aseries of motions consisting of moving one triangle of the tetrahedron in its plane it is seen that the tetra- hedron can be moved into any other having equal volume. These motions keep the volume constant and therefore k is an absolute constant. Hence choosing our unit so that ἢ = 1, we have (A BCD) (φ A) (φ B) @C) @D) From the definition we have ABCD=BCD-Aa=BCD-A(BCD) = 0 00-hy τ ὁ : ΠΡΟΣ ΟΠ 25 ID Te ΞΡ DUC peel De BG στ΄ 25) From (24) we see that if the vertex A lies in the plane A BC the volume is zero. Hence applying this to (25) we have AB-CD+AC-DB+AD-BC=0 ABCD= (24) as a relation connecting four points lying in a plane. This relation is seen to be identical with the relation connecting the Pliicker co- ordinates of a line. From this a theory of plane quadrilaterals could be built up. 20. Summary. We have defined a bilinear function of any two spaces in three dimensions. In case one of these spaces is a point we call this function a distance otherwise an angle. We have also defined certain areas determined by three elements and volumes determined by four. These functions are all invariant under a six parametered group of collineations projectively equivalent to the group of collineations leaving euclidean volume invariant. Under the correlation xy = const. each of these functions is equal to the dual function of the transformed elements. The expressions for these functions are aa (p A B) eee ᾧ 4) ᾧ Β) Πῶς -Φ πὶ (F a) (F B) 70 PROCEEDINGS OF THE AMERICAN ACADEMY. (p S:Fr) Tiga Ta CS ΜΌΝΟΣ Aa= ΠΡ : : : (16) ree (F Ar) Eigen ue) some OTe) eS NN τ ΣΕΡΊΟΙΞ τε} τος (Di) @ ΑἹ @B) 0) Fe ae Oa P= το σ᾿ (4 BCD) Aub) — : : 2: (@ A) @ B) ᾧ ΟἹ @D) ie ἀπ = Ὁ ἘΠῚ J sais “ (Fa) (FB) (Fy) (Fa) 21. Tetrahedron. The angles of a triangle will now be expressed in terms of the sides. For the angle C A B of the triangle A B C we have (¢ A B-FC A) (p A B) (pC A) TG A) EAB) — (pAB)(pC A) Replacing F by [¢ p] and applying (3) we have (φ A) (6 p-A BC) (p A B) (pC A) _ (6 A) LG A) WBC) + @ ΒΥ (PCA) + (@C) DAB)} (p A B) (pC A) Dividing numerator by (¢ A)? (¢ B) (¢ C), this becomes C26 CA 2 28 AB τ C A Angle CAB=CA-AB= CA, AB= CA; AB = PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. iil If we use A, B, C for the angles and a, b, ὁ for the sides opposite this becomes Hanke +b+e (26) be Similarly we have a= ete ει Τὴ In the tetrahedron if a, β, y, 6 are the planes opposite the vertices A, B, C, D we have for the angle apa 2 BCDCDA) _ @CD)(BCDA) (FBCD)(FCDA) (FBCD) (FCDA) _ODBCDA EOD. CA This gives for the volume BOD CMA ano BCD A —— CD 28) That is the volume of a tetrahedron is equal to the product of the areas of two faces and the dihedral angle between them divided by the length of the common edge. The trihedral angle a @ ¥ is given by aBy=af+By+ya _@D-BCDA DA-CDAB , DB-ADBC BCD-CDA'CDA-DAB ADB-CDB (PABCE 1 BCD DAF CDADB) Oo ΒΩ Cp ἢ ΠΡῸΣ, = BCDA This formula solved for B C D A will also express the volume in terms of the trihedral angle and the three face triangles and three edges which meet at its vertex. The volume can also be expressed in many other forms. 1 PROCEEDINGS OF THE AMERICAN ACADEMY. Linear distance in hyperspace. 22. The argument by which we derived the formula for the distance between two points in three dimensions applies without change to higher dimensions. The formula for distance is then always (q A B) (φ A) (φ B) where 4 is a complex matrix of order n —2 and ¢ ahyperplane. Sim- ilarly the angle between two hyperplanes is ΕΝ " Fa) BY where p is a complex matrix of order two and F a point. We wish these quantities to be invariant under the same group of collineations. This will happen if ¢ and q are determined by F and p and conversely. We shall therefore consider the system of complexes determined by a point F and a complex p of the second order. The details of this discussion depend somewhat on whether the space is of even or odd order. We consequently consider these cases separately. 23. Space of order ἡ = 2m. The progressive products of a complex p with itself give a system of complexes [p p], [p p p] ete. we shall denote these by the symbols p’, p? ete. In the present case p™ is represented by a sum of determinants of order n and hence is a scalar. We assume that this quantity is not zero. Such for example is the case if WE = oA Ay Ag Aste = eo ee lo and the points 4; do not lie in a hyperplane. For then na Ala eA oan Am): Since p” is not zero none of the lower powers are zero. We take as a fundamental system the quantities E> p, 'F p, ee pee, consisting of the powers of p and those powers multiplied by F. We shall find that this system forms a group under progressive and regres- sive multiplication, in the sense that the product of any two is either zero or a numerical multiple of a third in the system. To form products it is sufficient to recall that p is a sum of products of two points and hence in linear (distributive) operations behaves like a simple product of two points. Furthermore to multiply regressively PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 73 R by a product of points S we take from S all combinations D of points such that D is complimentary to R, arrange the others in a product C such that S = CD, and form the sum 2(RD)C. To obtain the product [pp] by resolving the second factor, we must take the sum of products, of p by all but two letters of any term of the second. factor times the product of those two. Those letters will occur in a combination p? Ω [05 Ξ τς ") αν —(2 ie < ᾿ - (m— 1) ΝΕ Ὁ; the second term being subtracted because in ii 1 ee ( 9 ) ἘΠῚ ΐ eee ἘΠ 1] and this combination may be selected in ( ) ways Hence occur 2 % ἯΙ ᾿ terms of the form p”-p, whereas there should be m — 1 in the expansion of [p?-p” 1]. Simplifying the above expression we get, since ΡΞ 2} er m— 1 ie Ul oe CL m Similarly ae r -ῷ 9 τ) v teal a pip ps2 (5) ἘΠ}. Ὁ. = ἈΦΞΕΊΞ ΣΙ, ἘΞ ΤΕΣ. Dupe: m Since p” is a scalar, we may solve this last equation for p”!. Chang- ing r into r + 1 in the result we have, m |p" pr | p™ (r+ 1) (m—r)’ the equation holding for r = 0 if we take p? = 1. Thus we have an expression for p” in terms of p’!. Expressing p™! in terms of ρ 2, etc., we have finally r | ie = ( m ae »" pe [ὦ +1) r+ 2)... μι] (μι --- γ)! 74 PROCEEDINGS OF THE AMERICAN ACADEMY. If we choose the magnitude of p such that ™m p™ = m! m-) d let a aa ey) ail (m — 1)! : this equation may be written ὯΝ Grime = : : Ξ ς 90 r! (m — r)! Be) woerewa— ἡ d9 seam. Again we have [F p™ | = πὸ 1} Ὁ F-p] and [piel er pee aha Pp me. Hence τ 3 = r(m—r z [pF pl] = Fp" p']— [pr F- pt] = 29 om (Fp, Solving this for [F p”!] and changing r into r + 1, we have m [p™*-F pl), p™ (m —r—1)(r +1)’ a formula holding for r = 9. By continued application of this for- mula we finally get [F p"| = [F p’) 5 ει: F pes) ; Ἂ: r! (m —r—1)!(m —1)! Let ep] (m — 1)! i Then [F 71] φ ἡ ΤΣ = : : δ 91 r! (m—r—1)! oe where Ul ee aio Ale 24. Space of order n = 2m-+ 1. In this case p” is of order n—1 and hence represents a hyperplane. Since the product p” is progressive this product must contain p (i. e., p can be expressed as D Ax [A; A;], the points A; being contained in ¢). Hence, [p-p™] = 0. PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. We assume that LF pe] 4 0. Then p” and [F p"], r = m, are not zero. ~I Or Since [p:p”] = 0, there can be no terms in the expansion of [p-F p™| which have F outside the parenthesis. p (F p™) = m(F p™) p+ i Leper sp | τὸ ( Consequently m = 2 To 1 m Pip" p= 2 = Cp Zep. Similarly LEP De () Dry π acme Ree ial) 77} Solving this for p” ! and changing r into r + 1, we obtain om [F php r (5) — Gp") po Co) p= Pp Repeated use of this formula gives finally ~ (Fp™ (r+) (m—n) 7: Ὁ ε- τὴ 2] Ep ) If we choose the magnitudes of F and p such that (F p™) = m! and let aa (m — 1) ie jij ne m! we have Ὁ 3, πὸ φ] r! (m—r)! wherer = 0,1, .. 2m. Letting r = Ο in (32) we have m! = η7 -φ m! (am fal r)! (A) (B) 76 PROCEEDINGS OF THE AMERICAN ACADEMY. Hence q” is not zero. Now q” is a point, and since 4 contains F as a factor, must be the point F. Also ΠΠΞΞ Ὁ} Consequently σα στ ς 0) m! again from (32) we have [φ ἢ | -------». : : : Be (m— 1)! - On The equations (A), (B), (C) and (A’), (B’), (C’) show that F, p are related to ¢, gq in the same way that the latter are to the former. Hence pees aA UE Aa be Dues) r! (m—r)! Where πῆρ De nares The formulae (30), (31), (82), (83) show that the system of quanti- ties p’, F p” is generated in the same way from F, p or from 4, q. If the product of two of these quantities is progressive the factors can be associated and the result is either zero or equal to a third. If the product is regressive we replace p” and F p” by their expressions in terms of ¢ and q. The product in this form is represented by a sum of matrices (in hyperplane coordinates) having a smaller number of rows than columns. The factors can therefore be commuted and associated giving a result which is number times a quantity of the form q’ or ¢q’. Hence the product of any two quantities of the funda- mental system is a numerical multiple of a third. Let So, a - | γ ge τὰ 27:1 —_ Ἢ ᾿ where r = 0, 1, 2, ...m. Dually we have the quantities o; such that, ager a) σὰν; aT | bq’ } : : “ See a) O21 = | PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 77 where r= 0, 1, 2, ...m. Equations (30), (31), (82), (83) show that ΟἿ᾽ SS Str . δ . . δ (36). 25. Distance and angle. The distance between two points A, ‘Bris See (φ A) (¢ B) Similarly we define the angle between any two spaces R, T of the same order r by the equation ΞΑ (q- σ, 1 R-o,4 7) RT a R-o.1 1) (σ, Π) (σ, 1) σα being the complex which multiplied by R and T respectively give points. This expression can be put into two other forms which we shall now obtain. We consider three cases depending on the form of Or+1- yk (1) If or = δ we have Ἐ Πρ" ἬΝ alp peel: [o,,1 Εἰ = aie = =n = [p-o,4 ΠῚ. Then τ (p:oy4 Tos 1.) -- (0,41 R-o,-4 ἽΝ De Fok Or = ! 7 I then GPR) F [pF OR! εἰ Capit ἰσ, ει R| τε Since in this case ¢,_; also contains F, we have (peop I στ) ΞΞ (Gan Roa ΤῊ In both of the preceding cases πα, τῆ, 1) = (= (Gael). [¢q"] k (3) Hf σιμι is of the form τ Oleg by the dual of the preced- » UL. 78 PROCEEDINGS OF THE AMERICAN ACADEMY. ing reasoning we have, since the sign must be positive in the first and negative in the second case (Ga Ro T= (Ay Ss Ge Beg eae) = (= 1)" (σε R-o,.1 Τὴ: For every case the following equation holds: ies (pie hoa 17) ἣν (q:o -aR-o,1T) ae (G41 τσ» 1) (σ, R) (σ, Τ) (σ, R) (σ, 7) (σ, R) (σ, 1) It is evident from the definition that 0. eS hi This together with the linearity of the expression, the factored form and symmetry of the denominator, shows that three spaces R, R’, π΄ of a pencil determine angles such that PR RR ROR 0! To prove this directly it is only necessary to place RY ΞΧ ΠΕ eR, in the expression for the above sum and clear of fractions. 26. Distance and angle in a section of hyperspace. A space R of our space of order n intersects the complexes S; of the funda- mental system in a set of complexes. For spaces contained in R we can define distance and angle relative to these last complexes. We wish now to show the relation between those invariants and the cor- responding invariants relative to the complexes §;. First consider the section made by a hyperplane a. This deter- mines with the complex p a point F, = [a pl, and with the complex [F p], a complex pi = [a-F pl. We can write this last expression in the form pi = (a F) p — [a-p- F). If we multiply this by itself r times, since the last term is a line, this PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 79 last term cannot appear more than once as a factor of any term of the result. Hence : : presario (a) arp 5. 0.51} -Ξ- 8) (al) per la-p-F On the elytra of Anoplischius sp., Buenos Aires, No. 2028, La Plata, No. 1518. A well marked species most nearly related to S. virescens, but differ- ing In various essential points. The arrangement of the distal anther- idia recalls that seen in Helminthophana. Zeugandromyces, nov. gen. Receptacle consisting of two superposed cells, the upper bearing a perithecium and antheridial appendage. The appendage consisting of a stalk-cell and a series of superposed cells above it, the lower basal cells clearly distinguished, or not differentiated from those above it and like them, bearing on the inner side a vertical double series of 178 PROCEEDINGS OF THE AMERICAN ACADEMY. paired antheridia, the terminal cell or cells of the series sterile, or converted directly into antheridia. Perithecitum usually solitary, normal, with a well developed stalk-cell; the short trichogyne arising from the base of the prominent free portion of the trichophoric cell. Were it not that sufficient material is available of two other species of this genus which occur on allied staphylinids, one in Borneo and the other in New England, I should hesitate to separate this type from the very large and varied genus Stigmatomyces. The antheridia recall those of Idiomyces, in which I have described an arrangement of antheridia in three vertical rows. I have not felt satisfied, however, that this was the actual condition, and a reexamination of fresh material of this curious type may show that here also the antheridia are in two and not in three vertical rows. The Argentine material is for the most part in poor condition, only one of the dozen or so specimens being fully matured. The perithecia do not greatly resemble those of Stegmatomyces, having well developed stalk-cells, while the distinction between venter, neck and tip is not well marked. The apex, in all three species, is rather characteristi- eally shaped, flat-conical, without projections or papillae. There appear to be four ascogenic cells in all cases. Zeugandromyces australis nov. sp. Perithecium nearly symmetrical and straight, rather elongate, rich amber-brown, paler distally; the base inflated, tapering thence gradually to the blunt conical apex; the stalk-cell stout, broader distally, faintly yellowish or hyaline, in the type bent abruptly near the base. Receptacle subtriangular, nearly symmetrical, broader distally where the septum is horizontal; subbasal cell somewhat broader, much smaller, irregular. Appendage tinged with brown, the terminal and basal cells darker, the stalk-cell subtriangular, broader externally, the basal cell more or less clearly distinguished from the five to seven cells above it, and like them bearing relatively large antheridia with long appressed upcurved necks; the terminal cell sterile, subtriangular, turned inward, externally spiniferous. Peri- thecium 15544 μ; the stalk-cell 1627 μ (distally). Appendage, including stalk-cell, 44-54 uw. Antheridia about 20 μ. Total length to tip of appendage 90 μ; to tip of perithecium 250 μ. On Scopaeus laevis Sharp. No. 1695, Palermo. Found on a single specimen of the three hosts collected. ~J em) THAXTER.— ARGENTINE LABOULBENIALES 1 CORETHROMYCES ‘Th. A comparison of new material from various parts of the world has led me to the conclusion that the scope of this genus should be con- siderably extended. Although those forms which, like the type, occur on Cryptobia are all similar and are readily grouped in a section by themselves, owing to the uniform characters of the appendages, there are other closely related forms or groups of forms, like those on δε οὶ, as well as various undescribed species on somewhat varied hosts, that do not seem to be distinguished from the type with suffi- cient clearness to justify the erection of new genera for their reception. As a result of this extension, it seems desirable, moreover, to discard the genus Rhadinomyces, which, though sufficiently well defined in its typical conditions, varies to forms too near Corethromyces for proper separation. That this union might prove necessary, I have already mentioned in my second Monograph (p. 317). A further complication in this connection has been encountered in connection with the species of Sphaleromyces, a type in which the antheridial characters are little known. The genus was based on ὃ. Lathrobii in which the antheridia appear to be solitary, but in a majority of the species which have been described under this generic name these organs have not been seen at all, or have been but doubt- fully recognized: for the reason that the material has all been obtained from dried insects, and was consequently for the most part in poor condition. Among the South American forms are several which would have been placed in this genus had it not been possible to determine from the fresh alcoholic material, that the antheridial characters were those of Corethromyces. The striking form for example, described below from material growing on Pinophilus, is undoubtedly congeneric with the two species formerly discovered on hosts of this staphyline genus, namely S. occidentalis and S. indicus; but several of the younger specimens obtained, in which the antheridia still per- sist, show clearly the intercalary nature of the latter. S. Quedionuchi was also obtained both in Chile and in the Argentine, and although the appendages here are densely tufted and small, a seriate disposi- tion of the antheridia seems also to be present. Since, apart from the supposed antheridial distinction, there are no essential differences between Sphaleromyces and Corethromyces, the former genus must also be abandoned. The genus Corethromyces thus modified, may be considered to include those forms in which a two-celled receptacle gives rise to a free 180 PROCEEDINGS OF THE AMERICAN ACADEMY. stalked perithecium, normally solitary, and to a single appendage consisting of a main axis of several superposed cells from some of which ramiferous cells are separated on the inner-side, the branches variously developed, the subbasal cell and sometimes the cell above it bearing antheridial branches; the antheridial branchlets them- selves, which really form the distinctive feature of the genus, some- times associated with sterile branchlets and bearing antheridial cells typically arranged in series of two or more superposed members, one or more of which occupy an intercalary position in the series. That even this character may be obscured, or is at least not always recog- nizable, is evident from an examination of the peculiar series of forms parasitic on species of Stilicus of which several additions are herein included. Although in more than one species of this very individual and peculiar group of forms, the seriate arrangement is well marked, instances occur in which it is rarely or perhaps never present. Thus in Corethromyces Stilicolus, which I formerly referred provisionally to Stichomyces, it is only after the examination of much additional material, that examples have been found in which the characteristic seriate arrangement occurs, the antheridia usually tending to become solitary or at least free, even when grouped: although in the light of further knowledge of this type there can be no question that it is congeneric for example with C. Stilici and others of this series, in which one or more of the antheridia may be intercalary. The conclusion thus seems unavoidable that both Rhadinomyces and Sphaleromyces should no longer be maintained as distinct genera, but should be merged in Corethromyces, which, in addition to the species previously described under this name and the new forms described below, may be regarded as embracing the following spe- cies: Corethromyces cristatus and C. pallidus formerly placed in Rhadinomyces; C. Stilicolus formerly included in Stichomyces; C. Lathrobii, C. occidentalis, C. Indicus, C. atropurpureus, C. Brachyderi, C. Chiriquensis, C. Latonae, C. obtusus, C. pro- pinquus and C. Quedionuchi formerly placed in Sphaleromyces. That further changes in the disposition of the last mentioned forms may become necessary, when better material of the other species related to C. Quedionuchi has been: examined, is suggested by the characters of the new genus Mimeomyces described above, which are exactly those of the group referred to, except for the presence of well developed compound antheridia. C. atropwpureus, for example, might well belong to the new genus, but in the type material, no signs of compound antheridia can be found. THAXTER.— ARGENTINE LABOULBENIALES. 181 Owing to the difficulties which are met with in determining the exact nature and association of the antheridia in many forms included in the genus it may be assumed that all those in which a two-celled receptacle bears distally a single perithecium on the one hand and a single main appendage on the other, bearing branches on its inner face and terminally, should be sought under Corethromyces, when it pos- sesses no characters which would exclude it from the genus. Corethromyces Argentinus nov. sp. Perithecium becoming very large, elongate, asymmetrical; the outer margin more prominent; the region of the subbasal wall-cells greatly elongated, usually distinctly suffused with purple-brown, and more or less inflated; or the whole perithecium of nearly the same diameter to the tip; which is well distinguished, blunt-conical, the apex flat, papillate, subtended by a slight elevation: the basal cell- region relatively short and compact, concolorous with the part above, the stalk-cell hyaline, but externally opaque at its base, short and about twice as long as broad. Receptacle small, the basal cell trans- lucent, reddish, broader above than the opaque subbasal cell. Primary appendage opaque below and externally indistinguishable below from the subbasal cell of the receptacle; consisting of three superposed cells, the two lower translucent along their inner margins, their limits barely indicated externally by a slight elevation, the subbasal cell associated with two unequal cells on its inner side; the lower larger than the subbasal cell itself, inflated, and bearing paired erect branches, which produce branchlets arising near the base only, the two lowest, usually, short, opaque, contrasting, directed obliquely outward; the rest suberect, more or less suffused with purplish or nearly hyaline, coarse, straight or curved toward the perithecium, the tip of which they may exceed when unbroken, the longer branches not numerous (six or more), simple, stout, septate, tapering slightly to blunt tips: the third, terminal cell of the main axis, very small, mostly translu- cent, bearing distally one or two short branches. Perithecium 100-- 290 40-55 μ, ascigerous part 165-270, stalk-cell 40-60 20-30 μ. Spores 403.5 μ. Primary axis of appendage 50; total length to tip of branches, longest 370; larger branches 8 yu in diameter. Receptacle 40 X 8 μ. On legs and abdomen of Cryptobium sp. Palermo, Nos. 1703-4. This species was very common on a dark almost black Cryptobiwm with yellow legs which frequented the low ground in the park. It is 182 PROCEEDINGS OF THE AMERICAN ACADEMY. well distinguished by its very large and long perithecia, and the stout, erect and elongate simple branchlets of the appendage, certain short oblique branchlets below their origin being alone deeply suffused. Corethromyces Ophitis nov. sp. Perithecium rather slender, translucent reddish brown, tapering but slightly to the hyaline blunt papillate tip; the basal cell well developed, hyaline, distinguished above by a slight constriction, the lower large; the stalk-cell relatively small, narrow, hyaline distally, but otherwise rich red-brown, its insertion very oblique, its suffused portion united to the basal cell of the appendage. Basal cell of the receptacle translucent brown, pale, somewhat longer than broad, slightly bent; the subbasal cell somewhat narrower below than the basal, nearly or quite opaque. Basal cell of the appendage opaque like the upper portion of the receptacle, and distinguished from it only by an external well defined rounded prominence; its second and third cells also opaque, both distinguished by a similar rounded promi- nence: the subbasal separated by an oblique septum from the basal and associated with two cells which occupy its whole inner surface; a lower, subtriangular, nearly equalling it in size, extending from its base for about three fourths of its length and bearing a red-brown ramiferous cell on either side; the upper much smaller and ramiferous; all the branches arising from these cells hyaline, two to four times subdichotomously branched, the ultimate branchlets longer, tapering, erect, the tips often abruptly recurved, some of them extending beyond the tip of the perithecium; the third cell of the main append- age subisodiametric, darker and abruptly constricted externally above its subtending prominence, a crest-like series of branchlets (usually broken) arising from its broad distal surface, the most external opaque or basally suffused. Perithecium 175 X28 yu including basal cell-region (20 μὴ). Main appendage 70 μ, to tips of branches 170 μ. Receptacle including foot 50u. Total length to tip of perithecium 209 jh On Ophites Fauvelii, in the Museo Nacional Collection. Collected at Palermo by Dr. J. Bréthes. Several specimens, only one of which is well matured, have been examined. ‘The species belongs in the section of the genus the mem- bers of which occur on Cryptobia. It is most nearly allied to C. purpurascens, but is readily distinguished by the characters of its appendage. THAXTER.— ARGENTINE LABOULBENIALES. 183 Corethromyces Platensis nov. sp. Perithecium becoming translucent amber-brown; usually straight, subconical, tapering more or less from the variably swollen venter to the blunt hyaline apex; the tip more or less clearly distinguished above a slight enlargement; the basal cells rather large; the stalk-cell variably, often greatly, elongated, and tapering somewhat to its insertion. Appendage consisting primarily of three superposed cells; the basal, and sometimes also the others, more or less deeply black- ened; the subbasal cell bearing distally from its inner side a pair of antheridial branches, one or both of which often become more or less highly developed through monopodial branching, forming two main axes of obliquely superposed cells; the lowest producing on the inner side fan-like antheridial branches, the ultimate branchlets consisting of two or three superposed antheridial cells; the rest bearing externally simple or branched, sterile, upcurved, appressed branchlets, the lower mostly blackened: the third cell of the primary appendage variably developed; often very small bearing distally and from its inner face, which may become outcurved and recurved, a variable number of simple bristle-like black branches, the lowest external one originally terminal (usually broken off), one of the others often greatly developed by successive monopodial branching, replacing the main appendage and consisting of from three to twelve obliquely superposed cells, each of which bears distally and externally, usually simple branch- lets, for the most part short, three-celled, becoming more or less deeply suffused with black or blackish brown, upcurved, more or less closely appressed; the two or three uppermost hyaline, long, multiseptate. Basal and subbasal cells-of the receptacle hyaline, small, subequal, or the subbasal larger. Perithecium, including basal cell-region, 118-125X3440 yp, the sporiferous part 75-1004; the stalk-cell 40-60X 12-20 4. Spores 24X2.5y. Greatest length of whole ap- pendage 150-360 yu. Receptacle, including foot, 4020p. Total length to tip of perithecium 85-235 μ. var. gracilis nov. var. Perithecium and its stalk-cell longer and more slender than in the type. Appendage divergent, slender, its primary axis consisting of three superposed cells; the basal hyaline below, blackened and slightly constricted above; the subbasal hya- line, rarely externally suffused, nearly twice as long as the basal cell, a small cell separated from its inner distal angle forming a rounded prominence from which arise right and left paired antheridial bran- ches, wholly hyaline, spreading, several times closely branched; an- 184 PROCEEDINGS OF THE AMERICAN ACADEMY. theridial cells single or two to four of these superposed; the third cell bearing distally one to usually not more than three branches; the outer, primary branch, shorter, slender, hyaline; the others, if present, hyaline, stouter, longer, sometimes once furcate above the basal cell. Perithecium 100-156 X 20-35 μ, including basal cell-region; stalk-cell 175X20 yu. Greatest length of appendage 150-430 μ. Total length to tip of perithecium 180-385 μ. . On Lathrobium niti'um Er., Palermo, Temperley and Llavallol, Nos. 1687, 1688, 1998; The type of this species occurs on various parts of the host and when its appendage is well developed is a very striking form. It is very variable in size and in the development of its appendage, and near the tips of the legs assumes a small, compact stout habit quite unlike the usual form. The variety corresponds exactly to the type formerly distinguished as Rhadinomyces, and occurs on the elytra, usually, or at the base of thelegs. It differs from the type in its slender form, the absence of sterile branchlets on the antheridial branches, and of the black bristle-like branches of the rest of the appendage. The examination of a sufficient series, however, appears to show that the two are not specifically separable. Corethromyces Scopaei nov. sp. Perithecium hyaline becoming faintly tinged with yellowish, rela- tively rather large, usually slightly asymmetrical owing to an out- ward curvature, tapering but slightly above the basal portion which is not prominently inflated; the tip short, conical, subsymmetrical; the small rounded papillate apex prominent; the basal cells forming a short compact group not distinguished from the base of the perithecium, the stalk-cell broad hyaline narrower below, set obliquely or sidewise on the small nearly isodiametric hyaline subbasal cell of the recepta- cle; the basal cell of which is about the same size but of characteristic form, rounded outward, its thick outer wall passing into and not dis- tinguished from the broad undifferentiated hyaline or slightly purplish foot. Appendage wholly hyaline, the basal cell hardly longer than broad, the outer wall greatly thickened and in contact below with the basal cell of the receptacle; the subbasal cell somewhat narrower, the outer wall greatly thickened; the distal portion of the appendage occupied by a more or less crest-like series of hyaline branches derived from the end of the subbasal cell and from one or perhaps more terminal cells which become displaced and appear to be external, THAXTER.— ARGENTINE LABOULBENIALES. 185 their cavities obliterated by their thickened walls, the outer branches short, directed outward and upward, the inner (from the subbasal cell) stouter, longer, once or twice branched near the base and ex- tending not much beyond the middle of the perithecium. Peri- thecium 65-75: ascigerous portion 55-70; the stalk-cell 2812 yu. Receptacle 20X16 u. Total length of appendage including branch- lets 60-80 u. Total length to tip of perithecium 95-120 4. Spores 18 xK:3 in. AK On superior abdomen of Scopaeus frater Lyach. No. 1698 and No. 1702, Palermo. A small pale species chiefly peculiar from the fact that no foot is distinguished from the peculiar rocker-like basal cell of the receptacle, which is usually quite hyaline. The species bears more resemblance to the Stilicus-inhabiting forms than to the more typical members of the genus. Corethromyces brunneolus nov. sp. Perithecium pale reddish brown with a yellowish tinge, usually rather strongly bent inward distally; the basal cells very small not distinguished from the base of the ascigerous portion, which tapers but slightly to the blunt rounded hyaline apex; the tip not distin- guished; the small basal cell-region clearly distinguished by a distinct constriction from the stalk-cell, which may be nearly straight, or strongly curved, distally broader or slightly inflated, about twice as long as broad; the stalk-cell and the appendage very asymmetrical in their relation to one another and to the small receptacle; which consists of two subequal cells, concolorous with the perithecium. Basal cell of the appendage relatively large, symmetrically inflated; the subbasal cell, at maturity and through displacement, appearing to bear directly a more or less fan-like series of short, rather stout, some- what incurved hyaline branches, which may be once or twice branched near the base. Spores 222.5 yu. Perithecia 58-62204y; asci- gerous portion 54-58y; the stalk-cell 23-3012. Receptacle 24X16 uw including foot. Appendage, total length including branches, longer, 100 μ; the basal cell 20X16 μ. On the elytra of Stilicus sp., Nos. 1511 and 2012, Temperley. This pale species appears to be very rare, only a very few specimens having been obtained. It is quite unlike any of the other forms which occur on Stilicus and appears to be most nearly allied to the preceding species. 186 PROCEEDINGS OF THE AMERICAN ACADEMY. Corethromyces Stilicolus nov. comb. Stichomyces Stilicolus Thaxter. This species which, in view of its single free antheridia, I formerly placed provisionally in Stichomyces, was found frequently in the vicinity of Buenos Aires on several species of δέ οι, and an examina- tion of sufficient material shows that, although the species tends to produce its antheridia singly, or free in groups, the intercalary arrange- ment also occurs, and there can be no doubt but that the form is con- generic with the other Stilicus-inhabiting species of the genus. The Argentine specimens are similar in all respects to those first obtained on Stilicus at Arlington, Mass. Corethromyces pygmaeus nov. sp. Perithecium becoming rather deeply suffused with dull reddish amber-brown, asymmetrical; the basal cell-region small and hardly distinguished, one of its cells usually bulging externally to form a distinct prominence; the ascigerous portion, usually rather abruptly inflated externally, the apex of the curvature forming a more or less well distinguished hump, the inner margin usually straight; the tip broad not distinguished, the apex truncate, subtended externally by a rather abrupt rounded prominence: stalk-cell suffused, becoming concolorous with the perithecium, usually strongly curved inward, distally broader below the base of the perithecium, from which it is distinguished by a very slight constriction, and which it nearly equals in length. Axis of foot at right angles to that of the basal cell of the receptacle, which is twice as large as the somewhat flattened subbasal cell; externally strongly concave, its inner margin convex, sometimes distally constricted on its inner side, a deeply suffused outgrowth arising from its outer upper angle; almost uniform in width above its narrower base, extending outward then upward abruptly beside the two basal cells of the appendage, sometimes bent inward near its rounded tip. Basal cell of the appendage large, nearly spherical; the subbasal cell small and surmounted by several hyaline branches, one or two of which may extend nearly to the tip of the perithecium. Perithecium 58-66 X 24-28 μ: stalk-cell 40-6020 μ. Spores 26X 2.5 w (measured in perithecium). Receptacle 20X 12 μ, its outgrowth 20-305 uw. Total length of appendage 30-40 uw. Total length to tip of perithecium 100 μ. THAXTER.— ARGENTINE LABOULBENIALES. 187 On head and labium of Stilicus sp., No. 1963B, Palermo. This small species was found only once in the park at Palermo but was also obtained on a similar host at Corral, Chile, No. 1902. It is allied to C. Stilici, from which it differs in the form of its perithecium and receptacle, as well as in the character of the outgrowth from the latter. Corethromyces sigmoideus nov. sp. Axis from tip of perithecium to foot, describing an even sigmoid curve, the lower curvature much shorter. Perithecium strongly curved outward, translucent amber-brown; the basal cell-region concolorous, often slightly distinguished from the ascigerous part, the basal cells well. developed and triangular; the apparent apex formed by a blunt outgrowth directly continuous with the ascigerous portion, of which it forms the bluntly rounded slightly asymmetrical termina- tion; the apex proper having its pore lateral in position and hardly distinguishable: stalk-cell but faintly suffused, broader distally, and distinguished from the basal cell-region by a slight constriction; abruptly curved near the base, the axis of which is directly con- tinuous with the subbasal cell of the receptacle. The’ latter slightly suffused, relatively large, extending on the perithecial side downward nearly to the foot, and obliquely separated from the externally deeply suffused basal cell; which is of about the same diameter throughout, including its upward extension which, lying beside the subbasal cells, extends beyond the base of the first cell of the appendage to which it is adherent, forming a rounded prominence; the upgrowth larger than the basal cell proper, and not distinguished from it. The basal cell of the appendage subelliptical, concolorous with the subbasal cell of the receptacle, its long axis nearly at right angles to that of the rest of the appendage which is curved across the stalk-cell of the perithe- cium; the subbasal cell small, flattened or rounded, bearing on its inner surface a smaller ramiferous cell, and distally a much larger one, often several times longer than broad, and bearing distally numerous branches; the latter more or less branched, all the branches tapering somewhat, slightly suffused below, hyaline above; the two or three longer ones curved downwards. Perithecia 70-85 23-27 μ: stalk- cell 60X18 μ. Receptacle including foot 40 u. Total length to tip of perithecium 135-170 μ. Spores 263 μ. On the superior right lateral margin of the prothorax of Stilicus elegans Lynch. Llavallol, No. 1994. 188 PROCEEDINGS OF THE AMERICAN ACADEMY. Closely allied to the last species, which grows in a similar position on another species of Stilicus; but readily distinguished by its sigmoid habit, and the different structure of its appendage and perithecium. Corethromyces uncigerus nov. sp. Perithecium rather bright translucent reddish amber, somewhat concave and more deeply suffused on the inner side, rather strongly convex externally, the basal cells clearly defined, subtriangular in a compact group, the basal cell-region not distinguished from the asci- gerous portion, which tapers distally to its peculiarly modified tip, the blackish suffusions of which extend to an opaque, hook-like pro- longation which, bending at right angles, forms a lid immediately above and often partly concealing the hyaline apex: the stalk-cell nearly hyaline, variously, often greatly, elongated, curved, or often straight and erect; distally broader than the basal cell-region, from which it is thus separated by a more or less pronounced constriction. Subbasal cell of the receptacle relatively large, hyaline, subtriangular, the basal cell narrow below, smoky, extending obliquely upward to the base of the appendage where it is continued by a deeply suffused broad straight erect upgrowth, which is flattened against the ap- pendage, and extends to or beyond its subbasal cell. Basal and subbasal cells of the appendage subisodiametric and subequal, or the basal larger and longer, the subbasal appearing to bear from its broad distal surface, a small tuft of hyaline, rather short branches and branchlets. Spores 26X2.8 uw. Perithecia 70-85X20-26 u; its stalk-cell 50-125 15 μ, distally, 20 u broad. Appendages, we 75 wu. Receptacle, including foot, 30-40 μ, its outgrowth 30-60 μ Total length to tip of perithecium, 150-250 μ. On the posterior legs of Stilicus elegans Lynch, No. 1994, not uncommon at Llavallol, and easily distinguished by the peculiar tip of its perithecium which recalls that of Chitonomyces psittacopsis or of C. Bullardt. Corethromyces armatus nov. sp. Perithecium nearly uniform dull purplish amber-brown, the basal cell-region not distinguished, or somewhat paler and very slightly narrower than the ascigerous part above; the inner margin slightly convex, the outer strongly so distally, the tip broad undifferentiated; the apex broad, flat, subtended internally by a rounded projection THAXTER.— ARGENTINE LABOULBENIALES. 189 and externally by a prominent conical outgrowth extending obliquely upward and outward and narrower toward its blunt, often slightly contracted, apex: the stalk-cell hyaline, shorter than the perithecium, straight or outcurved, often slightly enlarged on the inner side below the perithecium. Subbasal cell of the receptacle triangular, hyaline, the basal cell abruptly curved at right angles, wholly suffused with blackish, but not opaque; obliquely related to the subbasal cell, and continued below and just beyond the base of the appendage by an external outgrowth which is not free, even at its tip, being adherent to the basal and subbasal cells of the appendage. The basal cell of the appendage nearly hyaline, bent almost at right angles, and thus turning the rest of the appendage across the stalk-cell of the perithe- cium; the subbasal cell often abruptly narrower, hardly twice as long as broad, bearing distally a few external branches and a large appen- diculate cell, from which arise elongate tapering branches, two or three of which may exceed the perithecium and its stalk-cell in length. Spores 32X3 u. Perithecium 60-70X 20-23 μ, its terminal projection, upper margin 28 μ, lower 40 μ; stalk-cell 30-45X12-18 μ. Recep- tacle 30-40 u. Longest appendage 175 yu. Total length to tip of perithecium 120-150 μ. On the upper surface of the prothorax near the right margin of a species of Stilicus, Palermo, No. 2012, and Temperley; No. 1992, Tucuman. This species, which was met with rarely, always occurred in exactly the same position, and is easily distinguished by its appendiculate perithecium, and the peculiar position of its appendage. Corethromyces rhinoceralis nov. sp. Perithecium dirty pale brownish amber, a deeper patch of amber- brown involving the subterminal wall-cell on the inner side; subclavate in form, the gradual distal enlargement extending to the subterminal wall-cell; distally curved outward to the subhyaline apex which is slightly cleft, and subtended on the inner side by a long, straight, rather slender unicellular spine-like process which tapers slightly to a blunt apex and projects at right angles; basal cell-region well devel- oped, concolorous, not distinguished from the ascigerous part, nar- rower below where it connects with the rather slender free, subcylin- drical stalk-cell. Receptacle concolorous with the appendage and perithecium, the basal and subbasal cells of about equal length, the subbasal cell half as broad as the basal, except immediately above the 190 PROCEEDINGS OF THE AMERICAN ACADEMY. latter, and obliquely separated by a curved septum from the basal cell of the appendage which lies beside it and extends but slightly above it: the rest of the appendage rather slender, rigid, its axis of four or five successively smaller superposed cells, each bearing distally, from the inner angle, a short hyaline branch, seldom persistent and producing large bottle shaped antheridia singly or in series of two, one terminal and the other intercalary. Spores (in perithecium) about 45X6y. Perithecium, including basal cell-region, 240-250 46 w: the subterminal spine 80-90 uX8-10 uw near base; the stalk-cell 6015 u. Receptacle including foot 704. Free portion of append- age 135 μ. On the inferior surface of the abdomen of Pinophilus suffusus Er., No. 1977, Llavallol. Closely allied to C. Indicus, from which it differs chiefly in the clavate form of the perithecium, and in the highly developed spine which springs from a projection of one of the subterminal wall- cells. The species appears to be very rare, for although very many specimens of its host were obtained it was found in only two instances. Corethromyces macropus nov. sp. Nearly hyaline. Perithecium asymmetrical; the outer margin convex, the inner straight below the incurved tip; the basal cell-region not distinguished from the slightly and symmetrically inflated body, which tapers slightly to the undifferentiated tip; the latter slightly suffused with brownish, and rather abruptly bent inward, one of its lateral wall-cells deeply suffused with brown, and forming a free truncate projection immediately beside the flat-conical, hyaline, slightly geniculate apex: stalk-cell small, not distinguished from the basal cells, one of which lies beside it extending nearly to its base. Receptacle relatively large more or less strongly curved, the foot large and long, tapering from a large bulbous portion to its pointed extremity: the basal cell more or less deeply suffused with smoky brown, paler above, rectangular, somewhat longer than broad, dis- tinguished by a horizontal septum from the small subbasal cell, from which the perithecium and appendage arise asymmetrically. The appendage consisting of about five superposed cells; rigid, straight, divergent, nearly hyaline; the basal and subbasal cells not appendicu- late, the rest bearing short branches distally on the inner side. Peri- thecia, including stalk- and basal cells, 100-110X25 μβ. Receptacle, including foot, 55X18 u. Appendage 50-55X8-10 yu. Total length to tip of perithecium 150-1804. Spores 30 μ. THAXTER.— ARGENTINE LABOULBENIALES. 191 On Heterothops nov. sp., No. 1987, Llavallol. This curious form is most clearly distinguished by the peculiar conformation of the tip of the perithecium and its relatively large receptacle and foot; but is included only provisionally in the present genus owing to the fact that the antheridia are not distinguishable in any of the specimens. The host has been determined as a new species by Dr. Bernhauer. Corethromyces rostratus nov. sp. Perithecium tinged with pale brownish, long, slender, erect and straight, symmetrical; the basal cell-region distinct from the more or less inflated basal ascigerous part; the mid-region sometimes rather abruptly narrower and elongate; the tip not distinguished, symmet- rical; the apex narrow subsymmetrical, hyaline, abruptly papillate: stalk-cell small, concolorous, rather broader than long. Receptacle externally prominent below the insertion of the appendage, the basal cell large, subtriangular, suffused with smoky brown, externally opaque, its broad distal surface obliquely separated from the small flattish subbasal cell. Appendage somewhat divergent, consisting of five or six superposed cells; the basal nearly hyaline; those above it more distinctly suffused, and each bearing a branch from its distal inner angle; the branches once to several times divided, the subbasal cell of the lowest branch, in conjunction with the bases of its two or three branchlets, rather characteristically inflated; the ultimate branchlets slender, hyaline, cylindrical, associated with usually single (?) antheridia. Perithecia, above basal cells, 120-135X20-22 μ: the stalk-cell 6XSu. Receptacle 55-58 u. Spores 30X3 yu. Append- age 95-100 12-14 μ its longest branches 155 uw. Total length to tip of perithecium 200-230 u. On various parts, usually the abdomen of [eterothops sp., Temperley, No. 2000, Llavallol, Nos. 1985 and 1987. . It seems difficult to obtain this species in very perfect condition, and though I have examined material from a number of different individuals, I have been unable, even in the younger specimens, to determine the exact nature of the antheridia which appear to be solitary near the bases of the lower branches of the appendage. It is possible that I have mistaken short branches for these organs, and in any case the reference of the form to Corethromyces as above emended must be considered provisional. A well marked variety was also found having a hyaline obconical 192 PROCEEDINGS OF THE AMERICAN ACADEMY. basal cell, separated by a straight horizontal septum from the small triangular cell above, its perithecium and appendage closely approxi- mated. Stichomyces Catalinae nov. sp. Perithecium rather stout, nearly hyaline; the basal cell-region well developed, slightly broader than the base of the ascigerous region; the latter becoming gradually and but slightly broader to the broadly conical, symmetrical, or slightly bent, distal region, from which it is distinguished by a slight double corrugation on one or both sides; the apex small, often bent sidewise, rathér abruptly distinguished, symmetrical, rounded, hyaline and subtended by dark brown suffu- sions which often appear like paired rings; the stalk-cell well dis- tinguished, broader than long, distally bent abruptly upward from its insertion which is lateral, from the distal end of the subbasal cell of the receptacle. Receptacle deeply suffused with brown, except its narrow hyaline base just above the small foot; the basal cell broader distally, hardly twice as long as the somewhat broader sub- basal cell. The appendage consisting of an axis of four superposed cells not distinguished from the receptacle, and concolorous with it; the subbasal cell bearing from its upper inner angle a group of hyaline branches, which reach to or beyond the tip of the perithe- cium; the terminal cell smaller, hyaline, and bearing a few hyaline branches. Spores 201.5 μ (measured in perithecium). Perithecium 50-60 X 15-20 μ. Receptacle, including foot, 30-55X9-12 uw. Main axis of appendage 30-35 12 μ; total length to tip of longest branch- lets, 75 uw. Total length to tip of perithecium, 90-125 μ. On Conosoma testaceum Lat., No. 1984, Llavallol. The branches of the appendage in this species are usually badly broken, and even in those which are still intact, are so beset by masses of bacteria, that it has not been possible to make out the antheridia with certainty, although they appear to arise in small groups some- what as in S. Conosomae. The character of the perithecium and of its apex, and the dark continuous axis formed by the receptacle and main appendage, are characteristic of the species, although a few specimens were obtained that are smaller and in which the successive cells of the receptacle and appendage are less evenly continuous. THAXTER.— ARGENTINE LABOULBENIALES. 193 Laboulbenia Lathropini nov. sp. Receptacle relatively stout and small, cells I and II faintly suffused, subequal in length; the latter broader, sometimes longer; the rest of the receptacle and the perithecium deeply suffused with dirty olivaceous brown; cells III and IV subequal; the upper angle of cell V free between the perithecium and the slightly oblique insertion-cell, which is thick but rather small. The simple outer appendage enor- mously elongated, distally hyaline, the cells several times longer than broad, all similar; the first three or four somewhat shorter than the rest; the basal cell of the inner appendage very small, bearing an antheridial branch consisting of one to two small cells, terminated by one to two antheridia, one of which may be replaced by a long simple sterile branch. Perithecium relatively large, not wholly free, slightly and evenly inflated; the wall-cells strongly spiral and marked by fine irregularly parallel lines; the tip deeply suffused, the lip-edges hyaline, subequal, the apex suleate and turned strongly inward. Spores 75X8 uw. Perithecium 150-175X45-50 yw. Receptacle 120- 155 μ. Longest appendage 90016 μ at base. Total length to tip of perithecium 90016 wu. On the upper surface of the abdomen of Lathropinus fulvipes Er., No. 1975, Llavallol. A species of the simpler “polyphaga”’ type, most nearly allied to L. Oedodactyli, and distinguished by its enormously elongated outer appendage and spirally twisted, longitudinally striate wall-cells. The host was found rarely in decaying wood. ‘ LABOULBENIA FUNEREA Speg. This form which is very abundant on species of Anaedus in the vicinity of Buenos Aires, especially in the woods at Santa Catalina, is, in my opinion, best regarded as a variety of L. polyphaga. It is characterized by its small size, averaging about 175 u to the tip of the perithecium, the receptacle being usually rather short, about 95-100 μ, although cell II is occasionally considerably enlarged. Cell I is always hyaline, cell II often so, though frequently in- volved by the characteristic blackish olive-brown suffusion of the rest of the receptacle, which is concolorous with the perithecium except for a small hyaline patch usually present below the insertion- cell. The outer appendage is usually furcate above its subbasal cell, the two branches distally hyaline and tapering; the small basal 194 PROCEEDINGS OF THE AMERICAN ACADEMY. cell of the inner appendage bearing one or two short branches, the lower cells of which bear a few antheridia. The perithecium is straight, very slightly inflated, the tip clearly distinguished, deeply blackened, the lips hyaline, turned slightly outward, separated by a slight apiculus. Laboulbenia hemipteralis nov. sp. Receptacle rather short and stout, the basal and subbasal cells subequal in length; the former hyaline; the rest of the receptacle more or less deeply tinged with olivaceous, especially the relatively broad distal portion; cell VI (stalk-cell) small, triangular, its oblique contact with cell II not extending to the end of the latter; the basal cells of the perithecium obsolete; the ascigerous cavity lying immedi- ately above the stalk-cell. Perithecium olivaceous, tapering, its distal half, only, free; the tip conspicuously blackened and bent slightly inward; the apex subsymmetrically rounded, or slightly pointed, concolorous with the tip; the pore turned inward. Insertion- cell relatively very broad, lying somewhat higher than the middle of the perithecium, the basal cell of the outer appendage bearing a single branch, consisting of a single cell externally suffused at its base, bent inward slightly, producing four or five closely successive branchlets externally, the lowest of which is distinguished by a thin darkened septum and bears about four secondary simple branchlets in a simi- lar fashion, the lowest of which is more slender and suffused especially at its base, usually projecting subhorizontally, the others hyaline; the remaining primary branchlets hyaline, simple or fureate, often spirally curved above: the basal cell of the inner appendage giving rise normally to an outer and an inner and two lateral branches, consisting of single short cells, each bearing a large terminal brown antheridium, which may be replaced by a sterile branch bearing hya- line branchlets like those above the base of the outer appendage. Peri- thecia 6620-23 uw. Spores 222.6 (in perithecia). Receptacle 85X23 uw. Appendages to tips of longest branchlets, 105 yu. Total length to tip of perithecium 100-120 μ. On the legs and inferior surface of Velia Platensis Berg., Palermo, near Belgrano,'No. 1951 along the margin of a pool. (Van Duzee det.) This very clearly distinguished form which was found with the fol- lowing species. is the first of the genus thus far reported on Hemiptera. The material is abundant and in good condition. THAXTER.— ARGENTINE LABOULBENIALES. 195 Laboulbenia Veliae nov. sp. Receptacle dirty olivaceous, concolorous with the perithecium, cells I and II forming a stout elongate stalk about five times as long as the scarcely broader distal portion. The insertion-cell broad and thick, deep reddish, not quite opaque; the outer and inner basal cells of the appendages subequal; the appendages but faintly suffused or subhyaline, once or twice somewhat irregularly branched; the branches divergent, the two or three lowest cells short, slightly in- flated, distinguished by dark thin septa. Perithecium not wholly free, narrow, geniculate below the tip, the pore lying laterally on the inner side in the angle formed between the small rounded hyaline prominent inner lips and the greatly enlarged outer lip-cells, which are deeply suffused externally on the side above the pore, above and beyond which they form a characteristic large blunt erect slightly bent process, which terminates the perithecium, Spores 507 xz. Perithecia 125-130X24 u. Receptacle 235-200 μ; cells I and II 20018 u. Appendages including longest branchlets, 2004. Total length to tip of perithecium, largest, 350 μ. On the superior surface of the thorax of Velia Platensis Berg., No. 1951, Palermo near Belgrano. A very distinct species, remotely resembling L. ceratophora and its allies. A small group of adult specimens was found on the same individual with L. hemipteralis. Laboulbenia Lacticae nov. sp. Receptacle hyaline, becoming very faintly tinged with brownish yellow; cells I and IJ subequal, nearly as broad as the much reduced distal portion; cells III, IV and VI not greatly different in size, the insertion-cell occupying but half of the distal surface of cell IV, the rounded outer half of which is free externally. Basal cells of the appendage involved by the opacity of the insertion-cell, and indis- tinguishable; the outer bearing a compact group of six or eight suberect branches in two radial rows, or more irregularly placed, which bear short branchlets on their inner sides, and consist of two parts; a basal, seated on an almost hyaline cell and composed of rather short cells deeply suffused with blackish brown and constricted at the septa, and a distal portion suffused only at its base, above which it is quite hyaline rigid and tapering: basal cell of the inner appendage bearing one or two short branches on which one or two antheridia 196 PROCEEDINGS OF THE AMERICAN ACADEMY. may be produced, the latter sometimes occurring on the inner branches of the outer appendage also. Perithecium wholly free, concolorous with the receptacle, narrow, but slightly inflated, the tip nearly as broad as the body, and clearly distinguished by blackish suffusions; the lip-cells large rounded and bent slightly inward. Spores 453.5 μ. Perithecium 90-100 X 24-28 μ. Receptacle 80 15-155 X22. Longer appendages 135-150 u. Total length to tip of perithecium 175-280 μ. On the tips of the elytra, wings and abdomen of Lactica varicornis Jac. or a closely allied species. Palermo, No. 1462. LABOULBENIA BLECHRI Spegazzini. Receptacle slender, hyaline, the basal cell not symmetrically adjusted to the subbasal, which is slightly prominent above it on the posterior side, while the basal bulges below the subbasal on the ante- rior side; the subbasal somewhat longer than the basal, hardly broader; cells III, IV and VI subequal and subisodiametric, cell V very small. The insertion-cell, black, rather thin, not very broad; the outer appendage erect, simple, its three lower cells rather deeply tinged with olivaceous, especially externally, subequal, each somewhat broader distally and thus rather abruptly distinguished from one another; the rest of the appendage quite hyaline, tapering slightly: basal cell of the inner appendage much smaller than that of the outer, producing the usual branch on either side, each once or twice branched; the whole forming a group of four to six branchlets olivaceous below, which are relatively very stout, short, bent inward or across the perithecium, the longest extending just above its tip, the lower cir- cinate distally. Perithecium colorless, straight, its axis somewhat divergent from that of the slender receptacle, the basal cell-region forming an external rounded prominence, the junction of the basal and subbasal wall-cells also prominent; the tip, rather stout, sub- tended by a slight external prominence, the apex broad, the hyaline lips outwardly oblique, subtended by an olivaceous patch on the inner side. Spores 35X3 yu. Perithecium 62-7020-22 μ. Receptacle 80-100 yp. Appendages, longer, inner 55 yw, outer 110 yw. Total length to tip of perithecium 140 μ. On Blechrus sp., at the tips of the elytra. Llavallol, No. 1979. A single specimen of the host was found bearing this species which is most readily distinguished by its relatively very large incurved inner appendages. The perithecium may become suffused with age, but in the specimens examined it is quite hyaline, although they are sufficiently mature to have produced spores. THAXTER.— ARGENTINE LABOULBENIALES. 197 Laboulbenia Monocrepidii nov. sp. Cells I and II hyaline or faintly olivaceous, narrow, cell IT rather abruptly broader distally, and obliquely separated from cell III by an incurved partition; the distal portion of the receptacle deeply suffused with olive-brown, deeper externally below the very thick dark insertion-cell; cell V paler. Basal cells of the appendage suffused, subequal, each bearing a short single simple rarely once- branched erect similar appendage, the basal cell of which is subhyaline or more faintly suffused, and distinguished above and below by a constriction and by a blackened septum, the rest of the appendage short hyaline, tapering to a blunt point, the inner appendage single short simple, replacing a single small short antheridium found in younger specimens. Perithecium about three quarters free, deeply tinged throughout with olive-brown, slightly inflated; the tip long, not abruptly distinguished, suffused with blackish, the black shades extending downward separated by pale areas; the lips asymmetrical, the edges irregular, outwardly oblique, hyaline. Spores 754.5 uw. Perithecia 120-135X40-45 μ. Receptacle 150-225. Longest ap- pendage 80-110 μ. Total length to tip of perithecium 250-325 μ. On the elytra etc. of Monocrepidius sp., Palermo, No. 1683 and also at Llavallol. A clearly distinguished species, the first as yet recorded on a mem- ber of this family (Elateridae). Laboulbenia fuscata nov. sp. Receptacle tapering evenly to the small foot, dirty olive brown, cells I and II paler, cell ΤΥ externally rounded and prominent below the rather broad insertion-cell which is but little darker than the cells below it. Basal cell of the outer appendage roundish or bell shaped, deep reddish brown, hardly larger than the inner, the appendage externally blackened and curved abruptly outward above it, short, separated by an opaque septum from its deeply suffused reddish brown basal cell, and bearing two to three suberect or incurved short branches; the inner basal cell bearing two deep reddish brown, somewhat bell-shaped cells, terminated by a single short erect usually simple appendage. Perithecium free, except at the very base, dark translucent yellowish olive, subsymmetrical, curved slightly outward, twisted one quarter so that the tip is viewed at right angles to its normal position; the tip large, characteristically and slightly inflated, 198 PROCEEDINGS OF THE AMERICAN ACADEMY. especially its inner basal half, externally margined with black, the apex nearly opaque, broad, symmetrically bilobed. Spores copious 75X4.5 μ. Perithecium 156X48-55 uw. Receptacle 20075 μ. Total length to tip of perithecium 330-350 u. Longest appendages 120 μ. On legs of a small species of Pterostichus taken on flats outside the docks at Buenos Aires, No. 1968. A peculiar form, of which four fully developed specimens were obtained, which does not appear to be nearly allied to any of the described species. Laboulbenia granulosa nov. sp. Receptacle becoming more or less uniformly tinged with dark olive, the suffused area coarsely granular-punctate, the dark granulation involving the distal portion of the otherwise hyaline basal cell; cell II narrow, very obliquely separated from cell VI which extends nearly to its base, cells ΠΠ| and IV subequal. Insertion-cell broad and thick; cell IV protruding but slightly below it; basal cell of the outer append- age sometimes twice as large as that of the inner, both becoming concolorous with the receptacle; the outer appendage usually furcate above its subbasal cell; the basal cell of the inner appendage produc- ing a branch on either side, usually once branched; the branchlets of both appendages hyaline, eventually curved inward across and beyond the terminal portion of the perithectum. Perithecium evenly olivaceous, a few coarse scattered maculations on the basal third; somewhat inflated in the middle, the tip not abruptly distinguished, rather stout and broad; the apex asymmetrical; the outer lip-cell somewhat more prominent, the inner subtended by a blackish suffu- sion. Perithecium 11040 u. Receptacle 13540 uw. Total length 21D) Me On the legs of Argutor Bonariense Dej. (thus named in the Museo Nacional) No. 1460, Isla de Santiago, near La Plata. This species bears a distant resemblance to L. scelophila, but is distinguished by its more slender abruptly curved appendages and the blackish powdery granulation of its suffused portions. The host appears to be the same which is called by Spegazzini Argutoridius oblitus, which Mr. Henshaw informs me should be placed in Ptero- stichus. THAXTER.— ARGENTINE LABOULBENIALES. 199 Laboulbenia subinflata nov. sp. Receptacle rather long but variable, cells III and IV becoming olivaceous, the rest pale dull yellowish, the upper half or more of cell II characteristically swollen, broader than the receptacle above it, from which it is separated by a distinct indentation on one or both sides; cell III relatively large, sometimes twice as large as cell IV, the outer half of which lies external to the insertion-cell, below which it is thus prominent and obliquely rounded outward. The insertion- cell black, rather thick and narrow; the basal cell of the outer append- age several times as large as that of the inner, the subbasal cell similar and subequal, both becoming olivaceous; the latter bearing regularly two parallel branches distally, the outer usually shorter; the whole appendage erect or slightly divergent and reaching a short distance beyond the tip of the perithecium: the small basal cell of the inner appendage bearing a short erect branch on either side, from the base of which arises a unicellular antheridial branchlet terminated by two to three antheridia. Perithecium relatively small, the lower wall-cells and the upper basal cells becoming tinged with olive, distinguished from the part above by a more or less pronounced elevation, later obliterated, from which a darker area of olive-brown extends hori- zontally across the perithecium, which above it is pale amber-brown; the tip relatively narrow, abruptly distinguished externally above a conspicuous rounded prominence, its concave external margin broadly blackened; the lips outwardly oblique, coarse, the inner more promin- nent, rounded, subtended by a blackish patch. Spores 555 u. Perithecium 175-185 45-50 uw. Receptacle 310-415 X 62-78 μ; larg- est subbasal cell 187X75y. Appendages 200 μ, longest 215 μ. Total length to tip of perithecium 350-585 μ. On the left margin of the prothorax, superior, of “ Argutor Bonarien- sis Dej.”; Buenos Aires, Nos. 1512 and 1962; Llavallol, No. 2032. This species was found on a number of individuals of its host, and always in exactly the same position, sometimes in company with all of the six other species, including L. polyphaga, which occur on this host, from which it may be easily distinguished by its perithecium, appendages and inflated subbasal cell. Laboulbenia Bonariensis nov. sp. Large, long, slender, and as a rule evenly curved from base to apex. Receptacle becoming more or less evenly suffused with olive brown, 200 PROCEEDINGS OF THE AMERICAN ACADEMY. the base of cell I hyaline, the distal part more deeply suffused than the rest of the receptacle; cell II somewhat longer anteriorly than cell I, cell IV somewhat obliquely prominent below the insertion-cell, which is relatively narrow and thick: appendages slender, the basal cell of the outer very slightly longer than broad, somewhat larger than that of the inner, becoming deeply suffused with age, bearing a single slightly divergent branch, the slightly smaller basal cell of which bears two to three branchlets distally, its deep external suffusion continu- ous with that of its short slender outer branchlet, its one or two inner branchlets radially placed, simple hyaline erect, extending to or above the tip of the perithecium: basal cell of the inner appendage bearing one or two branches, sometimes once branched, hyaline, erect, similar to the adjacent branches of the outer appendage. Perithecium bent inward, becoming rich brown with a slight olivaceous tinge when fully mature; the base, above the basal cells, sometimes rather abruptly distinguished and slightly paler; the tip rather long, broad, hardly distinguished, sometimes bent very slightly outward; the apex broad, blunt, often symmetrically rounded; or the lips slightly prominent, subhyaline and subtended by a deeper shade on the inner side. Spores 70X6y. Perithecium 13535 to 210*55y, average 17542 μ. Receptacle 235-335 50-70 uw. Longest appendage 200yu. Total length to tip of perithecium 300-500 wu. On “ Argutor Bonariense Dej.’’ Usually growing in a single group not far from the base of the outer margin of the left elytron, but occur- ring less frequently on the legs and inferior surface. Llavallol, No. 2032; Temperley, No. 1512; Buenos Aires, No. 1962; La Plata, No. 1460. A species usually distinguishable with a hand lens from its large size and localized position on the left elytron. In one group of indi- viduals examined there is some variation from the type described, cell I being short, cell II much enlarged and separated from cell VI by a conspicuous indentation, so that the receptacle is subgeniculate; the tip is more prominently distinguished and bent inward, the lips broader and more prominent. The variations in size are considerable and almost straight individuals of the normal type sometimes occur. Laboulbenia lutescens nov. sp. “ Laboulbenia fumosa,”’ Spegazzini, Fungi Chilenses, p. 135. Receptacle more or less deeply, though not uniformly suffused with clear olive brown, especially along the margin below the appendages, the basal cell small, hyaline below; cell II but slightly longer; cells THAXTER.— ARGENTINE LABOULBENIALES. 201 II and VI subequal, the latter somewhat shorter; cell IV abruptly prominent externally below the insertion-cell. Insertion-cell deeply suffused, rather thick; the basal cell of the outer appendage somewhat smaller than that of the inner, externally opaque, bearing distally two branches radially placed; the outer branch strongly divergent to horizontal or even slightly recurved, almost wholly opaque, its opacity continuous with that of the basal cell; bearing above several subhya- line branchlets; the inner branch erect, once or twice branched, its basal cell and the outer primary branchlet arising from it, more or less deeply suffused externally: basal cell of the inner appendage slightly longer than that of the outer, bearing two erect slightly olivaceous branches, one on either side, which are usually twice branched; the ultimate branchlets hyaline, rigid, bluntly tipped, the longest scarcely reaching the tip of the perithecium. Body of the perithecium slightly and more or less evenly inflated, broadest in the middle, rich amber yellow, sometimes becoming tinged with olivaceous; usually, but not invariably, twisted one quarter, so that the tip is viewed at right angles to the normal position; the tip more or less deeply suffused with black- ish olive, short, rather abruptly distinguished, bent distinctly inward, its outer margin nearly straight, its inner strongly indented, the apex usually broad, horizontal, symmetrically bilobed; the lip-edges hya- line and evenly rounded; if the twist is absent, oblique, or sometimes four-lobed if the twist is one eighth. Spores 78X7u, Perithecium 125-145 35-40 uw. Receptacle 100-135 4. Total length to tip of perithecium 225-275 μ, average 250 μ. On the outer margin of the left elytron of “Argutor Bonariense Dej.” Buenos Aires, No. 1962, No. 1431 in Museo Nacional; also at Temper- ley and Llavallol. This species does not appear to be nearly allied to L. fumosa to which it has been referred by Spegazzini who found it on “ Argutori- dius” at Santiago, Chile. It was found by me on the same host at the Bafios de Apoquindo, near Santiago. Laboulbenia asperata nov. sp. Hyaline becoming pale straw- or amber-yellow. Receptacle normal, the subbasal cell variably elongated, rarely minutely corru- gated; cell V parallel to cell IV and slightly longer. Appendages hyaline, the insertion-cell transparent, faintly suffused with reddish, the basal cell of the outer appendage usually distinctly larger than the inner, broader than long and forming a more or less prominent 202 PROCEEDINGS OF THE AMERICAN ACADEMY. rounded or angular external projection variably developed below the usually solitary elongate branch or simple appendage which arises from it and is erect, sometimes divergent or even pendent, especially if it is associated with a second branch within; the basal cell of this appendage, sometimes its subbasal cell, inflated, broader than long, more or less deeply constricted at the very faintly suffused septa: the basal cell of the inner appendage producing two branches which may be simple or once branched at the base, usually slightly exceeding the tip of the perithecium, and sometimes elongate like the outer appendage. Perithecium subhyaline to yellowish, rather narrow, slightly divergent distally, the external basal wall-cell more or less conspicuously roughened by fine transverse ridges; the tip hardly distinguished, tapering very slightly; the apex broad, sub- tended on the inner side by a small faintly suffused patch, the lips evenly oblique outward, hardly prominent. Perithecia 11040 μ. Longest appendage 250 μ. Receptacle 100-235 yp. Total length to tip of perithecium, 150-350 μ, average 235 μ. On the elytra ete. of Tachys sp., Palermo, No. 1696. This species is nearly allied to L. Tachyis and to L. marina Picard, but differs from both in the characters of its appendages and insertion- cell, as well as by the characteristic external roughening of the outer basal wall-cell of the perithecium. Laboulbenia australis nov. sp. Receptacle indistinctly punctate, cells I and II becoming dirty yellowish, often contrasting with the frequently deeply suffused yellow-brown distal portion which often becomes somewhat olivace- ous. Insertion-cell horizontal, rather thick; the appendages rather copiously branched the branches subparallel in a rather compact group, usually erect or the whole bent slightly toward the perithecium; the basal cell-of the outer appendage twice as long as the inner, not distinguished from the cells above it, the appendage once or twice branched or sometimes simple: the basal cell of the inner appendage producing an erect branch on either side each once or twice branched, the antheridia arising singly or two together even from the third cells of the branches, so that they may lie opposite the tip of the mature perithecium. Perithecium free, except at its very base, usually straight, or concave externally and strongly convex inwardly, especially immediately below the tip, so that the whole perithecium is bent strongly outward distally in a characteristic manner; the tip THAXTER.— ARGENTINE LABOULBENIALES. 203 short, abruptly distinguished, laterally deeply suffused especially externally; the lips rounded, more or less symmetrically, translucent or hyaline. Spores 45X3.5 μ. Perithecia 98X35 uw. Appendages to tips of longest branches 155 uw. Receptacle 125-275 μ. Total length to tip of perithecium average 250-275 μ (150-300 μ). On all parts of a species of Apenes. Tucuman, No. 1940 (P. Spegazzini). This species of which abundant material is available, is somewhat similar to L. Oopteri, but differs in its characteristically and more strongly curved perithecium, and in the absence of dark septa in the outer appendage, the basal cell of which is never as highly developed, in the present species. Individuals growing on the legs are smaller, stouter and darker. . Laboulbenia flexata nov. sp. Yellowish to hyaline, with variable brown shades; the perithecium becoming uniformly rich translucent brown. Form rather slender, evenly curved throughout, but more or less distinctly geniculate between the basal and subbasal cells of the receptacle which are rather long and about equal in dimensions. Cells [V and V somewhat enlarged and divergent, carrying the very broad and thick black insertion-cell free from the base of the perithecium. Appendage consisting of an outer and an inner branch of the type of L. Texana; the outer stout, or curved somewhat away from the inner, and con- sisting of four to six large subequal cells, each bearing a simple branch- let like those of L. Texana, subtended by a small cell from which it is separated by a deeply blackened septum; the small terminal cell of the series bearing two such branchlets: the inner appendage consisting of two branches which spring from a common basal cell; one of them unicellular and terminated by a single antheridium, the other strongly curved across the perithecium, and consisting of five or six small superposed cells, each bearing a simple branchlet similar to those of the outer appendage. Perithecium rather narrow, curved toward the appendage, its middle opposite the insertion-cell; its tip abruptly distinguished, narrow, prominent, opaque, contrasting abruptly with the hyaline symmetrically rounded apex. Perithecium 155-200 48-55 wu. Receptacle 275-390 uw. Outer appendage 135-155x40 μ at base, longest 20050 uw: inner appendage 50-6012 μ; longest branchlets 120-140 μ. On the inferior left margin of the prothorax of Brachinus sp., No. 204. PROCEEDINGS OF THE AMERICAN ACADEMY. 1457, Isla de Santiago, La Plata; No. 1426 in Museo Nacional, no locality; No. 2030, La Plata (P. Spegazzini). The present species adds still another form to the well marked series of the L. Texana group, all of which occur on the inferior surface or legs of species of Brachinus, and which I have hitherto preferred to treat as varieties of L. Texana. Sufficient material of several of these forms which is now available, indicates clearly that the members of this series are better regarded as species, which correspond among themselves in a fashion very similar to that which may be seen in the much more numerous species which have developed on the allied host-genus Galerita in the Western Hemisphere. Among these forms Laboulbenia Oaxacana, alone, has not been found in the Argentine region, although Laboulbenia pendula is known only from Monte- video, and but a single specimen of what appears to be the typical L. Texana was obtained at the Isla de Santiago. Of the other members of the group the following were obtained. Laboulbenia incurvata exactly resembling the types, was found on a large Brachinus in the Museo Nacional, No. 1427, labeled “Argentine”; on several specimens of a Brachinus taken on the Isla de Santiago, La Plata, and on a Brachinus collected in Tucuman by P. Spegazzini. Laboulbenia retusa, which was first found in Florida, was again obtained on Brachinus from the Isla de Santiago near La Plata, No. 1457, as well as from Tucuman No. 1939. Laboulbenia tibialis, also first obtained in Florida, occurred in good condition on a Brachinus collected by P. Spegazzini in Tucuman, No. 1939. All the seven species of this group occupy more or less definite positions on the host, and none of them ever occur, as far as has been observed, on the upper surface; although L. Brachini, which is often associated with them, may be found in any position. Laboulbenia inflecta nov. sp. Basal cell of the receptacle hyaline or faintly suffused above, much longer than broad, the receptacle above it uniformly dull yellowish olivaceous and compact, the cells not greatly different in size; cell III extending upward sometimes almost to the insertion-cell. Insertion- cell somewhat oblique, thick, deeply suffused; outer and inner basal cells of the appendage subequal, the outer externally rounded and suffused, the axis of the outer appendage consisting of about five obliquely placed cells; those above the basal cell small, their branches THAXTER.— ARGENTINE LABOULBENIALES. 205 stout, relatively short, divergent; the main axis of the inner appendage consisting of five cells, the lower bearing relatively small stalk-cells terminated by single large stout antheridia. Stalk of perithecium hyaline, contrasting, very short, constricted; its axis coincident with that of the perithecium and bent inward at a slight but definite angle to the axis of the receptacle; the body of the perithecium translucent, nearly symmetrical, becoming deeply suffused with clear, slightly reddish olive-brown, subsymmetrically inflated throughout, the tip rather narrow, abruptly distinguished, more deeply suffused; the apex hyaline or becoming suffused, nearly symmetrically rounded or slightly irregular. Perithecium above stalk 110-128X35-38 μ, the stalk 8X15-20 uw. Receptacle 9840-45 μ, its basal cell 45-5020 μ. Main appendages 20 μ, their branches 50-75 uw. Antheridia 20 μ, their stalk-cells 10-12 μ. On the mid left elytron of a black species of Galerita (from two speci- mens), La Plata No. 2021, P. Spegazzini. This species resembles small forms of L. punctata, but differs in the complete absence of maculation, as well as in other minor points. Laboulbenia marginata nov. sp. Basal cell of the receptacle hyaline, cells JI and III opaque and indistinguishable, forming above a broad black margin extending upward so that the free distal margin is on a level with the insertion- cell; cell IV inwardly yellowish, obliquely elongated, externally dark brown, separated from the upper part of cell III by a clear oblique septum; cell V triangular, similarly suffused externally; both these cells, as well as the rest of the receptacle, transversely punctate. Cell VI and the cells above it subhyaline, soiled with dirty brown: the stalk of the perithecium hyaline, the main body deeply suffused, ex- ternally nearly straight and translucent, indistinctly punctate below, inwardly distinctly convex and opaque; the tip abruptly distinguished on both sides, opaque below the asymmetrical suleate apex; the inner lips prominent, broad, rounded, the outer much smaller, lower, the pore turned obliquely outward. Insertion-cell indistinguishable from the opaque basal cells of the appendages, the blackened portion curved outward and upward and forming a free rounded prominence subtending the first outer branch; this blackened area larger than the hyaline compact main appendages, the cells of which are very narrow; those of the outer seven or eight in number, including the basal cell, somewhat obliquely associated in a but slightly oblique series; the 206 PROCEEDINGS OF THE AMERICAN ACADEMY. cells of the inner appendage more obliquely superposed, six or seven in number, the three lower bearing antheridial branches consisting of single basal cells terminated by single antheridia; the simple sterile branches of the upper cells extending to about the middle of the peri- thecium. Perithecium 250-275 Χ 52 uw exclusive of the stalk (58X30 μ). Receptacle 190-20090 μ. Appendages to tips of branches about 175 μ; the antheridia 24 μ, their basal cell 204. Total length to tip of perithecium average 500-510 μ. On the inferior surface of the abdomen of Galerita Lacordairii. Museo Nacional, No. 1428, “Argentina.” Laboulbenia sordida nov. sp. Resembling L. perplexa; rather slender; the basal cell of the recep- tacle hyaline, the rest becoming irregularly suffused with dirty olive brown; the region below the insertion-cell becoming nearly opaque, the subbasal cell sometimes lighter or hyaline distally; cell IV sepa- rated from cell III and V by parallel septa at an angle of 45° to the axis of the receptacle. Insertion-cell broad, thick, horizontal, opaque; the opacity involving the outer basal cell of the appendage which is externally prominent upward. The outer appendage consisting of a series of seven or eight obliquely superposed cells, coherent through- out with the inner appendage, short; all, including the basal cell, bear- ing erect branches, the two basal cells of which are dark brown, the rest of the branch nearly hyaline and extending to or slightly above the middle of the perithecium: the inner appendage consisting of a series of usually five cells on either side above the basal cell, the distal one bearing a short erect branch, while the four lower bear antheridial branches consisting of a well developed brown basal cell, bearing distally a pair of divergent, brown, somewhat curved antheri- dia. Stalk of the perithecium clearly distinguished, about as long as broad, hyaline, contrasting; the main body deep olive brown, straight, asymmetrical, very slightly inflated below; the tip slightly darker, short, asymmetrical, more or less well distinguished, its outer margin oblique; the apex translucent, obliquely rounded outward, subtended on the inner side by an opaque suffusion. Perithecium, exclusive of stalk, 215-235X45~47 μ, the’ stalk 27-31X27 uy. Receptacle 215X 66 μ. Appendages, to tips of branches, longest, 1604. Antheridia 23-27 X6 p. On the tips of the elytra of a black Galerita, La Plata, No. 2021. This species is most nearly related to L. perplexa, from which it is THAXTER.— ARGENTINE LABOULBENIALES. 207 best distinguished by the short coherent primary appendages, short branches, and numerous paired antheridia. Laboulbenia Heteroceratis nov. sp. Uniformly pale straw-yellow, very variable in form. Receptacle usually rather elongate, but sometimes short and stout, the subbasal cells larger than the basal, cells [IV and V subequal. Insertion-cell concolorous with the cells below it, the primary outer appendage short, simple, cylindrical, hyaline, becoming distally flaccid; the inner consisting of a few ill defined short flaccid branches; the in- sertion-cell becoming very variably modified by secondary divisions, which may also involve the basal cells of the appendages so that the primary outer appendage may even become completely surrounded by small cells bearing either branches or curved antheridia, the branches sometimes forming a tuft of some length. Perithecium asymmetrical, the inner margin usually straight or slightly concave, the outer strongly convex; tapering to a snout-like tip so turned (in the Argen- tine material) that it is viewed sidewise and shows a blunt symmetri- cally rounded apex, subtended by a purplish shade. Perithecium 110-120X35+40 uw. Receptacle 156-235 uw. Appendages 50-60 μ. Total length to tip of perithecium 220-340 wu. Growing in various positions on species of Heteroceros sent from La Plata by P. Spegazzini in 1907, Nos. 1679-80. Also found on species of Heteroceros sent from Kansas by Dr. A. Stewart. This very peculiar form varies greatly in general habit, and from the secondary divisions of its insertion-cell and the basal cells of its appendages may assume an appearance very similar to that of some of the aquatic forms on Gyrinidae. Its relationships seem to be evidently with the forms found on Clivina and its allies; although a similar production of sessile antheridia from proliferous cells such as occurs in the present instance is not seen in other forms. The above description is based in part on material obtained from American species of Heteroceros which were found among a small collection of beetles kindly procured for me by Mr. Alban Stewart in Kansas City. The measurements given above are from the Argentine material. The Kansas specimens show the slightly oblique asymmetrical tip of the perithecium from the usual point of view. 208 PROCEEDINGS OF THE AMERICAN ACADEMY. Laboulbenia funeralis nov. sp. Dull blackish olive becoming opaque, except the basal and subbasal cells of the receptacle which are translucent dull olive, subequal, forming a curved or sigmoid stalk not abruptly distinguished from the rest of the receptacle, which is relatively narrow; the basal cell- region of the perithecium bulging externally, and forming a rounded flat, but usually distinct, prominence; above which the narrow perithecium tapers very slightly and evenly to the very broad tip, which is not distinguished; the apex partly hyaline bearing an inner shorter tooth-like appendage, and an outer which is longer and usually irregularly fureate. Appendages not very numerous, erect, septate at the base; the hyaline slender tapering distal portion extending to or beyond the apex of the perithecium. Perithecium 110-155 35- 40 »; the longer terminal appendage (longest) 20 uw. Total length to tip of perithecium 235-350 μ; greatest width 38-66 uw including elevation at base of perithecium. On the margins of the elytra of a species of Gyrinus, No. 1957, in a pond near the railroad station at Palermo. This species which seems constant in specimens from a considerable number of different individuals, is very closely allied to L. Gyrinidarum from which it differs more especially in its smaller size, in the color and conformation of its basal and subbasal cells which have no yellow- brown tint, are similar and subequal; both being much longer than broad; in the marked prominence below the perithecium, the tip of which is not distinguished even on the inner side, as well as by its terminal usually furcate apical appendage. Rhachomyces Argentinus nov. sp. Rather slender. Cells of the receptacle tinged with pale brown, small, about as long as broad, ten or twelve of the lower visible; the remainder wholly concealed by the closely appressed, rather slender, copious black appendages; those about the base of the perithecium somewhat stouter with hyaline tips, closely appressed about the perithecium, nearly uniform in length, and extending nearly to its tip, which projects free beyond them. Perithecium straight, sym- metrical, brown, the tip nearly black, the apex subhyaline, flat- conical or bluntly pointed. Perithecium 12040-4383 yw. Longest appendages about 95 uw. Total length to tip of perithecium 310-425 μ (longest). THAXTER.— ARGENTINE LABOULBENIALES. 209 On the legs of a small carabid beetle resembling Casnonia. Jujuy, Northern Argentine, No. 1480, Museo Nacional. This species is most nearly allied to R. Javanicus, from which it is distinguished by its more slender, copious and closely appressed appendages, which conceal the axis of the receptacle distally, as well as by the somewhat pointed apex of its perithecium. The material includes two small specimens not more than 200 u in length. Scaphidiomyces nov. gen. Axis consisting of a primary receptacle of two superposed cells, the subbasal bearing a primary branched appendage terminally, and subterminally a secondary receptacle consisting of an indeterminate series of superposed cells, which give rise alternately to stalked perithecia and to branches similar to the primary appendage. An- theridia simple, terminal on short branches. Perithecia normal. This type, of which two other species are known on scaphidians, from the Argentine and West Africa, appears to be related to the Compsomycetaceae although the number of spores in the asci has not been definitely determined. Some of the branches of the second- ary receptacle when young, show the same peculiar oblique septation characteristic of one of the appendages in Compsomyces; but this may not be significant, and the perithecium has but a single stalk-cell; the alternate production of branches and perithecia, and their associa- tion on the indeterminate secondary axis, have no parallel in any other genus. The characters of this type are nevertheless not clearly defined, and a definite conception of its limitations cannot be arrived at until sufficient material of other species is available. Scaphidiomyces Baeocerae nov. sp. Colorless, the perithecia becoming amber-brown at maturity, rather short and stout, somewhat inflated, subsymmetrical, narrowed distally to the broad tip; its apex broad, bluntly rounded or sub- truncate; the basal cells similar, rather small, projecting slightly; the region hardly distinguished from the body, and concolorous with it: the stalk-cell hyaline, but slightly longer than broad, narrower below. Basal cell of primary receptacle longer than broad, narrowed and suffused with blackish brown just above the foot. The primary appendage consisting of two to three superposed cells, bearing dis- tally short few-celled branches and branchlets. Secondary receptacle 210 PROCEEDINGS OF THE AMERICAN ACADEMY. continuous with and not distinguished from the primary, its axis of similar cells of approximately the same size, superposed more or less regularly in a somewhat zigzag fashion, the successive cells bearing with more or less regularity appendages similar to the primary append- age, and stalked perithecia of which there may be from one to four or five in various stages of development produced on the same side or alternating on opposite sides of the axis. Perithecia 75X35 μ, the stalk-cells 15-18 u. Appendages to tips of branchlets 704. Total length to tip of primary perithecium 150-310 wu. On elytra of an undescribed species of Baeocera, a small scaphidian feeding on Corticia under moist logs. Llavallol. (Determined by Dr. Csiki.) Scelophoromyces nov. gen. Main axis consisting of a basal and subbasal cell forming a primary receptacle, and a series of cells superposed above it; the subbasal cell producing a lateral branch of several superposed cells,-terminated by the primary perithecium: the upper cells of the axis, above the sub- basal cell, producing more or less copious branches on the inner side and terminally; while one or more secondary perithecia with single stalk-cells may arise from the lower. The lower cells of the primary perithecial branch, and sometimes the subbasal cell of the receptacle, giving rise to slender supporting outgrowths, which curve down toward the substratum. Antheridia (?) simple, and formed terminally from the lower branchlets. This genus is erected with some reluctance, since the nature of the antheridia is somewhat doubtful. The latter appear to be terminal cells of short lower branchlets from the main branches that arise from the upper cells of the axis above the subbasal cell, and which may be regarded as a primary appendage, or, since it gives rise to perithecia, as a secondary receptacle. Although numerous specimens are avail- able, and the form has also been obtained from the Amazon region, the branches are for the most part not well preserved, even in the youngest individuals. The several-celled stalk of the primary peri- thecium would suggest that the relationships of the genus might be with the Compsomyceteae, while the production of what may be re- garded as a secondary axis suggests Clematomyces and Scaphidiomyces. The adventitious branches which grow downward from the lower cells toward the substratum undoubtedly act as buffers, like those of Cer- atomyces rhizophorus described below, and Hydrophilomyces digitatus, THAXTER.— ARGENTINE LABOULBENIALES. 211 described recently by Picard to which further reference is made below under Ecteinomyces. Scelophoromyces Osorianus nov. sp. Pale straw- or amber-yellow, concolorous, becoming dirty amber- brown with age. Perithecium subsymmetrical; main body distin- guished from the slightly broader basal cell-region; of nearly equal diameter throughout, or but slightly inflated, the short stout tip abruptly distinguished, bent slightly outward; the apex broad and nearly truncate; the basal cells subequal, large, slightly prominent; two to six cells superposed to form the perithecial branch; the sup- porting branches simple, septate, tapering throughout to pointed extremities; two to four in number, one of them usually derived from the subbasal cell of the receptacle on the side opposite the peri- thecial branch. Main appendage, or secondary receptacle, consisting of eight to ten superposed cells, terminated by a more slender portion similar to the branches, which arise distally from cells obliquely sepa- rated on one or both sides of the upper cells of the main appendage; the branches more or less copiously branched, the ultimate branchlets forming more or less characteristic tufts, and curved toward the main axis: one to three of the lower cells usually producing a corresponding number of secondary perithecia similar to the primary one. Dimen- sions very variable. Perithecia, above hasal cells, 95-110 30-40 yu, the perithecial branch 25-120 μ, total length, including branch, 130- 250 μ; basal cell-region 20-40 25-30 u. Total length to tip of long- est branchlets (largest) 400 4. Supporting outgrowths 100-275 μ. On abdomen and elytra of Osorius sexpunctatus Bernh., Palermo, No. 1693, and Isla de Santiago, La Plata, No. 1972. Also from the Amazon, (Mann), on a very large Osorius. EcTEINoMYCcES Thaxter. I have called attention in my second monograph to the uncertain position of this genus, as well as of Hydrophilomyces; and also to the similarity between these two and Misgomyces. Although the exami- nation of fresh American material of Misgomyces Dyschirii from Kansas, recently received in moderately good condition, appears to show that this is a distinct genus more nearly allied to Laboulbenia, a further study of forms allied to Ecteinomyces and Hydrophilomyces has forced me to the conclusion that it is inadvisable to retain both 212 PROCEEDINGS OF THE AMERICAN ACADEMY. these names, and that all the species are best united under the first. The antheridial characters are doubtful in all the species, and it is still uncertain whether the structures described as simple antheridia in both cases are actually functional as such; since no actual discharge has been observed from them. In these, as in other cases in which the antheridia are not clearly distinguished, either by their position or form, it is often very difficult to distinguish them from young sterile branchlets, unless the material is examined while still fresh, so that the discharge of sperm-cells can be observed. I have therefore con- cluded to drop the name Hydrophilomyces, using Ecteinomyces to include the three new forms below described, as well as E. rhynco- phorus and Εἰ, reflexus. Hydrophilomyces digitatus Picard on Ochtebius marinus from France described in the Bull. Myc. Soc. de France, Vol. X XV, p. 244, 1910, should also be changed to Ecteinomyces digitatus Picard, since it evidently belongs in this group. Ecteinomyces rhyncophorus was found at Palermo on a small hydro- philid, and has also been obtained from Guatamala; the material in both cases corresponding in all respects to that originally obtained from Florida. Ecteinomyces filarius nov. sp. Wholly hyaline. Perithecium rather long and narrow, straight, hardly inflated, the tip rather long-conical with straight margins, subtruncate or rounded, the apex symmetrical and subtended ex- ternally by a distinct prominence; the basal cell-region not distin- guished, its cells flattened around the ascogenic cells; borne on a distinct short stalk-cell. Receptacle filamentous, slender, elongate, consisting of many (about forty) superposed cells; the distal ones becoming slightly broader, and occasionally cutting off a small cell subterminally or laterally; the axis continuous with an erect primary appendage of similar character, consisting of about six superposed cells, and lying close beside the perithecium and slightly exceeding it in length, bearing distally the remains of one or two branchlets. Spores (in perithecium) 30-35X3 yu. Perithecium 70X14 uw; the stalk-cell 8X10 uw. Receptacle 230-275X7-9 μ. Total length 290-340 μ. On the elytra of Coproporus rutilus Er.; Tucuman, No. 1934, (P. Spegazzini). The antheridia of this species have not been seen, and the types show only the bases of what appear to have been rather short branches THAXTER.— ARGENTINE LABOULBENIALES. 213 from the end of the appendage. Its hypha-like receptacle is even more striking than that of EF. Trichopterophilus, from its greater length and more evenly cylindrical form. Ecteinomyces Thinocharinus nov. sp. Wholly hyaline. The receptacle usually tapering continuously from above to the minute foot, its axis continuous with that of the perithecium and consisting of from six to twelve more or less flattened cells, which may occasionally be divided longitudinally; the foot-cell of some individuals developing an upcurved appendage, deeply blackened except along its inner margin, of variable length, thicker and bluntly rounded at its tip. Perithecium clearly divided into a nearly symmetrical oval venter and a long, stout, nearly straight, isodiametric neck-portion, the base of which is subtended on the outer margin by a more or less distinct prominence formed by the slightly protruding extremity of the outer basal wall-cell; the tip hardly distinguished, tapering but slightly to the blunt symmetrical apex. Appendage slightly divergent, consisting of six or more superposed cells, the basal larger, angular, in contact on its inner side with the small basal and stalk-cells of the perithecium; the terminal cells bearing a group of rather coarse branches, once or twice branched, the ultimate branchlets not reaching to the tip of the perithecium. Spores, in perithecium, 20X2.5 w. Perithecia 120-130X 23-27 “μ. Receptacle 55-65 yw. Foot-appendage 18 uw. Appendage 35-50 μ, its branches 75-90 μ. On the abdomen ete. of Thinocharis exilis Er., Temperley, No. 2004, and Palermo, No. 1701. The curious black outgrowth from the foot of this species, occurs in about half the specimens; but while in these it is well developed, there is no trace of it in the others, even when fully matured and growing in the same position. Ecteinomyces Copropori nov. sp. Hyaline or faintly tinged with yellowish. Receptacle consisting of from ten to twenty superposed cells some of which may become irregularly divided by one or two longitudinal septa, the cells usually flattened, often irregular, the basal cell subtriangular and deeply suffused with blackish brown above the small foot. Appendage at first not distinguished from the receptacle and continuous with it, 2 214 PROCEEDINGS OF THE AMERICAN ACADEMY. slightly divergent when mature, consisting of a variable number (eight to twelve) of superposed cells, the series tapering distally, some or most of the cells cutting off one or two small cells on the inner side, sometimes also on the outer side from which branches arise as well as antheridia (?) which are irregularly flask-shaped, single and sessile or borne one or two together on short branchlets; the sterile branches usually broken and not copiously developed. Perithecium nearly straight, its axis usually continuous with that of the receptacle, a venter neck and tip more or less clearly distinguished, the latter bent very slightly inward, the apex blunt and usually becoming minutely six-papillate; the outer, lower wall-cell slightly prominent below the neck; the two upper basal cells extending upward beside the venter, the stalk-cell short and subtriangular. Perithecium 140-200X38- 44 μ, smallest 100X 25 μ, stalk-cells and lower basal cells 20 u. Spores in peritheclum 35X3.5 uw. Receptacle average 200 μι Appendage 60-100 μ. .Total length to tip of perithecium about 325 μ. On the abdomen of Coproporus rutilus Er.; Tucuman, No. 1933, P. Spegazzini. Also from Los Amates, Guatemala, No. 1614 (Keller- man). The material of this species is not in very good condition and it is difficult to determine the character of the appendages and antheridia from them. The Guatemalan material includes only three specimens in which the perithecia are mature, and in these the papillation of the apex is either indistinct or lacking; but, although the individuals are somewhat larger, the perithecia more divergent, and the cells of the receptacle shorter and broader than the Tucuman material, the two forms seem identical. Autoicomyces bicornis nov. sp. Pale yellowish with a smoky tinge, deepest at the base of the peri- thecium. Basal and subbasal cells of the receptacle rather large, of about equal length. Appendage usually straight, somewhat diver- gent, comparatively slender; consisting of six or more superposed cells, and bearing a few small branchlets. Perithecium nearly straight externally, its inner margin convex; the tip lying in the fork formed by two outgrowths which arise symmetrically just below it from the wall-cells on either side; the outer shorter, rather closely septate, tapering to a blunt apex, and curved inward; the inner two or three times as long, usually septate only at the base, curved away from the perithecium and tapering to a blunt point. Perithecium 95-110X THAXTER.— ARGENTINE LABOULBENIALEFS. 215 40-45 μ, its longer appendage 60-200 μ, the shorter 70-78 μ. Ap- pendage 1385. Receptacle 80X35 4. Total length to tip of peri- thecium 175-190 uw; to tip of inner appendage 310-370 μ. On the inferior surface of the abdomen of Berosus sp. or a closely allied genus. Palermo near Belgrano, No. 1944. A species readily distinguished by its paired perithecial appendages, but conforming strictly to the type so clearly marked in this genus. Ceratomyces rhizophorus nov. sp. Receptacle small, hyaline, normal; the second and third cells broad and much flattened. The appendage long, of nearly equal diameter throughout, composed of numerous short flattened cells bearing scattered branches. The basal cell, and one or more of the upper cells of the receptacle, developing short rigid curved simple outgrowths, which grow downward to the substratum. Perithecium stout, tapering distally to a well distinguished, abruptly narrower, bluntly rounded tip; each marginal row of wall-cells comprising about twenty cells. Perithecium 10040 μ. Appendage 135X 16 (broken). Receptacle 50 μ, the foot 204. Total length to tip of perithecium 150 μ. At the tip of the left anterior leg of Tropisternus sp. Palermo, near Belgrano, No. 1645. All but two specimens of this small and peculiar species were unfor- tunately destroyed by accident, while they were being mounted, so that it has been necessary to base the above description on a single nearly mature, and one younger individual. It is, however, so pecu- liar, and so well characterized by its supporting outgrowths that it has seemed safe to give it a name. The outgrowths are evidently buffers, similar in function to those described in Ecteinomyces (Hydrophilomyces) digitatus Picard, and of Scelophoromyces described above. Ceratomyces ventriosus nov. sp. Receptacle relatively long, the subbasal cell and the cell above it deeply blackened laterally, the suffusion extending upward and involv- ing the outer margin or half of the cell which subtends the appendage. Appendage long and relatively slender, bearing a few scattered branches, the lower cells somewhat flattened and becoming divided by a few oblique septa. The receptacle, appendage and base of perithecium pale yellowish, or with a reddish-amber tinge. Peri- 216 PROCEEDINGS OF THE AMERICAN ACADEMY. thecium relatively very large and long, about forty-five cells in each row of wall-cells; more or less evenly curved away from the append- age, deeply rich red amber-brown, except at its pale narrower base, of the lower half characterized by a belly-like enlargement; the upper half of nearly the same diameter throughout; the tip subtended externally by a vesicular enlargement of one of the wall-cells, its hyaline apex pointed and bent inward toward the concave base of the long appendage, which is usually abruptly curved at its base, more or less deeply suffused or opaque below, tapering very slightly, consisting of about twelve cells, the lowest of which is comparatively small, and not extending above the apex of the perithecium. Peri- thecium 550-700X 100-110 uw (lower half) and 65-75 uw (upper half), the appendage 250-350 30 μ. On the inferior surface of the abdomen, near the tip on the left side of Tropisternus sp.; Palermo, near Belgrano, No. 1949. The long appendage of this remarkable species is very similar to that of the last, to which it seems to be most nearly allied, but from which it is easily separated by the form of its receptacle and its enor- mous pot-bellied perithecium. Ceratomyces marginalis nov. sp. Uniform dirty translucent amber-brown. Receptacle small, the foot and basal cell opaque and indistinguishable; the two cells above greatly flattened, the subbasal partly involved below by the suffusion of the cells above. The appendage small, short, consisting of four or five superposed cells, terminated by a few branchlets, erect, appressed against the perithecium or but slightly divergent. Perithecium rel- atively large, about eight wall-cells in each row, straight, but slightly and rather evenly inflated; the tip not distinguished, but terminated by an erect hyaline nearly cylindrical slender blunt apical prolonga- tion, subtended by a relatively very large sigmoid appendage, which curves toward and beyond it, thence bending and tapering upward, and composed of a series of eight or nine superposed cells of about equal length, sometimes terminated by a few short colorless branch- lets. Perithecium 90-110 35-45 μ, the longest appendage 100 μ. The receptacle, including foot, 55-6030. Appendage 60X7 μ. Total length to tip of perithecium 135-150 μ, to tip of appendage ΦΦ Ὁ yt, Beneath the margin of the elytra of a small pale hydrophylid. Palermo, near Belgrano, No. 1952. THAXTER.— ARGENTINE LABOULBENIALES. 217 In general habit this species is not unlike C. minisculus from which it is at once distinguished by its large perithecial appendage. Ceratomyces intermedius nov. sp. Receptacle faintly tinged with amber-brown, rather short, externally opaque above the basal cell to the base of the appendage, the blacken- ing involving the outer half or less of the cells concerned; the cell sub- ‘tending the appendage slightly prominent externally, below the latter. The perithecium and appendage usually divergent at the base of the latter, which is faintly tinged with amber-brown, stout, curved out- ward; consisting of a series of cells smaller distally, about six of the lowest very broad and flattened, becoming divided more or less irregu- larly by oblique partitions, and bearing a few scattered branchlets on the inner side. Perithecium large, stout, deeply tinged with dull amber-brown, paler at the base where it is distinctly narrower, the distal two thirds of nearly the same diameter throughout, or the middle third somewhat inflated; the tip short abruptly distinguished exter- nally, being subtended by a rounded prominence in which the series of wall-cells below it ends, its apex hyaline, asymmetrically rounded or outwardly oblique; the simple perithecial appendage becoming deeply suffused or opaque except at its bluntly pointed tip, erect or bent inward, consisting of from about six to eight successively smaller cells, the lower becoming deeply suffused; the basal cell very large, concave within, convex externally, the whole assuming a sigmoid curvature as it matures. Perithecium 310-39080-105 yu, the base 50-60 4; the appendage 105-170yu. Receptacle 74-82X75-78 μ, without foot (304). Appendage 2004548 μ at base. Total length to tip of perithecial appendage 660 yu. On the left anterior margin of the thorax of Tropisternus sp.; Pal- lermo, near Belgrano, No. 1946. A large and clearly distinguished species, intermediate between C. mirabilis, which it more nearly resembles in its perithecial char- acters, and C. cladophorus, which has a similar though somewhat. more highly developed appendage. Synaptomyces nov. gen. Receptacle indeterminate, consisting of a series of superposed cells; the uppermost of this series followed by two cells placed side by side, one of which is separated by a single small cell from the basal cell of 218 PROCEEDINGS OF THE AMERICAN ACADEMY. the appendage, while the other forms the base of the outer series of wall-cells of the perithecium. The appendage consisting of a series of superposed cells bearing scattered branchlets. Perithecium many- celled, indeterminate, without distinction of venter and neck, ap- pendiculate on the inner side below the tip. This genus, of which two other species are known on Hydrocharis, one from North America, and another from Africa, appears to be intermediate between Ceratomyces, which it resembles most nearly in the characters of its perithecium, and Rhyncophoromyces, which ἡ possesses a similar indeterminate receptacle. Although in the present species, which is taken as the type, several appendages develop in a compact group below the apex of the perithecium, in the African form there is only one which is very similar to that seen in species of Cera- tomyces. ‘The North American form, of which I have only one un- developed individual, shows that the sperm-cells -are developed exogenously exactly as in Rhyncophoromyces. Synaptomyces Argentinus nov. sp. Receptacle consisting of a series of about twenty superposed, much flattened, cells; surmounted by two somewhat unequal cells separated from one another by an oblique septum; a transversely elongated rounded cell lying obliquely between the anterior of the two and the basal cell of the appendage, which is more or less conspicuously indented externally. The appendage somewhat broken in the types, its basal or subbasal cell giving rise to a simple branch, the main axis of undivided superposed cells proliferating to form several slender branches, which arise from its tip. Perithecium relatively large and stout, hardly inflated above the base, slightly narrower distally, the papillate tip abruptly distinguished; the apex broad and asymmetri- cally rounded, the perithecial appendages arising in a group just below the tip on the anterior side, usually three being superposed; their extremities free, their bases laterally coherent, some of them proli- ferating to form slender terminal hyaline branchlets: Perithecium 335 X80-390-105 μ; its appendage without terminal branchlets 110-- 120 μ. Receptacle 250-27 70-80 μ distally. Appendage (broken) 160 15-18 μ. Total length to tip of perithecium 700-750 μ. On the left inferior margin of the thorax of Hydrocharis sp., No. 948, Palermo, near Belgrano. THAXTER.— ARGENTINE LABOULBENIALES. 219 In addition to the new forms above described the following species were found, and also a few others that are not determinable. Acompsomyces brunneolus Th. 234. PROCEEDINGS OF THE AMERICAN ACADEMY. appeared, in his opinion, to be so closely related to the form described by Preuss that he placed it in the same genus; since it was, however, not associated with chlamydospores like those of Sepedonium, but with an Aspergillus-like fructification, he named it P. aspergilliformis. Two kinds of bulbils were described as connected with this fungus, which resembled each other in color but differed in their mode of development. Of these two types, one is said to be large, sclerotium- like, without any differentiation into central and cortical cells, while the other is small and consists of several large central cells surrounded by a row of colorless cortical cells resembling those of Helicosporan- gium parasiticum, mentioned in the same paper. In connection with this fungus Eidam described conidia which, he states, were produced on exceedingly delicate, colorless, conidio- phores resembling somewhat those of Aspergillus albus Wilhelm, but the sterigmata are usually flask-shaped. These conidia were also borne individually on the sides of ordinary hyphae, being abstricted in chains from flask-shaped sterigmata and resembling those described by Eidam as associated with the form which he referred to Heltco- sporangium parasiticum. “Chlamydospores”’ were also described by Eidam in connection with his P. aspergillformis. “This form of reproduction,” he says, “seems to be by far the most common one connected with Papulo- spora and often is the only one. I have found, in great abundance, mycelia with only chlamdospores and no trace of bulbils or conidio- phores.”” On account of the presence of these chlamydospores which resemble the spores of Acremoniella, Lindau (’07) has redescribed this species under the name of Hidamia acremonioides Harz. The criti- cism that was offered as to the reliability of Eidam’s investigation of Helicosporangium may equally well be applied here. Bainier (’07) is of the opinion that he mistook the conidia of Acremoniella atra Sace. (Acremonium atrum Corda) for chlamydospores belonging to Papulospora, as these two species are often found associated with each other. Bainier (’07) found a fungus abundantly on straw, paper, cardboard, etc., which he calls P. aspergilliformis. His description of the conidia and conidiophores is practically the same as that given by Eidam (88). His fungus, however, does not produce acremonium-like chlamydo- spores, as did that of Eidam, but, on the other hand, developed pari- thecia with long necks, which he refers to the genus Ceratostoma. The asci, which are very transitory, even disappearing before the maturity of the spores, are ovoid with eight simple brownish spores HOTSON.— CULTURE STUDIES OF FUNGI. 235 somewhat variable in shape and grouped together, forming a sort of ball. Moreover, he considers that the bulbils of Helicosporangiwm parasiticum described by Eidam are merely abnormal forms of P. aspergilliformis, such as are often found among other Mucedineae. Another Papulospora, which was found in the tubers of Dahlia, has been described under the name of P. dahliae by Costantin (᾽ 88). The bulbils of this fungus are spherical, brownish-red in color, with two or three large central cells. All the cells are said to contain granular protoplasmic material at first, but the central cells soon become strongly colored violet and more densely filled with granular material and oil globules, and eventually the peripheral cells become empty and transparent. There were found associated with this fungus colorless septate spores which taper at both ends and corres- pond very closely to those described by Saccardo (Michelia I, p. 20) under the genus Dactylaria. Here again there is little evidence that the investigation was carried on with pure cultures and it is doubtful that the conidia and the bulbils described belong to the same fungus, since they were only found associated and not actually connected. It would thus appear that the only contribution on Papulospora that shows any evidence of work with pure cultures is that of Bainier (07). (c) Pyrenomycetous Forms. The first evidence of the definite association of a bulbil with one of the Pyrenomycetes as an imperfect form, is found in the description of Melanospora Gibelliana, published by Mattirolo in 1886,— although Zukal (’86) a few months previously had announced that he had found bulbils in connection with Melanospora fimicola Hansen, and M. Zobelii Corda, but gave no description of them. The fungus studied by Mattirolo was found growing abundantly on decayed chestnuts and was said to produce not only perithecia of Melano- spora but also bulbils, conidia and chlamydospores. In appearance and development these bulbils are said to resemble closely those of Baryeidamia, but with more variations. Their color is pale yellow when young, brownish-yellow at maturity, and they are often 100 μ in diameter. Mattirolo considered them immature perithecia, but, although he employed the most varied methods of experimentation, he was unable to make them develop into melanosporous perithecia. The conidia said to be connected with this fungus are described as small, colorless, spherical spores, on bottle-shaped sterigmata, resem- bling closely those mentioned by Eidam as belonging to Baryeidamia. 236 PROCEEDINGS OF THE AMERICAN ACADEMY. . The chlamydospores referred to this fungus are said to have very rough, thick walls, resembling somewhat those of Sepedonium. Al- though Mattirolo is of the opinion that these chlamydospores form a phase of the life history of M. Gibelliana, he admits that he has not absolutely proven it. He states he has “cultivated these forms without ever being able to establish unquestionably their origin and relation.” Berlese (’92) described a bulbiferous fungus producing perithecia, which he named Sphaeroderma bulbilliferum. This fungus he found growing abundantly on dead leaves of Vitis, Cissus and Ampelopsis. It is said to have several modes of reproduction, such as (a) micro- conidia, which appear in chains and which resemble those figured by Mattirolo as belonging to Melanospora Gibelliana and by Eidam, to Helicosporangium parasiticum; (b) chlamydospores, which varied somewhat in size — (these were ovoid, usually smooth, and golden- yellow in color, each with a septum near the base, which divided the chlamydospore into two unequal cells); (c) golden-yellow bulbils, which resembled those described and figured by Mattirolo in Melano- spora Gibelliana and which seem to be short-lived and, under the most favorable conditions, could not be made to produce mycelia; (d) perithecia, which were represented as almost spherical and when mature measured from 400-500 uw in diameter. They remain without an ostiole almost to maturity and consequently there is no formation of a neck. The color of the young perithecium is yellowish but becomes darker as it grows older, until at maturity it is almost a tan color. The asci are club-shaped with deep smoke-colored spores, ovoid and prolonged at the poles into short obtuse papillae. Another pyrenomycetous form producing bulbils has been reported by Biffen (701, ’02), and is said to be connected with Acrospeira mirabilis Berk., which was originally found on sweet chestnuts (Castanea vesca, Gaertn.). By the use of pure cultures, Biffen claims to have succeeded in obtaining not only the chlamydospores, as de- scribed by Berkeley and Broome in the Annals and Magazine of Natural History for 1861, but also what he calls “spore-balls”’ (bulbils) and definite perithecia. The spore-balls, which he says so closely resemble Urocystis violae that he “could not find a single characteristic to separate them by,” were obtained by sowing the ‘chlamydospores’ on a watery extract of chestnuts. Greater difficulty was experienced in producing the perithecia, but finally, by sowing the chlamydospores and bulbils on sterilized chestnuts, he records the following results: — “The ‘ chlamy- HOTSON.— CULTURE STUDIES OF FUNGI. 237 dospore’ infections gave a crop of ‘chlamydospores’ only; the spore-balls gave spore-balls and small reddish-brown, hard-walled perithecia. The walls of the perithecia were smooth and without bristles and the ostiole was small and flush with the surface, i. e., not raised on a papilla or forming a neck... .Berkeley’s A. mirabilis thus turns out to be one of the stages in the life history of a Sphaeria.”’ The investigations on the pyrenomycetous forms show more careful work than those under the two preceding headings. In all these there is evidence that pure cultures were used more or less, but in most cases it is uncertain how far the results were thus obtained. (d) Discomycetous Forms. There have been two fungi described which produce bulbils asso- ciated with discomycetous fructifications, one by Zukal (’85, ’86) and the other by Morini (’88). Zukal found two kinds of primordia in connection with his fungus; one, he says, consisted of two or three small mycelial branches which wound about each other and eventually produced reddish-brown bulbils with a cortex of small colorless, almost transparent, cells. The other primordium was made up of a number of hyphae massing themselves together and becoming quite large and, under proper conditions of nutrition, developing into apothecia of the Peziza type; but he does not give a name to this form. This fungus produced conidia abundantly on erect, branched coni- diophores. The conidia are spoken of as colorless, ellipsoidal, smooth, and they appear in clusters upon the ends of short sterigmata. Zukal’s cultures were grown on absorbant paper saturated with Leibig’s extract, but there is no evidence in his article that these were pure cultures, or that the life history of the fungus was carefully traced from ascospore to bulbil. Morini (’88) describes “ bulbil-like”’ bodies associated with Lachnea theleboloides (A. & S.) Sace. in old cultures. Since these occurred only in cultures that had run for a long time, in which the nutrient was probably largely exhausted by the previous growth of the fungus, and since the development was largely the same as that of the apothe- cium, Morini considers that the bulbils of L. theleboloides are abortive apothecia and, further, that they are analagous to the similar struct- ures described by Eidam, Karsten, et al. He apparently has used pure cultures in his investigation, but to what extent his results were obtained from such cultures could not be determined from his paper. 238 PROCEEDINGS OF THE AMERICAN ACADEMY. (e) Basidiomycetous Forms. The only account, as far as the writer is aware, of the definite association of bulbils with Basidiomycetes is given by Lyman (’07) in connection with his culture-studies of Cortictum alutaceum (Schra- der) Bresadola, his results having been obtained from pure cultures made of the basidiospores of this fungus. ‘The bulbils,’”’ he says, “are reddish-brown or chocolate-colored clusters of cells, more or less globose in shape, and usually 65-80 μ in diameter, although ranging as high as 220 y....They are frequently very irregular in shape, due to the unsymmetrical arrangement of the cells, and to the bulging of the free outer walls. There is no distinction between internal and external cells of the cluster.”’ Besides the basidiospores and bulbils this Corticium also produces conidia which are of the Oidium-type. Occasionally whole hyphae break up into chains of spores of this type. Lyman also mentions two other bulbiferous fungi which were referred to the Basidiomycetes, being recognized as such by the clamp- connections of their hyphae, although the basidiospores were not obtained. : Lastly, it may be well to mention an article by Harz (’90), in which he describes a fungus found growing on material obtained from the reservoir of a factory and which he names Physomyces heterosporus (Monascus heterosporus (Harz) Schréter). Although this fungus is probably a true Monascus, as Schréter has indicated, yet since it has been associated with bulbils, and since the ascocarps of Monascus in general bear a superficial resemblance to them, it may be well at least to mention it in passing. Harz has associated this form closely with Helicosporangium parasiticum Karsten, and created a new family Physomycetes — for the reception of these two genera. As, however, these two forms will be referred to again in connection with H. para- siticum Karsten, a further consideration of them will be deferred until that time. It will be seen from the foregoing brief review of the literature that much of it is quite vague and untrustworthy. This perhaps is what one would expect from investigations which were carried on during a period prior to the adoption by mycologists of the bacterial methods of handling pure cultures. This is especially true with regard to polymorphic forms, like some of those under consideration, where it is so necessary to adopt these methods in order to be absolutely sure of the different steps in following the life history of the fungus from spore-form to spore-form. The contributions of Lyman and Biffen HOTSON.— CULTURE STUDIES OF FUNGI. 239 on this subject show undoubted evidence that their investigations were carried on with pure cultures and that the life history from spore to bulbil was closely traced. It is probable that Bainier, Morini, Berlese, and Mattirolo also used pure cultures more or less, but there is little evidence in their writings that there was careful tracing of the fungus from spore to bulbil. Sources OF MATERIAL. Before recording the results obtained from the study of the various bulbiferous fungi cultivated by the writer, it will be well to refer briefly to the sources of material and the methods used in this investigation. In 1907, at the suggestion of Dr. Thaxter and with a view to obtain- ing as much material as possible for examination, the writer began collecting substrata of various kinds from widely different localities. This material was placed in moist chambers in the laboratory and as bulbils appeared pure cultures were made of them. The methods employed in doing this will be referred to later. Most of the material from which bulbils were obtained was collected either in the vicinity of Cambridge, Mass., or Claremont, Calif.; but bulbils were also procured from substrata received from other portions of New England and California, from Kentucky, Canada, Mexico, Guatemala, Cuba, Jamaica, Bermuda Islands, the Argentine Republic, Italy, ete. The substrata on which these fungi were found were very diverse. The most productive were various kinds of excrement (dog, rat, mouse, rabbit, pig, horse, goose, goat, etc.), dead wood (Acer, Lathy- rus, Quercus, Eucalyptus, ete.), decaying vegetables (squash, onions, etc.), straw (wheat, oats, barley, rye, alfalfa, etc.). A number were found on paper and old cardboard, as well as on a variety of other substrata. Of many hundreds of such cultures about two hundred yielded bulbils. CuLTuRE METHODS. The moist chambers used for the cultivation of these materials were usually crystallizing dishes covered with pieces of glass. A large amount of this material was grown in the laboratory and from time to time was carefully examined through the glass top with a hand lens. When bulbils were observed, one of them was picked out by means of fine dissecting-needles under a dissecting microscope, and after thor- ough washing in sterilized water on a flamed slide, was transferred to a test-tube containing sterilized nutrient material — usually potato 240 PROCEEDINGS OF THE AMERICAN ACADEMY. agar. In the case of some melanosporous forms the transfer was made by carefully touching the long cirri of ascospores, produced by the perithecia of this genus, with a piece of nutrient agar on the end of a sterilized platinum needle. The ascospores adhering readily to the agar, a pure culture was easily obtained. Bacteria sometimes gave trouble in some transfers, but as a rule these were gotten rid of either by picking out separate bulbils carefully and washing several times before growing them in acidulated nutrient agar, or by keeping the impure tubes at a temperature of 15-20° C. The growth of the bacteria being retarded either by the cold or acid, the mycelium producing the bulbil soon grew out beyond the affected region, and by gouging out a few of the ends of the hyphae with some of the agar and transferring to another tube, a pure culture was readily obtained. When these were secured the fungus was cultivated on various kinds of nutrient agar media, some growing better on one medium and some on another. The following were used most frequently: potato, onions, sucrose of different percentages, bran, rice, cornmeal, straw, plums, prunes, grapes, figs, bread, squash, Spanish chestnuts, wood, various kinds of dung, etc. These were usually used with agar, but some materials like wood, dung, straw, nuts, etc., were sterilized in bulk with plenty of water and without using agar while in some instances decoctions were used. In Claremont, California, they were grown in the laboratory at an average temperature of 25-30° C. In Cambridge many were grown in an oven kept at various constant temperatures, 20-25° C. giving the best results. The vessels used for these cultures were usually medium sized test- tubes, Erlenmeyer flasks of one and two litres, or preserve-jars with cotton plugs. These were filled about one-third full of nutrient agar and usually slanted to give more surface. On this nutrient the fungus would usually grow well for several months, and results were often obtained from pure gross cultures which could not be secured from the smaller ones. In the germination of the spores and bulbils, Van Tieghem cells were used very freely. For this purpose cover glasses of one inch and two inches in diameter were used and carefully sealed, plenty of sterilized water having previously been put in the cells which corre- sponded in dimensions with that of the cover glasses. The large Van Tieghem cells afforded an opportunity of using cultures of con- siderable size which were usually composed of decoctions of different kinds of nutrient material, sometimes with agar to make them solid, while at other times the decoctions were used as hanging drops. HOTSON.— CULTURE STUDIES OF FUNGI. 241 In cases where the transfer of conidia, only, was desired, two methods were employed to avoid getting either bulbils or pieces of mycelium. If the conidia were quite plentiful or were on erect stalks so that they were somewhat separated from the rest of the mycelium, this could be accomplished by means of a piece of nutrient agar on the end of a sterilized platinum needle. By careful manipulation and with the aid of a dissecting microscope, they could be touched with the agar to which they adhered readily, and after exami- nation under a microscope to determine if there were only conidia present, they were immediately transferred to a new tube or a Van Tieghem cell, as the case required. In instances where the above method could not be used, or where cultures from individual conidia were required to verify the relation between a conidial form and the bulbil, Barber’s spore-picking apparatus (’07) was employed. Plate-cultures were also used to advantage in some instances for separating the conidia from the bulbils. Throughout this investigation, as already stated, the results ob- tained are based upon pure culture methods and every precaution has been taken to avoid error as a result of contamination. It perhaps should be mentioned at this point that it is the intention of the writer to deposit living cultures of most of the forms described with the Centralstelle fiir Pilzculturen. SYSTEMATIC CONSIDERATION OF THE ForMS STUDIED. As has already been indicated, “ Bulbils” must in all instances be regarded as representing imperfect conditions of the higher fungi; and like the members of other more or less clearly defined “ form- genera”’ may be associated with perfect conditions included in wholly unrelated genera of the Ascomycetes and Basidiomycetes. They may, moreover, not only represent conditions of such perfect forms, but may be further associated with one or more additional imperfect forms. There may thus be present in some instances a succession of three or even four distinct reproductive phases which together make up the individual life-cycle. It has been the aim of the present investigation, therefore, to endeavor not only to obtain further information as to the occurrence, morphology, and development of these comparatively little known structures, but by means of careful and extended work with pure cultures to make some further contribution to our knowledge of their actual relationship in different cases. 242 PROCEEDINGS OF THE AMERICAN ACADEMY. Bulbils, as a rule retain their vitality a long time so that they germinate readily after a year or more. Their maximum longevity has not been precisely determined, but in some instances, as in Grandinia and Corticium, they have been germinated after three years. This fact of the extensive longevity of bulbils is of immense importance to the fungus, enabling it to withstand long periods of unfavorable conditions, the perpetuation of the species being thus comparatively well assured. In arranging the materials available for systematic consideration it has been found most convenient to group the forms under four main divisions, namely: those which are known or supposed to be connected with perfect forms belonging to the Discomycetes; those thought to be connected with Pyrenomycetes; those which appear to be imperfect conditions of Basidiomycetes, and lastly those the actual relationships of which are still undetermined. It has seemed best to consider the last group under a single form-genus, Papulo- spora, this name having been the first which was applied to bodies of this nature, and the variations in the morphology and development in the different species being such that a separation into more than one form-genus does not seem advisable. DISCOMYCETOUS FORMS. Previous investigations have brought to light but two bulbiferous Discomycetes; an unnamed species of Peziza observed by Zukal (85, ’86), and Lachnea theleboloides (A. & 8S.) Sace. reported by Morini (’88). To these is added a species of Cubonia now reported for the first time, specimens of which were sent for identification to Professor Elias J. Durand of the University of Missouri, to whom the writer is indebted for the following diagnosis: Cubonia bulbifera n. sp. ῬΙΆΤΕ 1, Figures 1-28. “Plants single or gregarious, often crowded, sessile or narrowed to a stem-like base, turbinate, 3-10 mm. in diameter. Disk cupulate or saucer-shaped, the hymenium pale fawn-color, even when young, but in old specimens wrinkled in a cerebriform manner, externally much darker, becoming almost black with age, smooth or grumous; margin irregularly lacerate-dentate. Consistency subgelatinous, excipulum pseudoparenchymatous throughout, of nearly rounded cells, 20-25 μ HOTSON.— CULTURE STUDIES OF FUNGI. 243 in diameter, the cortical cells blackish, often protruding in groups. Asci clavate, apex rounded, not blue with iodine, 125 & 15 4. Spores 8, uniseriate, hyaline, smooth, spherical, 12 u diameter. Paraphyses slender, hyaline, only slightly thickened upward. . Mycelium giving rise to numerous rounded, black bulbils, 75-100 μ diameter, composed of rounded cells about 20 μ᾽ diameter.” Cultivated on nutrient agar. Found on dog dung from Jamaica, Paestum (Italy), Guatemala and California, and pig dung from Guatemala. This fungus was first obtained by Dr. Thaxter on dog dung from Jamaica and has been kept growing in pure tube-cultures for twenty years; since then he has found it on the same substratum from Paes- tum, Italy, and from Guatemala. It was also secured from gross cultures of pig dung and of dead flowers believed to be of the genus Criosanthes from the last named locality, while the writer has found it on gross cultures of dog-dung from Claremont, California, from which a pure culture was obtained in a manner similar to that already described. This was not difficult, since the mycelium grows with great rapidity and the bulbils are produced in abundance. The fungus Was grown, on a great variety of media until the mature perfect form was obtained. The mycelium grows well on nearly all media, pro- ducing numerous dark-colored, almost black, bulbils. The best sub- stratum for producing apothecia is bran, or rat or dog-dung, although they developed quite readily on sweet-potato agar or on Irish potato agar with a little sugar; but it was found that after the fungus had been cultivated for a long time on artificial media, it failed to produce mature apothecia. On appropriate substrata such as bran, dung, etc. the rate of growth of the mycelium is remarkably rapid. The average of several measurements made of this fungus, grown at the temperature of the laboratory is as follows: 1 em. in 24 hrs., 24 em. in 50 hrs., 33 em. in 74 hrs., and 5 em. in 120 hrs. It is white and somewhat flocculent, and does not grow in a “zonate fashion” like that of the Peziza de- seribed by Zukal, but spreads out quite evenly over the surface of the substratum. In older cultures the hyphae become quite large, often over 10 w in diameter, and densely filled with granular protoplasm, but, as they reach their limit of size, they lose their contents. Frequently when a hypha becomes broken or a portion of it is killed, there seems to be a stimulus for growth at the free end, somewhat similar to that in higher plants which are subjected to wounding. This injury of the hyphae appears to cause a sort of damming up of food material, which 244 PROCEEDINGS OF THE AMERICAN ACADEMY. is evident from the sprouting out of several small hyphae, not only from the end but also from the sides near the end of the injured part; and these often twine about each other in such numbers, that it gives the appearance of a broom-like structure. The bulbils— Often within forty-eight hours, dark bodies, which eventually become black, may be observed with a hand lens, scat- tered over the substratum or in it; they are most abundant near the point of inoculation, from this point extending out as the peripheral growth of the mycelium increases thus exhibiting a progressive forma- tion. These black bodies are bulbils which soon become very numer- ous, forming a blackish crust over the substratum and usually giving the whole culture a black aspect. This is especially true when it is grown on such media as potato agar made very hard with about forty grams of agar to the litre. In such cases the mycelium is quite scanty and procumbent, and the bulbils thus become very conspicuous; while on media like rat dung, where there is an abundance of myceli- um produced, they are not so readily seen, since they are usually formed on or in the substratum. In the development of these struc- tures which are produced so abundantly, two or three intercalary cells become enlarged and filled with granular nutrient material, as shown in Figures 11-14, Plate 1. From these cells others are produced . by budding, or short branches are formed which surround the prim- ordial cells, and which in turn become enlarged so that eventually there is produced an almost spherical bulbil somewhat flattened, 75-100 μ in diameter, the cells in the center, usually considerably larger, but all filled with protoplasm, without any definite differentia- tion of cell-contents between internal and external cells. Not infre- quently, however, the marginal cells of old bulbils lose their contents, although they retain the dark color in the wall, but this is probably due to age. As a result of the unequal production of marginal cells, the bulbils may vary considerably in size and some become quite irregular in outline. Frequently the bulbils or the primordia of im- perfect ones, especially as the cultures become old, heap together and form conspicuous dark elevations scattered over the substratum. These structures eventually assume a yellowish color, probably due partly to fading and partly to the immature bulbils that compose them. The apothectum.— Occasionally there is found a spiral primordium, as shown in Figure 1, Plate 1, produced on short lateral branches which usually divide dichotomously, sometimes of the second or third order, the ultimate branches of which coil up spirally (Figures 1+, — δικόν, HOTSON.— CULTURE STUDIES OF FUNGI. 245 Plate 1). Ordinarily there are about one and a half to two turns in the spiral, but occasionally there are as many as four. If a lateral branch fails to divide, as it often does, only one primordium is pro- duced (Figure 4, Plate 1). Frequently after the first dichotomy, one of the branches does not divide again, but coils up immediately, while the other may divide once or twice before coiling ‘(Figures 2-3, Plate 1). Thus, according to the number and regularity of these dichotomous divisions, there may appear one, two, or more primordia which are more or less closely related to each other. Usually, however, the pedicels on which they are formed elongate, and thus they may become separated from each other. When this primordium has made about two turns, sometimes as many as four, small branches are pro- duced from the sides of the coils (Figures 5-6, Plate 1), which at this stage often become separated from each other, as shown in Figure 6. It is, however, a very obscure structure, the further details of which are difficult to follow. Occasionally on media like potato, more frequently on bran, Spanish chestnuts, sweet potato, etc., and quite freely on rat and dog dung, little white patches of hyphae are seen scattered over the substratum. These are the young apothecia. The fine, white, wool-like hyphae become thickly matted together and form a white superficial dome- shaped structure with fine filaments growing out on all sides (Figure 7, Plate 1), and asthese become older, they lose their contents and as- sume a brownish color. Shortly a circular opening appears at the apex (Figure 8, Plate 1), apparently due to the rapid and extensive growth of the inner portion of the apothecium. This opening gradually increases in size, often exhibiting a conical depression in the center which, as the hymenium enlarges, becomes flat and then slightly con- vex. Microtome sections, made at the time of the opening of the apothecium or shortly before, show the upper region closely crowded with long narrow paraphyses, nearly uniform in thickness, which a little later, slightly enlarge at the ends, forming the somewhat even surface of the hymenium (Figures 9-10, Plate 1). A short distance below the center of the apothecium, when about the age of that represented in Figure 8, Plate 1, a large cell containing deeply staining material is seen in microtome sections. This appears to be the ascogonium and from it very narrow hyphae, which also stain deeply, grow up between the sterile cells of the apothecium, and eventually produce the asci. At maturity the apothecium is brown- ish, measuring 3-10 mm. in diameter and 3-5 mm. in height; often in groups and occasionally with a short stem-like base. 246 PROCEEDINGS OF THE AMERICAN ACADEMY. When a portion of the hymenium containing some of the large cells below the sub-hymenium was put in a sterilized Van Tieghem cell in an endeavor to induce the ascospores to germinate, it was found that frequently these large cells, which measure 20-25 yp in diameter, sent out germ tubes, or turned brown, secreted thick walls about themselves and resembled considerably chlamydospores (Figures 26, 27). Germination of the ascospore.-— The mature asci are quite uniform, clavate, with the apex rounded, opening by a lid, 125 w in length and 15 w in diameter at the widest place. The ascospores are hyaline, spherical, 12 μι in diameter, and arranged in a single row. At maturity all the spores from each ascus are ejected with considerable force blowing off the lid at the apex in a manner somewhat similar to that of Ascobolus, and thus are thrown in a bunch for several centimeters, and, by means of the protoplasmic material which surrounds them, adhere readily to any glass surface with which they may come in contact. These spores were allowed to strike a sterilized cover glass and then supplied with nutrient material and cultivated in a Van Tieghem cell, which had previously been thoroughly sterilized. Not only were the spores alone used as just stated, but frequently a por- tion of the hymenium with the asci was gouged out with a sterilized platinum needle and hanging drops made of 11. In an effort to get these spores to germinate, various kinds of media were used, such as — potato, prunes, bran, horse dung, dog dung, Spanish chestnuts, carrots, etc., either as a decoction, or more often solidified with agar. In spite of these varied efforts, the spores could not be made to germi- nate. The writer some time ago succeeded in getting the spores of Ascobolus to germinate in Van Tieghem cells by first crushing them lightly between two glass slides, and it occurred to him that the same method might be successful here also. Accordingly hanging drops were made as before, using different media, but the spores were first crushed with a sterilized platinum spatula on the cover-glass. This method proved successful. These spores are composed of a thick brittle episporium and a thin flexible endosporium; the object in crushing was to break the former without injuring the latter. Many of the spores thus crushed were totally destroyed, and broken por- tions of the episporium were scattered over the culture; but in a few cases, where the pressure was sufficient just to break the episporium without injuring the endosporium, it was found that germination took place in from 24 to 48 hours (Figures 22—24, Plate 1). When this occurs the endospore pushes out, forming a germ tube which is - HOTSON.— CULTURE STUDIES_OF FUNGI. 247 only a little smaller in diameter than that of the spore itself (Figure 22), and frequently when it has grown a short distance, broadens out as much as 14 uw in diameter (Figure 29). Thus the primary hypha from the ascospore is very large (7-14 uw in diameter), well filled with food material, and grows quite rapidly under favorable conditions. The culture of these germinating spores was carried on in Van Tieghem cells until bulbils were produced on the mycelium. Germination of the Bulbil— The bulbils, unlike the ascospores, germinate with great readiness within twenty-four hours and any of the cells that contain protoplasmic material may send out a germ tube, which shortly produces other bulbils from intercalary cells, as described above. When the bulbils are crushed, the contents of each of the large cells escapes surrounded by an endosporium (Figure 19) and germinates readily in Van Tieghem cells. Little significance can be attached to this fact, however, as not only are nearly all bulbils similar in this respect, but it is a common occurrence among spores which are surrounded by a thick episporium, such as the ascospores just considered. In prolonged cultures of this fungus no other spore forms have been observed. LACHNEA THELEBOLOIDES (A. & S.) Sace. The association of this species with bulbil-like bodies is reported by Morini (’88) but it is not clear from his account whether the structures seen were true bulbils, or abortive apothecia, as he believed them to be. The apothecia, which he describes and figures, are very similar to those of Cubonia bulbifera but the spherical spores of the latter distinguish it at once. The bulbil-like structures which he describes were found only in old cultures in which the nutriment was more or less exhausted, and are described as irregularly globose, 160-220 μ, and rather hard. In many cases large cells of somewhat spiral form were visible in these bodies which Morini considered “rudimentary ascogonia.” The protoplasm of the external cells, is said to be replaced by an aqueous liquid and the walls become thick and brownish-red in color. A large number of the superficial cells, as in the case of the developing apothecium, give rise to short, often septate setae, which cover nearly the whole surface. When these “bulbils’’ were transferred to fresh substrata, only those with better developed “ascogonia” continued their development until they formed apothecia identical in character with those produced normally. In all other cases, 248 PROCEEDINGS OF THE AMERICAN ACADEMY. especially those in which the so called “ascogonium” had completely disappeared, Morini observed no further development, except that in rare cases, a few paraphyses were found. He is of the opinion that these “bulbil-like” bodies are degenerate apothecia, analogous to the bulbils of Eidam, Karsten, etc., and concludes his article by saying that “the forms heretofore called ‘bulbils’ or ‘spore-bulbils’ are to be considered as exactly homologous to apothecia of which they represent forms more or less degenerate or modified during many generations of unfavorable conditions.” ῬΈΖΙΖΑ, species; not determined. A species of “Peziza’”’ found by Zukal growing on a laboratory culture may be here referred to, which according to his account is associated with small bulbils 30-40 uw in diameter, reddish brown in color, and produced by “two or three small hyphal branches which wind about one another like serpents or twist, screw-like.” The primordium of the apothecium is somewhat vaguely described. The ascospores are said to be elliptical, hyaline, smooth, about 9 X ὁ μ, obliquely monostichous, germinating readily in from twenty-four to thirty-six hours. Since this form does not appear to have been studied by means of pure cultures its connection with the bulbils described must be regarded as somewhat doubtful. PYRENOMYCETOUS FORMS. In the review of the literature a number of pyrenomycetous forms that produce bulbils were mentioned, which have been referred either to the genus Melanospora or to the allied genera Sphaero- derma or Ceratostoma. More than twenty different gross cultures made by the writer of various substrata, such as onions, straw of various kinds, paper, pasteboard, Live Oak chips, rotten planks, tubers of Dahlia, old leather gloves, ete., have produced bulbils which in pure cultures have yielded melanosporous perithecia. In a few cases the perithecial form was found on the original sub- stratum and cultures were made from the cirri of discharged asco- spores, which on nutrient agar produced bulbils. In addition to bulbils, all of these forms also produce ovoid, hyaline conidia borne on characteristic bottled-shaped sterigmata. ‘The ascospores are yellowish brown, becoming black or smoke-colored, asymmetrical, more or less crescent shaped. They vary but little HOTSON.— CULTURE STUDIES OF FUNGI. 249 in size, the measurements of Melanospora papillata and M. cervicula averaging 10 X 25 4 while those of M. anomala are slightly larger, 12 X 28 uw. These variations, however, are so small that they could not alone be considered as specific. The size and shape of the asco- pores also correspond quite closely with those of Melanospora Gibel- liana and Sphaeroderma bulbilliferum. At maturity the ascospores appear as an irregular black mass in the center of the perithecium. As in all the species of Melanospora the asci are very evanescent. The walls become gelatinous and swell by the absorption of water, which increases the volume to such an extent that the mucilaginous mass protrudes from the ostiole, carrying out with it the embedded spores. If the atmosphere is somewhat humid, this mass of spores, as they are forced out, aggregate in a spherical mass at the mouth of the ostiole; but if the air is dry as they are pushed out, they adhere to- gether into a long, twisted, tendril-like filament, something like the paint as it is squeezed out of an artist’s paint-tube. These cirrose structures may measure from 10-18 mm. in length, and twist up into a variety of shapes. The spores not infrequently germinate while still in the cirrus, giving it a white appearance. Microtome sections show no paraphyses between the asci, but from the walls there grow out more or less conspicuously into the cavity above the asci, numerous hyphal branches, as paraphyses, which con- verge radially and extend upwards towards the ostiole. These prob- ably aid in the formation of the neck when it is present. In general the culture methods used were the same for all. Gross cultures of the various substrata were made in crystallizing dishes which were half-filled with sphagnum and covered with white filter paper, on which the substratum was placed. The whole was then well supplied with water and covered with a piece of plain glass and set in a place in the laboratory where it would be protected from the direct sunlight. When bulbils were observed, individual ones were carefully picked out under a dissecting microscope and cultures made from them, until a pure culture was obtained. ‘These were grown on various kinds of media until perithecia with the characteristic long cirri of ascospores, were obtained. Transfers of the ascospores were then made by touching one of the aerial cirri with a piece of nutrient agar on the end of a sterilized needle. In all cases pure cultures of ascopores obtained in this way produced bulbils. The germination of the ascopores was followed in Van Tieghem cells until bulbils were again produced on the mycelium, thus demonstrat- ing the connection between the ascospore and the bulbil. 250 PROCEEDINGS OF THE AMERICAN ACADEMY. In these forms the very young perithecium can be readily distin- guished from the bulbil, not only by its mode of development when that is different, but also by the color. The bulbils turn brownish at a very early stage in their development, such as is represented, for example, in Figure 2, Plate 2, while on the other hand, the peri- thecia frequently remain colorless, or nearly so, until they are beyond the size of the average mature bulbil, and the ascogonium usually can be distinctly seen in the form of one or two large cells lying towards one side of the young perithecium. The question of sexuality in connection with the formation of the ascogenous primordia has not been worked out. Structures have been observed that might well be taken for antheridial branches, but their attachment was not constantly or certainly observed, so that this phase of the problem will have to be left for future consideration. Among the twenty bulbil cultures from different sources which have been found by the writer to produce melanosporous perithecia, at least three distinct species appear to be distinguishable. Although these forms possess ascospores that show little if any variation, the differences in their perithecia, bulbils and secondary spore-forms are such that they cannot be included in a single species. Moreover, the characteristics are believed to be sufficiently distinctive to warrant their consideration as separate species. They have therefore been named Melanospora papillata, M. cervicula, and M. anomala. There thus appear to be several closely related Melanospora-like forms, in- cluding Sphaeroderma bulbilliferum, Melanospora Gibelliana and M. globosa all of which give rise to bulbils. The differences which distinguish the perithecia of these forms may be summarized as follows: Melanospora Gibelliana; neck of perithecium long and tapering, with terminal setae, asymmetrical ascospores. M. globosa; neck of perithecium longer than M. Gibelliana, no well- defined terminal setae, symmetrical ascospores. M. papillata, τι. sp.; perithecium with a distinct papilla only with terminal setae, asymmetrical ascospores. M. cervicula, τι. sp.; perithecium with a short neck, terminal and lateral setae, asymmetrical ascospores. M. anomala, τι. sp.; perithecium more or less definitely papillate, with occasional indications of abortive terminal setae, asymmetrical ascospores. Sphaeroderma bulbilliferum; perithecia without papillae or setae. The species of “Sphaeria “mentioned by Biffen as associated with Acrospeira mirabilis and the species of “Ceratostoma’”’ connected HOTSON.— CULTURE STUDIES OF FUNGI. 201 with bulbils by Bainier may also be melanosporous and will be re- ferred to later on. Melanospora papillata n. sp. PLaTE 2, Figures 1-26. Perithecia scattered or gregarious, superficial, membranous, semi- translucent, straw-colored to light brown, globose to pyriform, 350-450 uw X 400-500 yp, papilla surmounted by erect, somewhat divergent, continuous setae, 100-170 ww in length; primordium a group of one or more intercalary cells; ascospores asymmetrical, somewhat crescent-shaped 10 X 25 yp, yellowish brown becoming black; conidia abundant, hyaline, spherical to ovoid, on flask-shaped sterigmata; bulbils yellowish brown, irregular in outline, 50-60 μι in diameter, sometimes considerably more than this. On Live Oak bark (Quercus agrifolia Née) from Pomona, Cali- fornia. A pure culture of this species was easily obtained by making a transfer of the ascospores in the manner already described, on rich nutrients, fairly soft, with about 20 gm. of agar to the litre, and both perithecia and bulbils were produced abundantly. On substrata, however, poorly supplied with nutrient material, such as sterilized agar-agar, or even on a medium well supplied with food material if made very hard (about 40-50 gm. of agar to the litre) the bulbils are very sparingly produced if at all, the mycelium is quite inconspicuous and the perithecia appear scattered over the surface more or less abundantly. In its capacity to retain its power of producing peri- thecia this species resembles M. cervicula, while it is in sharp contrast to some other melanosporous forms studied in which, after long artificial cultivation the bulbils tend to become the dominant mode of reproduction and the perithecia are produced sparingly if at all. The bulbils. The hyphae, which vary in diameter from 4-7 p, are hyaline, with numerous oil globules and prominent cross walls, and are usually very scantily developed. The bulbils make their appear- ance as small straw-colored bodies scattered somewhat sparingly and usually in small patches over the surface of the substratum. In the process of development, which was carefully followed in Van Tieghem cells and in pure cultures in test-tubes, hyphae divide up into short intercalary almost isodiametric cells, one or more of which enlarge (Figure 1, Plate 2) while the contents becomes densely granu- lar and filled with oil globules. At this stage these enlarged cells are 252 PROCEEDINGS OF THE AMERICAN ACADEMY. colorless or opalescent with a comparatively thick wall and look much like chlamydospores. The adjacent cells of the filament on either side of them become stimulated and also enlarge to some extent, but remain colorless longer than the others, although they are eventually incorporated into the bulbil. The primordial cell or cells soon become brownish and produce others by gemmation, which in turn produce still others (Figures 2-5, Plate 2), so that the mature bulbil finally consists of one or two, occasionally more, large central cells with shghtly thickened walls, surrounded by a number of smaller less highly colored ones, with thinner walls. The mature bulbils measure from 50-60 uw in diameter, although they may vary consider- ably. Sometimes three or four intercalary cells enlarge and take part in this process, producing an elongated, somewhat irregular bulbil, while at other times there are as many as eight or ten such cells; but in this latter case they seldom go farther than the production of a few lateral cells which soon become empty and colorless, as is shown in Figure 7, Plate 2. Not infrequently the terminal cell or series of terminal cells becomes the primordium (Figures 24-25), the further development of which is the same as the one already described. In Van Tieghem cell- cultures, bulbils are sometimes produced with more central cells than ordinarily occur in tube-cultures, and these, which are usually spheri- eal, contain oil globules which give them a peculiar, somewhat opalescent appearance. The cortical cells in such cases are somewhat flattened, as indicated in Figure 22, Plate 2, a condition which may be due to the pressure exerted by the increased number of the central cells. The perithectum.— The form of the primordium of the perithecium is essentially the same as that of the bulbil but the former, as has already been stated, can, even in the early stages of its formation, be readily distinguished from the latter by the fact that it is colorless. It can be distinguished also from the primordium of the perithecium of M. cervicula, which in many respects it resembles, by the fact that the latter turns brownish at a much earlier stage in its development, producing a large number of radiating hyphae, so that its outline is soon indistinguishable. Usually one, rarely two, intercalary cells take part in its formation, and from these, two or three large cells are produced laterally by budding (Figure 8, Plate 2). From the intercalary cells, or, more frequently, from the adjacent ones of the hypha, branches are sent HOTSON.— CULTURE STUDIES OF FUNGI. 252 up which eventually enclose this group of large cells. These branches which divide up into short cells, form the wall of the perithecium. Sometimes, as in the case of the bulbi!, a terminal cell may become the primordium, as is evidently the case in Figure 10, Plate 2, where there are two large cells which have originated from a terminal one. The mature perithecium is straw-colored, globose or slightly pyri- form, measuring 400-500 uw in diameter, but often much smaller than this, the variations in size are largely due to the character of the medium on which it grows. It is surmounted by a crown of setae which surround the ostiole and are colorless, 100-170 j in length, stiff, erect, straight, and tapering to a point. There are no lateral setae of this nature, but frequently superficial cells near the base of the perithecium may send out filaments which serve as attachments to the substratum. The perithecia often occur grouped in consider- able numbers and not infrequently two or three are found which have more or less fused during their development, having no doubt arisen from primordia which were in close contact with each other. Some time after their formation the cirri of ascospores begin to assume a whitish appearance which is due to the presence of numerous germi- nating spores producing many abnormalities. A very common form in such cases is shown in Figure 14, Plate 2 where, instead of a regular germ tube, a large opalescent, spherical body is formed at the end of the spore, which contains a great deal of granular material and stains deeply. Occasionally a second such body is produced, and from these one or more lateral branches may arise (Figures 18-20, Plate 2). Not infrequently a series of these swollen cells appears terminating a branch and these become spherical and form a bulbil-like structure (Figure 17) such as issometimes met with in Van Tieghem cell cultures (Figure 21). One of the most striking features of these germinations is the copious formation on the germ tubes of ovoid conidia which arise from bottle-shaped sterigmata and usually adhere in short chains, although they sometimes cohere at the tips of the sterigmata in a spherical mass. As already mentioned conidia similar to these are also quite frequently met with on the mycelium in all parts of the culture, and when the spores collect in masses the fructification might readily be mistaken for that of Hyalopus. In some cases the outer cells of the bulbils increase in numbers until the whole structure is about half the size of a perithecium, although very irregular and sclerotium-like. In each case, however, the cells of the original bulbil retain their deep tan-color, while those which have resulted from this secondary growth are distinguished 254 PROCEEDINGS OF THE AMERICAN ACADEMY. by light colored walls resembling those of the typical perithecium. The occurrence of such abnormal forms, which may be quite frequently produced on media rich in nutriment such as bran-agar for example, and their resemblance to young perithecia, suggested the possibility of a direct development of perithecia from bulbils similar to that suggested by Bainier (’07), and an effort was accordingly made to determine this point. Individual bulbils showing this tendency were isolated and their further development watched in Van Tieghem cells, while others were transferred to different kinds of media, moist cotton, moist filter paper, ete., but in no instance could they be in- duced to develop into perithecia, although when the moisture was sufficient, they produced numerous germ tubes which grew out forming the typical mycelium. Melanospora cervicula, n. sp. PLATE 3, Ficures 16-24. Perithecia scattered or gregarious, superficial, membranous, semi- transparent, straw-colored to brownish, globose to pyriform, 350- 450 Χ 450-550 μ, with a definite neck 85-140y in length, terminal setae 100-170 uw in length, erect, somewhat divergent, continuous, sharp, subulate; lateral setae on the neck and upper part of the peri- thecium; ascospores asymmetrical, somewhat crescent-shaped 10 Χ 25 μ, yellowish brown becoming black; conidia common in tube cultures, hyaline, spherical to ovoid, on flask-shaped sterigmata; bulbils yellowish brown, irregular, normally 50-60 mw in diameter, sometimes 100 yu, primordium one or more intercalary cells. This form is also said to produce conidia on secondary “ Harzia-like”’ heads, and chlamydospores resembling those of Acremoniella atra. On rabbit dung, Cambridge, Mass. This melanosporous fungus was obtained from Dr. Thaxter who had grown it for some time as a pure culture. It was originally found on a gross culture of rabbit dung from the vicinity of Cambridge, Mass., and has proved to be of special interest on account of its differ- ent methods of reproduction. In addition to perithecia and bulbils, this fungus seems to have associated with it two other spore forms, chlamydospores resembling those of Acremoniella atra and also conidia produced on secondary heads resembling those of the genus Harzia. Alcoholic material furnished by Dr. Thaxter was used for the study of these two modes of reproduction. This material was the result of a series of transfers HOTSON.— CULTURE STUDIES OF FUNGI. 255 of the cirri of ascospores and therefore probably pure. The writer has under cultivation transfers of this same fungus but although it has been grown on various kinds of media, both very rich and very poor in nutrient material, and hard and soft, ete., yet thus far he has not succeeded in obtaining either the chlamydospores or the “ Harzia- like” fructification. This is probably due to the fact that the pro- duction of these structures is secured under certain peculiar conditions not readily controlled. In general this fungus resembles M. papillata in form and habit of growth. The predominant type of reproduction in both is by asco- spores the production of bulbils being scanty, while in some cases, as on attenuated agar cultures, they are not produced at all. The peri- thecium of MW. cervicula which is usually 400-500 μι in diameter, has a definite neck 85-140 yw in length, while MW. papillata which is slightly smaller, seldom reaching 500 in diameter, has no neck but often a papilla-like structure from which the setae arise. Moreover, the former probably produces conidiophores of the “Harzia type’? and also chlamydospores which resemble those of Acremoniella atra. The Bulbils— The mycelium is colorless, procumbent or only slightly aerial, growing evenly over the surface of the substratum. The hyphae, which are copiously septate, measure 5-7 μ in diameter, but often large swellings occur in them which seem to act as storage organs and from which several branches may grow out as shown in Figure 21, Plate 8. These are found not infrequently on attenuated artificial media such as agar alone without any nutriment, on which the mycel- ium is very scanty, being barely visible even with the aid of a hand lens. On such media, it should also be noted that as in M. papil- lata, bulbils are not produced. It further resembles the latter in the mode of development of the bulbils, the primordium consisting of a group of intercalary cells. It is, however, subject to considerably greater variation and many irregular, incomplete or imperfect forms appear. Since the mode of development is essentially the same as that described for M. papillata, it will be unnecessary to repeat the description here. They are, however, produced very sparingly on most media, and on some, such as that just mentioned, do not occur at all, although on a rich substratum not too hard, such as sugar, chestnut or bran agar they are produced quite abundantly. The perithecium.— In general the perithecium resembles that of M. papillata, but is clearly distinguished by having a definite neck. They, however, vary considerably in size, sometimes reaching 550 μ in diameter, their form often being somewhat contorted, with only 256 PROCEEDINGS OF THE AMERICAN ACADEMY. a slight difference in size between the neck and body, while at other times several may be grown together. The neck is short, 85- 140 uw in length surmounted by a group of terminal setae of about 100- 170 μ in length. The mode of development of the perithecia is some- what variable. Although at times they seem to be produced from in- tercalary cells, yet more frequently a short lateral branch is produced which may form a close coil of one or two turns, and occasionally even a definite spiral is found as is shown in Figure 19, Plate 3. The young perithecia turn brownish at a much earlier stage of their devel- opment than either those of M. papillata or M. anomala. This fact, together with the large number of radiating hyphae that are produced from the initial cells, a condition not occurring in either’ species just mentioned, make it very difficult to follow the early development. When the perithecium is young before the neck is produced, filaments with thick brownish walls, apparently stiff and with prominent septa, are seen scattered sparingly over the surface and radiating from it. They are formed by the outgrowth of some of the peripheral cells, and as the perithecium becomes older, as has already been stated, their number increases and some grow down into the substratum and act as hold fasts. The “ Harzia-type”’ of reproduction.— This mode of reproduction which was studied from material preserved in alcohol appears in small tufts scattered over the surface of the substratum. Short lateral branches become swollen at the end after the fashion of Oedocephalum or Aspergillus, and from this head a number of flask- shaped sterigmata are produced, on the ends of which occur secondary heads crowded with hyaline conidia which are usually spherical and sessile but occasionally more or less ovoid and furnished with short stalks (Figure 24, Plate 3). The secondary heads seem to vary con- siderably in size, and being so completely covered with conidia it was difficult to determine at all times the exact relation of the different parts of this fructification. In several cases there appeared to be little or no swelling of the secondary head, but with the limited amount of material at hand this could not be determined with certainty. Occasionally the head instead of being spherical is somewhat elongated, and the bottle shaped stalks, on which the secondary heads are formed, are scattered along the margin of this as shown in Figure 23, Plate 8. This fungus also ‘produces numerous spherical conidia on bottle-shaped sterigmata along the margin of the hyphae, similar to those described for the other melanosporous forms. The chlamydospores.— On the preserved material already referred HOTSON.— CULTURE STUDIES OF FUNGI. 257 to, there were also found associated with the “ Harzia-like”’ fructifiea- tion, chlamydospores which are ovoid, smooth, brownish, thick- walled, and have the distal end rounded. They are produced usually on short lateral branches which taper towards the tips and may be continuous or septate. ‘The mature spores are quite uniform in size, about 17 X 21 μ, although there were some that appeared to be mature, which were slightly smaller than this. These spores resemble both in color and form those of Acremoniella atra Sace. There are certain other fungi that produce imperfect forms of the “Harzia”’ and “Acremoniella”’ type which will be further considered below in connection with P. aspergilliformis. Melanospora anomala n. sp. PuLaTE 2, Figures 27-30; Puate 3, Figures 1-15. Perithecia scattered or gregarious, superficial membranous, straw- colored or light brown, globose or subglobose, 250-350 μ. Χ 350- 450 μ, ostiole formed in connection with a definite but inconspicuous papilla without setae, primordium a spiral of 4 or 5 coils; ascospores asymmetrical, somewhat crescent-shaped 14 Χ 28 μ, yellowish brown becoming brownish black; conidia, hyaline, spherical to ovoid, on flask-shaped sterigmata: bulbils yellowish brown, variable in size 70-140 u in diameter, sometimes elongated ones 1804 in length, primordium a group of intercalary cells. On Spanish chestnuts in laboratory culture. Gross cultures of Spanish chestnuts, which were imported probably from Spain obtained by the writer in the Boston market, produced numerous brownish colored bulbils when cultivated in moist chambers. By using the general methods already described, separate bulbils were transferred to sterilized nutrient-agar tubes and, after a few transfers, were obtained pure. The mycelium of this fungus is white and more or less aerial, vary- ing according to the media in which it is grown. When grown on soft chestnut agar, it becomes quite flocculent, while on chestnut decoc- tion it forms a more or less felted layer over the surface, assuming the brownish color of the liquid; but on potato agar its growth is rather scanty. The diameter of the hyphae varies from 2.5-7 μ. The bulbils.— Scattered over the aerial hyphae and on the substra- tum are seen numerous small yellowish-brown bulbils, which, when examined microscopically, are found to vary considerably in size and outline, many of them nearly spherical, others somewhat elongated. 258 PROCEEDINGS OF THE AMERICAN ACADEMY. Usually there is no differentiation between the cortical and central cells, but in old bulbils several empty cells, which may or may not be colorless, are often found loosely attached to the periphery. The central cells are often larger than the more superficial ones, but this is not always true, since in many instances they are perfectly uniform throughout. These bulbils are usually developed from a lateral branch which divides up into short cells. These produce short secondary branches (Figures 27, 29, 30, Plate 2) which also divide up into short cells and may produce others by a process of gemmation. Sometimes the primordium consists of a group of intercalary cells (Figures 28, Plate 2, and Figure 14, Plate 3) which may produce other cells by budding in a manner somewhat similar to that of M. papil- lata. At maturity the bulbils are irregularly spherical, about 70- 140 μ in diameter, but where several interealary cells have taken part in its formation, the long axis frequently measures 180 μ. This bulbil may be distinguished from M. papillata or M. cervicula by the fact that the cells are usually homogeneous throughout, while in the latter two there is a more or less definite cortex. The margin is also often more irregular in the bulbil under consideration as is shown in Figure 15, Plate 8. In the immature bulbils which show this uneven outline more markedly than the mature ones do, there sometimes appear short branches of two or three seriate cells which extend beyond the others. The perithectum.—In an effort to induce this fungus to produce the perfect form, it was grown on various kinds of media. Decoctions of potato, bran, corn meal, Spanish chestnuts, ete., were hardened with agar-agar, some hard, some soft, but nothing except variations in the size and development of the bulbil could be obtained. Finally, after removing the shells of some fresh, sound chestnuts, the kernels were sliced up and used for cultures. On this medium perithecia were produced in abundance. These are almost spherical in form and vary from 300-400 u in diameter, no ostiole being developed until they are nearly mature, at which time a few cells about the opening form a definite, though inconspicuous papilla. Terminal setae are wholly absent, and only rarely do the superficial cells produce lateral filaments. Frequently, however, short projections are observed from some of the cells that compose the papilla, as if an attempt were being made to produce setae. The perithecia are light yellowish-brown in color, much lighter than that of the bulbils, and so translucent that the spores can be readily seen grouped together in a black mass in the center (Figure 12, Plate 8). HOTSON.— CULTURE STUDIES OF FUNGI. 259 Development of the perithectum.— The primordium of the perithecium is quite different from that of the bulbil. In this case a short lateral branch coils up spirally, usually making about four or five turns, but in some cases as many as eight. Figures 1 to 8, Plate 3, represent successive stages in the development of the spiral. Usually the second and part of the the third turn become enlarged while branches are given off from the first or from the cells below it. These branches grow up around the spiral and often send secondary branches in between the swollen lower coils so that they are forced apart (Figures 7, Plate 3). The branches continue to grow until they have enveloped the whole spiral, which soon loses its characteristic form. It would appear that the upper portion of the spiral either becomes a disorganized mass of mucilaginous material or not infrequently seems to be pinched off and ejected during the formation of the wall of the young perithecium, as is shown in Figure 7, Plate 8. By the time the wall is completed all that can be recognized of the spiral are two or three large cells which come to lie free in a cavity usually towards one side of the peri- thecium and which stain deeply (Figures 9-10, Plate 3). Sometimes branches seem to come off from each of the coils, so that one finds the spiral with a number of very short lateral branches produced from its outer surface. Occasionally also the lateral branch that produces the spiral, while making its first coil, divides into short cells and sends off secondary branches from these, as shown in Figure 3, Plate 3. Whether either or both of these develop into perithecia or bulbils, or are to be regarded as abnormalities, could not be determined, since they were of rare occurrence. Conidia on bottle-shaped sterigmata, similar to those produced by M. papillata also occur in this species (Figure 13, Plate 3). Germi- nating ascospores particularly, produce them abundantly in a dry atmosphere, but they are more sparingly developed on the mycelium. This fungus resembles somewhat a form described by Berlese (92) under the name of Sphaeroderma bulbilliferum, which is referred to below. The former has, however, a slightly smaller perithecium (300-400 yw in diameter) with a papilla about the ostiole, while the latter is 400-500 yu in diameter, and has no papilla, the ostiole being flush with the surface. The Sphaeroderma moreover is said to have connected with it large two-celled chlamydospores, which have not been found associated with M. anomala although the writer has re- peatedly searched for them. Berlese does not describe the method of development of the bulbils, but states that “the sporeballs resemble those described by Mattirolo as belonging to Melanospora Gibelliana,”’ 260 PROCEEDINGS OF THE AMERICAN ACADEMY. The bulbils of the latter are not unlike those of M. anomala in size, color and mode of development. The species of “Sphaeria,”’ referred to by Biffen (’02) in connection with Acrospeira mirabilis, also resembles somewhat M. anomala. It differs from the latter, however, in several important respects. The perithecium has no papilla about the ostiole, the ascospores are sym- metrical and the primordium of the bulbil is a spiral. Again the mode of development of the perithecium from a spiral primordium resembles somewhat that of Melanospora stysanophora described and figured by Mattirolo (’86). The mature perithecia however, are different, M. stysanophora having a distinct neck. The latter is also said to be associated with a Stysanus-like fructification. MELANOSPORA GIBELLIANA Mattirolo. This species was found by Mattirolo on a gross culture of decayed chestnuts in moist sand, and besides melanosporous perithecia and bulbils it also produced chlamydospores and conidia on bottle-shaped sterigmata. The perithecium, which develops from a spiral primordium, 15 somewhat pyriform with a long neck surmounted by terminal setae. The neck, however, is considerably longer than that described for M. cervicula. The ascospores are brownish-black and asymmetrical, somewhat similar to those described for the other melanosporous forms. The bulbils are said to be nearly spherical, pale yellow to brownish- yellow, and often 100 μ in diameter, with a colorless cortical layer of cells resembling somewhat the appearance of Papulospora coprophila. In its development a short lateral branch divides and forms a number of short secondary branches which intertwine forming an irregular spherical body varying considerably in size. This species also is said to have associated with it chlamydospores somewhat resembling Sepedonium, as well as conidia on bottle- shaped sterigmata. MELANOSPORA GLOBOSA Berl. In the same article in which he describes Sphaeroderma bulbilli- ferum (’92) Berlese also describes Melanospora globosa which he found growing on small pieces of decaying wood and herbaceous material. The perithecium of this species is, as the name indicates, globose, 250-280 μι in diameter and 360-450 uw (rarely 500 μὴ long. The neck HOTSON.— CULTURE STUDIES OF FUNGI. 201 is well developed, 110-200 μ in length. The ascospores differ from those, already described, in being symmetrical. The other forms have asymmetrical ascospores which are somewhat crescent-shaped. Besides the perfect form this species is said to have: microconidia which resemble those of Acrostalagmus; chlamydospores that are of the type of Acremoniella atra; and bulbils which he considers of the same nature as similar structures described by Mattirolo. Berlese succeeded in obtaining bulbils on the mycelium produced from asco- spores but he failed to find any perfectly developed. SPHAERODERMA BULBILLIFERUM Berl. This species which is described by Berlese (’92) was found growing on dead leaves of Vitis, Cissus, and Ampelopsis. It is said to have several kinds of reproductive bodies, such as ascospores, bulbils, conidia and chlamydospores. The perithecium is globose or sub-globose, 400-500 uw in diameter, without any neck, setae or papilla. These characteristics distinguish it from any of the melanosporous forms already referred to. It resembles M. anomala but is slightly larger and has no papilla. The ascospores are brownish-black and asymmetrical. The bulbils are yellowish, nearly spherical, 80-150 μι in diameter, consisting of polyhedral cells and surrounded by a layer of empty cortical cells. They are said to resemble quite closely those described in connection with Melanospora Gibelliana. The conidia occur in chains on bottle-shaped sterigmata resembling those of the melanosporous forms already referred to. The chlamydospores, which measure 32-40 X 24-25 μ, are described as yellow, oval, smooth, composed of two unequal cells, and formed terminally on the ends of short lateral branches. “CERATOSTOMA”’ sp. indet. Bainier (’07) has reported that he has determined the connection of a perithecium of the genus Ceratostoma with Papulospora asper- gilliformis. He is of the opinion that the bulbils in this instance are immature perithecia and that, under proper conditions as regards nutriment and moisture, they may be induced to complete their development. In this form, the bulbil is produced by a short lateral branch which coils up spirally, the coils becoming quite compact. One or more of the terminal cells enlarge and eventually become filled with 262 PROCEEDINGS OF THE AMERICAN ACADEMY. conspicuous food material. The cells below the spiral send out branches which divide and may, in turn, produce others. These grow up around the spiral and completely envelop it, thus forming a somewhat spherical mass of cells. In a moist atmosphere these are said to develop into sclerotium-like bodies. By transferring these large bulbils to pieces of moist bread, Bainier succeeded in inducing them to develop into perithecia which he refers to the genus Ceratostoma, although it is not evident why this form should not also be referred té Melanospora. This subject will be further dealt with below under Papulospora aspergilliformis. In connection with pyrenomycetous forms it will be well to con- sider briefly two additional species which may be regarded as doubt- fully pyrenomycetous. FORMS DOUBTFULLY REFERRED TO PYRENOMYCETES. Papulospora candida Sace., parasitic on Geoglossum, has been re- ported by Dr. Thaxter to be connected with hypocreaceous perithecia found on specimens of the host obtained in South Carolina; but this material was, unfortunately, not available for examination, and since pure cultures of this fungus grown on different media have thus far failed to produce any perfect form, its position must, for the present at least, remain more or less uncertain. The fact, however, that the bulbil is definitely connected with a Verticillium would seem to afford strong evidence of its hypocreaceous nature. A second doubtful form is Acrospeira mirabilis (Beck ἃ Br.), with which Biffen (’02) has associated a species of “Sphaeria,’’ but since he was unable to obtain the bulbils or “chlamydospores” as he terms them, of Acro- speira from pure cultures of the ascospores, his conclusions must be accepted with some reserve. PAPULOSPORA CANDIDA Sacc. Piate 4, Figures 1-47. This fungus was first found by Ellis who collected it in New. Jersey and distributed it by N. A. F. No. 3673. The species appears to be common and distributed from N. Carolina to Maine. The material for the present investigation was found growing abundantly as a parasite on Geoglossum glabrum in a maple Sphagnum swamp near Walnut Hill, Mass. It was first described (Mich. II, p. 576) as Papulospora candida, by Saccardo who also mentions that Verti- HOTSON.— CULTURE STUDIES OF FUNGI. 263 cillium agaricinum_Link, var. clavisedum (Mich. II, p. 577) is asso- ciated with it. A large number of specimens of Geoglossum, with plenty of Sphag- num and leaf mould about each, were collected — some infected, others not — and were grown under bell jars or in a large germinating vessel with a glass top. It was thus kept growing for nearly two months, until it could be determined whether the Papulospora would grow as a saprophyte on artificial media. A number of tube cultures were made of the bulbils on various kinds of media, the most success- ful of which were the ascoma of Geoglossum itself. About a dozen large specimens of these with long stalks were selected and each put in a test-tube which had previously been supplied with about half an inch agar. These were then sterilized in an autoclave, the object of the agar being simply to hold the specimen in place and thus lessen the chances of contamination in making the transfers, ete. On this medium a pure culture was eventually obtained, which was then transferred to other media such as potato, corn meal, chestnut, horse dung, ete., hardened with agar. This fungus grows fairly well as a saprophyte, better on hard than on soft media such as potato and bran, but very slowly on horse dung, on which, after a month, it had not grown much more than an inch from the point of inocula- tion. Associated with the Papulospora on the ascoma were found, among other fungi, specimens of Plewrage anserina (Rabh) Kuntze and Verticillium agaricinum Link, the latter producing in pure cul- tures very large and conspicuous, brownish sclerotia. On its natural host Papulospora candida forms conspicuous white blotches spreading over the upper portion of the ascoma (Figure 47, Plate 4), and if not too wet, extending down the stem. Although the host is usually found in damp sphagnum swamps, the parasite is largely confined to those specimens that grow tall, so that their tops are comparatively dry. The mycelium is white, procumbent, branch- ing copiously, but soon becoming indistinguishable as such, even with a good hand lens, mainly on account of the large number of bulbils that are formed which give the whole fungus a powdery appearance. When examined under a microscope the mycelium is opalescent, owing to the presence of numerous oil globules (Figures 42, 44, Plate 4) and other colorless material in the cells. The cultures become com- pletely covered with the white powdery bulbils which a little later assume a characteristic cream color. The bulbils— During the process of development of the bulbil a short lateral branch divides up into a number of cells and the end 264 PROCEEDINGS OF THE AMERICAN ACADEMY. one enlarges and usually also the second or third (Figures 29-37, Plate 4). From these, other cells are then produced by budding, the lateral walls of which eventually adhere closely to those ad- jacent, so that there comes to be from two to six large central cells surrounded by a number of smaller ones, all filled with granular proto- plasm, the only apparent difference being in their size. As they mature, however, the inner and outer cells become markedly differ- entiated. The former, which are large with conspicuously granular contents and with numerous oil globules, secrete a thick hyaline wall, while the latter, which become empty and spherical, adhere to each other loosely, their contents probably being absorbed by the central cells (Figure 41, Plate 4). Although the terminal cell is usually the most prominent in producing the larger central cells, yet one or both of the two adjacent cells may take the lead and, owing to their lateral growth a somewhat crosier-like coil may even occasionally be produced by one or more of these secondary branches. Germination of the bulbils.— For the purpose of studying the germi- nation, bulbils in different stages of development were placed in Van Tieghem cells. In about twelve or fifteen days the marginal cells of those that were immature — that is, those whose superficial cells still contained protoplasm — began to send out vegetative branches, one or two from each cell (Figure 42, Plate 4); but the central cells were not observed to produce tubes at this stage. After about a month the mature forms begin to germinate, but very sparingly, each of the large central cells usually sends out a single germ tube which readily pushes aside the loosely adhering peripheral cells. The germ tubes or vegetative hyphae, as the case may be, usually divide up into short cells which become swollen with the protoplasmic contents and more or less constricted at the partitions (Figure 42, Plate 4). The conidia.— The erect septate conidiophores of the so-called Verticilium agaricinum (Link) Corda, var. clavisedum Sacc., already referred to, are invariably associated with the bulbils in pure cultures, and are thus shown to be not, as Saccardo supposed, accidentally concomitant but a regular phase of the life cycle. Figure 45, Plate 4, shows bulbils and the Verticillium fructification definitely connected on the same erect hypha. This phenomenon 15 of so frequent occur- rence that there is no possibility of error. The conidiophores are simple or branched, with the sterigmata in whorls, varying greatly in number, commonly in threes and frequently clustered at the apex. The mature conidia are ellipsoidal to oblong and rounded at both ends, varying considerably in size, the average measurements being HOTSON.— CULTURE STUDIES OF FUNGI. 265 14 X 15 μ, although the length may vary from 12 to 15 yu. In this respect it differs from [΄. agaricinwm in which the conidia are smaller and ovoid in shape. Both of these forms have been cultivated in pure cultures for some time and seem to be absolutely distinct, the one, V’. agaricinum, producing ovoid conidia often clustered at the apex of the sterigmata as well as an abundance of large brownish sclerotia not associated with bulbils, while the other has oblong coni- dia, rounded at both ends, somewhat larger than the former, and on germination the mycelium invariably gives rise to bulbils, without any trace of the sclerotia. The germination of the conidia of P. candida was carefully followed in Van Tieghem cells, using different kinds of nutrient media. In these cultures many interesting variations were observed, as is shown in Figures 1-12 and 15-27, Plate 4, all of which have the same magnifi- eation. Figures | and 2 show the variation in the size of the conidia. During the first twenty-four hours they enlarge by the absorption of water, becoming almost spherical (Figure 4), in which condition they are ready to germinate, the diameter at this stage varies from 12-18 μ. The germ tube, which may appear at one or both ends (Figures 7, 20) or from one or both sides of the conidium (Figures 6, 8), sometimes grows out to form a mycelium (Figure 10) on which bulbils and the conidial fructifications are produced; but more often, in Van Tiegham cells at least, it rounds up and forms another large cell. Several large cells may be produced in a similar way, which become almost spherical in shape and densely filled with granular protoplasm and oil globules, and from these acting as central cells, other smaller ones are formed laterally by budding, and in about sixteen days a bulbil consisting of two to six large central cells surrounded by a layer of smaller ones, all containing protoplasm, results. ACROSPEIRA MIRABILIS (Berk. and Br.). Puate 5, Figures 18-23. Acrospeira mirabilis (Berk. and Br.) appeared on a gross culture of Spanish chestnuts obtained from the Boston market. It was from this same material that Melanospora anomala was obtained but from other gross cultures. The former was first described by Berkeley and Broome in 1861, a more detailed account being given by Berkeley in his “Introduction to Cryptogamic Botany.’ Massee (03) refers to it as a very destructive parasite doing a great deal of damage to chestnuts in Spain, but states that “nothing as to the life 266 PROCEEDINGS OF THE AMERICAN ACADEMY. history of the parasite is known.” Before Biffen (’02) examined this species, the only method of reproduction known was by its so-called “chlamydospores”’ which at maturity consist usually of one large, thick-walled, chocolate-brown, warty cell and three or more colorless cells adhering closely to it. By the use of pure cultures Biffen claims to have succeeded in obtaining not only the “chlamydospores,” as described by Berkeley and Broome, but also what he calls “spore balls”? (bulbils) and definite perithecia. The mycelium of Acrospeira is fine, colorless, procumbent, more or less sparingly developed, and produces large numbers of reproductive bodies, which, in their development and structure, are bulbils rather than “chlamydospores.” They are so abundant that the whole surface of a culture, which would otherwise be white, assumes a brownish aspect. The readiness with which these bulbils are pro- duced makes it comparatively easy to trace their development, which, in brief, is as follows: an erect lateral branch usually divides into three secondary branches (Figure 18, Plate 5) each of which coils up much like that of Papulospora parasitica, to be considered below. They make about one to one-and-a-half coils and divide into three cells by cross septa. The middle one of these three, as a rule, en- larges rapidly, forming the functional spore (Figure 21, Plate 5) (the central cell of P. parasitica), but occasionally the end cell (Figure 20, Plate 5) more rarely the third, is the one that functions in this respect; while the other cells of the coil, ordinarily three or more in number, grow less rapidly and eventually lose their contents, become colorless, and adhere to the side of the large cell. If the marginal cells should increase in number so as to enclose the large cell completely, there would be practically the same condition as exists in P. parasitica (Figures 16, 17, Plate 5). In the present form, however, the large cell becomes dark brown in color and develops a thick wall, which eventually becomes warty, and measures 25-30 uw in diameter. Fig- ures 18-23, Plate 5, illustrate the stages in the development of this bulbil. Thus in Acrospeira we have a structure that is only slightly -Jess complex than that seen in P. parasitica, a form in which many imperfect bulbils can with difficulty be distinguished from some of those of Acrospeira, their only difference being due to the absence of a warty episporium. ‘These bulbils were grown on various kinds of sterilized nutrient material, and most of the experiments described by Biffen were repeated. The culture conditions were varied with regard to media and other conditions of growth, in many. of these experiments, but more bulbils of the same kind were always produced HOTSON.— CULTURE STUDIES OF FUNGI. 267 and never, so far as the writer has observed, have any indications been seen of the development of “spore balls,” or perithecia such as have been described by Biffen. BASIDIOMYCETOUS FORMS. As has already been mentioned (p. 238), bulbils were first reported among the Basidiomycetes by Lyman (707), who not only definitely connected one form with Corticium alutaceum (Schrader) Bresadola, which is dealt with briefly below, but also refers to two other kinds of bulbils, the mycelia of which have well marked clamp-connections; but basidiosporic fructifications were not produced abundantly enough to allow of their identification. Dr. Lyman has kindly supplied the writer with specimens of these forms for the purpose of comparison, which will be referred to under their respective species. The methods used here were much the same as those already de- scribed, except that more gross cultures of wood were used with different amounts of moisture. The best results were obtained from decoctions of bran in one or two litre Erlenmeyer flasks with pieces of rotten wood that extended considerably above the liquid, so that the mycelium could obtain the degree of moisture that best suited it. In order to keep the pieces of wood in place and thus lessen the chances of contamination a quantity of agar was sometimes put in the bottom of the flasks. GRANDINIA CRUSTOSA (Pers.) Fr. Puate 6, Ficures 1-10. Bulbils of this species were obtained from at least ten different sources, mostly on substrata such as rotten chips of Live Oak (Quercus agrifolia Née), old canvas, paper, cardboard, ete., from Claremont, California. It has been found also by Dr. Thaxter on gross cultures of rabbit dung from Mass. and on rotten wood from Buenos Ayres, and is probably the same as that referred to by Lyman (ΟἿ, p. 166), which was obtained by Mr. A. H. Chivers on a gross culture of bits of wood, paper, etc. The mycelium, which shows quite marked clamp-connections, is colorless, procumbent, producing numerous white fibrous, rope- like strands of hyphae which radiate conspicuously in all directions from the point of inoculation. The white mycelium, however, soon takes on a light straw-colored aspect, owing to the formation of bul- 268 PROCEEDINGS OF THE AMERICAN ACADEMY. bils in large numbers, which gradually become darker as they mature. When grown on nutrient agar in large receptacles like Erlenmeyer flasks, after the mycelium has covered the whole substratum with powdery bulbils, new centers of growth-activity occur at different points on the surface of the culture, and the radiate development of the hyphae and the subsequent formation of bulbils are repeated on the top of those first formed. If the flasks have plenty of nutrient and do not dry up, this process may be repeated two or three times, the amount of mycelium, and consequently the number of bulbils formed, decreasing each time, so that eventually there appears a thick powdery mass with here and there large, white, rope-like strands of hyphae persisting, which is all that can be distinguished of the mycelium. The bulbils are usually more or less spherical in shape, varying from 52 to 88 uw in diameter, although often exceeding this size, espe- cially when the primordia of two happen to be so close together that their hyphae intertwine, thus forming a large irregular body. The individual cells are large, densely filled with granular material and oil globules, spherical at first; but the central .ones soon become angular by pressure, while the marginal ones retain more or less their original form. There is no differentiation of a cortical layer; the cell wall and contents are uniform throughout, except that occasion- ally some of the peripheral cells which project beyond the others lose their contents, but this is the exception and is probably due to age. The bulbils— The hyphae which take part in the formation of the bulbils become enlarged, conspicuous, and more or less contorted on account of the prominence and swollen nature of the clamp-connec- tions, which often occur at short intervals. The lateral branches from these divide up into short cells, so that there comes to be a number of almost spherical hyaline cells with fairly thick walls and filled with granular material and oil globules (Figures 4-9, Plate 6). During the formation of new cells, which are also spherical in shape and produced by budding from the marginal ones, the central cells gradually lose their original form and become angular, as a result of the lateral pressure or resistance offered by the outer cells. When the bulbils are nearly mature, they assume a light straw or “rusty- cinnamon” color. Figure 10, Plate 6, represents a mature bulbil, drawn on the same scale as the other mature forms. This method of development follows very closely that described by Lyman (707) in connection with Corticium alutaceum, considered briefly below. Formation of basidiospores.— The basidiosporic fructification of OE .ἀδνν ἁανα ee τὰ ΘΟ δ. οι νὰ. HOTSON.— CULTURE STUDIES OF FUNGI. 269 Grandinia has been produced on gross wood cultures of this bulbil and also on test-tube cultures of bran-agar of about 40 gm. of agar to the litre, by three or four of the ten cultures from different sources under cultivation. Preparatory to its formation, the mycelium ceases to produce bulbils and forms a sort of incrustation, chalk-white in color and becoming pustulate by the time the spores are formed, Figure 1, Plate 6. The pustules on examination are found to be made up of more or less thickly interwoven branching hyphae, which have become enlarged and densely filled with granular material and oil globules, the ultimate ramifications of which form the hymenium (Figure 2, Plate 6). The basidia, which form a somewhat loose hy- menium, each produce four spores, which are ellipsoidal to oblong in shape, measuring about 4 X Sy. These spores were germinated in Van Tieghem cells and the growth of the mycelium followed until the formation of new bulbils, which were transferred. to nutrient agar media, where they produced mycelia and bulbils like the original culture. On tube cultures this fungus occasionally produces typical sclerotia, which are formed by the massing together of many hyphal branches which remain colorless for some time and thus are easily distinguished from the bulbils. Moreover, they are larger, 400-500 u in diameter, irregular in shape, somewhat darker in color at maturity, and com- posed of smaller, compact cells. Grandinia also produces conidia of the Oidium-type on slender clampless conidiophores, such as are described by Lyman (’07) for Corticium alutaceum. CorRTICUM ALUTACEUM (Schrader) Bresadola. The bulbils of this species were obtained from Dr. Farlow, who found them on a piece of rotten oak bark collected at Chocorua, N. H. It was comparatively easy to get a pure culture, as the bulbils are pro- duced in large numbers and germinate readily. This form has been carefully compared with specimens of Corticiwm alutaceum obtained from Dr. Lyman and they proved to be the same. The development of the bulbil and the character of the conidia are practically identical with those described for Grandinia and, as these have been well worked out in pure cultures by Lyman (’07), it is not necessary to repeat the results here, a detailed description of which may be obtained by con- sulting his article, pp. 160 and 196. The mode of development of the bulbils and the character of the conidia, however, have been carefully 270 PROCEEDINGS OF THE AMERICAN ACADEMY. verified. Lyman obtained his cultures from the basidiospores collected on old rotten oak logs in the field and pure cultures from these produced bulbils. The writer began his cultures with bulbils, also collected in the field, and, after a great number of unsuccessful attempts, finally succeeded in obtaining a basidiosporic fructification similar to that described by Lyman. This was accomplished by using gross cultures of partly decayed wood in two litre Erlenmeyer flasks with sufficient agar to hold them in place. The mycelium, as usual, produced bul- bils profusely on the agar and wood, but after six or eight weeks near the top of the pieces of wood conspicuous patches of white mycelium appeared, which eventually produced the hymenium and basidiospores of C. alutaceum. Papulospora anomala n. sp. Plate 6, Figures 11-19. This form, which was obtained from four different localities,— three from the vicinity of Claremont, California, found on Live Oak chips, and one on an old paper from Cambridge, Mass.,— has been grown on a variety of substrata in the hope that it would produce its perfect form, but thus far all these efforts have failed. That it belongs to the Basidiomycetes is shown by its clamp-connections, which, however, are not so prominent as those in the two preceding forms, from which it is further distinguished by the dark brown, opaque, almost black color of the bulbils, the compact nature of their cells, and their mode of development. The mycelium is white, procumbent, scanty, slightly aerial on some substrata, with a large number of con- spicuous oil globules, and not infrequently contains swollen intercalary cells, which are also densely filled with food material and probably act as storage organs. The bulbils— The primary hyphae are small, seldom more than 3 uw in diameter, and do not produce bulbils; but scattered over the secondary hyphae, which vary greatly in width, often reaching 10 μ and under some abnormal conditions 14 μ, are seen slightly swollen, colorless, intercalary cells, quite different from those mentioned above, about 4 or 5 uw in diameter, sometimes projecting considerably and resembling short stunted branches; at other times the base of a short lateral hypha swells slightly and forms the primordium (Figure 12, Plate 6). From the primordial cell or cells branches are sent out in different directions, the basal cells of which become spherical and in turn may produce other similar branches (Figures 13-15, Plate 6). HOTSON.— CULTURE STUDIES OF FUNGI. pain The lateral walls of these basal cells adhere firmly to each other and the cells become incorporated into the bulbil. Figures 11-15, Plate 6, illustrate the early stages in the develop- ment, and Figures 14 and 15 show the formation of the spherical cells at the center, around the initial cell or cells, while Figure 16 represents a little later stage, which is composed of small hyaline cells with very indistinct walls and forming almost a spherical body with few, if any, cells projecting beyond the others. About this stage, or usually a little later, it would appear that the bulbils cease to form new cells, or, if any, very few, and that the further increase in its size is chiefly due to the enlargement of the individual cells which compose it and which, up to this period, have been small, hyaline, with in- distinct walls. As these cells enlarge, there is quite a strong lateral pressure exerted, which tends to make the walls angular, which in the meantime have become more prominent and gradually assumed a brownish tint, that later becomes a dark brown, almost black. As a result of this mode of development, the bulbil at maturity has a clear-cut, even margin, without any appendages or sharp projections, nearly spherical in form, except where some cells in the process of enlargement increased faster than others or in cases where two pri- mordia were formed close together and their early branches became intertwined, forming an elongated, compound structure. The color, which becomes so deep that even the cell walls cannot be distin- guished, may be bleached out by placing them in potassium hydroxide for a few hours. The mature bulbils (Figure 17, Plate 6) vary in size, usually measuring from 125 to 175 μ in diameter, although occa- sionally some are even larger. BuxBit “No. 200.” This form was obtained from Dr. G. R. Lyman and was originally found by Dr. G. P. Clinton in the vicinity of Cambridge, Massachu- setts, on a fragment of an old newspaper in a field. In general this species resembles Grandinia in the mode of development of the bulbils, the presence of conidia and the clamp-connections of the hyphae. The bulbils, however, are much darker and the mycelium does not form the white, fibrous, radiating strands that are so charac- teristic of Grandinia. On gross cultures, especially of wood or horse dung agar, the hyphae mass together in conspicuous papilla-like elevations, which are much more prominent than the fructification of Grandinia. These 2.7. PROCEEDINGS OF THE AMERICAN ACADEMY. elevations are composed of closely compacted basidia-like structures. Unfortunately thus far the writer has observed only a few scattered basidia with basidiospores so that it has been impossible to obtain a specific determination. BULBILS NOT YET CONNECTED WITH A PERFECT FORM AND INCLUDED IN THE FORM-GENUS PAPULOSPORA. Key to the Species of Papulospora. I. Primordium interealary. AC vibulloils. folache ks tl ete aire Se Bins ote ners eee cas P. pannosa n. sp. B. Bulbils yellowish to dark brown. 1. Bulbils, brownish-yellow, central cells 28-55 » in diameter. P. immersa τι. sp. 2. Bulbils straw-color, central cells 10-20 μ in diameter. P. irregularis τι. sp. Bulbils dark psoray hyphae with clamp-connections. P. anomala τι. sp. ΕΣ vo. II. Primordium one or more lateral branches. A. Primordium normally a single lateral branch. 1. Primordium a spiral. a. Cells of bulbil heterogenous, definite cortex. i. One central cell. a.’ ‘Cortex* completes, 21.24 Aes eee P. parasitica. τ imcomplete 2% 4.2.45 Acrospeira mirabilis. ii. More than one central! cell. a. Spiral in one plane, cortical cells spinulose. P. spinulosa τι. sp. 8. Spiral in more than one plane, 2-6 central cells. (a) Bulbils a dark brown......... P. coprophila. (b) ΠΕΡΙ ΘΕ. P. rubidan. sp. b. Cells of bulbil homogenous. i. Bulbils brown 21-36 win diam... P.sporotrichoides τι. sp. il. ‘‘ steel gray 21-36 » in diam... ..P. cinerea τι. sp. 2. Primordium not a spiral. a. Bulbils large, 100-7504 in diam.....P. aspergilliformis. |b). = ΘΟ ΞΘΡ π ἴπτα. cream colors eee eee P. candida. B. Primordium two or more lateral branches forming a spherical aggre- ration of:cells.at the tops can: <2. epee teen P. polyspora n. sp. Heretofore fungi producing bulbils have been referred chiefly to the form-genera Papulospora and Helicosporangium, but the characters on which these two have been based are not clearly defined, and as already stated, it does not seem desirable to recognize more than one form-genus.. Since Papulospora was the name first employed to represent bodies of this nature, all the fungi that the writer has ex- amined that produce bulbils, the perfect form of which has not been determined, are placed in this form-genus which may be described as follows. HOTSON.— CULTURE STUDIES OF FUNGI. 273 Papulospora. Mycelium extensive or scanty, flocculent or procumbent, usually white but sometimes dark colored. Reproduction by means of bulbils, i. e., reproductive bodies of more or less definite form, com- posed of a compact mass of homogeneous or heterogeneous cells which may be few or many, but are always developed from primordia of more than one cell. Other modes of reproduction may be present. For convenience bulbils may be grouped under three heads: those which form an intercalary primordium of several cells; those which typically originate from a primary spiral; and those that are pro- duced by a perpendicular branch or branches which do not form a spiral. As has already been pointed out the distinction between simple bul- bils and compound spores on the one hand, and the more complex bul- bils and sclerotia on the other, is not always definite, and in certain instances it is difficult to determine to which category a given struc- ture belongs. Compound spores are reproductive bodies of more than one cell, having a more or less definite form, and are usually the result of a successive or simultaneous division of a single cell. On the other hand, sclerotia are compact bodies capable of reproducing the plant and formed rather by the massing together of vegetative filaments, forming a pseudoparenchymatous tissue, but not developed from a group of more or less definitely related cells. Moreover, the individual cells of a sclerotium are not at all spore-like or independent of each other. Bulbils, are reproductive bodies, more or less definite in form and mode of development, and normally derived from primordia of more than one cell, rather than the result of successive or simultaneous divisions of a single cell, and their individual cells are more or less independent and spore-like. Papulospora immersa n. sp. PuaTE 10, Figures 17-25. Mycelium white, septate, scanty, procumbent, growing in or on the substratum; bulbils, light brownish-yellow, irregular, 88-150 μ in diameter, but very variable, sometimes the long axis exceeding 260 μ, often immersed; central cells large 28-55 w in diameter, angular, with conspicuous oil globules; 50-70 cells in surface view, but in irregular forms 100 cells, no differentiation of internal and external cells. No other mode of reproduction at present known. 274 PROCEEDINGS OF THE AMERICAN ACADEMY. On horse and dog dung from Cambridge, Massachusetts, and rabbit dung from Innerkip, Ontario. Both the bulbils and the mycelium usually grow more or less below the surface of the substratum. The former are often found immersed more than a centimeter. It is easily distinguished from P. polyspora by its mode of development and from P. pannosa by its color, the latter being black. It resembles most nearly P. irregularis, from which it may be distinguished by its darker color, the size and con- spicuous contents of the cells of the bulbils and the fact that the latter become more or less imbedded in the substratum. The mycelium, since it is formed largely in the substratum, is in- conspicuous in tube-cultures and is composed of large swollen hyaline cells, densely filled with oil globules and often much contorted (Figure 17, Plate 10). In older cultures the cells lose their contents. This fungus was grown on different kinds of media, but could not be induced to develop any other mode of reproduction. It grows well on bran and horse dung agar, the bulbils often becoming very large and numerous just below the surface of the substratum, forming almost a continuous layer, and often producing a more or less hard crust. In contrasts of mycelia in plate cultures, a marked heaping of the hyphae occurs where the two mycelia come together, and the bulbils seem to be somewhat larger, and more irregular in this region, but no other marked difference was observed. The bulbils— The primordium of the bulbil consists of one or more intercalary cells which become much enlarged. For example, Figure 17, Plate 10, a later stage of which is seen in Figure 23, shows several such cells, all of which would have taken part in the formation of a somewhat elongated irregular bulbil, such as is shown in Figure 23. On the other hand, Figure 18 represents a primordium which consists of a single cell, and Figures 19-22 are further stages in its develop- ment. In the latter case a more or less spherical bulbil is the result (110-148 » in diameter), while in the former it is more irregular, often exceeding 260 μ through the long axis. The method of enlarge- ment, however, is exactly alike in both cases, that is, short lateral branches are produced from the bases of which are cut off a series of short cells which enlarge, becoming spherical at first and later, as the bulbil increases in size and the cells are subjected to lateral pres- sure, forming a compact angular mass in the center. Occasionally the branches are replaced by cells which, arising as lateral buds, become spherical and in turn give rise to other buds, the lateral walls of which adhere closely and ultimately form a more or less ore HOTSON.— CULTURE STUDIES OF FUNGI. Sle spherical or elongated bulbil with a fairly even margin, the central cells of which soon become angular. In either case all the cells are filled with conspicuous oil globules. At maturity there is no differ- entiation of central and cortical cells, but all are uniformly filled with food material, the central ones being larger, 28-35 μι in diameter, “and more angular than those nearer the periphery. Papulospora pannosa n. sp. Puate 6, Figures 20-25; Piate 8, Figures 28-31; Pate 9, Figures 18-20. Mycelium white at first, becoming dark smoke-colored, 8-10 u in diameter, somewhat shaggy; bulbils black, irregular, variable in size and outline, sometimes 350 μι in diameter, but usually consider- ably less; cells homogeneous throughout, 200-300 cells in surface view; primordium, a group of intercalary or terminal cells. No coni- dia observed. On laboratory cultures of rabbit and goat dung, and on corn-cobs from Claremont, California. Pure cultures of this fungus from about fifteen different sources were obtained and grown on various kinds of media and the mycelium from the different sources contrasted with each other, but thus far it has not developed any other mode of reproduction than the bulbils. This species is easily distinguished from most of the others by the color of its bulbils. The only other black form is that of Cubonia bulbifera from which it differs in size and the character of its outline, which is quite even and regular in the latter, as well by the fact that the hyphae are black at maturity. The bulbils—The mycelium which grows well on a variety of media in tube-cultures, appears somewhat shaggy, is white at first, gradually becoming dark smoke-colored, with prominent cross walls which remain rigid when the cells collapse (Figure 31, Plate 8). The hyphae which are 3-4 » in diameter when young and hyaline, gradually increase in size until they are 8-10 μ in diameter, and have already become dark in color at the time the black bulbils are produced. During the formation of the latter, the hyphae become much dis- torted, and divide into a series of short, somewhat inflated cells which are separated by constriction at the septa (Figure 24, Plate 6), some- what after the fashion of Cubonia bulbifera, but the successive cells of these series are much more irregular and of greater diameter. These enlarged cells send out lateral branches (Figure 18, Plate 9), from 276 PROCEEDINGS OF THE AMERICAN ACADEMY. which are cut off short basal cells which assume a spherical form, ‘become swollen and may produce other branches similar to the primary ‘ones. This mode of development is illustrated by Figures 20-24, Plate 6, and Figures 18-19, Plate 9. Instead of the enlarged cells producing branches, however, other cells may arise laterally from them by gemmation, become spherical, and may in turn give rise to others in a similar fashion. In either case the lateral walls of adjacent cells eventually adhere firmly, thus forming a compact group, each cell of which is almost spherical at first, but later be- comes irregular. The further multiplication of the peripheral cells is subject to considerable variations. Not infrequently the primary or secondary branches, owing to local variation, grow much faster than others and thus produce more cells in that region of the bulbil. If there are several of these points of special activity, the mature bul- bils may be quite irregular in outline. Occasionally a bulbil is formed from a single lateral branch (Figures 28-30, Plate 8), new cells being formed by a process of budding or by short branches as in the other cases. Ordinarily, at maturity, they are more or less spherical or somewhat elongated, their margins roughened by projecting cells (Figure 20, Plate 9) and are very variable in size, sometimes as large as 350 in diameter. There is no differentiation between the inter- nal and external cells as far as contents are concerned. The central cells are, however, as a rule, larger and more angular. Papulospora irregularis n. sp. Pirate 9, Figures 11-17. Mycelium white, more or less procumbent; bulbils hyaline, be- coming light straw-color, somewhat spherical (140-170 μ᾽ in diam.) to irregular in outline (250-300 uw in diam.), margin very uneven; primordium a group of intercalary cells. On rat dung, Kittery Point, Maine. A pure culture of this species was comparatively easy to obtain. In the hyphae, which are hyaline, procumbent and inconspicuous, certain intercalary cells become enlarged and, by a process of budding, these give rise to other cells which in turn may produce still others. Sometimes short lateral branches are produced, the basal cells of which enlarge and take part in the formation of the bulbil (Figure 15, Plate 9). The young bulbils are colorless, covering the substratum, but in older cultures they turn light straw-color. They are usually somewhat spherical in form, measuring 140-170 » in diameter, but HOTSON.— CULTURE STUDIES OF FUNGI. 277 frequently run into irregular sclerotium-like bodies, 250-300 w in diameter. In old cultures the hyphae often form a felted mass over the substratum. This mode of development is similar to that of P. pannosa, from which, however, it is easily distinguished by the color of the mycelium and bulbils, those of the latter species being black. It also resembles P. immersa, but it is lighter in color and does not have such large cells with conspicuous oil globules and the bulbils are not immersed in the substratum. Figures 11-17, Plate 9, illus- trate the mode of development of this bulbil. Papulospora spinulosa, n. sp. PLATE 9, Figures 1-10. Mycelium white, scanty, septate, procumbent, becoming slightly brownish when old, 3.5 uw in diameter, the old hyphae somewhat larger; bulbils hyaline until well developed, at maturity light choco- late-brown, somewhat spherical, 55-88 » in diameter, 50-60 cells in surface view; primordium a coiled lateral branch which remains prominent throughout the development, becoming empty and show- ing slight thickenings in the walls. No other means of reproduction known. On rat dung, Kittery Point, Maine. This fungus was found on a gross culture of rat dung obtained from Kittery Point, Maine, and has been grown for about three years on various media without producing any reproductive body other than bulbils. The mycelium is white and grows quite -sparingly on most media. It has been found that bran agar or rat dung agar is the best nutriment on which this species will grow. The bulbils.— During their early stages of development the bulbils are hyaline until they are about half grown, at which time they begin to turn a light brown and at maturity assume a chocolate-brown color, often covering the whole substratum with several layers, so that all appearance of hyphae is lost sight of, except around the margin where a white zone about 5 mm. in width indicates the actively growing region of the mycelium and the formation of new bulbils. In the process of development a short lateral branch coils up, usually crosier fashion (Figures 1-4, Plate 9), although ocecasion- ally the tip somewhat overlaps, as shown in Figure 3, Plate 9. The primary loop varies greatly in size, as may be seen from a compari- son of Figure 1 with the other figures representing the development, all of which are drawn on the same scale, but even these large open 278 PROCEEDINGS OF THE AMERICAN ACADEMY. primordia form eventually quite close coils. The helix which consists of one to one and one-half turns, divides into cells from which short lateral branches are produced, usually growing towards the center, rarely outward (Figures 5-7, Plate 9). These branches twine and intertwine, the lateral walls adhering firmly so that eventually a somewhat spherical body is formed which superficially resembles the sporangium of afern. The cells of the original spiral are more promi- nent than the others, usually slightly elevated with well marked walls, and correspond to the annulus, as will be seen from Figures 9-10, Plate 9. Figure 10 is a view of an immature bulbil, looking down on the “annulus,” while Figure 9 is a side view of the same. At maturity the bulbil, which is nearly spherical, is 55-88 yp in diameter. The cells of the primary coil usually become empty and lighter colored, showing slight thickenings scattered over their surface, oc- casionally projecting slightly, thus giving the’ appearance of minute spines. Sometimes a lateral hypha divides dichotomously and each branch coils up and produces a bulbil. Similar branches may be produced directly from the superficial cells of a bulbil (Figure 8, Plate 9). The mode of development in this form resembles that of certain species of Urocystis, such as U. cepulae, the common onion smut, in which a lateral branch coils up, making about one turn, and this divides into cells from which secondary branches are given off. Figures 4, 5, 6 and even 7, Plate 9, might almost equally well illustrate the development of Urocystis cepulae. Papulospora coprophila, nov. comb. Helicosporangium coprophilum Zukal (’96). PLATE 10, Figures 1-16. Mycelium white, septate, flocculent, abundant, persistent; bulbils, dark brown, more or less spherical, 30-40 uw (rarely 60 μ) in diameter, with one to four (sometimes as many as 10) large central cells sur- rounded by a cortex of empty colorless or slightly brownish ones; primordium spiral, of one to four turns, the end cell usually becoming a central cell. Conidia on bottle shaped sterigmata, frequently in white tufts scattered over the surface of the substratum. On onions, straw, horse dung, ete., Cambridge, Massachusetts, and California. Onions have proved very productive as a substratum for bulbils. Some onions obtained from the Boston market which had been shipped a HOTSON.— CULTURE STUDIES OF FUNGI. 279 from New York State, produced several different kinds and among them P. coprophila which has been secured from at least ten differ- ent sources, not only on onions, but frequently on horse dung and straw. It grows readily on potato and bran agar, but, like many of the other species, after continued artificial cultivation the mycelium becomes scanty and the bulbils few. In such cases it can be re- juvenated by growing on a gross culture of sterilized fresh horse dung, on which the mycelium is developed luxuriantly and becomes floccu- lent, producing bulbils and conidia abundantly. This species appears to be the same as that described by Zukal (’86) under the name of Helicosporangium coprophilum which he found growing on horse dung. The general appearance of the bulbils of these two forms, their size, color, and at least one phase of their development seem to be identical. The form under consideration, however, differs from the description given by Zukal in producing a copious supply of flocculent hyphae. This may be due to the differ- ences in the conditions of cultivation. P. coprophila resembles in mode of development the species referred by Eidam to Helicosporan- gium parasiticum Karsten, but the bulbils of the latter are brick-red, with yellowish cortical cells which, judging from the figures, are much less prominent than in the present form. The only other close allies are P. parasitica and P. spinulosa, the former easily distinguished by its single large central cell, the latter by its mode of development, and the presence of slight thickenings in the walls of the cortical cells. This form develops sparingly on very moist substrata. On nutrient potato agar containing sugar, however, or on fresh horse dung, it grows well. Contrast cultures of mycelia from different sources yielded nothing more than additional variations in the filaments and bulbils. The former grew much more luxuriantly at the points of contact of the two sets of mycelia. The bulbils— A short lateral branch coils up, making about one or one and a half turns, the end cell enlarges, becomes spherical and frequently turns brownish. As it continues to increase in size its two lateral faces protrude more or less conspicuously and may even become subpendent, as in P. parasitica (Figure 4, Plate 5). These projections, however, often behave differently from those of the latter, since they are frequently cut off and thus form other enlarged central cells. Sometimes the second or even the third cell of the coil enlarges and takes part in the formation of the central cells. Those that do not enlarge grow out laterally over the surface of the central cell or cells and eventually completely enclose them. Figures 13-15, 280 PROCEEDINGS OF THE AMERICAN ACADEMY. Plate 10, show what appear to be arrested forms of this mode of development, all of which have brownish walls. These conditions resemble somewhat the mode of development figured by Zukal (’86). About three or four days after inoculation on fresh nutrient agar which contains sugar, there frequently appears a spiral primordium of three or four turns, as shown in Figures 1-6, Plate 10, which divides into cells from which short secondary branches are produced, or other cells are formed by gemmation, so that eventually the spiral is enclosed by them. The cells of the spiral enlarge and usually lose their characteristic form. The lateral walls of the superficial cells adhere firmly together, so that eventually there comes to be one to four (sometimes as many as ten) large central cells, surrounded by a cortical layer of empty and often colorless cells (Figures 10-11, Plate 10). The development of the spiral may be checked at nearly any stage of its formation and thus certain variations in the form and number of the central cells of the bulbil may result. This variability in the formation of the spiral seems to be largely due to the character of the medium which, when favorable, usually produces quite regular primordia with the maximum number of coils, while under less favora- ble conditions, or after the substratum has been once run over with the hyphae, many variations are found. Some of the spirals are loosely coiled (Figures 1-2, Plate 10), while others are close and compact (Figures 4, 6, Plate 10). Although the primordium usually loses its spiral form early in its development, it is occasionally found surrounded by an irregular layer of cells, as shown in Figure 8, Plate 10. These bulbils resemble somewhat the primordium of a perithecium, like that of Melanospora as shown in Figures 5-6, Plate 3. On account of this resemblance an effort was made to induce them to develop into some perfect form, but although many and varied kinds of experi- mentation as to media, moisture and temperature, were tried, all efforts proved unsuccessful. There are also associated with this bulbil spherical or slightly ovoid conidia, on bottle shaped sterigmata, identical with those found in connection with the melanosporous forms. These conidia, which frequently appear on conspicuous white tufts of hyphae scattered over the surface of the substratum, may be formed individually, in chains, or occasionally in a moist atmosphere may cohere at the ends of the sterigmata in a spherical mass. Although, as a rule, the sterigmata occur laterally on the walls of the hyphae, they are often found clustered on irregularly swollen branches and exhibit all the variations referred to below in connection with P. aspergilliformis, HOTSON.— CULTURE STUDIES OF FUNGI. 281 although the characteristic ‘“ Aspergillus-like” fructification illus- trated in connection with.the latter has never been observed. These conidia were picked out with Barber’s apparatus and transferred to nutrient tubes where they germinated and produced mycelium on which bulbils developed. In this respect they differed from those of P. aspergilliformis, which, although repeated efforts were made, could not be induced to germinate. When these bulbils are crushed the contents of the large central cells escape, surrounded by a thick endosporium (Figure 11, Plate 10). These cells germinate readily in Van Tieghem cells (Figure 12, Plate 10). Papulospora rubida n. sp. PuaTE 8, Figures 12-27. Mycelium white, procumbent or slightly aerial on some media; bulbils more or less spherical, 30-40 yw in diameter, with 2-5 large central cells surrounded by a layer of empty cells which usually retain their yellowish red color, at maturity the whole culture has a brick-red aspect; primordium a spiral, with many modifications; conidia on bottle-shaped sterigmata, but not formed in white tufts. On dog dung from Buenos Ayres. This species was obtained from a pure culture received from Dr. Thaxter, which he has had growing for a number of years. It was originally found on dog dung from Buenos Ayres. In general it resembles P. coprophila in size, form, and mode of development. It is easily distinguished, however, bythe appearance of the culture. The mycelium is more or less procumbent and the bulbils give the whole substratum a brick-red aspect, in old cultures forming a leathery incrustation which often cracks as the medium dries up. The my- celium of P. coprophila, on the other hand, is flocculent, filling the whole lower part of the test-tubes in slant cultures, and the bulbils give the culture a dark brown appearance. The cortical layer is colorless and more definitely marked in the latter species. The hyphae of the form under consideration vary from 3-14 μ in diameter and, especially in old cultures, have well marked cross walls. Large swollen intercalary cells (Figure 24, Plate 8), are often formed, which seem to act as storage cells, as they are densely filled with granular, protoplasmic material and oil globules. The bulbils— A short lateral branch coils up Gili usually mak- ing one to one and a half turns (Figures 12-15, 21, 22, 27, 25a, Plate 8) and divides up into cells all of which become more or less swollen. 282 PROCEEDINGS OF THE AMERICAN ACADEMY. One or more of these cells, as a rule the first or second or both of them, increase in size beyond the rest, becoming densely filled with granular material and oil globules, while the other cells grow out laterally (Figure 16, Plate 8) and eventually enclose the enlarged cells in a manner similar to that of P. coprophila and P. parasitica. It some- times happens that when the end cell enlarges, protuberances are pro- duced from the lateral sides, which may even become subpendent, as in P. parasitica (Figure 26, Plate 8). The development of the cortical cells is shown in Figures 16, 21, 22 and 27, while Figure 25 is a median section and Figure 18 a surface view of the mature bulbil. Thus at maturity the bulbil is more or less spherical, 30-40 w in diameter with 1-5 (usually 2 or 3) large central cells each of which varies from 10-14 w in diameter (Figures 16, 25, Plate 8), surrounded by a cortex consisting of a single layer of empty cells, rarely more, which is often incomplete. The walls of the cells of this cortical layer usually retain their color. Occasionally the short lateral branch instead of making but one or one and a half turns continues the spiral until from three to five turns are formed (Figures 17, 20, Plate 8). From the cells of the spiral are produced others laterally by budding, which eventually adhere to each other laterally, thus forming a wall about the spiral. This is similar to the process observed in connection with P. coprophila. This species also produces conidia on bottle-shaped sterigmata similar to those described in P. coprophila, but they do not, as far as the writer has observed, occur in white tufts scattered over the sub- stratum as they do in the last named species. Papulospora sporotrichoides n. sp. PLaTE 12, Figures 1-41. Mycelium white, procumbent, usually scanty; bulbils dark choco- late colored, somewhat spherical or flattened, 21-36 u in diameter, primordium a spiral of one to two turns, with conspicuous oil globules, the spiral sometimes not well marked. Conidia and conidiophores of the Sporotrichum type. On Live Oak chips (Quercus agrifolia) and corn cobs from Clare- mont, California, and Maple chips from Newton, Massachusetts. The bulbils—In the development of the bulbil a short lateral or terminal branch coils up, divides into a number of short cells with walls well distinguished, forming a close spiral of two or, rarely, three turns. This process is illustrated by Figures 1-9, Plate 12. During Ee ee were ee Oe HOTSON.— CULTURE STUDIES OF FUNGI. 283 the very early stages of development, the primordia are colorless, somewhat larger than the ordinary hyphal threads with more granular material. The walls, however, begin to turn brown shortly after division takes place. In Figure 5, for example, the walls are dis- tinctly colored. In the mature bulbil the spiral form can sometimes be recognized (Figure 8, Plate 12), but more frequently, owing to the unequal enlargement of the cells composing the coils, or some modi- fication in the development which will be spoken of later, all trace of it is lost. The development of these bulbils was carefully followed in pure Van Tieghem cell cultures, and many interesting modifications were observed. Quite frequently, as illustrated in Figures 12-14, Plate 12, before the spiral has completed one turn or the walls of the individual cells thickened, one of the cells, usually the third or fourth from the tip, grows out into a vertical branch and coiling divides into cells similar to the first. The second coil may repeat this same process, so that two or three or even four coils like that which is shown in Figure 14, Plate 12, are formed one above the other, each producing a separate bulbil. These usually continue their development inde- pendently of each other, but not infrequently the primordia overlap and a single “compound” bulbil of two or three spirals, as the case may be, is the result. Occasionally this secondary branch is_pro- duced on the opposite side of the cell so that it grows into the concave portion of the first coil as shown in Figure 15, Plate 12. In some in- stances a single coil only may be formed, the cells of which enlarge as usual (Figures 19-25, Plate 12) becoming divided during the process, by thin cross partitions which are at first hardly visible without stain- ing. The multicellular bulbil thus produced, does not become dark at once like the normal type but remains hyaline for some time, slowly changing color and only after it has become fully mature does it assume the dark brown tint of the more common type from which, however, it is eventually indistinguishable. The Conidia——A conidial form of reproduction, which usually appears on old cultures after a large number of bulbils have been produced, is also connected with this fungus. These conidia are of the Sporotrichum type and were obtained from pure cultures by the transfer of individual bulbils. It seemed desirable, however, to obtain the bulbil-type from germinating conidia in order to eliminate all chance of error; but this was found unexpectedly difficult for the reason that single spores isolated by Barber’s apparatus refused to germinate although cultivated in varied media. The conidial form 284 PROCEEDINGS OF THE AMERICAN ACADEMY. is as a rule scantily developed in older cultures only, but by using a special nutrient composed of a decoction of bran, Spanish chestnuts,. horse dung and rotten wood hardened with agar, an abundant pro- duction of conidia was obtained after two months, the conidiophores (Figures 35-36, Plate 12) rising well above the substratum at the mar- gin of the culture, so that large quantities of spores were readily obtained in an absolutely pure condition. Cultures of these yielded about two per cent of germinations after twenty days. The development of these germinating conidia (Figures 38-41, Plate 12) was continuously followed in Van Tieghem cells until bulbils were produced on the mycelium derived from them. The conidiophores (Figures 35-36, Plate 12) which are colorless at first but become light grayish brown at maturity, are larger (3.54 μ in diameter) than the other hyphae from which they arise, with quite irregular walls producing numerous lateral conidia which rest either upon short stalks or upon little projections of the wall of the conidio- phore, or are completely sessile. The conidia, which are also colorless at first, but become the same color as the conidiophore, are ovoid, 4X 7 u, with smooth, fairly thick walls. During germination, they swell so as to be almost spherical in shape (Figures 39-41, Plate 12). Papulospora cinerea n. sp. PLATE 8, Figures 1-11. Mycelium white, septate, procumbent, forming a felted mass over the substratum; bulbils steel-gray or slate-colored, somewhat spheri- cal and flattened, 21-36 μ in diameter, with three or four large angu- lar central cells, and a layer of fairly regular cells forming a cortex, but of the same color as the others; the primordium a spiral of one or two coils. No conidia known. On gross culture in the laboratory, Cambridge, Mass. This fungus was found running over a gross culture in the Crypto- gamic Laboratories at Harvard University by Dr. Thaxter and has been kept growing as a pure culture for more than ten years. It is easily distinguished from any of the others by the steel gray or slate- color of the bulbils, which are round, somewhat flattened in form, and measure 21-36 μ in diameter, in which respects they resemble those of Papulospora sporotrichoides. The mycelium is white, procumbent, forming a felted mass over the substratum, the slate-colored bulbils being scattered among the white hyphal filaments, finally giving the whole culture a bluish gray or steel-gray appearance. When young HOSTON.— CULTURE STUDIES OF FUNGI. 285 the hyphae are closely packed with oil globules which escape into the water when the filament is ruptured, and might be mistaken for spores. The bulbils— A short lateral branch coils up, usually making one or two turns, rarely more, and frequently less than two, and divides into a number of short cells from which secondary branches are pro- duced, or from which individual cells are formed by budding (Figures 7-8, Plate 8). In either case, spherical cells which gradually increase in size, are developed, and the lateral walls adhere closely to each other. The original coil, the cells of which in the meantime have become much enlarged and filled with granular material and oil globules, is thus eventually completely surrounded. At maturity three or four large central cells may be distinguished which have become angular by pressure, surrounded by a layer of fairly regular cells which are also usually somewhat angular except the outer walls. It often happens that when one turn is made by the primordial coil, the secondary branches begin to form, while at other times two or more turns are formed before this happens. Between these two. extremes a number of variations are found. Not infrequently the lateral branch becomes divided into four to eight cells and may or may not be coiled at the end, and from these, secondary branches are produced which coil around each other and around the original branch, dividing and subdividing, the lateral edges eventually adher- ing closely, and producing a more or less elongated bulbil (Figures 4—6, Plate 8). This process also inhibits the further growth of the coil. An extreme instance of this is shown in Figure 6, Plate 8, where several cells are seen to take part in the formation of lateral branches. Bulbils formed from a primordium of this type are elon- gated, irregular, and larger than those formed in the usual way. Although this species was grown on a great variety of nutrient media, it could not be induced to develop any perfect form or even another imperfect type. Papulospora parasitica nov. comb. Syn.: Helicosporangium parasiticum Karsten. (nec Eidam.) ῬΙΑΤΕ 5, Fiagures 1-17. Mycelium septate, white, flocculent; bulbils light brown, nearly spherical, 15-21 μι in diameter, with a single large central cell sur- rounded by a single layer of empty colorless cells; primordium a spiral, coiled crosier-fashion. 286 PROCEEDINGS OF THE AMERICAN ACADEMY. On bread, Cambridge, Massachusetts; mouse dung, Duarte, Cali- fornia. This form which appears to be identical with Helicosporangium parasiticum Karst. was found by Dr. Thaxter on bread in Cambridge, Massachusetts, and kept as an herbarium specimen, but was too old to be resuscitated. The writer also found it on a gross culture of mouse dung in an old paper bag obtained from Duarte, California. This culture was so overgrown with Penicillium and other foreign material which grew so much more rapidly than the bulbiferous fungus that it was difficult to get it pure. This was finally accomplished by using a gross culture of sterilized peas on which the mycelium of the bulbil grows quite rapidly. The bulbils— The development of the bulbils, which are produced in large numbers, agrees in all essential points with the original de- scription and figures of Karsten (65). Short lateral branches of the hyphae coil up crosier-fashion and, although quite open at first, soon close up, forming a close coil which divides into short cells, all of which increase in size to a certain degree. One of these, usually the end cell, but not infrequently the second, enlarges more rapidly than the others and becomes a “central cell,” the remaining members of the coil forming a ring or “annulus” around it and becoming firmly at- tached to the side of the original lateral branch. As this central cell increases in size more rapidly than those of the coil, considerable lateral pressure is exerted and consequently protuberances usually appear on each side of it which usually becomes subpendent and subsequently may divide into two or three lobes (Figures 4, 5, 9, 10, Plate 5). As this tension is released, probably through the inerease in size of the “annulus,” the large central cell loses its lobed appear- ance and assumes a spherical form (Figure 11, Plate 5) and may later become somewhat angular. In the meantime the cells composing the out laterally, extending over the surface of the large central cell, and in the mature bulbil completely corticating it, the walls adjacent adhering laterally. Sometimes there is a small pore left at one or both of the centers of the lateral faces of the central cell and through them at germination the germ tube grows, but this is the exception and is probably one of the incomplete stages of development that will be spoken of later. During the early stages of development and even until they have almost reached their full development these bulbils are cclorless, but eventually they become light brown. At maturity they are nearly ‘ ‘annulus”’ begin to grow HOTSON.— CULTURE STUDIES OF FUNGI. 287 spherical in form, consisting usually of a single large central cell about 10-14 μ in diameter, densely filled with granular material and oil globules, and surrounded by a single layer of empty colorless cells, the whole bulbil measuring 15-21 μ in diameter. Although the foregoing description of the mode of development of the bulbil is the characteristic one, the process may vary considerably in differ- ent cases. Occasionally there appears a tendency to form a helix, at other times a protuberance from the central cell develops only on one side or not at all, and quite frequently the “annulus” is incomplete, or the cortical cells that are derived from it fail to cover the whole central cell. It would thus appear that the development of the bulbil may be arrested at nearly any stage, and these arrested forms, under proper conditions, will germinate almost immediately. In Van Tieghem cells these bulbils germinate in 24-36 hours and send out one or two germ tubes, as shown in Figures 15-16, Plate 5, which arise from the central cell only. The germ tubes usually proceed from that region where the marginal cells meet or, as some- times happens fail to meet, leaving two small pores, as already men- tioned. In incompletely developed bulbils, the germ tube seems to come out from any point offering the least resistance. Conidia-like bodies were occasionally found connected with this fungus when grown on straw. A short lateral branch, which not infrequently becomes septate (Figure 17b, Plate 5), enlarges at the end and from it an ovoid cell (4.5 XK 6.5 μ) is abjointed. Unfortu- nately these were produced so rarely that their germination and further development could not be observed. Figure 17, Plate 5, however, shows a direct connection between these “conidia’’ and a bulbil. This form agrees in all respects with the original description and figures of Helicosporangium parasiticum (Karsten ’65) except that it is saprophytic and that no “endospores” are found in the central cell. As already stated, Karsten was of the opinion that the contents of the cortical cells passed into the central cell, either directly or by diffusion and as a result of the union of these different protoplasmic bodies the spores were formed. If the account given by Karsten is correct, in all its details he was not dealing with a bulbiferous form at all. It would seem, however, that later writers are probably correct in considering them as such, since Karsten may have been misled by the presence of more or less regular oil globules, such as occur in this and other species and which might easily have been mistaken for endo- spores. On the other hand, it is by no means impossible that he was 288 PROCEEDINGS OF THE AMERICAN ACADEMY. dealing with a form related to Monascus, which has not been recog- nized by subsequent investigators. Since, however, the morphology and development of his “Helicosporangium”’ corresponds so exactly with that of the bulbil under consideration and since also the “ para- sitism”’ of his plant on “beets,” seems at least very questionable, the writer feels little hesitation in concluding that he was dealing with a bulbil, in all probability identical with the one under consideration. Harz (’90), in his account of Physomyces heterosporus (Monascus heterosporus (Harz) Schréter), is of the opinion that this plant is closely related to Helicosporangium parasiticum Karsten, and further suggests that Papulospora sepedonioides Preuss, belongs near this fungus also, the difference consisting in the fact that the central cell of the latter is said to contain but one or only a few “ endospores.”’ The bulbils described and figured by Zukal (86), under the name of Dendryphium bulbiferum, also resemble this form in appearance and mode of development, except that it does not produce the lateral protuberances from the developing central cell, at least they are not mentioned or figured, and that it is described and illustrated as being intimately connected with hyphae producing spores of the genus Dendryphium. In this connection it may also be mentioned that the spores of Stephanoma strigosum Wallr. (Asterophora pezizae Corda, Syntheto- spora electa Morgan, Asterothecitum strigosum Wallr.) show stages that resemble quite closely certain. conditions in the development of P. parasitica. Figure 35, Plate 5, for example, is an abnormal spore of Stephanoma and, except for its size and color, might easily be taken for an imperfectly developed bulbil of the form under con- sideration, such as is represented by Figure 14, Plate 5. A corresponding resemblance may also be seen between imperfectly developed bulbils of the present species, in which the cortical cells have failed to surround the central cell completely, and the immature bulbils of Acrospeira mirabilis described above. PAPULOSPORA ASPERGILLIFORMIS Eidam. PLATE 7, Figures 1-20. This bulbil was obtained from several different sources, chiefly on onion leaves, wheat chaff, and oat straw from the vicinity of Cambridge, also on straw from Claremont, California. It is not at all rare and can easily be obtained by placing straw in a moist chamber. It is readily distinguished by its relatively large, irregular, sclerotium-like HOTSON.— CULTURE STUDIES OF FUNGI. 289 bulbils. Pure cultures from a half-dozen different sources were made by the methods already described, and kept under cultivation on a variety of media. The septate mycelium grows very slowly on nearly all substrata, producing the best results on bran agar, and on sterilized fresh horse dung on which it becomes somewhat flocculent. The primary mycelium grows on the top of the substratum, or just below the surface, and sends up lateral branches into the air. It is these lateral branches that produce its peculiar Aspergillus-like fructification. The primary mycelium becomes very large, usually somewhat contorted and packed full of granular material and oil globules. The hyphae, which an- astomose readily often forming a sort of network, measure as much as 11 μ in diameter, and some of the swollen lateral branches 17 μ (Figure 4, Plate 7). Occasionally, especially in the young hyphae, there occur large swollen intercalary cells containing oil globules and other food material (Figures 17-18, Plate 7). These seem to be cells for the storage of food. The bulbils—The mycelium grows out evenly in all directions from the point of inoculation. In about two or three weeks (on horse dung, in about a week), small brownish-red spots appear near the margin of the mycelial growth. These are young bulbils, and on closer examination they are found to develop as follows. A short lateral branch (Figures 2-3, Plate 7) well filled with nutrient material, sends out branches which twine about each other. The former sometimes coils at the tip but this seems to be incidental. These secondary branches may come off near the base of the lateral branch (Figure 3, Plate 7), and by twining about the primary hypha may incorporate it into the bulbil. More often, however, the secondary branches come off a short distance from the hypha (Figures 2, 4, 6, Plate 7), so that, especially in the early stages, it is evident that they are on short pedicels. The secondary branches intertwine with each other, and divide up into short cells, their lateral walls adhering firmly to those of their neighbors and eventually forming a compact mass of uniform cells. At maturity these bodies superficially resemble true sclerotia perhaps more nearly than they do typical bulbils, but they are developed from a group of cells composing the primordia, and not from a mass of interwoven hyphae from different sources. They vary considerably in size and shape, some of them being nearly spherical, about 100 in diameter; but most of them are irregular in form, reaching in old cultures 570 X 750 4. There is no differentia- tion between the marginal cells and the central cells. Microtome 290 PROCEEDINGS OF THE AMERICAN ACADEMY. sections show that the bulbil is uniform throughout (Figure 20, Plate 7) all the cells containing protoplasm, and under favorable conditions capable of sending out germ tubes. In this respect it differs from the typical sclerotium, which usually has a compact layer of several cells in thickness (the rind) which forms the margin. The primordia are colorless at first (Figures 2-4, Plate 7), then light-yellow, later ruby- red, and finally reddish brown and opaque. In this as in most other bulbils the process of development may vary greatly. Figure 1, Plate 7, shows the primordia of three bul- bils, two of which and possibly the third also, would probably have grown together, forming a large, irregular, sclerotium-like body. This phenomenon occurs quite frequently, giving rise to a variety of forms, which vary with the number of the initial primordia taking part in their development, their proximity, and the inequality of their development. In such cases each primordium develops in- dependently, until its lateral branches intertwine with those of one or more that lie adjacent to it, a compound bulbil finally resulting, in which the several origins are indistinguishable. Aspergillus-like fructification. Conidia are frequently produced both on Aspergillus-like heads and also laterally, on the sides of the hyphae (Figures 10-11, Plate 7). The latter are usually isolated, sometimes irregularly grouped. The conidiophores arise from erect lateral branches, and are frequently septate; rarely branched. They are very minute, so that one can detect them only with difficulty, even with a good hand lens. The length of the conidiophore varies greatly, some being quite short, others so long that it is difficult to trace them to their origin. The swollen head of the conidiophore is usually spherical, or nearly so, and on it are arranged somewhat irregularly numerous simple sterigmata. These vary slightly in size and shape, but always have a broad base and taper more or less gradually, often to a point, at the distal end. The relative length of the vertical and transverse diameters of the swollen base varies some- what, so that one may find gradations in shape from almost spheri- cal to napiform. The conidia are nearly spherical, sometimes ovoid, smooth, colorless, minute, occurring in chains, and dropping off very readily; but in moist atmosphere the conidia, instead of being pro- duced in a chain, frequently adhere and form clusters much like those of Hyalopus. There are many variations in the arrangement of these conidia, which may, for example, arise, as is shown in Figure 9, Plate 7, termi- nally and laterally on irregularly clavate extremities of hyphae. HOTSON.— CULTURE STUDIES OF FUNGI. 291 Occasionally a conidiophore may form an intercalary swelling with conidia on it, as if it were a secondary head (Figure 10, Plate 7). Chlamydospore-like bodies occur quite frequently. They are mostly intercalary but sometimes terminal (Figures 13-16, Plate 7). When young they are colorless, or opalescent, slightly swollen, ovoid cells, filled with granular material. At maturity they are usually more spherical and have thick brown walls (Figures 13, 15, Plate 7). Occasionally more than one cell takes part in the formation of these spore-like bodies. Figure 16, Plate 7, shows two such cells and Figure 5, Plate 7, a large number of “ chlamydospores”’ closely packed together. There are several forms that have Aspergillus-like fructifications, similar to those just described and which may be considered briefly at this point. As has already been noted, Eidam (’83) describes these structures in his account of Papulospora aspergilliformis, and also chlamydospores resembling those of Acremoniella atra Sace. (Acremonium atrum Corda.) such as are produced by Melanospora cervicula. Eidam, however, described two types of bulbils in P. aspergilliformis, a small one that develops in a manner similar to the form examined by the writer, and a large one, the primordium of which is spiral, resembling that described by Bainier (07). Τ is quite possible that Eidam has here confused the primordia of two species the larger of which corresponds in all essentials to that studied by the writer. On the other hand his smaller bulbil would correspond more closely with that studied by Bainier. Bainier (07), in his article on Papulospora aspergilliformis also refers to its “Aspergillus-like” conidial fructification. According to his account the primordium of the bulbil consists of a short lateral branch which coils up spirally and eventually produces a more or less spherical bulbil. Under certain conditions of nutrition and moisture, however, the latter are said to produce large sclerotium-like bodies, which in turn may be induced to develop further and form perithecia, which are referred to the genus Ceratostoma. This form described by Bainier seems to be different from the one under consideration, since the bulbils of the latter do not develop by means of a spiral and are large and sclerotium-like. The present form, moreover, has been grown for nearly three years and during that time it has never been observed to produce any other type of bulbil than the one de- scribed. It has, however, produced in abundance conidia on Asper- gillus-like conidiophores which sometimes occur in direct connection with the bulbil (Figure 8, Plate 7). This species has been compared 292 PROCEEDINGS OF THE AMERICAN ACADEMY. with material received from Professor Bainier by Dr. Thaxter, and the two forms have been grown on many and varied kinds of nutrient material for nearly three years during which time, as already men- tioned, the American material has never been observed to produce small spherical bulbils; nor has the form received from Bainier developed the large sclerotium-like bodies which he describes, al- though every effort has been made to obtain them. There is also a marked difference in the method of growth in these two forms. The mycelium of the American form grows very slowly on bran or corn agar, but fairly rapidly on horse dung, while Bainier’s species grows rapidly on a variety of media. There is also a marked difference in the general appearance of the two while growing in cultures; the mycelium of the former being quite inconspicuous at first and often two or three weeks elapse before bulbils are produced. The two forms thus appear to be very probably distinct and there seems little doubt but that Bainier was mistaken in referring his species to P. aspergilliformis. Neither of these forms has associated with it Acremoniella-like Chlamydospores, such as Eidam describes and it seems not improbable that Bainier is right in believing that these spores do not belong to P. aspergilliformis, but are those of “Acremonium atrum” which although frequently associated with it are not a part of its life cycle. The writer has under cultivation about a dozen pure cultures of Acremoniella atra obtained from different sources, some of which were closely associated with bulbils, and these have been grown for nearly three years under varying conditions of temperature, moisture, and nutrient material, the different mycelia having been contrasted on plate-cultures under various conditions. In no instance, however, have bulbils or Aspergillus-like conidiophores been produced. Harz (11) has described a form under the name of Monosporium acremonioides that produces chlamydospores and _ conidiophores similar to those of P. aspergilliformis Eidam, but not associated with bulbils, and states that the conidia were produced on secondary heads either sessile or short-stalked, like those of Melanospora cer- vicula. This latter character has been used by Costantin (88) as the basis of a new genus, Harzia, into which he puts the foregoing species under the name of Harzia acremonioides. Later, in referring to Papulospora aspergilliformis Harz (’90) calls attention to the striking resemblance between the two spore-forms of this fungus and those of Monosporium acremonioides Harz, and suggests that, if they are the same, the name should at least be Papulospora acremonioides, although i iat ti 7. δὰ HOTSON.— CULTURE STUDIES OF FUNGI. 293 he takes exception to the generic name on the ground, as will be seen later, that it does not correspond with the description of the genus by Preuss. Lindau (’07) apparently is of the opinion that these two forms are the same and he creates a new genus, Eidamia, for their reception under the name EF. acremonioides (Harz). The conidial form of Melanospora cervicula resembles quite closely Harzia acremonioides in having its conidia on secondary heads and in producing Acremoniella-like chlamydospores, but:differs in possessing bulbils and melanosporous perithecia. It is quite possible, however, that the two are identical. It is possible also that the so-called “Harzia type” of fructification, as seen in M. cervicula and the “ Aspergillus-like” type as seen in P. aspergilliformis, are modifica- tions of one and the same mode of reproduction: since on several occasions the writer has found in connection with the conidial fructifi- eation of M. cervicula instances in which secondary heads seemed to be lacking, but, owing to the fact that there was only a limited amount of material available, this point could not be absolutely determined. The perithecium of this form, however, is clearly of the melanosporous type, and can hardly be the same as the Cerato- stoma described by Bainier. The writer has under cultivation the Mycogone ulmaniae Potebnia, (07) (Chlamydomyces diffusus Bainier) obtained by Dr. Thaxter from Liberia and kept in cultivation for over fifteen years. In addi- tion to its large two-celled, warty, chlamydospores, this species also produces conidia on “Aspergillus-like” conidiophores similar to those of P. aspergilliformis. Conidial forms similar to those above mentioned are also described by Moller (98) in connection with the garden fungi of certain species of ants in the tropics. Again, large chlamydospores, somewhat similar to those of Melano- spora cervicula except that they are divided into two unequal cells, have been described by Berlese (92) in connection with Sphaeroderma bulbilliferum. They differ from those of Mycogone ulmaniae, how- ever, in being smooth. Papulospora polyspora, n. sp. PuaTeE 11, Figures 1-13. Hyphae septate, hyaline, scanty, procumbent, 5-7 μι in diameter (sometimes as much as 9 μὴ); bulbils dark red-brown usually with a 294. PROCEEDINGS OF THE AMERICAN ACADEMY. thin mucilaginous film about each, eventually becoming a dry powdery mass, completely concealing the mycelium, more or less spherical, 119-122 μ in diameter, composed of closely compact angular cells, 150-200 cells visible in a surface view; cells homogeneous throughout. Individual cells of the bulbil eventually forming spherical spores, 17— 22 win diameter loosely held together. No other spore-form known. On straw, old paper, from California and cotton flowers from Cuba. This fungus has been obtained from at least three different sources. It was found by Dr. 'Thaxter running over a gross culture of the flowers of Cuban cotton and also by the writer on gross cultures of barley straw from Claremont, California, and on old paper from Duarte, California. The usual methods of obtaining a pure culture were employed here, after which the fungus was grown on various kinds of nutrient material, but it could not be made to produce any perfect form. Mycelia from widely different sources were contrasted in Petri dishes but no results were obtained except the production of certain abnormal enlargements and contortions of the hyphae, such as may frequently be observed in contrasting forms of even widely different species. The mycelium of this fungus is white, inconspicuous, procum- bent, the hyphae densely filled with coarse granules or oil globules. At a short distance from the margin of growth small white pustules are seen, which gradually become larger and more frequent as they approach the point of inoculation. These soon turn tan-colored, and are frequently associated with small drops of liquid of nearly the same color, which may often be seen surrounding a bulbil. At maturity these bulbils are almost spherical, 119-122 μ in diameter, composed of closely compacted angular, often irregular cells, uniform throughout, there being no distinction of a definite cortex. They occur in large numbers heaped together, covering the whole substratum and obliter- ating completely the naturally scanty mycelium. In older cultures they become a dry powdery mass. The bulbils.— The formation of this bulbil is different from that of any of the others thus far considered, since they result not from the development of a.single primordium but from the combined activities of several primary branches. One or more procumbent hyphae send up vertical branches which twine about each other (Figures 1+, Plate 11). Usually several of these branches arise simultaneously at a given point (Figure 3, Plate 11) and as the bulbil increases in size, more and more of these take part in its formation, their extremi- ties combining to produce the bulbil proper, while just above the substratum there may form a sterile supporting base, often with a HOTSON.— CULTURE STUDIES OF FUNGI. 295 diameter nearly equal to that of the bulbil itself and composed of interlacing hyphal strands, which are partly made up of branches from the procumbent hyphae and partly by the branching of the original vertical ones. These supports or “stalk-like” structures vary in length, some being quite long (100 μὴ, while at other times the bulbils appear to be almost sessile on the horizontal branches. The primor- dia that are produced later, are hindered in their upward growth by the presence of the first formed bulbils, which, however, are soon broken away from their attachments and pushed up so that eventually several irregular layers of independent spherical bodies are produced, the oldest ones being on the surface. Whether the vertical hyphae first formed fuse at the apex could not be determined. They evidently receive some stimulus, for they begin to send out short branches in different directions, which in turn divide and subdivide, and these intertwine among themselves and, with other hyphae that grow up from the original horizontal branches, form an interlacing weft which becomes more and more compact, producing a hyaline, spherical body in which the walls are very thin and almost indistinguishable except after staining. As they increase in size they assume a brownish tint and finally a rich tan-color, during which time the walls gradually become more definite and eventually are well marked. Since liquid media appeared to have a peculiar affect on the develop- ment of these bulbils, cultures were tried in large flasks on pieces of wood partly immersed in bran decoction, so that the effect of different degrees of moisture might be observed, as the mycelium spread from the liquid medium toward the dryer portions of the wood. Under these conditions it was found that the bulbils formed on the wood about three or four inches above the liquid, began to assume a paler aspect and soon became light straw-colored, instead of the dark tan of the normal bulbil. On examination it was found that the cells com- posing these pale bulbils, instead of being compact with angular walls as in the normal form, had rounded up and become spherical (17-22 μ in diameter), adhering very loosely by means of a mucilaginous mate- rial that had evidently been secreted by them, so that a very slight pressure would separate them into individual spores (Figure ὃ, Plate 11). The germination of these “spore-masses” was followed carefully in Van Tieghem cells — some crushed, others not — and it was found that nearly all the spores germinated in twenty-four hours, some producing one, others two germ tubes, which were hyaline and septate, becoming much branched (Figures 9-10, Plate 11). When allowed to remain adherent, the spore-mass sent out germ tubes in all 296 PROCEEDINGS OF THE AMERICAN ACADEMY. directions which shortly forced the individual spores apart. The bulbils were also germinated in Van Tieghem cells, but their germi- nation was much slower and they produced comparatively few germ tubes which seemed to be chiefly from the superficial cells. In water cultures the hyphae are usually larger and more densely filled with granular material, with numerous large swollen intercalary or terminal cells (Figures 9b-13, Plate 11). These cells are grouped together irregularly as if attempts were being made to form bulbils but they do not become compact. ‘They often grow very large, as may be seen by a comparison of Figures 90--[8, Plate 11, all of which have the same magnification. This development and final fate of the bulbil of P. polyspora, suggest a similar condition that is found in Aegerita. In Aegerita Webbert Fawcett (10) the “sporodochia”’ which measure 60-90 μ in diameter, consist of an “aggregation of conidia-like, inflated, spherical, cells, 12-18 μ in diameter,” resembling the conditions described for P. polyspora. The development of the latter on the other hand recalls also that of the sporodochium of A. candida Persoon (Penio- phora candida Persoon) as described and figured by Lyman (07) and it is possible that the two structures may be similar in nature. OTHER RECORDED BULBIFEROUS FORMS. In addition to those above enumerated several other bulbils or bulbiferous forms have been recorded, some of which have already been referred to, but which may here be again mentioned. Papulospora Dahliae Costantin (88). This species was found by Costantin on roots of Dahlia. Its bulbils appear to be somewhat like those of P. coprophila, brownish-red in color, with two or three large central cells surrounded by a layer of empty cortical cells. Conidia belonging to the genus Dactylaria are, however, said to be associated with these bulbils, although it is not evident that the species was cultivated in a pure condition. Dendryphium bulbiferum Zukal (86) has been mentioned on page 233, and also in connection with P. parasitica. The bulbils described and figured by Zukal are said to be directly associated with the conidia of a Dendryphium; but here, as in other forms studied by this author, there is no evidence that pure culture methods were used in studying the fungus. “ Haplotrichum roseum Lk.” is also stated by the same author (’86) to be associated with bulbils said to be very similar to those of the HOTSON.— CULTURE STUDIES OF FUNGI. 297 Dendryphium just mentioned; but here again pure cultures do not appear to have been used. As far as the writer is aware, moreover, this common hyphomycete has never been seen to be thus associated by any other observer. Papulospora (Stemphylium) Magnusianum (Sacc.), (Michelia, I, 132) a form collected by Magnus in the Tyrol, distributed in Vester- gren, Micr. Sel., No. 1150, and also figured by Saccardo in Fungi Italici, No. 934, should be mentioned in the present connection, since it is a typical bulbil and by no means a compound spore like that of species of Stemphylium. Clathrosphaera spirifera Zalewski (88), is a form which the author, although his observations are concealed in Polish text, appears to regard as bulbiferous, or as producing bodies comparable to bulbils, which are also associated with a species of Helicoon. The writer has himself observed various other more or less ill de- fined types of bulbils, which have not been above enumerated, since they do not appear to be sufficiently well marked to warrant a definite name. “No. 170” for example (Figures 24-34, Plate 5), was found in California on straw from Claremont, and on old paper from Duarte. The fungus is characterized by an abundant white mycelium, the hyphae of which produce bulbil-like bodies consisting of a few cells each, as indicated in the figures. Their characters and development, however, are not constant and their exact nature is somewhat doubt- ful. COMPOUND SPORES AND OTHER REPRODUCTIVE STRUCTURES WHICH RESEMBLE BULBILS. Reference has already been made to the close resemblance which exists between the so called “spore-balls”’ of some of the Ustilaginales, and the structures under consideration; in fact it would be quite impossible to differentiate the spore-balls of Urocystis or Tubercinia from bulbils, as far as concerns their gross structure and method of development which may be exactly similar. They are, however, clearly distinguished in other ways; since in bulbils, spore formation is never preceded by any nuclear fusion, so far as is known; and further- more the germination of bulbils in no way resembles that of the smuts; and there is never any indication of the formation of anything corre- sponding to a promycelium. Attention has also been called to the fact that the compound spores 298 PROCEEDINGS OF THE AMERICAN ACADEMY. which are associated with the imperfect forms of many of the higher fungi, may bear a close resemblance to bulbils. Although compound spores may in general be distinguished by the fact that they normally arise as the result of the septation of a single cell, while in the pro- duction of bulbils two or more cells are primarily involved, to which others are added by a process of budding which may also be combined with secondary septation, it is not always possible to separate them with certainty. Spores like those of Stephanoma, referred to else- where, in which the empty superficial cells arise by budding, serve, however, to break down this distinction. On the other hand, the more complicated types of bulbils are easily comparable to the simpler types of sclerotia, such as occur for example in Penecillium Italicum, Verticilltum agaricinum and similar forms. Such sclerotia, however, result from the irregular and indefinite massing together of vegetative filaments, the densely compacted cells of which do not partake of the nature of spores, while the func- tional cells of bulbils are usually spore-like and act independently of one another at the period of germination. Among the compound spores formed in connection with the imper- fect conditions of higher fungi, several may be mentioned which have bulbil-like characteristics. Stephanoma strigosum Wallr. a parasite on Peziza hemispherica which, as Dr. Thaxter informs the writer, occurs also on Genea hispidula in this country and is connected with an undescribed hypo- creaceous perithecial form, might very well be regarded as a bulbil of a simple type, since not only are its spores similar in their develop- ment, but, when mature, are hardly distinguishable from the more simple bulbils which are often produced, for example, by Papulospora parasitica. Stemphylium macros poroideum Sace., which has been examined from cultures kept in the Cryptogamie Laboratories, produces a compound spore consisting of one large functional cell to which, at maturity, two or more empty ones are attached. In this condition it resembles very closely the bulbil of Acrospeira mirabilis; but in view of the fact that it develops as a result of the successive divisions of a single terminal cell, it must be regarded as a compound spore. Certain other forms also of Stemphylium as well as of Mystrosporium might well be mis- taken for bulbils. Hyalodema Evansu P. Magn., which von Hohnel has referred to Coniodyctium Chevaliert H. & Pat., produces a hymenium-like layer bearing compound spores which, except in color, are very like the HOTSON.— CULTURE STUDIES OF FUNGI. 299 bulbils of Papulospora sporotrichoides. Their development, however, is clearly that of compound spores and not of bulbils. Eleomyces olei Kirchner (’88) a fungus found growing in poppy oil, produces a compound spore which consists at maturity of a large thick-walled functional cell, surrounded by several empty coherent cells, the whole resembling the bulbil of Acrospeira. If, as suggested by Kirchner, this body results from the coherence of several adjacent cells, it might well be regarded as a bulbil and not a compound spore. Various other spore-forms might be mentioned which bear more or less resemblance to bulbils, but those above enumerated are sufficient for purposes of illustration. Before leaving bulbil-like forms, how- ever, two or three additional types may be mentioned, the nature of which is not altogether clear, since they are neither compound spores nor typical sclerotia. Aegerita Webberi Fawcett (10), a fungus attacking scales on Citrus, produces, under certain conditions, bulbil-like bodies which consist of loosely coherent spore-masses closely comparable to those of the aberrant Papulospora polyspora, the development of which, under moist conditions, has been described above. Sorosporella Agrotidis Sorokin (’88, ’89), which attacks the larvae of Agrotis, fills the latter with loosely but definitely coherent cell- groups which might also be compared to those of P. polyspora. Lastly, among structures which bear a striking resemblance to bul- bils, the peculiar spore-balls of Spongospora subterranea (Wallr.) Johnson should be mentioned; which, although they might readily be taken for a species of Papulospora, have been shown to belong to the life-cycle of one of the Mycetozoa. THE MORPHOLOGICAL SIGNIFICANCE OF BULBILS. Opinions concerning the morphological significance of bulbils differ widely. Preuss (’51), Eidam (’83), DeBary (’86), Mattirolo (86) all regarded them as normal structures which function as auxiliary methods of reproduction; while Karsten (65), Zukal (’86), Morini (88), and Baineir (07) looked upon them as immature ascogenous fructifications of either perithecial or apothecial forms, believing that their arrested growth was due to unfavorable environment, and that, with proper nutriment, they might be able to complete their development. Although it is possible that the last mentioned view may be correct in some instances, it is quite certain that in many cases, where both 300 PROCEEDINGS OF THE AMERICAN ACADEMY. bulbils and ascocarps are present, this cannot be the case, since the primordia and development of the two are widely different. Thus in Cubonia bulbifera, for example, the bulbil is produced from a group of intercalary cells, while the primordium of the apothecium is a spiral. In like manner Melanospora anomala develops bulbils which arise from intercalary cells, somewhat as in Cubonia, while the perithecia arise from free spirals. It is quite possible, however, that in other cases, as for example in M. papillata, where the primordium of the bulbil and that of the perithecium are similar, they may be homologous. But even in such cases, the two primordia are distinguishable so early in their development, that it is more than probable that here, also, they cannot be regarded as immature ascocarps. Various attempts have been made by the writer to induce the bulbils of various species to continue their development and produce ascocarps. Many bulbils of ἢ. papillata for example, that had grown larger than the more normal types, were isolated and placed on different media where they were exposed to different degrees of moisture, with this end in view. Simi- lar attempts were also made with the bulbils of P. coprophila, in which the spiral bulbil-primordium might be supposed to suggest its ascogonial nature. In no instance, however, was any evidence ob- tained that would seem to point to the conclusion that they were to be regarded as anything but independent non-sexual propagative bodies, except that, in some instances they increased in size, sometimes be- coming approximately half as large as perithecia. This enlargement, however, was unassociated with any structural differentiation such as always characterizes the developing perithecium. Although Bainier reports that he was successful in inducing the bulbils of Papulospora aspergilliformis to develop directly into peri- thecia which he refers to Ceratostoma, the writer has been as un- successful with this species as with others, even when using material derived from a living culture received from Bainier by Dr. Thaxter. In view of the careful and long continued experiments made by the writer in this connection, and his entire failure to obtain positive results, the assumption seems justified that ordinarily, at least, bulbils are not to be regarded as abortive ascocarps, but rather as an auxil- iary method of reproduction that has been interpolated in the life history of certain fungi without definite relation to other forms of reproduction which they may possess; or if they have in reality been derived from some other reproductive body, that this was more probably some type of compound non-sexual spore, rather than the primordium of an ascocarp. HOTSON.— CULTURE STUDIES OF FUNGI. 301 DISTRIBUTION AND OCCURRENCE OF BULBILS. It is evident from the foregoing account that bulbiferous types are not only widely distributed, but are very readily obtained if sought for, and, like so many other types among the Fungi Imperfecti, have been independently developed by a variety of species wholly unrelated and belonging to widely separated groups among the Pyrenomy- cetes, the Discomycetes and the Basidiomycetes. Such bulbiferous conditions, therefore, cannot in any sense be regarded as forming any- thing in the nature of a Natural Group. If one may judge from our actual knowledge of these forms, it would appear, on the contrary, that the bulbiferous condition was a specific one, the habit having been developed by certain species, only, in genera, the other members of which have no such secondary means of propagation: just as the habit of producing sclerotia of a characteristic type, has arisen in a few species, only, of Penecillium, like P. Jtalicum. The same princi- ple is well illustrated in the large genus Corticium many species of which have been tested by means of pure cultures. Here again one finds a single species, only, which possesses the bulbiferous habit, namely C. alutaceum, pure cultures of which become completely covered by its dark brown bulbils. In view of the wide distribution and common occurrence of bulbil- producing forms, it is not a little surprising to find such scanty refer- ences to them in mycological literature; and from the experiences of the writer in studying them, it seems certain that further attention to this subject will not only yield numerous other forms, but will show connections with “perfect”? conditions even more varied than is at present indicated. KEY TO THE SPECIES OF BULBILS HEREIN CONSIDERED. According to their method of development bulbils may be grouped in three more or less well defined categories namely: those which originate from a primary spiral; those which develop from an inter- calary primordium of several cells, and those which arise from a group of vertical hyphae. Using these characters as a fundamental basis for separation, the species above enumerated may be distinguished as follows. 302 PROCEEDINGS OF THE AMERICAN ACADEMY. Key to the Species of Bulbiferous Fungi. ὦ A. Primordium normally involving more than one cell. I. Bulbils black or smoke-colored. 1. Bulbils 75-100 in diam. margin even........ Cubonia bulbifera. 2. ~ 200-3002." — * ‘irregular. . Papulospora pannosa. Il. Bulbils yellowish red to dark brown. 1. Hyphae showing clamp-connections. 1. Bulbils dark brown or chocolate colored. i. Bulbils 65-80. in diam. clamps conspicuous. Corticium alutaceum. 11. “125-1754 “ “5. margin even, clamps incon- SPICUOUSs rare cesta Ce eee ke Papulospora anomala. 2. Bulbils yellowish or hight brown. i. Bulbils light yellow, hyphae radiating conspicuously. Grandinia crustosa. ii. Bulbils brownish yellow, hyphae formed evenly. “No. 200.” 2. Hyphae not showing clamp-connections. 1. Bulbils scanty, perithecia usually present. i. Perithecia with neck, lateral and terminal setae. Melanospora cervicula. papilla and terminal setae. Melanospora papillata. 2. Bulbils abundant, perithecia usually absent. 1. Primordium intercalary. (i). Bulbils brownish-yellow, dente cells 28-55 up ae {{ “ec SLOW RL OT: Vat να ν Papulospora immersa. (1). Bulbils straw-colored, central cells 10-20% in Gian, ate eee Soe Papulospora irregularis. il. Primordium one or more lateral branches. (i). Primordium normally a single lateral branch. a. Primordium a spiral. § Cells heterogenous, definite cortex. A. One central cell. x Cortex complete. Papulospora parasitica. x * Cortex incomplete. Acrospeira mirabilis. B. More than one central cell. Spiral in one plane, cortical cells spinulose Papulospora spinulosa. * % Spiral normally in more than one plane, 2-6 central cells. a Bulbils dark brown. Papulospora coprophila. 8 Bulbils brick red. Papulospora rubida. §§ Cells homogenous, bulbils 21-36 yu in diam. brownish producing sporo- trichum spores. Papulospora sporotrichoides. HOTSON.— CULTURE STUDIES OF FUNGI. 303 b. Primordium not a spiral. § Bulbils large, 100-750 wu, irregular. Papulospora aspergilliformis. 70-150 μι, somewhat spherical, producing perithecia with slight pap- illa..........Melanospora anomala. (ii). Primordium two or more lateral branches forming a spherical aggregation of cells at the top. Papulospora polyspora. III. Bulbils white to cream colored, 30-35 » in diam. Papulospora candida. IV. «steel gray, 21-86 in diam......... Papulospora cinerea, ia HARVARD UNIVERSITY April, 1911. LITERATURE. Bainier, G. 07. Evolution du Papulospora aspergilliformis et étude de deux Ascodesmis nouveaux. Bul. Trimestriel de la Société Myc. de France. Tome XXIII, p. 132. 1907. Barber, M. A. 07. On Heredity in Certain Micro-organisms. Kansas Univ. Sei. Bulls, Vol TV, Ὁ 1907. Bary, A. de and Woronin, M. ’66. Ascobolus pulcherrimus. Beitr. z. Morph. u. Phys. der Pilze. Taf. IV. 1866. 81. Comparative Morphology and Biology of Fungi, ete. Trans. by Garnsey and Balfour; Oxford. 1887. Berkeley, M. J. ’46. Observations, Botanical and Physiological, on the Potato Murrain. Jour. Hort. Soc. of London, Vol. I, p. 9. 1846. ’57. Acrospeira mirabilis. Intr. Crypt. Bot., p. 805. 1857. 60. Papulospora, Preuss. Outlines of British Fungology, p. 354. 1860. Berlese, A. N. 92, Intorno allo sviluppo di due nuovi Ipocreacei. Malpighia. Anno V, p. 386. 1892. Biffen, R. H. 01. Notes on some factors in the spore-formation of Acro- speira mirabilis (Berk. and Br.). Proc. Cambridge Philo. Soc. Vol. XI; Pt: I, p. 136s 901. 304 PROCEEDINGS OF THE AMERICAN ACADEMY. 703. On some facts in the Life History of Acrospeira mirabilis (Berk. and Br.). Trans. British Mycol. Soc., Vol. II, p. 17. March, 1903. Claypole, Mrs. E. W. 91. Baryeidamia parasitica Karst. Bot. Gaz. Vol. XVI, 263. 1891. Costantin, J. 88. Note sur un Papulospora. Jour. de Bot., Vol. II, p. 91. 1888. "88a. Les Mucédinées simples, p. 82. 1888. ’88b. Notes sur quelques parasites des Champignons supérieurs. Bull. Soc. Bot., pp. 251-256. 1888. Eidam, E. ὙΠ. Ueber die Entwickelung des Helicosporangium parasiticum Karst. Jahrb. schles. ges. f. vaterl. cult. Breslau, Vol. LY, pp: 1225 1989: 877: ᾽88.. Zur Kenntniss der Entwickelung bei den Ascomyceten. Cohn’s Beitrage. Zur. Biol. 4. Pflanz. Vol. If, pp: 377-483; pl. 19-28. 1888. - Engler und Prantl. 90. Die Natiirlichen Pflanzenfamilien. 1 Teil. 1 Abth. p. 148. Farlow, G. W. ’77. Note on Papulospora sepedonioides Preuss. Rept. Mass. Board of Agric., Vol. XXIV, pt. 2, p. 176 (15). 1877. Fawcett, H. S. 10. An Important Entomogenous Fungus. Mycologia, Vol. II, No. 4, p. 164. 1910. Fischer, Ed. ’97. Rabenhorst’s Kryptogamen-flora. Vol. I, abth. V, p. 127. Harz, C. O. 11. Einige neue Hyphomyceten Berlins und Wiens nebst Beitr. zur Systematik derselben. Bull. Soc. Impér. de Moscou, Vol. XLIV, p. 88. 1871. "90. Physomyces heteroporus, n. sp. Bot. Centralb., Vol. XLI, pp. 405-411. 1890. Hohnel, Franz von. 10. Uber die Gattung Hyalodema. Annales Mycologici, Vol. Viti. No. 6, p. 590. 191): Johnson, T. 708. Spongospora Solani, Brunch. (Corky Seab). Econ. Proe. Roy. Dublin Soc., Vol. I, p. 453. 1908. OL ΝΣ συ Ὁ συν βιιἐεὺ. ὑπο π HOTSON.— CULTURE STUDIES OF FUNGI. 305 Johnson, T. 09. Further observations on Powdery Potato Scab, Spongo- spora subterranea (Wallr). Sci. Proc. Roy. Dublin Soe., Vol. XII, p. 165. No. 16. 1909. Karsten, H. ’65. Ursache einer Mohrriibenkrankheit. Bot. unters. a. ἃ. phys. Lab. landwirt. Berlin. Heft I, pp. 76-83. 1865. ’80 Helicosporangium Karst. Deutschen Flora, p. 123. 1880. ’88. Bary’s “ Zweifelhafte Ascomyceten.” Hedwigia, Vol. XXVII, pp. 182-144. 2 Figs. 1888. Kirchner, O. ’88. Ueber einen im Mohndél lebenden Pilz. Ber. deutsch. bot. Gesell. General-Versammlung, Vol. 6, p. CI. 1888. Lindau, G. 07. Rabenhorst’s Kryptogamen flora, 1° p. 123. Lief. 93. 1907. Eidamia acremonioides Harz. Lyman, G. R. 07. Culture Studies on Polymorphism of Hymenomycetes. Proc. Boston Soc. Nat. Hist., Vol. XX XIII, No. 4, pp. 125-209, plates 18-26. 1907. Magnus, Ρ. 10. Ein neuer krebsartige Auswuchse an der Wirtspflanze veranlassender Pilz aus Transvaal. Berichten d. deutsch. botan. Ges. 28 Bd., p. 377. 1910. Massee, G. 99. A Text-Book of Plant Diseases. p. 305. 1899. Mattirolo, O. ’86. Sullo sviluppo di due nuovi Hypocreacei e sulle spore- bulbilli degli Ascomiceti. Nuovo Giorn. bot. Ital., Vol. XVIII, pp. 121-154, 2 plates. 1886. Moller, Alfred. ’93. Die Pilzgarten einiger sudamerikanischer Ameisen. Bot. Mittheilungen aus den Tropen von Dr. A. F. W. Schimper, Heft 6. 1893. Morgan, A. P. ’92. Synthetospora electa Morg. Bot. Gaz., Vol. XVII, p. 192. 1892. 92a. North American. Helicosporeae. Jour. of Cincinnati Soc. Nat. Hist., Vol. XV, p. 39. 1892. 306 PROCEFDINGS OF THE AMERICAN ACADEMY. Morini, F. ’88. Biografia degli apoteci della Lachnea theleboloides (A. et 5.) Sace. Mem. R. Ace. Scienze ἃ. Istituto di Bologna, Ser. 4, tom. 9, p. 611. 1888. Potebnia, A. 07. Mycogone Ulmariae Potebnia, Annales Micologici, Vol. V, p. 21. 1907. Preuss, C. G. T. ’51. Papulospora Preuss. Sturm’s Deutchlands Flora, Abth. Til; Pilze, Heft: 30,’p. 89. Vat. 25: 1851: Saccardo, P. A. ’86. Sylloge Fungorum. Vol. IV. 1886. Schroter, J. 97. In Engler τι. Prantl’s Die Naturlichen Pflanzenfamilien. I Teil. 1 Abth. p. 149. 1897. Sorokin, N. ’88. Parasitologische Skizzen. Centralblatt. f. Bakter. u. Para- sitenkunde. Bd. IV, No. 21, pp. 644-647. 1888. ’89. Un Nouveau Parasite de la Chenille de la Betterave, Soro- sporella agrotidis. Bull. Scientifique d. France et d. Belgique, Vol. XX, p. 76. 1889. Ule, E. 701. Ameisengarten 1m Amazonasgebeit, Engler’s’ Bot. Jahrb. Vol. XXX. Beiblatt 68 : 45-52. 1901. Wallroth, F. W. ’42. Die Naturgeschichte der Erysibe subterranea Wallr. Beit. zur. Bot., p. 118. 1842. ’42a. Linnaea, Vol. XVI, pt. 2. p.332. 1842. Woronin, M. 82. Beitrag zur Kenntniss der Ustilagineen. In De Bary and Woronin, Beitr. Morph. u. Phys. der Pilze, Ser. 5, p. 5. taf. 2. 1882. Abhandl. d. Senckenb. naturf. Ges. 12: 559. Zalewski, A. 88. Prayezynki zycioznawstwa grzybow przez. Krakow. Dru- karnia uniwersytetn jagiellonskiego. 1888. Zukal, H. ’85. Mycologische Untersuchungen. Denkschriften d. k. Aka- demie d. Wissen. (Wien), Bd. 51, pt. 2, pp. 21-26, Taf. 2, Figs. 1-4. 1885. ’86. Untersuchungen iiber den biologischen und morpholo- gischen Werth der Pilzbulbillen. Verh. k. k. Zool. bot. Ges. Wien, Vol. XXXVI, pp. 123-135, plate 4. 1886. EXPLANATION OF PLATES. The figures of Plates 1-12 were drawn with the aid of a camera lucida using different combinations of the Bausch and Lomb lenses, All the mature bulbils were drawn with the same magnification, namely 4 mm. objective and 3 eye piece, and for the stages of development of the bulbils, 4 mm. objec- tive and 12 eye piece were used. The plates have been reduced in reproduc- tion about three-quarters. PLATE 1. CUBONIA BULBIFERA. Ficures 1-6. Different forms of the primordium of the apothecium. Figures 7,8. Young apothecia. Figure 9. Section of the mature apothecium. Ficure 10. Asci and paraphyses. Fiaures 11-16. Stages in the development of the bulbil. Figure 17. Mature bulbil. Figure 18. Contortions of the hyphae. Figure 19. Portion of a crushed bulbil with the contents of the cells escaping. Ficure 20. Ascospore. Ficure 21. The endosporium broken off. Ficures 22-24. Germinating Ascospores. Ficures 26, 27. Sprouting vegetative cells from the inner portion of the apothecium. Fiaure 28. Germinating bulbil producing spiral primordia directly. Hotson. —Cucture Stupies oF Funai PLaTe 1. Proc. Amer. Acapo. Arts ANd Sciences. Vor. XLVIII. PLATE 2. MELANOSPORA PAPILLATA. Figures 1-6. Stages in the development of the bulbil. Figure 7. A group of Chlamydospore-like intercalary cells. Fiaures 8-10. Stages in the development of the perithecium. Ficure 11. pune ot a mature perithecium showing the relative size of the ulbils. Figure 12. A group of asci crushed from a young perithecium. Ficurss 13-20. Germinating ascospores. Figures 21, 22. Forms produced in Van Tiegham cell cultures. Fiaure 23. Conidia on flask-shaped sterigmata produced on a hypha. Ficures 24, 25. Stages in the development of a terminal bulbil. Fiaure 26. An intercalary bulbil with three large central cells. MELANOSPORA ANOMALA. Figures 27-30. Stages in the development of the bulbil. PLATE 3. MELANOSPORA ANOMALA. Fiaures 1-12. Stages in the development of the perithecium. Figure 12. Mature perithecium. Figure 13. (a) Germinating ascospore showing a bottle-shaped sterigma. ; (0) Bottle-shaped sterigma on a hypha. Ficures 14, 15. Other stages in the formation of the bulbil. Figure 15. A mature bulbil. MELANOSPORA CERVICULA. Ficures 16, 17. Primordia of the bulbil. Figure 18. A bulbil produced from a group of terminal cells. Figure 19. Primordium of the perithecium and conidia on flask-shaped sterigmata. Figure 20. Mature perithecium. Figure 21. Abnormal forms common among the hyphae. Figure 22. Chlamydospores of the Acremoniella type. Figures 23, 24. “Harzia-like”’ fructification. Hotson. — Cucture Stupies oF Funei. Piate 3. Sse : SSE Proc. Amer. Acav. Arts ano Sciences. Vor. XLVIII. ee ane O47 ᾿ ἜΣ Ὁ χροὶ apts κι ity a μ᾿ Ἷ te ἢ ye ye ay Mees epee / ΧΑ δ μ᾽" pase : τὰς wit ~~ ye ae Wea Pac eo A hai ΝΙΝ i » wh ae f ai m/ ἡ “ail ie i ea ped Mie ΓΝ ΕΝ Ἐν “ae [ο΄ “4 εἰ ye Py ἡ ge av as ews aya ᾿ “a Wei Fy. é ᾿ ae _ ἢ fies si hike” ““ ce, δε tog “ER a ᾺΣ 4 wil bu PLATE 4. PAPULOSPORA CANDIDA. Ficures 1,2. Variation in the size of the conidia. Ficures 3-12, and 15-27. Stages in the germination of the conidia and the development of the bulbil from them. Ficures 28-41. Stages in the development of the bulbil from a lateral FIGuRE 42. Ficure 48. FIGURE 44. Fi@ureE 45. FicureE 46. FIGURE 47. branch of the hyphae. Germination of the superficial cells of the bulbil. Conidiophores of Verticillium agaricinum var. clavisedum. Portion of the hyphae showing large oil globules. Showing intimate connection between the bulbil and the Verticillium. An irregular primordium of a bulbil. Ascoma of Geoglossum glabrum attacked by the parasite. > ἐν ςΣ Θ =) EY 9: το 3 a nd - tS] CD Or) ous 5 - \i ED) AS 3 . | : ΣΝ PD. A fost BN) gD spat EN Ot SF s. Vor. XLVIII. PLATE δ. PAPULOSPORA PARASITICA. Ficures 1-14. Show various stages in the development of the bulbil. Ficures 4, 5, & 9, 10. Show the protuberance from the lateral surface of the large central cell. Ficures 15, 16. Germinating bulbils. Figure 17. Conidia-like bodies connected with the bulbil. Ficures 35b, 36. Swollen intercalary cells. ACROSPEIRA MIRABILIS. Fiaures 18-23. Stages in the development of the bulbil. Figure 20. The end-cell has enlarged to form the central cell. Friaur& 21. The second cell has enlarged to form the central cell. FiaureE 22. Several empty cortical cells are shown. REPRODUCTIVE Bopirs RESEMBLING BULBILS. Fiaure 24-34. Irregular forms of a doubtful bulbil (No. 170). Figure 35. Spore of Stephanoma strigosum Wallr. Hotson. — Cucture Stuoies oF Funai. Prate 5. Proc. Amer. Acapo. Arts ano Sciences. Vor. XLVIII. « “,ἦ * ᾽ ᾿ "Ὁ "ye ᾿ aed A ἘΠ ἐν 4 5 ᾿ ¥ i ᾿ οἷ δ Re ας ι ive δι 6.2 te! ¥ ᾿ προ ν F “lle ΓΝ re) Aa aly χω Μ 4 fi δὼ ΤΕΥ αὐ ΟΣ Νω ᾿ - 24 ᾿ As SIO any ὙΦ ἡ oe A ᾿ me “ “; ᾿ r - “« > ἜΣ, ae, f are i. Soh ty eer iad ab Tia (ene § 1 PLATE 6. GRANDINIA CRUSTOSA. Ficure 1. Pustulate habit of the fructification. Figure 2. Hymenium with basidiospores. Figure 3. Basidiospore. Ficures 4-10. Stages in the development of the bulbil. Fieure 10. Mature bulbil with the same magnification as all the other mature bulbils. PAPULOSPORA ANOMALA. FicurE 11-17. Stages in the development of the bulbil. Ficure 17. Mature bulbil. Ficure 18. Two primordia close together. Fiaure 19. Large intercalary cells densely filled with oil globules. PAPULOSPORA PANNOSA. Fraures 20-24. Stages in the development of the bulbil from intercalary cells. Fiacure 25. Occasional mode of formation of intercalary primordia. Pirate 6. Ευνοι. Hotson. —Cucture Srtubdle€s ΟΕ νοι. XLVIII. Proc. Amer. AcAv. Arts AND SCIENCES. ΓΝ Ἂν Bad ot : ; Pe ΟΝ ᾿ μ: a et 5 ie, ai ) : be aw io « an ae = Ἦν 92 hl oo eee x "ἡ, - ᾽ & ' Α ᾿ Ι + -- ἘΣ 5 τ τὰν " i 1 ~ , a ¥ ἃ ε . Ἷ » { - ” | ’ ‘ i é ; “ - ‘ a A " - ¥, . ᾿ f ἊΣ =i J “aS > et J 9 5 i fe ΝΣ PLATE 7. PAPULOSPORA ASPERGILLIFORMIS. Ficures 1-4, & 6. Stages in the development of the bulbil. Figure 5. A group of Chlamydospore-like bodies. Ficure 7. A primordium that produces a very irregular bulbil. Ficure 8. ‘Aspergillus-like’’ heads produced directly from the bulbil. Ficures 9-12. Different forms of the “ Aspergillus-like” fructification. Fraure 12. Abnormal conditions. Ficures 13-16. Chlamydospores. Ficures 17, 18. Large swollen cells, likely storage cells. Figure 19. Bulbil forming from terminal cells. Ficure 20. Section of a mature bulbil. Hotson. — Cucture Stupies of Funai. PLate 7. Proc. Amer. Acapo. Arts Ano Sciences. Vor. XLVIII. PLATE 8. PAPULOSPORA CINEREA. Ficures 1-10. Stages in the development of the bulbil. Figures 4, 6, &9. Modifications of the regular mode of development. Fiaures 10, 11. Mature bulbils. PAPULOSPORA RUBIDA. Figures 12-16. Stages in the development of the bulbil. Fiaures 25a-27a, 21, 22. Other stages in the development of the bulbil. Figures 17, 20. The spiral primordium that sometimes occurs. Figure 25. Section of a mature bulbil showing five large central cells. Figure 18. Surface view of a mature bulbil. PAPULOSPORA PANNOSA. Figures 28-30. The development of a bulbil from a lateral branch. Figure 31. A collapsed hypha showing rigid septa. wr ef PLAT Proc. Amer. Acapv. Arts Ano Sciences. Vor. XLVIII. PLATE 9. PAPULOSPORA SPINULOSA. Figures 1-7. Stages in the development of the bulbil. Ficure ὃ. Primordia produced from a superficial cell of an immature bulbil. Ficure 9. Section of a mature bulbil showing the ‘ Annulus.” Fiaure 10. A surface view of the same looking down on the ‘‘ Annulus.” PAPULOSPORA IRREGULARIS. Ficures 11-17. Stages in the development of the bulbil. Figure 17. A mature bulbil. PAPULOSPORA PANNOSA. Ficures 18-20. Stages in the development of the bulbil. Fiaure 20. A mature bulbil. Hotson. — Cucture Stupies oF Funai. Pate 9. Proc. Amer. Acav. Arts ano Sciences. Vor. XLVIII. renee PLATE 10. PAPULOSPORA COPROPHILA. Figures 1-8. Stages in the development of a bulbil from a spiral. Fieure 6. An υδύειι condition, the production of conidia directly from the spiral. Figure 8. A spiral primordium surrounded by an irregular layer of cells. Ficure 9. Immature bulbil that has developed like Figs. 14 and 15, and also a spiral primordium. Ἷ Fiaure 10. Median section of a mature bulbil with two large central cells. Figure 1l. A Sg pes with the contents of the large cells crushed out (Fig. 11b). Ficure 12. Germination of one of these cells. Fiaures 13-15. Forms arrested in the process of development. Ficures 16. Surface view of the mature bulbil. PAPULOSPORA IMMERSA. Fieure 17. Irregular hypha densely filled with protoplasm. The primor- dium of the bulbil. Figure 18. Primordium consisting of a single intercalary cell. Figure 19-25. Stages in the development of the bulbil. 10. sy Vols Ce VE PLATE 11. PAPULOSPORA POLYSPORA. Ficures 1-7. Stages in the development of the bulbil. Figure 7. A mature bulbil. FicurE 8. Group of spores adhering loosely together. Ficures 9 & 10. Germinating spores. Ficures 9b, 10b, 11-13. Modifications that occur when grown in liquid media, Pirate 11. Hotson. — Cucture Stupies oF Funct. XLVI. Proc. Amer. Acao. Arts AND Sciences. VOL. ΙΑ ΤΟΣ PAPULOSPORA SPOROTRICHOIDES. Ficurms 1-9. Stages in the development of the bulbil. Fiaure 8. A mature bulbil. Ficure 9. A side view of an immature bulbil. Ficures 10, 11. Abortive forms. Figures 12-16. Modifications in the formation of the spiral. Figure 17. Anirregular bulbil germinating, magnified more than the others. Figure 18. Branch of the hyphae showing primordia of the bulbils. Fiaures 19-25. Modifications in the development of the bulbils which are hyaline. Figures 26-28. Semi-diagrammatie representation of the mode of cell formation in the development of the hyaline bulbils. Fiaure 29. A section of a mature bulbil. Figures 30, 31. Large interealary and terminal cells found in the hyphae. Figures 32-34. Germinating bulbils. Fiaures 25-26. Conidiophores with conidia. Figure 37. Conidiophore produced directly from the bulbil in a Van Tieg- hem cell culture. 5 Fiaure 38. Conidium. Figure 39. The form the conidia usually assume before germinating. Fiaures 40, 41. Germinating conidia. Plate 12. Hotson. — Cutture Stupies OF Funat. Vor. XLVI. Proc. Amer. Acav. Arts AND SCIENCES. a ang Proceedings of the American Academy of Arts and Sciences. Vou. XLVIII. No. 9.—Srprremper, 1912. CONTRIBUTIONS FROM THE JEFFERSON PHYSICAL LABORATORY, HARVARD UNIVERSITY. THERMODYNAMIC PROPERTIES OF LIQUID WATER TO 80° AND 12000 KGM. By P. W. Bripeman. i μν -. μ. 2 ᾿ εἶν i wy , ‘ 2 ῃ ve fe 7 1 e 7 ᾿ , τ “s ὗ ; ΓΑ _ Ν " (e ᾿ + . 4 Ἰ > hd υ ij δ ' τ oe, i” ᾿ i - ἡ! Ὁ om Ε ι ,» - . ἃ - το ‘ ὃ ᾿ " fe Ψ Ἧ ᾿ ~ - ὃ Ad ca - Υ rey . vy Z ΓῺ 5 ᾿ on... F \ ~ “Ὁ. : bs ᾿ ΄ 7 ‘ oa ΕΗ ΕῚ . fe Us i ᾿ “ΟΝ ’ ἐ ᾿ . i ΄ Ἵ CONTRIBUTIONS FROM THE JEFFERSON PHYSICAL LABORATORY, HARVARD UNIVERSITY. THERMODYNAMIC PROPERTIES OF LIQUID WATER TO 80° AND 12000 KGM. By P. W. Brip@Man. Received June 26, 1912. TABLE OF CONTENTS. Pace. πο RTOS pee τιν εἰπε Rn Rn Bums Roe Se Fr meres, Slt τι 510 Method . . Sand |, ease g a Gata ote aoe: Sth IED Previous Use of the Method ............ . B12 Description of the Apparatus . τυ tree Se ro! μ, Correction for the Distortion of the Vessel. . . . . . . . . 316 Experimental Procedure . . σι τα ad che” 135140) In Determining Compressibility Raison ene lac Merah Fra tomes Ce) Calibration of Meneeun Calling i 2 gs ees See oa Formulas . . ἀντ Shy ane TN OE In Determining ilacitionte: ΘΕ νυ Os eee eet ae a 326 he ataan ἢ εἰν AP eer * PAE a aw Aaer Ole Compr essibility Af law Pressuteas:, )-/ το ον το θέν χα τ: ΠΤ ΠτΟ τ ΘΠ Ow, ΕΥΘΕΒΌΓΘΗ ο΄... si hea eel nd epee ce yt 390 Compresaibility at High*Pressures: « (.- 4 a, Se, 391 ΤΠ ΡΠ 5 ποτ athens Pressunes: Gs y5\ Gs. Ssh, es, PRS ἢ 5554 PHBE URSIONAGI Le; FueHIMES’ 7) ck Se ae Pues LG GN ate 986 Table of Volumes . . Pe δέον ἡ πλὴν cert ote τ ων OOS Method of Consttuctiony cal) &: . ΝΣ ας ΤῊΣ 336 Wanous; PhemhodynamicQuantitess). Ut i. wo. kw 357 eS Ov Compressibility, [ | ον ΡΣ το 840 Dilatation, (2) αν ioc eee ae OT p Work of Compression, W = — “Ὁ (5) Cpr. Vliet Dee ony 46 4" Heat of Compression, Q = — τ AS), AD AN ap fae Σ Oe! Change of Internal Energy, AE = W a OUI Wes fais ah προ.) 948 Pressure Coefficient, (32 See) NED στο ον 1540 OT /v Specific Heat at Constant Pressure, Cp τ΄. <=. . 4. . dol Specific Heat at Constant Volume, (Cs. . . .... . . 9852 Thermal Effect of Compression, (= Ss} eet me mR a φ Adiabatic Compressibility, (5) Oe oe Ae τπνῸς λει σοῦ Volume of Kerosene as a Function of Temperature and Pressure . . 356 Compressibility and Dilatation of Ice VI re Cl Pate Se ake lie. ai eee 310 PROCEEDINGS OF THE AMERICAN ACADEMY. INTRODUCTION. Tuts paper is in the nature of a supplement to a former paper on the properties of water in the liquid and the solid forms.1_ The solid forms were studied over a range of 20,000 kgm. /cm.?, and from —80° to +76°, but the study of the liquid reached only from the lowest temperature of its existence to about +20°. Above 0°, measurements were made on the liquid at only 20°. The two measurements, at 0° and 20° were sufficient to give the mean dilatation between 0° and 20°, but not the variation of dilatation with temperature. It was assumed in the earlier paper that the variation of dilatation with temperature became negligible at high pressures, since this seemed to be the most plausible assumption in view of all the data then available. In this present paper the study of the liquid has been continued from 20° to 80°, and to 12000 kgm. The pressure range is greater than that of the preceding paper by about 2,500 kgm. The range is not great enough to entirely cover the region of stability of the liquid, but it is as great as it was convenient to cover with the method used here, which is different from that of the former work. It has the advantage of very much greater rapidity of operation, but since it depends on the complete elastic integrity of the steel pressure cylinders it is not possible to reach so high pressures with it as with the former method. [The former limit of 9500 kgm. was set by the freezing of the liquid and was not due to any limitation of the method.] Never- theless, it may be hoped that the present temperature and pressure ranges are both wide enough to give a fairly complete idea of the nature of the effects to be expected at high pressures with varying tempera- ture. Measurements of the dilatation have been made at four tempera- tures, so that it has been possible to find the variation of dilatation with temperature at any pressure. Perhaps the most unlooked for feature disclosed by the measurements is the fact, contrary to the assumption of the first paper, that the variation of dilatation with temperature does not become vanishingly small at high pressures, but reverses in sign. This means that while at low pressures the volume increases more and more rapidly with rising temperature, at high pressures the expansion becomes more slow at high temperatures. \@ The data of this paper are sufficient to completely map out the p-v-t surface over the domain in question: Both the first and second 1 Bridgman, These Proceedings, 47, 439-558 (1912). ΟἹ BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. Old derivatives are therefore completely determined, so that we now have all the data at hand for the determination of any one of the thermo- dynamic properties of the liquid. This means that we are in a posi- tion to find such quantities as the specific heats, change of internal energy, adiabatic temperature rise etc., as well as the more easily determined compressibility and thermal dilatation. The latter part of the paper, after the discussion of the method and the presentation of the data in the first part, is occupied with the computation of these various thermodynamic quantities. The accuracy of some of these is probably not very great, because the error in the second derivative of an experimental quantity may be considerable. It has, therefore, seemed best to give the general view of the nature of the quantities which is offered by a graphical representation, rather than to give tables, with the tacit assumption of greater accuracy which usually goes with a set of tables. In spite of the lower order of accuracy of some of these thermodynamic quantities, it has still seemed well worth while to give them, since even the general trend of some of the quantities, such as the specific heats, has not been hitherto known with relation to pressure. The data presented here are only the beginning of a projected study of the characteristic surface under high pressures for a number of liquids. The measurements have already been carried through for twelve other liquids beside water. The purpose of this study is ultimately the development of a theory of liquids, since it would seem that a much more intimate grasp of the nature of the forces at work in a liquid would be afforded by a study over a wide pressure range, than over the comparatively low pressures hitherto used. It must be admitted, however, that this broader purpose is not particularly furthered by this work on water, because of the well known abnor- malities of this substance. In the previous paper several abnormali- ties had been shown to exist at low pressures. In this paper, new abnormalities are found at higher pressures. Water gives the ap- pearance of becoming completely normal only at the higher tempera- tures and pressures of the range used here, but of course whether this is really normal or not cannot be told until the behavior of normal liquids has been discovered. The full significance of the present data, in their bearing on such questions as the polymerization of the liquid, for example, cannot appear until after the discovery of the laws for entirely normal liquids. The investigation of water before that of normal liquids was undertaken for two reasons; firstly because of the desire to complete the work for water already begun, and 312 PROCEEDINGS OF THE AMERICAN ACADEMY. secondly because in this and the following investigation a new method for determining the compressibility was to be used, which had not yet been proved to be reliable, but which could be tested by a com- parison of the results obtained by this method with those already obtained by another method at lower temperatures for water. In addition to the data for liquid water, two other quantities were determined incidentally in the course of the work, and are given at the end of the paper. One of these is the experimental measurement of the compressibility and thermal dilatation of ice VI between 0° and 20° and 6360 and 10,000 kgm. The other is the measurement of the volume of kerosene up to 12,000 kgm. and between 20° and 80°. THe ΜΈΤΗΟΡ. The method in its fundamental idea is as simple as it would well be possible to devise. The substance, whose compressibility or thermal dilatation is to be measured, is placed in a heavy steel cylinder in which pressure is produced by the advance of a piston of known cross section. The change of volume, given by the distance of advance of the piston, is measured as a function of the pressure. The method is simple, rapid, and above all, applicable to the highest pressures. But there are a number of corrections which must be made, often difficult to determine, which doubtless account for the slight use which has been made hitherto of the method. Apparently, with the excep- tion of the present work, it has been used recently only by Tammann,? and by Parsons and Cook. Tammann and Parsons and Cook applied it only to the measurement of compressibility, reaching pressures of about 4000 kgm. The author has previously applied it to the measurement of the thermal dilatation of water at tempera- tures below 0° C. over a pressure range of about 6500 kgm. The most serious of the errors which readily occur to one is that of leak. It is almost essential to the success of the method to secure a piston absolutely free from leak, and this has hitherto been a matter of some difficulty at high pressures. Tammann did not entirely secure this freedom from leak, but avoided it in large measure by the use of a very heavy oil, such as castor oil, and still further lessened the error by correcting for the slight amount of leak by measuring the amount of liquid which escaped past the piston in a given time. This method would not be applicable to the highest pressures, however, because 2 A. D. Cowper and G. Tammann, ZS. Phys. Chem., 68, 281-288 (1909). 3 Parsons and Cook, Proc. Roy. Soe. A, 85, 332-349 (1911). BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 313 of the freezing of the oil. Parsons and Cook were able to secure entire freedom from leak up to 4000 kgm. by the employment of a cupped leather washer combined with a brass dise of special design. It has been the experience of all those who have worked with high pressures, however, that no leather washer is capable of standing pressures very much in excess of the limit of 4500 kgm., since the leather rapidly disintegrates under the pressure. In the present work the same form of packing was used which was used in the pre- vious work on the freezing of water and mercury under pressure. This has been proved in the previous paper to be absolutely free from leak up to the highest pressures which can be sustained by the steel containing vessels. In the present work this same packing has proved itself to be reliable for the purposes of this method. The question of the method of measuring pressure is also of con- siderable importance in using this method, since the usual measuring devices, such as a Bourdon gauge, cannot be applied, for reasons to be discussed later, and attempts to calculate the pressure directly from the force required to produce motion of the piston are likely to be in error because of the friction of the packing. Parsons and Cook did, however, adopt this latter method, and computed the pressure from the known force required to move the piston. The effect of the friction of the packings was allowed for in as large a degree as possible by taking the mean of the readings during increasing and decreasing pressure, assuming that the friction remained constant. The results obtained by Parsons and Cook in this way were surprisingly good. That the friction did remain fairly constant was indicated by the constancy of the results and the fact that the curve nearly always returned to the starting point; but it is doubtful if the method would work at very much higher pressures because of the increase of friction due te the flow of the softer parts of the piston. The brass washers used by Parsons and Cook would almost certainly have upset under two or three thousand more kgm., and it is the experience of the author that it is difficult to obtain even steel washers which will stand much more than 8000 kgm. without taking some set. In fact, at high pressure there must necessarily be some plastic yield, in order to follow the expansion of the cylinder. The result of this set in the washers is that the friction becomes very irregular, and cannot be assumed to be the same during increasing and decreasing pressure. Variations in the amount of friction due to this cause of as much as 200 or 300% have been found at the higher pressures of this work. _ The only escape from the difficulty seems to be to measure the 314 PROCEEDINGS OF THE AMERICAN ACADEMY. pressure directly inside the cylinder. This was done by Tammann by connecting a Bourdon gauge directly to the cylinder. But it is known that the errors of the Bourdon gauge become rapidly more serious at higher pressures,* due to the increase of hysteresis, so that this gauge could not be used for the pressures of this experiment. Furthermore, no Bourdon gauge has up to the present been made of sufficient sensitiveness which is capable of standing more than 6500 kgm. In the present work the pressure was measured inside the cylinder by inserting directly into it a coil of manganin wire, which had been already calibrated against an absolute gauge. This method of measuring pressure has been fully described in a previous paper.® It was necessary for the purposes of the present work, however, to make a somewhat more careful determination of the temperature coefficient than was done formerly, and this determination will be described in detail later. The method has shown itself perfectly satisfactory and reliable in every respect. One coil of wire has been used almost continuously for over six months, and occasional calibra- tions have shown no change. These calibrations were made by measuring with the coil certain fixed temperature-pressure points, such as the freezing pressure of mercury or of ice VI, at some fixed temperature. The apparatus used in the present work is the same in most features as that used in the former work, a detailed account of which has already been given in the papers mentioned. Only the points in which this has been changed will be mentioned here. It was a disadvantage of the former method that the apparatus consisted of two parts; the lower part, a cylinder containing the liquid to be measured, was placed in a thermostat, and the upper part, a cylinder in which pressure was produced, was exposed to the temperature of the room. When tem- perature was changed in the thermostat below or pressure was changed in the cylinder above, liquid passed from the one cylinder to the other, experiencing in the transition a change of temperature, and so a change of volume also. This change of volume accompanying a known change of temperature varies in an unknown way with the pressure, and to apply the correction it was necessary to make an independent, set of experiments. In the present form of apparatus the difficulty was avoided by including everything in one cylinder. This cylinder contained the liquid under investigation, the pressure measuring coil, and the piston by which pressure was produced. It 4 Bridgman, These Proceedings, 44, 201-217 (1909). 5 Bridgman, These Proceedings, 47, 319-343 (1911). BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 315 was placed in the lower part of the hydraulic press and, together with the lower part of the press, was placed in the thermostat. The di- mensions were so small that this could be done without increasing to an unwieldly bulk the size of the apparatus, the four tie rods of the press being 1 1/8” in diameter and their centers 6” apart. It is the same form of apparatus which was used for the measurements on ice VI up to 20,500 kgm. The present experiments run to only 12,000 kgm., however, since it is evidently an absolute essential to the success of the method that there should be no permanent distortion of the eylinder. It would be easily possible to reach pressures much higher than those reached in this experiment, but it was felt that the risk and the extra time involved in the probable construction of new apparatus was not justified at present, when it seemed that the most important work was to map out the field, obtain data for as many liquids as possible, and determine the general nature of the significant problems. Later, if there are crucial points which need the use of much higher pressures, it will be a comparatively easy matter to obtain them. The cylinder used in this experiment was not the same as that used in the previous work on water. This new cylinder is from a piece of chrome-vanadium steel made in the electric furnace by the Haleomb Steel Co., of Syracuse, N. Y. The steel itself is a wonderful product, and without it the present investigation would not have been so easily possible. It shows a tensile strength of 300,000 lbs. per sq. in. when hardened in oil, and an elastic limit of about 250,000 lbs. These figures are considerably in excess of those for the steel used in the previous investigation. The steel furthermore is remarkably homo- geneous, because of its production in the electrical furnace. One of these pieces was pierced with a hole 1/8’ diameter and 13” long, and the drill came through concentrically without any variation from the straight line. The dimensions of the cylinder used in the present work were 4 1/2” outside diameter, 13’’ long, inside diameter 17/32” for the greater part of its length, with an enlargement to 3/4’ at the lower end for the reception of the manganin coil. The original inside diameter was 7/16’’. The cylinder was prepared for use by hardening in oil and then subjecting to a pressure much in excess of that con- templated for the actual experiment. The seasoning pressure was over 30,000 kgm. Even under this high seasoning pressure the cylinder showed very little permanent change of internal dimensions, not stretching as much as 1/32.’ This is less than the amount of stretch which has been found for any other grade of steel. The 316 PROCEEDINGS OF THE AMERICAN ACADEMY. effectiveness of the treatment is shown furthermore in the fact that in over six months of continual use the inside has not stretched by so much as an additional 1/10000’’.. The hole was enlarged to a final size of 17/32”, instead of keeping it as small as possible, because of the difficulty of reaming out the hole so as to give a satisfactorily smooth surface after the seasoning process. The difficulty was occasioned by the hardness of the steel, and several attempts were necessary before the desired result was produced. The pressure measuring coil was the same as that used in the last part of the work on ice VI. The construction of the insulating plug was also the same as that used there. During the course of the work it was necessary to take this plug apart several timess, because water had reached the mica washers, and once or twice the mica washers themselves have given way. These mica washers are the weakest part of the entire apparatus as at present used, since they gradually disintegrate and fail by shear after prolonged use, but it is a matter of only a few hours to replace them. Every time after the insulating plug has been freshly set up it has been tested for insulation resis- tance, both during application of pressure and after release. The resistance was in all cases as high as several hundred megohms, the limit of the measuring devise. The steel of the insulating plug has also failed once or twice by the “pinching-off effect”’® after long use. This also is an easy matter to repair. Failure of this type is attended with some danger, however, because of the violence of the explosion with which the ruptured plug is expelled. The surest way of avoiding this danger is to so mount the apparatus that the plug points at the floor or other indestructible object. The hydraulic press, the method of measuring the displacement of the piston, and the details of the packing of the moving piston, were the same as that used in the former paper. In the use of the apparatus to determine compressibility there is one serious error which did not enter into its use in the determination of the change of volume during change of state, namely the correction for the distortion of the cylinder in which the piston moves. At low pressure the correction is relatively unimportant, and may be com- puted from the theory of elasticity, if one is willing to assume that the theory is sufficiently accurate for this type of stress. But at higher pressures the correction becomes more important, increasing in percentage value directly with the pressure, and is almost certainly 6 Bridgman, Phil. Mag., 24, 63-79 (1912). BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 917 not calculable by the theory of elasticity, because of the entrance of such effects as hysteresis. ΤῸ determine the correction an auxiliary set of experiments is necessary. Evidently if the true value of the compressibility of some one substance were sufficiently well known, then the apparent compressibility as determined by this method would give the correction for the distortion of the cylinder. No such com- pressibilities are known with any high percentage accuracy, but this is not necessary, provided only that the uncertainty in the standard compressibility is small in comparison with the distortion of the vessel. The substance which most readily suggests itself because of its small compressibility is steel, but this is a solid, whereas the method is applicable directly only to liquids, so that some modifica- tion of the procedure is necessary. Such a modification readily sug- gests itself, and has been used by the author in the previous determi- nations of the thermal dilatation of water at temperatures below 0°, and has also been used by Parsons and Cook. The modification is to replace part of the liquid under investigation by a steel cylinder, and determine the compressibility of the liquid and the steel together. The difference of two determinations, the one for the liquid alone, the other for the liquid and the steel, gives a value for the difference of compressibility between the liquid and the steel from which the effect of the distortion of the vessel has been almost entirely elimi- nated. Furthermore, the compressibility of the steel is so small in comparison with that of the liquid that the slight uncertainty in the value for the steel is of no account, so that the compressibility of the liquid is given directly. The application of this method would demand, then, that the inte- rior of the cylinder be filled first with water and the apparent compressi- bility determined, and then part of the water replaced by steel and the apparent compressibility determined again. But this demands that the coil of manganin with which the pressure is to be measured come directly in contact with the water, which evidently cannot be allowed because of the short circuiting produced by the water. It seemed to be necessary, then, to devise some sort of protection for the coil, which should not occupy so much volume as to introduce a serious correction, and which should at the same time transmit the pressure readily to the innermost parts of the coil. Considerable time was spent in trying to devise such a protection. The scheme adopted was to surround the coil with a small mass of vaseline enclosed in a flexible sac, formed from the finger of a silk glove, and rendered impervious to water by painting it over with several coats of the col- 318 PROCEEDINGS OF THE AMERICAN ACADEMY. lodion of surgeons. This sac was tied with silk thread directly over the end of the insulating plug. It was proved by trial that the vaseline did not become so viscous under pressure as to refuse to trans- mit the pressure with sufficient freedom, but the arrangement did not prove itself as trustworthy as was to be desired. The collodion might leak after several applications of pressure, which made it necessary to reassemble the insulating plug and redetermine the elastic constants of the apparatus, because the distortion included in the plug itself was sufficient to introduce appreciable error. The device probably could have been made usable with a little more effort, but it would always have been more or less unsatisfactory, and would have been applicable only to those liquids which do not attack the collodion, whereas most of the organic liquids which it was desired to use in the future do so attack the collodion. The attempt to protect the coil was abandoned after a month’s work, therefore, and the method re- placed by another, which at first sight introduced additional com- plications, but which is really just as simple as the first, and has the advantage of being applicable with only slight modifications to the investigation of other liquids. The modified method used two liquids in every determination, one beside the one whose compressibility is to be measured. The water under investigation is placed in a thin shell of steel fitting the inside of the cylinder. This shell, when in position in the cylinder, is sur- rounded on all sides and above and below by kerosene, which below transmits pressure to the manganin coil, and above reaches to the moving piston with which pressure is produced. In the auxiliary experiment to eliminate the effect of the distortion of the cylinder, the shell with water is replaced by a solid cylinder of steel, and the quan- tity of kerosene remains the same as before. The motion of the piston due to the change of volume of the kerosene remains the same in the two experiments, therefore, and the difference of readings of the two sets gives directly the difference of compressibility between the water and the steel. The disadvantage of the method is that it is not possible to use so large quantities of water as in the former method, because the steel shell containing the water remains invariable in length under pressure, and enough kerosene must be introduced origi- nally to take up the change of volume of the water in this shell as well as the distortion of the other parts of the apparatus. - The reduction in the quantity of water under experiment is not greater than 30%, however, and the other advantages more than outweigh this com- paratively small loss of accuracy. BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 319 The procedure in using the apparatus in this finally modified form is as follows. The manganin coil is first screwed into the lower part of the cylinder. The rubber washer used to make this plug tight is one cut with a standard set of cutters, so that all the washers used for this purpose are always the same in size. This insures that the distortion due to the compression of the washers shall always be the same. The steel shell with the water in it is next introduced from above. The quantity of water is previously determined by weighing. It is desirable not to fill the shell to closer than 1/4” of the top, ex- perience having shown that otherwise water is likely to spill out and find its way to the manganin coil. The kerosene is next introduced into the cylinder from above. To ensure entire filling of all parts of the apparatus and the exclusion of air, only part of the kerosene is at first poured in, the air is then exhausted by attaching the mouth of the cylinder to an air pump, or simply by exhausting with the lungs, and then the remainder of the kerosene poured in. The amount of kero- sene is determined by weighing the dish from which it is poured before and after filling. Because of the wetting of the dish by the kerosene it is not always possible to obtain exactly the amount of kerosene desired each time, but the variation is seldom over 0.02 gm., and the very slight effect of this discrepancy may be corrected for, as will be described later. Finally the movable plug is introduced into the top of the cylinder, taking particular pains not to allow any of the kerosene to escape in the process. Here again the rubber washer used has been cut with standard cutters, so that the amount of rubber used here is also the same in all the experiments. The cylinder is then placed in the thermostat, and the zero of the manganin coil read at the temperature of the room. The thermostat is then adjusted for the desired temperature and the cylinder seasoned for the run by the application of pressure. A preliminary seasoning is necessary because of the hysteresis shown by the cylinder, and this hysteresis is shown with respect to both pressure and temperature. Many of the early results were somewhat in error because the necessity of this seasoning for tempera- ture as well as for pressure was not clearly recognized. The method of seasoning to be adopted depends on the kind of data which it is desired to obtain from the run, whether the compressibility at con- stant temperature or the thermal dilatation at constant pressure. If it is desired to determine the isothermal compressibility, the season- ing consists simply in raising the pressure through the entire range and releasing several times. It was found by experiment that three 320 PROCEEDINGS OF THE AMERICAN ACADEMY. such preliminary excursions were sufficient; after this the cylinder settles down into a state in which the normal hysteresis cycles are retraced with perfect regularity. Of course it is necessary to make the compressibility determinations immediately after this seasoning, as the effect gradually disappears with time. The time occupied in making the final readings to 12,000 kgm. and back with increasing and decreasing pressure, making in all 20 readings, might vary from two to three hours. After every change of pressure it was necessary to wait for the temperature effect of compression to disappear; this time was from 5 to 7 minutes. If the thermal dilatation under constant mean pressure is to be determined, the seasoning consists simply in taking the cylinder once through the temperature range contemplated as well as through the pressure range. A word of description as to the general procedure in determining the thermal dilatation at constant mean pressure will not be out of place. The general plan is to change the temperature while the piston is kept invariable in position, and therefore while the volume is also approximately constant. The rise of temperature produces a rise of pressure, so that after the rise of temperature it is necessary to bring the pressure back to the former value by with- drawing the piston if the change of temperature has been an increase, or advancing it if the change of temperature has been a decrease. The amount, by which the piston is withdrawn, as also the new final pressure, is noted. The temperature is then changed again, and the same set of readings made again. Thus every observation at any given temperature involves two readings of the position of the piston and the corresponding pressure. The slight change of pressure during the changes of temperature carries with it hysteresis effects, which it is necessary to avoid by previous seasoning, exactly as for pressure changes over a wider range. ‘Two processes of seasoning are necessary for temperature, therefore, one a larger one for the entire temperature range, and another smaller one for the slight changes of pressure incident to the changes of temperature. This second seasoning is made after the first more extensive seasoning simply by running the pressure back and forth several times through the small range of pressure to be met with during the temperature changes. This small range was determined by preliminary experiment. In the actual calculation of the results there are a number of corrections to be applied. These will now be discussed in detail separately. In the first place the temperature coefficient of the manganin coil has to be determined with particular care. This is BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 321 because the pressure changes brought about by changes of temperature during the determinations of the thermal dilatation are comparatively slight, so that any change of the pressure coefficient of the coil brought about by the change of temperature appears in the result greatly magnified. Thus for the sake of example, we will suppose that a change of temperature of 20° produces a change of pressure of 400 kgm. at 10,000 kgm. total pressure. This figure is a fair average of the results to be met with in practice. If now the pressure coefficient of the coil is changed by 1% by this same rise of temperature, the pressure will thereby appear to have risen 500 kgm. instead of the actual 400, introducing an error of 25% for a change in the constant of the coil of only 1%. In addition to the effect of the temperature coefficient of the coil, there is an effect due to the change of the zero of the coil with temperature, but this change can be determined by observations of the temperature coefficient of the coil at atmospheric pressure and is easy to measure with the requisite accuracy. The change in the pressure coefficient of the coil with temperature is more difficult to determine with the desired accuracy. It would not be possible to determine this by a direct calibration against the absolute gauge with which the mean value of the coefficient has been determined, for the reason that the absolute gauge itself is not accu- rate to better than 1/10%, and this would still leave a possible error in the thermal dilatation of 2.5%. To affect the desired calibration, some standard of pressure must be used which can be relied on to remain absolutely constant. Such a standard pressure is evidently afforded by the transition point of the liquid to the solid form of any convenient substance at some fixed temperature. In previous work the transition points of both water and mercury have been determined at various temperatures with an accuracy in the absolute pressure of 1/10%. To make the calibration it is only necessary to keep the pres- sure constant automatically at this known value by placing in com- munication with the chamber in which is the manganin coil to be calibrated another chamber in which are the liquid and solid forms of the substance whose transition temperature and pressure are known. This second chamber is to be kept at constant temperature accurately enough so that slight changes in this temperature will not produce changes of more than the allowed amount in the transition pressure. For this purpose the most convenient fixed temperature seems to be that of melting ice at atmospheric pressure, and the most convenient substance to use mercury, because of the sharpness of the freezing, and the ease with which it can be obtained pure. aoe PROCEEDINGS OF THE AMERICAN ACADEMY. The actual arrangements in making this calibration for the tempera- ture coefficient of the pressure coefficient of the coil were as follows. The upper cylinder of the hydraulic press in which pressure was produced contained in addition to the moving plunger a steel shell in which was as large a quantity of mercury as convenient, about 150 gm. This upper cylinder as well as the entire lower part of the press was surrounded by a tank containing ice and water, by which the temperature of the mercury could be kept continuously and accurately at 0°. A heavy nickel steel tube led out of the lower end of the upper cylinder through the bottom of the tank, and connected with the lower cylinder in which was the manganin coil under exami- nation. This lower cylinder was placed in an oil bath with thermo- static regulation, by which the temperature could be set at and retained at any desired value. The experimental procedure was as follows. The temperature of the lower bath was set at any desired value, and the pressure increased until the freezing point of mercury at 0° was slightly passed. The mercury then froze, with decrease of volume, thus bringing the pressure back to the known equilibrium value at 0°. After equilibrium had been reached, the resistance of the manganin coil was read. The pressure was then lowered slightly by withdrawing the piston. This was followed by automatic restora- tion of the equilibrium pressure, brought about by melting of the frozen mercury with increase of volume. The transition point was always so sharp that no difference could be detected in the equilibrium pressure whether approached from above or below. The temperature in the lower cylinder containing the manganin was then changed to another desired value. This change of temperature, if it were an increase, would naturally carry with it a rise of pressure, but the pressure is then automatically lowered by the freezing of the mercury. After a steady state is reached, the new value of the manganin re- sistance is read, and then the pressure lowered again by slightly withdrawing the piston, and the value of the resistance noted again after the equilibrium conditions have been restored from below. In this way the coil can be calibrated over the entire temperature range contemplated for the experiments. Of course this calibration is good only for one fixed pressure, but in view of the proved linearity of the pressure-resistance relation within 1/10% from 0° to 50°, it seemed safe to let the calibration go at this one determination, particu- larly since no effect could be found. The calibration of the manganin was carried out at five tempera- tures; 25°, 45°, 65°, 85° and 110°. No appreciable change of the BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 323 coefficient could be found for the four lower temperatures, but be- tween 85° and 110° there is a very perceptible change of 1%. But since the range of temperature of the actual experiment did reach over 80°, no correction was applied to the observations for this effect. It is to be noticed that this result is valid only for this one coil, since previous work, both by Lisell 7 and by the author, have shown that different pieces from the same spool of wire may show slight variations - in the temperature coefficient, which is sometimes positive and sometimes negative. In addition to this special calibration for slight relative changes in the pressure coefficient with temperature, the absolute value of the pressure coefficient has been checked from time to time during the course of the experiments. This could be done conveniently with the apparatus as used for the compressibility determinations by determin- ing the transition point of ice VI, or of mercury at known temperatures. These calibrations have shown no change whatever in the pressure constant of the coil. It has already been stated that the actual measurements involve two sets of readings, one with the apparatus filled with water, kerosene and asmall amount of bessemer steel, and a second set with additional - steel replacing the water. By subtracting the piston displacement at any given pressure for these two sets of experiments a value is obtained which gives approximately the piston displacement for the water alone, and from which the effect of the distortion of the vessel has in large measure been eliminated. But a moment’s consideration will show that the effect of distortion has not been entirely eliminated, and it is necessary to apply a correction for the slight residual effect. The correction comes because of the fact that the position of the piston at corresponding pressures is not the same in the two sets of experi- ments, so that the subtraction leaves still uncorrected the distortion due to the part of the cylinder exposed to pressure in the one set of experiments and not so exposed in the other. This correction can- not be determined directly, and the only way seems to be to calculate it by the ordinary theory of elasticity, taking for the constant of the steel the values under ordinary conditions, which are known not to vary much even for the most different varieties of steel. There is undoubtedly some error in the correction as so determined, but the total value of the correction is at best small, and any such error is relatively unimportant. . 7 Lisell, Om Tryckets Inflytande p& det Elektriska Ledingsmotstandet hos Metaller samt en ny Metod att Mita Héga Tryck (Diss. Upsala, 1909). 324 PROCEEDINGS OF THE AMERICAN ACADEMY. The compressibility of the steel replacing the water also evidently enters as a correction factor. This compressibility is relatively slight, and it has been previously determined over a range of 10,000 kgm. The value of the compressibility of the steel also changes with the temperature, but this change has also been shown by direct experi- ment to be slight, so shght that it can be neglected. In the present work the value was assumed to be constant, in- dependent of temperature and pressure, having the value 58 Χ 10% per kgm. per sq. em. There is also a correction to be applied for the compressibility of the kerosene, if the amount does not happen to be the same in the two sets of ex- periments, and it was seldom that the amount was exactly the same. The variation was very small, however, and the correction is easy to apply if the Figure 1. Diagramshowing compressibility of the kerosene itself the position of the piston. To jis known. This was determined with Fh ae τ ce be ΣΝ sufficient accuracy for the purpose by the compressibility. an independent set of experiments, exactly the same in principle as those for determining the compressibility of water. The results of these in- dependent experiments are given at the end of the paper. The following formulas were used in making the corrections, and include all the corrections mentioned qualitatively above. Figure 1 shows the position of the piston at different times in the course of the experiment. The left hand part of the diagram (denoted by the suffix 1) is for the cylinder when it is filled with kerosene and bessemer steel only, and the right hand part (denoted by the suffix 2) is for the cylinder when it contains water, kerosene, and bessemer steel. A and C are the positions of the piston at the arbitrary zero of pressure in these two sets of experiments (this arbitrary zero was usually taken in the neighborhood of 2000 kgm. and will be denoted by p), and B and D indicate the position at some higher pressure, the same in the two sets, which will be denoted by p’. We now write down the expressions for the total volume of the cylinder beneath the piston. it 2 BRI DGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 329 ra at eee ae an ee Ag BOVE ΝΞ ΕΝ At C, V2 = Κὰ + Vo no + V2. At D, Vol = Vox! + Vo! no + Τὼ where the suffixes Καὶ, H»O, or S indicate that the volume is for the kerosene, the water, or the steel respectively. Subtracting the equations above from each other, we obtain (Vy = ‘ee (Vy - V,’ = (Viz = γι) = (Vo, — Vx") Ἐ τ ( Η0-- Γ' H,0) ΞΕ Fa ar V5’) =< (Vo, ag to ° We now denote by Al the difference of displacements at the two positions A and C, and by Al’ the corresponding difference at the positions B and D. We now assume that V; and V2 differ only by the volume of the cylinder of length Al, and similarly Κι and V2’ differ only by the cylinder of length Al’. This assumption is justified if only the positions of the pistons at A and C are so far removed from the end of the cylinder that the end effects in the distortion of the interior are the same in the two cases. This condition has been shown _ by the theory to be satisfied when the distance is two or three diameters, as it always was in these experiments. Hence we may write, ha va 80 (1+ ap) Al Vy’ -- V./ = δῇ ( + ap’) Al where so is the initial section of the cylinder at atmospheric pressure, and a is the factor of proportionality by which this is changed with pressure. Now if we call the displacement form A to B, D; and from C to D, Do, then. 1. - Al = D; + AF and the above equation may be thrown into the form Vi — V2 — (Vy — V2!) = — 89(D2 — Dy) (1 + a. p’) + 59 Ala (p — p’) We now make use of the fact that the total change of volume of any substance under pressure is proportional to its mass. If Av (positive for a decrease) is taken as the change of volume of 1 gm. between p and p’, then, Vix -ς Vu — (Vo, --- Voy’) ΞΟ ἊΝ UE (muy, -- m2},) Vo πὸ — Vo π,0 = AtH,0™ Β,0 = Vay = (ie, = Vo, = A 0; (rr, — m2.) 326 PROCEEDINGS OF THE AMERICAN ACADEMY. This enables us to solve the equations for the compressibility of the water and the kerosene, giving, 1 AvH,o = Feet (Dz — Dx) (1 + ap’) — δολία (p — p’) + Av, (my, — mez) + Av, (m4, — mes)} and for the kerosene, when the two runs are both made with kerosene, as in determining the data for kerosene given at the end of the paper, 1 Av, = ———— {5 (D2 — D1) (1 + ap’) — soAla (p — p’) — Ar, Mor — Mik (m2, — m5) } The considerations so far apply only to the measurement of com- pressibility at constant temperature. The thermal dilatation is deter- mined in the same way as the compressibility from the difference of the thermal dilatation as given by two sets of experiments, one with the water replaced by steel. The piston displacement is not the same at corresponding pressures here, either, and a correction is to be applied for the thermal dilatation of the part of the cylinder which is exposed to pressure in the one experiment and not so exposed in the other. But this portion of the cylinder to which the correction is to be applied was seldom more than 1” in length, and the correction for this amount of steel is negligible in comparison with the thermal dilatation of the total quantity of water. There is also a correction to be applied for the dilatation of the steel replacing the water, and this correction is small but not negligible. It was assumed that the dilatation of the steel remains independent of the pressure over the pressure range used, and the value for ordinary mild’steels at atmos- pheric pressure was employed. This value is 0.000039 for the cubic expansion per degree Centigrade. The corrections to the measurements of the thermal dilatation are not so serious or so important as those for the compressibility, since the total effect is much smaller and most of the corrections become negligible. The method of determining the thermal dilatation has already been explained to be that of observing the change of pressure brought about at constant volume by a known change of temperature. From this the change of volume with temperature at constant pres- sure can be immediately determined if the slope of the p-v curve at set a3 (x) (= ) Op ov\ . that point is known, for \ ar = —( — ( Mg ee ers BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 327 dently given directly from the curves for compressibility at constant temperature. The slope of this curve changes somewhat with the temperature, so that a correction should be applied for this, but the change is so slight at the higher pressures that for this purpose the compressibility can be assumed constant. At the lower pressures, below 2500 kgm., the change cannot be neglected, and another method of computation must be applied. The thermal dilatation at low pressures was.determined by taking directly the difference between the isothermals traced out at different temperatures. This method is not applicable at high pressures be- cause the irregularities of isothermals traced at different times is sufficient to make their difference an inaccurate measure of the slight change of volume with temperature, but at the low pressures, the errors introduced by hysteresis and other irregular action of the steel cylinder are so slight that the method may be used directly to give the value of the compressibility, and by taking the differences, the value of the thermal dilatation. In fact it would seem that the method would be applicable with slight modifications to the determination of the compressibility of a great variety of substances at low pressures, and it is very much more rapid than the methods hitherto used. A special setting up of the apparatus was necessary for the experi- ments at low pressures, because in order to be able to reach low pres- sure on release of pressure it is necessary that the friction in the movable plug be not too high, and if the pressure has once been run to so high a value as to upset the plug, the friction becomes so great as not to permit release of pressure to much below 1500 kgm. For these experiments, then, the plug was made initially a push fit for the hole, by making it about 0.0015’’ smaller than when used for the higher pressures, and in performing the experiment the pressure was never pushed beyond 2500 kgm. In other respects the experiments at low pressures were the same as those at higher pressures. It was not necessary to take quite so elaborate seasoning precautions at these low pressures, however. With regard to the amount of hysteresis or elastic after-effects to be met in the experiments, the difference of the displacment with increasing or decreasing pressure usually amounted at the middle of the range to 0.03 in. This amount was very uniformly consistent, indicating that the cylinder had really settled down to a steady be- havior. The piston always returned to the starting point to within the limits of accuracy of reading, indicating that there was no leak or permanent set, or wearing of the packing in appreciable amount. 328 PROCEEDINGS OF THE AMERICAN ACADEMY. Of course the experiments at low pressures showed very much less hysteresis, in fact it was so small as to be almost imperceptible. The effect of hysteresis was eliminated as far as possible by using for the displacement at any pressure the mean of the results with increasing and decreasing pressure. The hysteresis was so constant that it would probably have been sufficient to have used consistently the results either at increasing or decreasing pressure. The actual pro- cedure has, therefore, the weight of two independent determinations. In the determinations of thermal dilatation, on the other hand, the hysteresis effects were so much smaller, that except for one run initially to show that there was no effect of this kind, the readings were always made either only with increase or only with decrease of temperature for any mean pressure, never with both increase and decrease. THE Data. Three independent sets of experiments were performed to give the change of volume with temperature and pressure over the entire range; namely the isothermal compressibility at pressures over 2500 kgm., the isothermal compressibility and the thermal dilatation at pres- sures below 2500 kgm., and the thermal dilatation at pressures over 2500 kgm. ‘This is the actual order of experiment, but for the pur- poses of presentation it will be better to use the natural order, pro- ceeding from low to higher pressures. COMPRESSIBILITY AT Low PRESSURES. The method with the present form of apparatus is not very sensitive at the low pressures, and not many measurements were made over this range. Two sets of determinations of compressibility were made, the first at 20°, 40°, 60°, and 80°, and the second at only 20° and 80°. Here, just as for the measurements at the higher pressures, there is always sufficient friction in the packing after the pressure has once been applied not to permit of close enough approach to the zero to make an extrapolation back to the zero justifiable. And if the extra- polation to the zero is to be made from the readings during first appli- cation of pressure, special effort has to be made to design the washers so as to avoid small initial distortions. For this reason only the second of the above sets could be used by extrapolation back to the zero of pressure. The readings of volume at 20° and 80° were corrected back to 40° from the thermal dilatation as determined by this same set of experiments, so that we have from the above two values for the BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 329 compressibility at 40° up to 2200 kgm. The first set of readings at five temperatures is consistent with this latter set above 1000 kgm., but at the lower pressures gives values for the compressibility which are doubtless too high. To find the best value for the change of volume at low pressures-we now have three sets of data, those of the TABLE I. VoLUuME OF WATER AT 40° AND Low PRESSURES BY DIFFERENT Meruops. Pressure, kgm. ΟῚ." Piston. Amagat. Change of Volume, cm.*/gm. Final Mean. .0000 0000 .0000 .0203 .0376 .0532 .0673 present determination, those of the previous work by the method of the steel piezometers, and the results of Amagat. The most probable value for the change of volume has been found by comparing these three sets of values. These values are given in Table I, as also the mean selected from them as the most probable value from the data at present in hand. In taking this mean, the greater weight has been given to the values of Amagat at the lower pressures, since his method of measurement was doubtless more accurate for the low pressures than the present method, which was intended only for high pressures, but at the upper end of the range in the neighborhood of 2000 kgm., more weight has been given to the present determinations. It is to be noticed that the mean value taken as final is lower than that found by Amagat. This divergence is in the same direction as that found by Parsons and Cook, who worked with a method like the present one. The deviation found by them from the results of Amagat is greater than that adopted here. 330 PROCEEDINGS OF THE AMERICAN ACADEMY.. DILATATION AT Low PRESSURES. For the thermal dilatation at low pressures, two sets of determina- tions were made; one was the series of isotherms at four different temperatures already mentioned, and the second was by the method adopted for the higher pressures, namely variation of temperature at constant mean pressure. The method of calculation for this lower .002 -012 ΗΕΞΕΞΗΞΕΞ ἘΞ 5 : ett : = ξξξξξεξ He He 0 ae ἘΞΈΒΗ :- ο ἘΞΈΞΞΞΗΣ ἘΞ: }ΞΞ ae o Ste Ἐπ Ξ ΕΞΕΙΕΕΙ 5, -O10f ΞΕ ᾿ς ΞΞΞΞΕΞΞΕΞΕΞΞΈΞΞΙΞΕΞΕ Ἑ ἘΞ τι: SHE : Ht Ό * stirs = aH safe He ὦ 00s Ξ Ὄ Es SS 7 = Ξ ets vo Z [5555 = ¥ : = .006 τ ο [ HH ΞΘ fo 4: Ξ a is - . HEE [} --Ξ 004 Φ » a Φ Vv on = i) = 9 οἵ: iis ΠΈΞΞΕ ἘΞΈΈΞΕΞΕ fe : : HESTE: O. 1 25. 5. 4.756] 7 <8)" Ὁ 10 Meas Pressure, kgm. / cm.’ x 10° Ficure 2. The change of volume of water for intervals of 20° plotted against pressure. range was not the same as that employed for the higher pressures, as already explained, due to the fact that the slope of the isothermals is not sufficiently independent of temperature at the lower pressures. The method of computation adopted here was a graphical one, by plotting the observed volume and pressure points for the different temperatures and taking the difference between adjacent curves graphically. The temperatures at which the different determina- BRIDGMAN.—— THERMODYNAMIC PROPERTIES OF WATER. 991 .tions were made were not exactly the even temperatures desired, namely 20°, 40°, and 60°, and 80°, but they were in all cases within a few tenths of a degree of these temperatures. The results were corrected to these even temperatures by assuming the mean variation with temperature over the whole temperature range to hold for the few tenths of a degree on either side. The final result given by the data is the total change of volume for an interval of 20°; from 20° to 40°, from 40° to 60°, and from 60° to 80°. The mean of the results of the two sets of experiments is shown with satisfactory accuracy in Figure 2, on which are plotted all the values obtained by the different methods. The results for the low pressures are shown in the full black circles. These values are seen to extrapolate, without forcing, to the values already found by other observers for atmospheric pres- sure, and they also make fairly good connections with the values found by the other method for the higher pressures. In view of this agree- ment it did not seem to be necessary to make further determinations of this quantity. CoMPRESSIBILITY AT HicH PRESSURE. The determinations of the isothermal compressibility at higher pressures extended over a considerable interval of time and are more numerous than any of the other determinations. In all, twelve deter- minations of this quantity were made, at five different temperatures. These determinations include those made during the early course of the experiment, when the attempt was being made to find the thermal dilatation directly from the difference of compressibility at different temperatures. A little work with the method showed that it was not sufficiently accurate for the purpose, but the results obtained then can be used to give the compressibility at the standard temperature, 40°, by applying the temperature correction found from the later more accurate results. The temperature of 40° was chosen as the standard because this is the lowest of the 20° intervals at which the water is liquid up to 12000 kgm. The results of these twelve determinations, extending over a period of three months, are shown in Table II. The results as given are reduced to 40°, but the temperature at which the original measure- ments were made is given also in the table. Two of these sets of determinations differ considerably from the others, and were discarded in taking the mean, although as it happens one of these discarded sets is too high and the other too low, so that it makes very little difference PROCEEDINGS OF THE AMERICAN ACADEMY. 992 ‘UBOU ΘΠ) 8] UL popavosicqy κα ἽΠΒΘΤΑΙ [eur SOFT” 68ST" OSTT LLOT οθ΄ OTSO° 6990" 9970" OSZ0° 0000" re) Gd) POET” 0.1 550Τ᾽ Τ000᾽ ἀετῷ “AON δ180᾽ δ600᾽ ΤΙΝ 9960" 0000" T6G1° O8TT CLOT LY60- 9080" 090° 1270᾽ 226 } | ΘΠ: 801 | GOST” 8.11 | LOGh: POOL’ | 9801" | F6OT™ θοῦ | 9960° | 0260" v6L0° | €180° | 0880" Gs90 | cS90° | 0290° ογ΄ | 8970° | 180° €7c0° | LSZ0° | δυσί" 0000" 0000" | 0000° (SR) SONS “GCO-FG “AON | “AON | “AON | | ‘SS-LE 1 99d ‘9-¥ “AON |-6G AON] ‘09d 0000° “66-61 ‘00d 1270. σοζθ᾽ ΟΟ000᾽ “ς-ς uve “τα )εἶταο ‘aUIN[OA 10 osueyOD ‘OF LY σαν κα 1ὸ AWOTOA AO ΠΌΝΨΗΩ JO SNOILVNINYALACG OCNGUAdHIGC] AO SLTIASAY Tl ΠΊΘΥΙ, ‘OInssod q BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 333 in the final result whether they are included in the mean or not. For convenience in making the computations the pressure was taken in units given conveniently by the changes of the manganin resistance, the intervals of pressure corresponding to a displacement of the slider of the bridge wire of 5 em. TABLE III. CoMPARISON OF REsuLts BY Two MetTuHops ror CHANGE OF VOLUME OF WATER AT 20°. Pressure, : Pressure, kgm. Piston. Piezometer. cm.” Piston. | Piezometer. «Ο000 .0814 .0821 .0954 0964 . L078 .1105 . 1190 . 1229 These results, reduced to 20° are shown compared with the results of the previous determination in Table III. It is seen that the newer results are lower than the former ones, the difference being about 1%, except at the higher pressures, where the difference is greater. The agreement is perhaps not as close as could be desired, but at present there seems to be no way of choosing between the results. There is no consistent discrepancy, which would indicate a fundamental error in the present method, such as in the correction applied for the dis- tortion of the steél cylinder, for example. If there were any such error it could be eliminated by so choosing the correction as to make the present results agree with the former ones. In the absence of any means of deciding between the two methods therefore, and since the results by the present method reach over a wider temperature and pressure range, and since also the method has been used much more extensively than the former one and with no greater discrepancy in the individual results, these present results have been accepted as the best ones. But it must be remembered that the absolute com- pressibility given here may be m error by as much as 1% at the higher pressures. This error, however, will not be found to invalidate any of the conclusions drawn from the data. 334 PROCEEDINGS OF THE AMERICAN ACADEMY. DILATATION AT HicH PRESSURES. The determinations of the thermal dilatation at the higher pressures were made on four occasions. The first two of these were preliminary, during which was discovered the necessity of seasoning for tempera- ture as well as for pressure, and also the necessity for the secondary pressure seasoning over the small range of pressure accompanying the changes of temperature. These first two determinations, while confirming the results of the two later ones, were not given much weight in selecting the final value. The method of computation adopted in finding the thermal expansion from the data requires mention. At first an attempt was made to apply the same graphical method which has been already explained in its application to the determinations at the lower pressures. This method involves the drawing of a curve of the same general slope as the compressibility curve through the two points giving piston displacement against pressure at each temperature. But it was found that even after the seasoning for the small pressure range involved here, the points were too irregular to give good results by this method. The irregularities may be due to residual hysteresis, but are more probably due to slight irregularities brought about by the motion of the piston itself. These irregularities are too minute to have any effect on the com- pressibility determinations. The best way to avoid them is to utilize in the computations only those readings during which the piston remains stationary. This means that only the change of pressure accompanying a change of temperature is used in making the computa- tions, the second reading at any temperature by which the pressure is brought back to the mean value being ignored. The change of volume at constant pressure for the given change of temperature is then computed from the known change of pressure at constant volume and the previously determined change of volume with pressure at constant temperature. In making this computation it is generally necessary to make two corrections; one to bring the temperature interval to the exact 20° desired for the final results, and the second to correct for the very slight change of measured piston displacement accompanying the change of temperature. This change of displace- ment is seldom over 0.003”. It is probably not due entirely to actual motion of the piston, but partly to temperature changes in the bars of the press which dip into the thermostat. That this method of computing the results is preferable to the graphical one previously mentioned is shown by the fact that this method gives very much BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 330 more uniform and consistent results when applied to the same set of data than the graphical method. The method of computation adopted was first to calculate inde- pendently from the individual observations of each set of readings the thermal dilatation at six mean pressures between 2200 and 12,000 kgm. Then smooth curves were drawn through these points for each set of readings, the curves being spaced in the best way so as to give regular variations with both pressure and temperature. The values given by the smooth curves of each set of readings were then combined into the grand mean. In taking this grand mean, as already explained, almost the entire weight was given to the last two sets of readings. The agreement between the different sets was best at the higher temperatures, 60° to 80°, and about equally good between 20° and 40° and 40° and 60°. ΑἹ] four sets of curves, while not agreeing very well as to the numerical value of the coefficient, do agree as to the general character of the results, which are, perhaps, not quite what would be expected. The unexpected feature is the change in the sign of the temperature derivative of the dilatation at the higher pressures. At the low pressures the dilatation is greater at the higher temperatures, but at the higher pressures the thermal dilatation becomes less at the higher temperatures. This essential feature is verified on all four sets of curves. There are indications that it may be an essential characteristic of the behavior of any normal liquid at high pressures, and that it is not peculiar to water alone. This is shown by the work on kerosene, and is also indicated by the work at present being done on still other liquids. This will be taken up in greater detail later. The other feature not to be expected is the increase in the value of the thermal expansion between 20° and 40° at the higher pressures. It is to be distinctly expected that the ther- mal dilatation will decrease with rise of pressure, as indeed it does for all the other intervals of temperature, but this rise between 20° and 40° is shown by all the sets of determinations and seems to be an un- doubted fact. It is probably connected with some new abnormality in the behavior of water at the higher pressures, which may be con- nected in some way with the appearance of the new variety of ice. The values finally taken as the best values for the thermal dilatation are the mean of the results of the four determinations, much the greater weight being given, as already explained, to the two latter determina- tions. Figure 2 gives these results, as also those of the other methods at the lower pressures. The agreement of the two best determina- tions at the higher pressures is about 5% for the lower temperature 336 PROCEEDINGS OF THE AMERICAN ACADEMY. interval from 20° to 40°, 3% for the interval 40° to 60°, and 2% from 60° to 80°. The order of accuracy to be expected in these thermal measurements is not so great as that in the compressibility determina- tions, therefore, but perhaps the accuracy is as great as could be expected when one considers the smallness of the quantities involved, and the difficulty of making such measurements at high pressures. At any rate the absolute value of the coefficient cannot be very much inerror. This is made probable by the agreement with the known values at atmospheric pressure. The accuracy is at least high enough to enable us to expect a fairly good quantitative description of the vari- ous thermodynamic quantities under high pressure, even those most sensitive to error. The calculation seems to be worth while carrying through in some detail, because such calculations seem never to have been undertaken for any substance, even for the low pressure range up to 3000 kgm., which is the range over which compressibility determi- nations have been previously made. Discussion OF RESULTS. The first necessity for a calculation of the various thermodynamic quantities is as accurate as possible a knowledge of the relation between pressure, temperature and volume over the entire pressure- temperature plane. It may be shown that this is sufficient to com- pletely determine the thermodynamic behavior of the substance if in addition the behavior of the specific heat at constant pressure, for example, is known in its dependence on temperature at atmospheric pressure. This may be assumed to be known well enough for the present purpose. The first and the most important outcome of the present data is, therefore, the construction of a table giving pressure, volume, and temperature at sufficiently close intervals. In con- structing this table the basis of computation was the compressibility as determined at 40°. This, together with the known value of the volume at 40° and atmospheric pressure, gave the volume as a function of the pressure down a line through the middle of the table at 40°. The values of the volume were tabulated for intervals of the pressure of 500 kgm., the values found graphically from smooth curves through the experimental points being so smoothed as to give smooth second differences. The values of the change of volume for intervals of 20° now were combined directly with these values to give the volume as a function of the pressure at 0°, 20°, 60°, and 80°. To find the intermediate values of the volume, smooth curves were drawn through BRIDGMAN.—— THERMODYNAMIC PROPERTIES OF WATER. 337 these five points at every constant pressure, and the intermediate values so chosen as to given smooth values for the second differences over the entire temperature range. The values for the points below zero, Which are also given in the table, were taken directly from the previous work, the values for the dilatation found there being kept without modification, but the present value for the compressibility at 0° being used. The differences so introduced may be seen by com- parison of the two tables to be only slight. The table gives the volume to only four significant figures, since this is as many as the variations in the values of the compressibility would entitle one to, but in making the calculations of the thermal expansion it was necessary to keep three significant figures for the expansion, which would mean five figures in the table. The thermal dilatation per degree rise of temperature was deter- mined from the values used in the construction of the table for the differences of volume at 5° intervals by dividing by 5, and using the result as the thermal expansion at the mean temperature. The values of the total change of volume for five degree intervals had been smoothed so as to give smooth second differences, so that the dilata- tion as found in this way was smooth also with respect to the second differences, and could be used directly to give the second tempera- ture derivative of the volume at constant pressure. The difference of thermal dilatation at different temperatures can evidently be combined with the known compressibility at 40° to give the compressibility as a function of the temperature. These several quantities so determined; the compressibility, the thermal expansion, and the second temperature derivative of the volume, in their dependence on temperature and pressure, are the basis of most of the calculations of the quantities of thermodynamic interest to be given presently. The accuracy of most of these quan- tites is not so high but that they can be shown as well in figures as in tables, and this manner of presenting them has been chosen as giving the most ready general survey of the facts. The tables and figures follow. The results are given simply for themselves, without much comment, except to call attention to the unexpected features, or those properties which seem to be peculiarly characteristic of high pressures. It would not be safe to generalize from the behavior of this one liquid, abnormal at low pressures, to the general behavior to be expected for any liquid for high pressures and the bearing on a possible theory of liquids. 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SCS τ τ" τὸ ΤΠ ὉΠ ἢ ΠΠ τοἢν» ο85 «οὐ 090 οὐ 981 G08" 0LE8" 888° S2t8° 60F8" 6098" 648° O8h8° F9F8° cscs’ 6998" #998" 2868᾽ 9998" 6F98° ξ08᾽ O198" TSZ8° F&L8° LTL8° 9698" €F88° S288 L088° 9828" €h68° 268° 2068° 8888" 0906° 806° ST06° 2668" 1916" OST6° €&16° ZIT6 €626° 9126° 0926° Zeb" 686° γῦρο O1F6° λΟΡθ᾽ 6096" 6896° 8296° #896" 9826° 8216° TLL6° 8226 6698" p898" PLL8° 9918" 9188° F988" F868" GL68" FOT6” § F606" £h06° SEC6" 9TF6° FO0FG" 8696" S616" 1000°T 6666°0 0000°T 9000°T LT00°T £968" $806" 8666" “01 “oG τοῦ “of «οὐ oS -— 005 - “HUOLVURdNaA 1, GNV DTAOssaud JO NOILONOY V SV ὙΠ} Δ FO ANOTIOA “AI WTEV.L 308 “m3 “gINss01 J BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 339 In presenting the results, the quantities have been arranged in order of simplicity of the thermodynamic formulae, which is also the order of directness with which they are derived from the experi- mental data. Volume, cm.? per gm. 4 oo aoe 8 60} X "Wo / ᾿πϑν ‘aunssotg 4 a Π ΟἿ 6 8 Ficure 3. Isothermal lines for water, showing volume against pressure. In Table IV are given the values of the volume for intervals of pressure of 500 kgm., and intervals of temperature of 5°. The table does not require comment. It was computed in the way already described. The values of the volume at intervals of temperature of 20° are shown as a function of the pressure in Fig. 3. The figure does not show the results as accurately as the table, but enables one to form a clearer mental picture of the nature of the results. The curves, on the scale of the figure, do not show any abnormalities to the eye, except in the neighborhood of the origin, where the well known negative expansion at 0° results in the curves drawing together. 340 PROCEEDINGS OF THE AMERICAN ACADEMY. There are various abnormalities besides those in the neighborhood of 0°, however, as will be shown by the other figures. | With regard to the compressibility there seems to be some variance of usage, so that it will be well to call attention to the fact that the quantity used throughout this paper in the sense of compressibility is Isothermal Compressibility 0.0 42 ἘΠΕ : tes HE ΠΗ i ἘΠΕΒΕΙΤΕ : tt 0.0.41 d et =4t'} ΤΗΣ Tf iff SHG ἘΒΘΕΤΗ ΒΗ ἢ ἸΞΈΙ ΠΗ ; ἘΠΕ ΠΗΞΕΙΕ ΠΗ ΠΗ ΕΣ te $923 4 - BS Ol δ ν 4 νὴ G72) 28 9 aa 11 5 Pressure, kgm. / cm.’ x 10° Figure 4 ‘Theisothermal compressibility of water, (=) against pressure. Op /t the derivative (Fe) : Op t same sense. Figure 4 shows the compressibility, that is, the analytic Sometimes the expression : (=) is used in the t expression (1 , as a function of the pressure at 0°, 20°, and 80°. ι It would have made the figure too crowded to have tried to show the values for 40° and 60° also. The complete values for the five standard temperatures are shown in Table V separately, however. The figure shows the well known abnormality in the compressibility at the low pressures, namely a higher compressibility at the lower than at the higher temperatures. This abnormality disappears above 50°, and from here on the compressibility increases with rising temperature. The figure shows that at 80° the initial compressibility is higher than OE ee BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 341 at 20°, although it has not yet risen to the value at 0°. In addition to the abnormality at low pressures, the curve shows also a slight TABLE V. ComPRESSIBILITY OF H:O. (2), em.3/gm. 40°. 60°. Pressure, kgm./em,? | — | abnormality at the higher pressures in the neighborhood of 6500 kgm. Here the compressibility at 20° rises and at the melting point of ice VI, it has become higher than the compressibility at 80°. The thermal dilatation shows abnormality in the same locality; it would seem to be 342 PROCEEDINGS OF THE AMERICAN ACADEMY. connected in some way with the appearance of the new variety of ice, but the exact connection cannot at present be stated. The large change in the value of the compressibility brought about by pressure should be noticed, amounting at 12,000 kgm. to a decrease of five fold. Furthermore the rapid flattening of the curve at the higher pressures also should be commented on. The curve gives the appearance, for the pressure ranges used here, of becoming asymp- totic to some value greater than zero. Of course this cannot really be the case for infinite pressures, for otherwise we should have the volume completely disappearing for some finite value of the pressure, but it may indicate the entrance of another effect at the higher pres- sures, which may persist in comparative constancy for a greater range of pressure than will ever be open to direct experiment, such an effect as the compressibility of the atom, for example. This possibility has been already mentioned and made plausible from the data of the preceding paper. If instead of the compressibility as defined above, the quantity . (=) , which in this paper will be called the relative compressibility, t is plotted, a curve of the same general character as that shown will be obtained. The compressibility may also be plotted against a different argument than the pressure. For many purposes the pressure is perhaps not the most significant independent variable that might be chosen. This is because the external pressure is not a measure of what is happening inside of the liquid. We conceive a liquid as composed of molecules in a state of constant motion and of collision with each other, acted on also by attractive forces between each other. The effect of these attractive forces is to produce at the interior points a pressure which may be much higher than the external pressure. The external pressure is equal to the interior pressure diminished by the amount of the attractive pressure drawing the molecules to the interior at the exterior surface, where the attraction is an unbalanced action in one direction. The amount of the unbalanced pressure at the outside depends in a complicated way on the law of attraction between the molecules, on their mean distance apart in this surface layer, and on the distribution of velocities in this layer. The external pressure required to hold the liquid in equilibrium is, therefore, largely a sur- face phenomonon, and is connected in a complicated way with the state of affairs at inside points. A more significant independent variable, therefore, would be one involving only the condition of the BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 343 molecules on the average throughout the mass, and not one depend- ent on the surface layer. There are only a few such quantities de- pending on the state of the liquid at interior points. Any quantities involving in any way the constancy of pressure or of entropy, for ex- ample, do depend on the complicated action of the surface layer. One of the quantities which is independent of this surface layer, however, is the volume. In many theoretical considerations the use of the vol- ume as an independent variable is known to produce simplifications. If the volume, instead of the pressure is taken as the independent variable for the compressibility, curves are obtained of the same general appearance as when the pressure is used forthe variable. The compressibility falls with decreasing volume, and the curvature is in the same direction as when the pressure is the independent vari- able. The same general characteristics are also shown if the relative compressibility instead of the compressibility is plotted against the volume. The two sets of curves, for the compressibility and the relative compressibility, do show one feature in common, however, different from the curves when the pressure is used as the variable. This is the fact that the compressibility is always lower for the same volume at the higher temperature. This is true throughout the entire range of volume used; there is no crossing of the curves indicating abnormalities, such as is the case when the pressure is used as the variable. This is what one would expect on the kinetic theory. A liquid, at two different temperatures but at invariable volume, differs only in the violence of the motion of its molecules. At the higher temperature, the kinetic pressure due to the motion is greater, and so the resistance offered to change of volume under a given increase of external pressure 15 greater when the temperature is higher. Fig. 5 shows the thermal dilatation as a function of pressure at various temperatures. The thermal dilatation plotted in the figure is the expression (2 instead of the expression : (=) , which is some- » » ot times used as the dilatation. The usage adopted here for the dilata- tion is analagous to that explained above for the compressibility. The values listed in the figure were obtained from the table of volumes in the manner already described. The curve at 0° was obtained from the data of the previous paper for the low temperatures, but in that paper the mean value of the thermal expansion for the range 0°-20° was given, whereas here the instantaneous value at 0° is given instead. The substitution of the instantaneous for the mean dilatation produces no change in the general character of the curves, however. 344 PROCEEDINGS OF THE AMERICAN ACADEMY. The points at the higher temperatures were obtained from the data of this paper alone. There are two striking features that call for special comment. ‘The first of these is the abnormal behavior of the curve for 20°. In the initial stages, the dilatation rises with increasing pressure, unlike normal liquids, but this merely indicates the return of water to the normal behavior to be expected at high pressures. At about 3500 kgm. the curve at 20° has reached a maximum and begins -to descend with increasing pressure, as it does for the curve at 0°. But the descent continues for only a little way, and at 5500 kgm. the curve begins to rise again, indicating the entrance of a new abnor- = T [Sanu et + ; + Senses ται : =: + = ΞΕΞΞΞΞ: rte ht ἘΞ ΞΕ ἘΞ ΞΕ gee ae errs arse a Ξε : Fatt nine Ft Ξ Ξ ΣἘΕΞΞΞΞΞΞΞΞΞΞΕΙ = th Ht + tH pesees ἘΕΞ port 9. geseaeassazeze: ΞΕΞΞΞΞΕΞΞΒΞΞΞΞΞΞΞΈΞΞΞ: ΞΕ ΞΞΞΞΒΞΞΞΞΞΞΕΣ ett Ὁ Sete + ΤΙ ΕΗΞΕΞΞΕΕΕΞΉΞΗΤΕ iy} + a 1353503 + ra 2: res Ὁ — = 3 . --- τ-ας--- = ++ — = ΕΣ SE SESE - oy = 55: is] + Ξ = ΓΕ = psesess i Ξε Vv pi = posses = adi tuassnasqraetitasvteseerefertae ΤΥ as 3° 4 B67 PS 9. 10 aaa eee Pressure, kgm. / cm.’ x 10° Figure 5. The dilatation of water, (=) , against pressure. Pp mality. The abnormality is not so striking or so great in amount as that in the neighborhood of 0° and atmospheric pressure. The ab- normality at 20° continues for about 2500 kgm., up to 8000, where the curve is terminated by the entrance of the solid phase, but the direc- tion of the curve has already begun to change, indicating that if it could be continued, this abnormality also would probably disappear at higher pressures. As to the question of experimental error here, there would seem to be no room for doubt as to the actual existence of this new abnormality, for it was shown by all four of the dilatation curves, even those taken before the method was got to running satis- _ ΎΨΜ ΥΥΥ ΨῃΨῃ0ΟΙΝ BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. B45 factorily, and in which the accuracy was not very high. The curves at the higher temperatures behave as one would be prepared to expect in the region of low pressures. The curve for 40° shows vestiges of the abnormal behavior at the low pressures, namely slight initial rise of dilatation with rising pressure, followed by a fall, but the curves at the higher temperatures, 60° and 80°, show the regular initial decrease with rising pressure shown by all normal liquids. But at higher pressures, the behavior of all three of these curves, for 40°, 60° and 80° is different from what might be expected. The unexpected feature consists in the crossing of the curves, all in the vicinity of the same pressure, 5500 kgm., so that at higher pressures the thermal dilatation at the higher temperatures is lower than it is at the lower temperatures. It has been already remarked that there are indica- tions, both from the present work and from that of Amagat, that this - may be the behavior for any normal liquid at sufficiently high pressures. The comparative constancy of the thermal dilatation at the higher pressures, fs also a matter perhaps not to be expected. Thus the expansion at 40° remains nearly constant over the entire range of pressure, while the compressibility has in the same range dropped from 44 to 9. It was distinctly expected, before these measurements were taken, that the dilatation would show the greater variation with pressure, so that the effect of temperature on the volume would tend to disappear at the higher pressures, but such is not the case. The relative thermal dilatation may be plotted against pressure, as was the relative compressibility. The curve shows no striking features. The curve plotting relative dilatation against volume has also been plotted, and this is the same in general character as the others. The slight differences consist in an accentuation of the ab- normalities in the neighborhood of 5500 kgm., and the fact that at the lower volumes, that is at the higher pressures, the dilatation against volume increases with decreasing volume for 40° and 60°, but decreases for 80°. These figures for the thermal dilatation and the compressibility complete those which are obtainable directly from the table. Other quantities of thermodynamic interest may be obtained by combining these, however. Perhaps the simplest of these quantities are those connected with the absorption of energy when the pressure is changed at constant temperature. The first of these is the actual mechani- cal work done by the external pressure in compressing the liquid at constant temperature. This of course is simply the expression 346 PROCEEDINGS OF THE AMERICAN ACADEMY. W= ul Dp (= dp+ It was obtained by a mechanical integration t from curves similar to the volume curves of Figure 3, drawn on a larger scale. For this purpose the integrating machine owned by the mathematical Department of Harvard University was used. The sae + 9 8 7 6 δ 4 8 2 7 8 9 10 Pressure, kgm. / cm.” x 10° Work of compression, kgm. m. per gm. Ficure 6. The mechanical work of compression at 60°. actual value of the mechanical work at any pressure is of course de- pendent on the temperature, but since the variation is so slight that it would have been impossible to show it in the figure (see Figure 6), the work of compression is plotted for only the one temperature, 60°. Although the change of external work with temperature was too slight to show in the diagram, the change with temperature was nevertheless taken account of in making the calculations of the quantities depend- ing on it to be described immediately. After the first 4000 kgm. it is seen that the curve becomes very approximately linear. The curve for a substance which retains the same compressibility unchanged over a wide pressure range, as steel for example, is a parabola, the work increasing directly as the square of the pressure. That this curve for water becomes linear, means that the compressibility decreases so fast with increasing pressure that the decrease in the yield “ΠΑ. “ὠὰ. σι δ BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 347 of the liquid for a given increment of pressure decreases almost at the same rate that the pressure itself increases. The total heat given out during an isothermal compression may be derived from the formula (=) =—T (5) . This quantity is shown Op/, OT/p in Figure 7. The figure does not call for especial comment. The peseasess Ἑ 40 + eth Be a se es o a ἢ 3 3) 30 at ἘΠῚ : = 20 Seas 3 δ 2 satsescssesccsenseees 3 ἩΣΉΉΤΗΣ ἢ Θ᾽ ees a ols ξ ΠΕ + HESS: « 1 ΕΞ : 559.» (οἵ τ + 4 | >= jan ee a8 ΤΡ se 3 srt : c= Ἶ : HTH ΤῊΣ HEE ptt ae Ol eohes 4 “are 7 85... 9 10 11 15 ᾽ Pressure, kgm. / cm.” x 10° Figure 7. The heat given out by water during an isothermal compression. rapid change in the direction of the isothermal lines in the vicinity of the origin due to the abnormal behavior at low temperatures and pres- sures is manifest from the figure, as also the slight abnormalities at the upper ends of the 0° and the 20° curves, already commented upon in other connections. Beyond 5000 kgm. the curves for all tempera- tures tend to become linear and parallel to each other. These two quantities, the heat liberated in compression and the mechanical work, combine to give the change of internal energy along an isothermal, this change of energy being equal to the difference of the heat and the mechanical work. The change of energy so calcu- lated is shown in Figure 8. The change is a decrease, which continues at all temperatures up to the highest pressures. In the previous paper a value of this quantity was given, confessedly inaccurate, since in the computation the mean thermal dilatation between 0° and 20° had been used instead of the actual dilatation at 0° or 205, The 348 PROCEEDINGS OF THE AMERICAN ACADEMY. curve so obtained had the characteristics of the curve now given for 0°, but the maximum at the top was much more strongly accentuated than in the present figure. It was surmised in the previous paper that at high enough pressures the internal energy of all liquids would probably increase instead of decrease along an isothermal. This surmise seemed Ἢ t t He : 4 tf agesas +H : ΕΗ : Ht segssesecesssessss 18 : ἘΞ ΕΞ sasas HoH itt + ἜΤΗ +H 1 2 8 “ΜΠ 667776284 99 10. 11 10 Pressure, kgm. / cm.” x 10° Change of Internal Energy, gm. cal. per gm. Fiaure 8. The decrease of internal energy of water during an isothermal compression. plausible because one would expect that at high enough pressures the energy stored up as strain in the interior of the molecules in virtue of the extremely high pressures would more than counterbalance the work done by the attractive forces of the molecules themselves as they were brought closer together by the action of the pressure. This present figure shows that this is not the case, however, for the range of pressure reached here. The lower temperature, 0°, is the only one at which this reversal of the direction of the change of internal energy manifests itself, and this change, in comparison with the other curves, is now seen probably to be an effect of the other abnormalities shown at low pressures and temperatures. Nevertheless it would still seem as if at very high pressures the energy must increase instead of de- crease along an isothermal, but the only indication of it from the present curves is in the direction of curvature, which is in the direction © Ὡς ἐπα i a i i i il i i i .... BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 349 to indicate the possibility of a maximum and a reversal of direction at higher pressures. The pressure for a maximum, however, if there is one, is much beyond the reach of any at present attainable. Within the pressure range of these measurements, the attraction between the molecules still remains the dominant feature, so that the work done by the attractive forces and liberated as heat much more than suffices to overbalance the mechanical work of compression. The internal energy of a substance is one of those quantities which depend only on the properties of the mass of the substance at interior points and do not involve the action of the surface layer. Change of energy plotted against volume shows in the first place that the change of internal energy is much more nearly a linear function of the volume than it is of the pressure. The average slope of the isothermal lines of energy increases rapidly with rising temperature for the lower temperatures, but the two curves for 60° and 80° run nearly parallel to each other for their length. Abnormalities are shown at the upper ends of the 0°, 20° and the 40° curves, and the 0° curve shows the same maximum as it does when plotted against pressure. The origin, of course, for the curves at different temperatures does not coincide as it does for the same quantities when plotted against pressure. One other quantity may be simply determined in terms of the compressibility and the thermal dilatation alone, the so-called pres- sure coefficient, that is, the change of pressure following a rise of temperature when the temperature is raised by 1° at constant volume. This quantity is given immediately in terms of the compressibility and the thermal dilatation by the well known formula, (se). — &), Ge) It is shown plotted in Figure 9. The curves for 0° and 20° show anomalies, as is indicated by the unexpected direction of curvature. The other curves for the higher temperatures seem to be regular enough, though of course it cannot be told whether the course of these curves is the same as that which would be shown by a normal liquid or not. At the upper ends of the high temperature curves, the curva- ture is in such a direction that if they were continued far enough the pressure coefficient would decrease instead of increasing with rising pressure. This quantity, the pressure coefficient, has been made the basis of theoretical speculation. It has been enunciated as a law, approxi- mately true, by Ramsay and Shields, that the pressure coefficient 350 PROCEEDINGS OF THE AMERICAN ACADEMY. is.a function of the volume only. This means that if the coefficient were plotted against volume instead of pressure the curves for all five temperatures would fall together. That this is not the case for water at high pressures is shown very distinctly in Figure 10. At the lower pressures and the larger volumes, the curves for the different tempera- ζὸ οι (ve) (=) Coefficient of Pressure + + aon: Beoossas + +H saan #4 Ἔ jausegges passaassas es mas eg ze: ΘΟ 1 5 τ π΄, Π XG 7 8, eo Opie 1 Pressure, Kgm. / cm.” x 10° Figure 9. The pressure coefficient, that is the change of pressure accom- panying a rise of temperature of one degree, as a function of the pressure. tures are very widely separated. The abnormality on the curve at 0° in the neighborhood of the locality where the new variety of ice makes its appearance is very striking. At the higher pressures the curves do draw together, but they are not approaching coincidence, for they cross in the neighborhood of a volume of about 0.85. It does not seem likely that the entire failure of coincidence throughout the whole range of pressure can be due to abnormalities, since even at low pressures water is nearly normal at the higher temperatures, and certainly at the higher pressures and temperatures we have every reason to expect that its behavior is quite like that of other liquids. This completes the list of quantities which can be deduced directly from the compressibility and the thermal dilatation. Other quanti- BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 351 ties of thermodynamic interest involve the specific heats, and these in turn involve the second temperature derivative of the volume. The first of these quantities is the specific heat at constant pressure. Y As This is given by the thermodynamic equation (=) =—T (=) ἘΠῚ ΠῚ ? : ap), δῖ, will be seen that only the derivative of the specific heat is given by the data as directly determined. In order to obtain the specific heat itself, the derivative, obtained from the ta- bles in a manner already described, must be integrated. This integration was performed mechanically, in the same manner as the integration for the mechanical work of compression. The results are shown in Figure 11. The values for the specific heat as a 100 90 80 Volume, cm.° per gm. function of temperature at atmos- Fieure 10. The pressure coeffi- pheric pressure were taken from the cient of water as a function of the Coefficient of Pressure ΠΗ aed Danse These values seem to be open to some slight question at the present time due to experimental work done by Bousfield ° since the publica- tion of the tables, but in any event the possible error is slight, too slight to be visible on the scale of the figure. The curves show the now expected abnormalities at 0° and 20°. The striking feature about the curves for the higher temperatures is the very rapid increase of the specific heat with rising temperature at the higher pressures. The specific heat at first decreases on all the curves except at 0°, but passes through a minimum, and then increases. The pressure of the minimum rapidly becomes less with rising temperature, and is situated at 6500 kgm. for 40°, 5500 kgm. for 60°, and at 1100 kgm. for 80°. At 80° the specific heat rises rapidly beyond the minimum, reaching the value 1.17 at 12000 kgm. Any valid characteristic equation should predict the behavior of the specific heat at high pressures as well as giving the volume in terms of pressure and temperature, since from the equation the second tem- perature derivative of the volume may be found. The equation of Tumlirz 19 has been mentioned in the preceding paper as giving per- haps as good agreement as any with the previously known facts over 8 Marks and Davis, Steam Tables. (Longmans, Green, and Co.) 9 W. R. and W. E. Bousfield, Trans. Roy. Soc. (A), 211, 199-251 (1911). 10 Tumlirz, Sitzber. Wien, Bd. 68, Abt. Ila (Feb., 1909), pp. 39. 352 PROCEEDINGS OF THE AMERICAN ACADEMY. a pressure range of 3000 kgm. This equation would predict a con- tinuous diminution in the specific heat up to infinite pressures, the limiting value being very approximately 0.5. It was shown in the preceding paper that there is some new effect introduced at the high pressures which does not make itself felt at the low pressures, with the a HEEEG ἘΣ gine Ty le ἘΞ ἘΠΕῚ ΘΟ 1 5. 8. ἡ τ Ὁ ΒΕ 9. 10 Π| 10 Pressure, kgm. / cm.’ x 10° Figure 11. The specific heat at constant pressure of water as a function of the pressure. result that an extrapolation to infinite pressures from the behavior for the first 3000 kgm. is not safe. This was shown in that paper by the behavior of the volume, which tended to decrease more rapidly at the high pressures than was predicted by the formula. The present data also show that there is a new effect at the high pressures, and indicate that the effect, whatever it is, is such as to have a much greater influence on the specific heats than on the volume itself. The specific heat at constant volume may be found from the specific (5), τς δῚ ; (χω, This quantity, so calculated, is shown in Figure 12. The same ab- normalities are shown at 0° and 20° as were shown in the curves for C,. The curves for 40° and 60° decrease for nearly their entire lengths, although they are just beginning to rise at the very highest pressures, but the curve for 80° shows the same sharp turning point and the same rise through the greater part of its length as the curve heat at constant pressure by means of the formula, C,—C,= —r EEE ee eee ee eee BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 353 for C,. This quantity, the specific heat at constant volume, has more theoretical significance than the other specific heat, since this repre- sents the heat going into the rise of internal energy of the liquid when the temperature rises, and does not involve the work done against external pressure in expanding the liquid. The external work in- C,, gm. cal. per ok ee foe 4 Bee - 6 wR τὸ 0.115 Pressure, kgm. / cm.’ x 10° Figure 12. The specific heat at constant volume of water as a function of the pressure. volves in a complicated and at present unknown way the action of the surface layer, while the specific heat at constant volume does not contain this surface effect. This specific heat is therefore one of the quantities mentioned in the beginning as having significance be- cause it does not involve the unknown attractive forces between the molecules as displayed in the surface layer. In order to show this independence of the surface layer, of course C, should be plotted against a variable not itself involving the action of the layer. It is evidently not adequate, therefore, to plot C, against the pressure as as been done in Figure 12. C, plotted against volume may be ex- pected to show this independence of the action of the surface layer. It is shown so plotted in Figure 13. The figure is of the same general character as that in which it is plotted against pressure, but the separation of the curves for the different temperatures is greater, partly because the curves do not start from a common origin. The minimum on the curves for 40° and 60° comes at a lower pressure than it does in the former figure, and the upper end of the 80° curve is perhaps a trifle steeper at the upper end, but there are no essential differences. The entire behavior of the curves is not what one would 354 PROCEEDINGS OF THE AMERICAN ACADEMY. expect from the ordinary theoretical considerations, however. It is usually considered that when the volume of a substance is kept in- variable all, or else a fixed fraction, of the heat put in during a rise of temperature goes toward increasing the kinetic energy of the mole- cules. This is because the temperature is supposed to be proportional ΞΈΞΞΕΕΣΕΣΕΡΕΣΕΕΣΕΕΣ ΕΕΕΕΣΕΕΣ εετεετεεεέξεας C,, gm. cal. per gm. re) =) 105. LOO +> 9007 30° 85 ? 30 Volume, cm.? per gm. Figure 13. The specific heat at constant volume of water as a function of the volume. to the energy of translation of the molecules, and therefore, because of the law of the equipartition of energy, to the total energy of the molecules. We should expect, therefore, that the input of energy required to raise the temperature by a specified amount would in- volve only the interval of temperature, and would be independent of the absolute value of the temperature and of the volume. The curves show most convincingly that this is not the case. This sug- gests that in formulating a theory of liquids it would be well to scrutinize pretty carefully several assumptions that underlie the above considerations, namely that the temperature is proportional to the kinetic energy, that a fixed fraction of the total energy of the molecules is kinetic, and that the law of the distribution of velocities is independent of temperature. Another quantity of thermodynamic interest which may be found in terms of the specific heats is the thermal effect of compression, that is the rise of temperature in degrees accompanying a change of pressure adiabatically of one kgm. per sq.cm. This may be computed Ov ΠῚ ἣν, τί a) φ by the thermodynamic formula (Ξ The results so Cp a — δϑδ BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 355 calculated are shown in Figure 14 for the five standard temperature intervals. The character of the curves is the same as that shown so many times before, namely a rise to a maximum and then a fall at 0°, the abnormal behavior at the upper end of the 20° curve, and the more or less regular behavior of the three curves for the higher + ΤῊ ἢ Rut ταῖν ΠΡ ἢ 8, Sin4p ee Ρ 6. ἢ ἢ Pressure, kgm. / cm.’ x 10° Figure 14. The adiabatic rise of temperature of water against pressure. temperatures, with the crossing of the high temperature curves below the low temperature curves at the higher pressures. In the preced- ing paper only the approximate values for the very lowest tempera- ture interval could be found. The calculation was based on the mean value of the dilatation between 0° and 20°. The general character of the curve was the same as that found here for 0°, namely a rise to a maximum and then a fall. Finally it is possible to compute from the quantities in hand the difference between the isothermal and the adiabatic compressibilities. This is found from the formula (=) — (=) pe) (ey. The Ip)» Op/)- Cp \Or/p results are shown in Figure 15. The general character of the results is exactly the same as those previously given for the temperature effect of compression. Here again, the results at the lowest tem- perature agree with those of the previous paper which were based on a mean value for the dilatation. 356 PROCEEDINGS OF THE AMERICAN ACADEMY. PROPERTIES OF KEROSENE UNDER PRESSURE. In the course of the experiment other data were gathered inci- dentally which are of interest for themselves, and which will now be given. First of these is the compressibility and the thermal dilatation of kerosene. It was not necessary to determine this quantity in 0.0.8 0.0,2 Qo. 1 Se 8) 4 ἰδ 6252 OS. θὲ 10; 115 }5 Pressure, kgm. / cm.’ x 10° Figure 15. The difference between the adiabutic and the isothermal compressibilities of water. order to find the corrections to be made for the distortion of the vessel, but since half the work was already done in determining the effect with the cylinder partly filled with kerosene and the other part filled with bessemer steel, it seemed worth while to make the additional run necessary to determine the pressure and temperature effects on the kerosene. Not so many determinations were made of these quantities for kerosene as were made for the water. The results are given in Table VI. The curves showing the total thermal change of volume for 20° intervals are shown in Figure 16. This figure is the analog of Figure 2 for water. The results are very different. At the ‘lower pressures the dilatation is greater at the higher temperatures, as it is for all normal substances, but with rising pressure the effect is reversed, the dilatation becoming greater for the lower temperatures. This is the same behavior which takes place for water at higher temperatures after it has regained normality. But above 5000 kgm. the kerosene shows other abnormalities quite different in their charac- OO ΨΚ BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 307 ter from those of water. This is shown plainly in the figure as a separation and then a drawing together again of the curves. The curve for 20°-40° between 6000 and 8000 and the curve for 60°-80° beyond 9000 accomplish this separation and drawing together again TABLE VI. VoLUME OF KEROSENE AS A FUNCTION OF TEMPERATURE AND PRESSURE. (The volume at 0° and atmos. pressure is taken as unity.) Pressure, Volume. kgm. by rising with rising pressure, exactly as do some of the curves for water. The abnormality is doubtless due to an entirely different cause, however. In this case the effect is to be explained by the delayed freezing of the kerosene. Kerosene is not a simple pure substance, but is a mixture of several components with different melting points. Freezing under these conditions is not sharp, but is spread out over a considerable interval of temperature or pressure as the case may be. Neither is there any necessity that the freezing 358 PROCEEDINGS OF THE AMERICAN ACADEMY. should ever be perfectly complete, as indeed it is probably not. This may be shown at atmospheric pressure by plunging the kerosene into solid CO:. The effect is to change the kerosene to a white pasty mass, like white vaseline. The pressure at which this transition occurs will rise with increasing pressure. The existence of a transi- ΤΗ agegun gs τὴ: rf. sit: τε ἘΞΕΗΕ > + £ ; Ba gS page T ΤῊΣ .024 ἩΤΗΞΕΗΞΗΞΗ ΡΞ ΣΕΤΗ £ + 020 Rss : ΞΕΗΗῚ ΤῊΣ Change of Vol. at 20° Intervals paps az ΤΉ ΞΈΣΕΤΗΣΤΕ 11 12 ἐξ 004E= ad eben ee O° 4 5 Fane δ’ δ 8. ano Pressure, kgm. / cm.’ x 10° Fiaure 16. The change of volume of kerosene at constant pressure for a rise of temperature of 20°. tion point, if there were one perfectly sharp, would be shown by an abrupt rise of the curve by an amount corresponding to the change of volume on freezing. But with the delayed freezing which takes place here due to the separation out of the separate components from a solution of varying strength, this abrupt rise becomes con- verted into a gradual rise extending over a fairly wide pressure range. Furthermore, the mean pressure at which this rise takes place in- creases with rising pressure, just as the ordinary freezing point is raised by increasing pressure. These features are all clearly shown in the diagram. At the extreme right of the diagram, at pressures over 12,000 kgm., there is evident the beginning of the reversal of the effect, ον BRIDGMAN.—- THERMODYNAMIC PROPERTIES OF WATER. 359 that is, the curves are going to cross again, and the thermal dilation become greater at the higher temperatures. This may possibly indicate a reversal of the reversal of the effect mentioned above for liquids, but more probably the meaning is simply that at pressure above 12,000 the substance is practically a solid, and that for solids the reversal of the effect found in liquids at high pressures does not occur. There is one bearing which these observations have on the previous data which should perhaps be mentioned. This is in connection with the delayed freezing. Whenever freezing takes place there is usually the possibility of subcooling before separation to the solid form takes place. The amount of subcooling usually taking place depends on the nature of the liquid. In some it is very considerable, while in others it is negligible. If such subcooling took place here, it would produce irregular results, because the change of volume in the kero- sene transmitting pressure to the water would not always be the same under the same pressure. The only answer to be made to this ob- jection is that in this experiment the subcooling was not great enough to produce sensible irregularity. No discrepancies were found in the data suggesting that they were due to this effect. It was feared in the beginning of the work that the effect might be very troublesome, but such did not turn out to be the case. Also with respect to the solidification of the kerosene, the experi- ments showed that the solidification could not be complete, but the kerosene, even at the highest pressures, must remain a pasty mass like vaseline in nature, always capable of transmitting pressure nearly hydrostatically. But that on the other hand the kerosene does undoubtedly become pretty stiff under pressure has been already shown in the course of some measurements on the linear compressi- bility of steel rods. The second bit of data collected incidentally in the course of the work was a measurement of the expansion and the thermal dilatation of the high temperature variety of ice. Ture CoMPRESSIBILITY AND THERMAL DILATATION OF Ice VI. Although these data are not directly concerned with the properties of liquid water, which forms the subject matter of this paper, still it was so easy to obtain them without any modification in the arrange- ment of the apparatus, that it was thought worth while to measure them. In the previous paper on the properties of water and the 360 PROCEEDINGS OF THE AMERICAN ACADEMY. several varieties of ice, a very rough experimental value for the com- pressibility was given, as also a computation of the approximate compressibility, neglecting the thermal dilatation of the ice, for which no experimental value was found at that time. These measurements here include a direct measurement of the thermal dilatation, and two different determinations of the compressibility by two different methods. The value for the dilatation may be combined with the already determined values for the volume of the liquid and the change of volume when ice VI separates out, to give a third independent value for the compressibility. The determinations of the dilatation will first be described. This was found in the same manner as the dilatation of the liquid, by chang- ing the temperature at constant mean pressure, and measuring the change of pressure brought about thereby. Three determinations of this were made for the combination of ice and kerosene, and two for the combination of kerosene and bessemer. The agreement of the different determinations was within 2% of the mean. The dilatation was found between 0° and 20° at a mean pressure of 10,000 kgm. The correction introduced by the thermal dilatation of the bessemer cylinder in the control experiment is fairly large here, being about 25% of the entire effect. The value assumed for the cubic dilatation was 0.000036, which is the value for atmospheric pressure. The effect of pressure is to decrease this number slightly, which would result in a larger value for dilatation of the ice. The effect of pressure on this quantity is, however, very small, and the error so introduced is probably negligible. The mean dilatation found in this way for the 20° above 0° at 10,000 kgm. was 0.00241 cm.3/ gm., or 0.000120 em.3/ gm. per degree. This is considerably less than the dilatation of the liquid in this neighborhood, for which the value 0.00040 has been found previously. This value for the dilatation may now be combined with the other data for the liquid and the solid to give the compressibility of the solid along the equilibrium curve. For this we have the following data: vol. of 1 gm. of water at 0° and 6360 kgm., 0.8428 em.%, and at 20° and 9000 kgm. (these are the equilibrium pressures at these temperatures) 0.8160 cm.%. For the change of volume when the liquid freezes to the solid we have at 0°, 0.0916, and at 20°, 0.0751. This gives for the volume of ice at the equilibrium pressures at 0° and 20° the values 0.7512 and 0.7409 respectively. The decrease of volume of the ice along the equilibrium curve is 0.0103. Part of this is an increase due to rise of temperature, which according to BRIDGMAN.— THERMODYNAMIC PROPERTIES OF WATER. 361 the above data is 0.0024. This leaves a decrease of 0.0127 to be accounted for by the increase of pressure of 2640 kgm. which gives a mean compressibility over this range of 0.0000048, a little more than one third of the compressibility of the liquid over the same range. The direct determination of the compressibility of the ice was made by two different methods. One of these was the same as that used roughly in the preceding paper, that is by finding the difference of the slope of the curves plotting piston displacement against pressure above and below the transition point to the solid. The values obtained in the preceding paper for this were very rough. In these determina- tions the cylinder was very much more carefully seasoned, and the readings were made with all the precautions which had been sug- gested by all the experience of this paper. Two determinations of this quantity were made at 0° and also two determinations at 20°. The two values for the difference of compressibility differed by 2.5% at 0° and by 0.7% at 205. The value found for the difference was 0.0000087 at 0° and 0.0000067 at 20°. Combining with the values given already for the compressibility of the liquid, this gives for the compressibility of ice VI 0.0;49 at 0° and 6360 kgm., and 0.0,;43 at 20° and 9000 kgm. Mean 0.0;46. The second method for determining the compressibility was exactly the same as that for finding the same quantity for the liquid, com- paring the displacements when the apparatus was filled with ice and kerosene with those when the ice was replaced by bessemer steel. This determination was made over a wider pressure range, to find if possible the variation of compressibility with pressure. No variation with pressure could be found over a range of 4500 kgm. at 0° and 3300 kgm. at 20°. The absolute values do not agree with those found by the two other methods, however, the figures being 0.0;31 at 0° and 0.0,35 at 205. The cause of the discrepancy is not clear, but is probably connected in some way with the hysteresis of the cylinder. The hysteresis was not regular for these small pressure ranges, being at times almost negligible, and again being as large as for almost the entire pressure range from atmospheric pressure to the maximum. There seems little question but that the greater weight is to be attached to the values found by the first two methods. This third determina- tion does show, however, that the variation of the compressibility with pressure and temperature over this range is so small as to be beyond the accuracy of these measurements. In selecting the best probable value for the compressibility the only weight that will be — ee 362 PROCEEDINGS OF THE AMERICAN ACADEMY. ee assigned to this third determination is in slightly lowering the mean of the other two. The final most probable values for Ice VI are as follows: for the compressibility 0.0,45, and for the thermal dilatation 0.000120 cm.?/ gm. over the range 6360-10,000 kgm. and 0° to 20°. The cost of much of the apparatus used in this investigation was defrayed by an appropriation from the Rumford Fund of see American Academy. JEFFERSON PHysIcAL LABORATORY, Harvarp UNIVERSITY, CAMBRIDGE, Mass. Ἢ Proceedings of the American Academy of Arts and Sciences. Vout. XLVIII. No. 10.— Srpremser, 1912. CONTRIBUTIONS FROM THE CRYPTOGAMIC LABORATORY OF HARVARD UNIVERSITY. LXXI.— PRELIMINARY DESCRIPTIONS OF NEW SPECIES OF RICKIA AND TRENOMYCES., By Routanp THAXTER. - bd CONTRIBUTIONS FROM THE CRYPTOGAMIC LABORATORY OF HARVARD UNIVERSITY. LXXI.— PRELIMINARY DESCRIPTIONS OF NEW SPECIES OF RICKIA AND TRENOMYCES. By ον THAXTER. Received August 19, 1912. RICKIA. THE genus Rickia has proved to be a large and varied one, and although I have enumerated below only those forms parasitic on Acari which have come under my notice, many others are known to me on a variety of hosts, an account of which I have reserved for a future paper. The general habit appears to be very variable, includ- ing in addition to the condition seen in the type form, others in which the median cell-series is undeveloped, as well as various species with a more or less complicated system of branches. . The antheridial characters, moreover, appear to be equally variable. Not only do the antheridia which are extraordinarily abundant in some species seem wholly lacking in others, but their character may vary in differ- ent cases. In some there may be a single antheridium, only, similar to that of Peyritschiella, definitely placed at the base of the perithe- cium; or an antheridium of this type may be associated with others of the normal habit variously disposed. Again even in forms having the three characteristic cell series, antheridia may be present like those of the genus formerly separated as Distichomyces, each anther- idial cell becoming more or less free in a compact group. Since both the antheridial characters and those of the receptacle thus appear to be so variable, it has not seemed desirable to limit the genus to the type form as illustrated by Rickia Wasmanni, and I have therefore given it a more liberal interpretation; including under it forms with two or with three cell-series, whether they be simple or branched, and whether their antheridia be of the Rickia or the Distichomyces type. The latter genus is, therefore, abandoned, one species only, Rickia Leptochiri, being involved in-this change. 366 PROCEEDINGS OF THE AMERICAN ACADEMY. The only American form, R. minuta, thus far recorded on Acari, has been described by Paoli (“Redia,” Vol. VII, fase. 2, 1911, repub- lished in Malpighia, Vol. XXIV, 1912) from immature specimens with undeveloped perithecia, a practice which it is surely most desir- able to avoid in the systematic study of a group which presents such great difficulties as do the Laboulbeniales. I have been fortunate, however, in obtaining abundant material of this species, fully matured, from the Amazon region, for which as well as for other hosts, I am indebted to the kindness of Mr. W. H. Mann who has allowed me to look over his collections made on the Leland Stanford Expedition in 1911. I am further greatly indebted to the kindness of Messrs. T. Petch, Geo. Schwab and J. B. Rorer who have most generously collected or caused to be collected for me numerous insects, in Ceylon, Kamerun and Trinidad respectively, from among which a majority of the following hosts were obtained. I am also indebted for two species of Acari collected in Grenada to Mr. C. T. Brues and kindly placed at my disposal; while lastly I am much indebted to Mr. Nathan Banks for his determinations of the host-genera. In the following diagnoses I have assumed that the side bearing the perithecium is “anterior.””’ The spore measurements are for the most part made within the perithecium. Rickia furcata nov. sp. Furcate, sometimes irregularly branched. Basal cell short and rather stout, the receptacle above it dividing in two straight divergent branches; an anterior, bearing a perithecium, and aposterior. An- terior branch consisting of a series of usually eleven cells, the lower superposed horizontally, the upper obliquely; all cutting off appen- diculate cells externally; the series extending nearly to the apex of the perithecium, to which it is united throughout its length; the second cell of the series extending inward below the base of the latter, the outline of which is symmetrically subfusiform, the inner lip-cell protruding as a finger-like process. Posterior branch indeterminate, formed by a double series of cells which are more or less regularly paired above the second cell of the outer row, the third cell bearing the primary appendage on its narrow subtending and long cylindrical basal cell; many, but not all of the cells above in both rows cutting off distally and externally small cells which bear well-developed appressed appendages or antheridia (?). Appendages subcylindrical, 8-16 X 2.5u. Perithecium 30-40 X 8-104, including terminal projection THAXTER.— RICKIA AND TRENOMYCES. 367 (2.5-3 μ). Spores about 25 X 2.54. Total length to tip of perithe- cium 40-70 μ, to tip of posterior branch 50-175 μ. On Euzerconspp. No. 2481, Trinidad; No. 2236, Manaos, Amazon; No. 2058, Grenada, W. I. This species, and to a more marked degree the following, depart greatly from the normal type, and would be placed in a new genus with little hesitation were it not for the structure which characterizes various others of the many undescribed species known to me. It is evident that the “posterior branch”’ is an indeterminate proliferation beyond the primary appendage, which appears to involve both the “median” and the “posterior” marginal series of the more normal forms. The receptacle, especially when a primary perithecium fails to develop, may become variously branched and more than one secondary perithecium may be produced. Antheridia of a type like that of Distichomyces appear to be developed externally on the posterior branch nearer the base. The specimens from Brazil and Trinidad seem to be identical, although those from Grenada, though otherwise similar, are constantly somewhat smaller. Rickia arachnoidea nov. sp. Basal cell rather short and stout, the receptacle above it dividing into two usually fureate arachnoid branches; an anterior on which a perithecium is produced, and a posterior. Anterior branch indetermi- nate, consisting of two parallel series of cells usually not opposite, irregularly appendiculate, furcate at a variable distance from its base; one of the branchlets sterile, often greatly elongated; the other short but variable, bearing a perithecium which on one side is usually united to the upper six cells, some of them appendiculate, which continue one of the two series forming the perithecial branchlet which thus extends to the apex of the perithecium, beside which it terminates in a short brown appendage: the perithecium long, slightly and nearly symmetrically inflated, the tip bent distally abruptly sidewise; the other row of the perithecial branchlet ending horizon- tally or obliquely below the base of the perithecium and consisting of from three to eight cells, some of which are appendiculate. Pos- terior branch indeterminate, furcate, usually, just above its first to fifth pair of cells, the cells of the two indeterminate branchlets not paired, irregularly appendiculate, indeterminate, usually greatly elongated: the second cell of the main receptacle below its furcation bearing the large long nearly cylindrical basal and subtending cells 368 PROCEEDINGS OF THE AMERICAN ACADEMY. of the primary appendage, which may be on either side. Appendages suffused with brownish, mostly rather short and stout, 7-18 X 4μ. Spores 30 X 3y. Perithecia 70 X 18-20. Diameter of branches 8-10 μ, greatest length 460-520 μ, in largest specimens. Basal and subtending cell of primary appendage 18-20 X 4 μ, the former rarely divided. On Discopoma sp. Trinidad, No. 2433; on Trachyuropoda sp. Trinidad, No. 2429; also an immature specimen from the Amazon on same host; on Euzercon sp., Trinidad, No. 2482. When normally developed this curious form appears to be more or less regular in its structure, as above described, but especially when injured or when the first perithecium aborts, secondary branching takes place, and more than one perithecium may be formed. That there is no significance in “anterior”? and “posterior” as applied to the main branches of this form, is indicated by the variable position of the primary appendage beyond which they proliferate. The plant has a characteristic sprawling habit, its branches resting on the upper surface of its host, which is its usual position of growth. Unless it is viewed sidewise, the cell-series bordering the perithecium is not visible, and may thus be wholly overlooked. The appendages, as in all the species, are borne from small subtending cells. Among described species it is most nearly allied to R. furcata. Rickia anomala nov. sp. Hyaline, rather strongly curved throughout above the basal cell. Median cell-series wanting. Basal cell wholly free, longer than broad, of nearly the same diameter throughout. Anterior series consisting of three or rarely four cells, subisodiametric, externally convex, subequal, without appendages. Posterior series of usually nine cells, the two or three lower larger, rounded; the rest smaller, subequal, irregularly rounded; the first, third, fifth, and seventh cells separating distally small cells which subtend appendages, the second cell subtending the basal cell of the primary appendage, which is relatively very large, wholly free, constricted at the base, terminated by a small cell which subtends the appendage proper; the latter somewhat smaller than the others, but otherwise similar, faintly brownish, bladder-like, roundish, or somewhat longer than broad. Perithecium directly continuous with the anterior series, externally wholly free, rather long and narrow, the tip well distinguished, nar- rowed, its lower half united on the inner side to the distal cell of THAXTER.— RICKIA AND TRENOMYCES. 369 the posterior series, which ends in a minute suffused roundish hardly distinguishable cell; the inner lip-cell forming a finger-like straight free process. Spores about 25 X 3 yu (in perithecium). Perithecia 30- 35 X 8-10.54. Basal cell 14-18 Χ 5-6.5y. Basal and subtending cell of primary appendage 16-17 X 7μ. Appendages 9X 4.5- 7X 6u. Total length 48-56 Χ 14-16 xu. On a minute mite belonging to a new genus, near Iphiopsis. Trinidad, No. 2440. Although there are fourteen specimens of this peculiar species in various stages of development, none of them show any indication of the presence of an antheridium. Rickia Discopomae nov. sp. Hyaline, becoming slightly soiled with dirty brownish throughout. Basal cell large, twice as long as broad. Main body of the receptacle of about the same diameter throughout, broadening slightly below the perithecium, usually rather strongly curved. Cells of the three cell-series small, subequal, squarish or subisodiametric, arranged in tiers of three cells each with some regularity; the middle series extend- ing half way along the tip of the perithecium, its two or three terminal cells free beyond the base of the primary appendage, which terminates the posterior marginal row. Cells of the median row fifty to sixty in number, sometimes less; those of the anterior marginal row thirty to fifty; of the posterior marginal row fifty to sixty, the cells of both marginal rows cutting off appendiculate cells irregularly, except those of the posterior row opposite the perithecium which produce them uninterruptedly; the appendages and antheridia thus irregularly and rather sparingly distributed along the margins. Appendages short and usually inflated. Perithecium rather short and _ stout, the tip often somewhat bent outward, the apex blunt. Spores 30 X δμ. Perithecium 48-52 X 18-25. Total length 250-350 Χ 18- 32 μ, measured below the perithecium. Appendages 7-10 X 3-4 u. On superior surface of Discopoma sp. Peradenyia, Ceylon, No. 2111. The antheridia of this species are not certainly recognized, but appear to be of the type seen in “ Distichomyces.”’ The appendages appear to branch occasionally, becoming fureate, a condition possibly resulting from the proliferation of old antheridia. 370 PROCEEDINGS OF THE AMERICAN ACADEMY. Rickia elegans nov. sp. Basal cell hyaline; cells of median row small, rounded; those of marginal rows horizontally elongated or their axes directed upward somewhat obliquely, more than fifty cells in the posterior row, about twenty-five in the anterior; the cells at maturity in all the rows be- coming deeply suffused with rich blackish brown and quite indis- tinguishable; all the cells of the marginal rows cutting off small cells which remain almost wholly hyaline and bear short appendages, their cup like bases rich brown, the distal portion hyaline. Peri- thecium wholly united on its inner side to the median row, the last two or three free cells of which reach to the middle of the short stout deeply suffused rather broad tip, which is bent rather abruptly out- ward; the apex hyaline, or translucent; the body nearly straight, about the same diameter throughout, rather narrow, rich brown, not as deep as the tip, the outer margin somewhat irregular, continuous with that of the receptacle below. The whole plant straight or curved, tapering gradually from apex to base. Perithecium 65-85 X 20 yu. Appendages about 15 X 4y. Total length 200-220 x 35- 40 μ. On legs and margin of body of Discopoma sp. Peradeniya, Ceylon, No. 2110. This species is very closely allied to R. Berlestana Paoli (Bac.), differing chiefly in its much more numerous cells, which are smaller and differently arranged and the total suffusion of the receptacle. In fully mature specimens, the perithecium is concolorous with the receptacle, and not distinguishable from it. Rickia cristata nov. sp. Basal cell three times as long as broad, its upper half or less included between the lower cells of the marginal rows. Posterior row crest- like, the cells radially elongated, each separating several appendiculate cells, the pointed bases of which are intruded between them nearly to their bases, the appendiculate cells becoming so multiplied, where the series curves over against the tip of the perithecium, that the primary cells are obliterated; the primary cells of this series about eighteen, the appendiculate cells thirty-six to forty: the anterior series extending slightly beyond the middle of the perithecium, the base of which it incloses, consisting of three or four cells from which a number of appendiculate cells are cut off, as in the posterior series, ee eee ee ee THAXTER.— RICKIA AND TRENOMYCES. 371 one or two of the uppermost bearing pointed antheridia: the append- ages six to eight: the middle series of six flattened cells lying along the inner margin of the perithecium for a little more than two thirds of its length. Perithecium rather short and stout, slightly curved, the apex blunt and opposite the bases of the distal appendages of the posterior series, the tip well distinguished externally. Spores 30 X 4u. Perithecium 45 Χ 18 μ. Free portion of the basal cell about 18 u; the rest of the plant 60-75 X 48-52 u. Appendages 16-25 X 4, becoming brownish and subtended by the usual dark cup-like base. On the inferior surface of a mite parasitic on Prioscelis sp. (?) and belonging to a new genus near Cilliba. Kamerun, No. 2438. A species closely allied to R. Coleopterophagi Paoli and R. minuta Paoli, differing in the form of its appendages and the arrangement of its cell-series. The single type of R. Coleopterophagi as well as those of R. minuta, are immature, so that it is not possible to judge of the perithecial characters in these species. The latter, however, has been received from Brazil (Mann) on various mites parasitic on Scara- beidae, and an abundance of well matured individuals are available for comparison. The species though very variable is quite well distin- guished from the one above described. The tip of its perithecium is wholly free; the cells of the middle series vary considerably in number and extend as far as those of the posterior series, which is more nearly vertical, the general habit of the plant being more slender; the basal cell is not intruded between the lower cells of the anterior and pos- terior series and there are other differences. Rickia pulchra nov. sp. Basal cell variably developed, more often short, the upper half enclosed by the lower cells of the marginal series; or long and very stout distally. Posterior marginal series consisting usually of four cells, the lower opaque blackish brown bearing distally a very minute rounded appendage, the next above somewhat rounded and cutting off a small cell which subtends an antheridium, the third large tri- angular, its pointed end directed upward, and cutting off three to five appendiculate cells which lie external to it; the uppermost small, flattened, distally pointed, separating a single minute cell which lies external to it and subtends a small short brownish spine-like append- age: the anterior series consisting of three cells, similar to and sym- metrical with the corresponding cells of the posterior series, and 372 PROCEEDINGS OF THE AMERICAN ACADEMY. bearing an antheridium and appendages in a similar fashion so that the individual is bilaterally subsymmetrical: the middle series con- sisting of but two flattened cells, the upper, its broader extremity free beyond the distal cell of the posterior series, nearly twice as long as the lower, which is opaque below and forms with the two lower cells of the two other series a suffused area in which cell-divisions are not visible and which extends upward so as to involve the lower half of the perithecium; the tip of which is nearly free, usually bent slightly toward the anterior series, and subtended anteriorly by a straight appendage about 15 Χ 3 μ, suffused towards the base, and apparently the indurated base of the trichogyne. Appendages nearly symmetrical on either side, long and slender, hyaline, becoming deeply suffused at and towards the base, cylindrical, tapering slightly at base and apex. Antheridia normally solitary, borne distally from the subbasal cells of the two marginal series, hyaline, the necks pur- plish, curved outward. Spores, in perithecium about 22 Χ 3.5 μ. Perithecia 35-40 X 15y. Basal cell 18-50 X 6-15. Appendages 35-60 Χ 4-6. Total length exclusive of stalk 48-56 Χ 35-38 μ. On the inferior surface and legs of Macrocheles sp. and Celaenopsis sp. Kamerun, Nos. 2488, 2439. A very beautiful species, quite unlike any other known form. The specimens on Celaenopsis are somewhat smaller. Rickia obcordata nov. sp. Hyaline. Basal cell bent, its pomted upper half filling the sinus of the slightly asymmetrical obcordate body. The marginal series consisting of typically six cells each and subsymmetrical with one another, the posterior shorter, terminated by the slender basal cell of the primary appendage which, like all the appendages and the antheridia, projects radially in a more or less regular fashion: basal cells of the marginal series radially extended, broad and rounded externally, separating a small triangular cell above, which subtends an appendage symmetrically placed on either side of the body, the second and third cells of both series separating externally three to four small cells which subtend each an antheridium, the necks quite hyaline projecting more or less radially, usually straight, the third cell on the posterior side usually bearing an appendage distally: the fourth and fifth an antheridium and an appendage, or an appendage only in both series, except in cases where there are but five cells in the posterior series, the uppermost of which always subtends the δον EE ee δὰ, ee τ μὰ μιν... THAXTER.— RICKIA AND TRENOMYCES. 373 primary appendage; the sixth cell of the anterior series producing neither appendage nor antheridium. Appendages subcylindrical, several times as long as broad, faintly suffused aboye the conspicuous blackened slightly constricted base. Median series consisting of five cells successively smaller from below upward, the three lower rounded, the uppermost triangular, its upper face free below the slightly projecting truncate or bluntly rounded free tip of the peri- thecium. Thelatter otherwise completely enclosed, vertical or slightly oblique, and lying almost wholly anterior to the median axis. Peri- thecium 60 Χ 254. Body 90-100 X 78-85 μ. Basal cell including foot 28-35 X 15-18 yu. Appendages 24-35 X 5y. Projecting an- theridia 12 μ. On a minute mite. Kamerun, No. 2441. A very minute form characteristic from its obcordate almost sym- metrical form and radiating antheridia and appendages. Rickia elliptica nov. sp. Hyaline, elliptical to nearly circular in outline. Basal cell very short, sometimes entirely included in the angle between the inner surfaces of the basal cells of the marginal rows, the foot, only, project- ing beyond the general outline of the main body. Anterior marginal row consisting of from five to eight cells subradially elongated, the two uppermost extending downward to sharp points, all or nearly all cutting off distally a small triangular appendiculate cell; the append- age which terminates the distal cell appressed against the free anterior face of the tip of the perithecitum: posterior marginal row consisting of from seven to nine cells, similar to the anterior series except that the upper cells are smaller, the uppermost much smaller, bearing distally the basal cell of the primary appendage which is small, narrow, free, not greatly longer than the subtending cell of the very small appendage; other appendages stouter, short, irregu- lar with slightly suffused bases. Median series of six to eight cells, one to three of the terminal ones externally free above the basal cell of the primary appendage, the successive cells subisodiametric, some- what irregular in outline, and not greatly differing in size. Peri- thecium almost wholly inclosed, the tip free externally, slightly bent outward below the apex which is subtended on its inner side by an erect finger-like upgrowth, geniculate at its base; body of the perithecium rather long and narrow, subsymmetrical. Spores (in perithecium) 22 Χ 2.5. Perithecium 30-40 Χ 10-12 μ, not includ- 374 PROCEEDINGS OF THE AMERICAN ACADEMY. ing the projection which is 7 X 2u. Basal cell, including foot, 8-16 u. Total length of body 50-66 Χ 35-40 un. On legs of Discopoma sp. Trinidad, No. 2433. Although seven specimens in perfect condition and of various ages have been examined, I have seen no indication of an antheridium. The base of the trichogyne persists as a minute dark rounded body below the base of the upgrowth from the inner terminal wall-cell. Rickia inclinata nov. sp. Minute, hyaline, of irregularly rounded form. Basal cell forming a well defined slender stalk, the upper third or half inserted in the angle between the two basal cells of the marginal rows. Anterior marginal row not extending above the base of the perithecium, con- sisting of two radially elongated cells which are subequal and cut off distally and externally two to three appendiculate cells: posterior marginal row consisting of seven cells like those of the anterior, exter- nally convex, the second to the fourth more radially elongate than those above, which are successively smaller; the basal usually sepa- rating one, the rest two appendiculate cells distally and externally; the terminal cell much flattened followed by the broad basal cell of the primary appendage, which appears to be a member of the series, its inner margin in contact with the fourth cell of the median series: median series of four subequal irregularly rounded cells. Perithe- cium stout, its axis straight and characteristically tilted inward at a slight angle to that of the receptacle, its base in contact with the distal cell of the anterior series, externally wholly free; the tip quite free, bent very slightly outward, the apex broad, flat, each lip-cell projecting very slightly and somewhat irregularly. Spores 25 X 3y (in perithecium). Perithecium 38-40 X 11 yu. Basal cell, including foot, 25 X Su. Total length of body to tip of perithecium 50 X 41- 44. Appendages hyaline, tapering very slightly, 16 X 34, with clearly defined dark basal septa. On a minute mite, as yet undetermined. Trinidad, No. 2307. A characteristic and minute species, distinguished by its tilted perithecium, which is externally free. It is closely allied to R. Celae- nopsis, from which it differs in the number and arrangement of its cells, etc. I have been unable to dete mine the presence of an an- theridium in either of the two adult types. a δι νων μ.μ..... σι THAXTER.— RICKIA AND TRENOMYCES. 37 Rickia Celaenopsis nov. sp. Hyaline, minute, somewhat angular in outline. Basal cell forming a well developed stalk, the upper third or less inserted in the angle between the two basal cells of the marginal rows. Anterior series consisting of two cells, the lower characteristically triangular in form, its outer margin straight and evenly continuous with that of the upper cell, which is radially elongated and cuts off distally an appendiculate cell which is relatively very long, its distal half or two thirds pro- jecting free beyond the margin and subtending a relatively very large and long antheridium which projects above it just at the base of the perithecium: posterior series consisting of typically six cells, the basal like that of the anterior series, triangular, but cutting off distally a slightly prominent appendiculate cell; the four cells above obliquely elongated, lying subparallel, and separating distally a conspicuously protruding upturned appendiculate cell; the terminal cell triangular, subtending the wholly enclosed sublenticular basal cell of the primary appendage, the subtending cell of which is free, bell- or dome-shaped, bearing a rather stout appendage. The appendages subcylindrical, several times longer than broad, rarely furcate, with the usual dark subtending base: median series consisting of usually six cells, the basal and distal somewhat larger, the rest squarish or slightly com- pressed, subequal, the upper margin of the distal cell free, its oblique wall very thick and directly continuous with the margin of the tip and the distal portion of the venter of the perithecium which rise erect beyond it. Perithecium thick walled, somewhat inflated, quite free and convex externally, erect or nearly so, the tip symmetrical, ‘truncate conical, the apex flattened or slightly rounded. Spores 20 X 3m (in perithecium). Perithecium 40 X 20μ. Basal cell including foot 25 X 8yu. Total length of body to tip of perithecium, 50 X 88 μ, largest. Antheridium about 12 uw long. On legs of Celaenopsis sp. Trinidad, No. 2426. Closely allied to R. inclinata, but differing in many details of struc- ture, the triangular form of the two basal cells of the lateral series giving it a characteristic appearance. Rickia discreta nov. sp. Hyaline, rather elongate. Basal cell relatively large and long, distally symmetrical, but slightly intruded between the lower cells of the marginal series. Anterior marginal series consisting of three 376 PROCEEDINGS OF THE AMERICAN ACADEMY. to four subequal obliquely separated cells, the lowest cutting off an appendiculate cell distally and externally, the upper an antheridium of the Peyritschiella-type, which subtends the base of the perithecium from which its hyaline sharply pointed stout extremity projects obliquely upward: posterior marginal series consisting of usually seven obliquely separated cells, usually the first, third and fifth, only, separating a rather large appendiculate cell; the uppermost cell triangular, its upper margin continuous with that of the distal cell of the median series, subtending the basal and large subtending cell of the primary appendage, the two latter subequal, the basal somewhat broader: median series consisting of normally six successively smaller, vertically slightly elongated cells. Perithecium erect, slightly curved outward distally, the tip free, the apex symmetrical, truncate, slightly papillate. Appendages relatively long and stout, yellowish, sub- cylindrical, the basal ring black and conspicuous; 15-25 X 3.5 y, the primary one 30-45 μ, its basal and subtending cells 10 XK 4μ. Peri- thecium 25 X 9u. Basal cell including foot 20 X 7u. Total length to tip of perithecium 55-65 X 18-22 μ. On an undetermined gamasid mite. Trinidad, No. 2308. This species is well distinguished by its large discrete yellowish appendages, somewhat elongate form, and large single antheridium. In one of the nine specimens examined a second antheridium is devel- oped just below the first. Rickia spathulata nov. sp. General form spathulate except for the projecting tip of the peri- thecium. Basal cell rather stout, its upper half or less inserted in the sharp angle between the lower cells of the marginal series. An- terior series consisting of six to eight cells, the lowest irregularly triangular, externally slightly concave, and without appendage, the rest usually but not always appendiculate, radially elongated, and shghtly oblique upward; the subterminal cell bearing also an an- theridium, the basal cell of which penetrates three fourths of its length; the terminal cell sometimes separating a second antheridium, its inner margin in contact with the lower two thirds of the perithecium, narrow, its extremity broader and convex: posterior series consisting of ten to thirteen cells, usually eleven, the lowest externally convex like the rest, the other members of the series each usually cutting off an appendiculate cell about half their length and lying between them; the upper ones successively narrower and more elongated radially; ee THAXTER.— RICKIA AND TRENOMYCES. oLL the cells above the second or third curved inward in a somewhat crest-like series which lies parallel to the median series and the inner margin of the perithecium, the terminal cell of the series small, triangular, bearing the large basal cell of the primary appendage which, with the small subtending cell, forms a free straight projection, its axis bent inward at an angle of about 45° to that of the receptacle: median series consisting of eight to ten cells, the two or three lowest enclosed by the marginal series, the rest lying against the strongly convex inner margin of the perithecium, the free slightly convex margin of the uppermost reaching almost to the base of the free tip. Perithecium rather stout, its outer margin nearly straight, its inner convex, the outcurved tip, and externally a small portion of the body, free; the apex flat, protruding slightly externally. Spores 28 Χ 3 y, in perithecium. Perithecium 40-46 X 16-20 yu. Basal cell, including foot, 28-33 X 9-llyu. Total length, not including primary ap- pendage base, 12-16 X 6-Su. Appendages 6 X 2u or smaller, wholly smoky brown, usually broken off, the dark base not conspicu- ous. On legs of Celaenopsis sp. No. 2229, Amazon, “M. ἃ M.” (Mann No. 41). A very well marked species peculiar for its more or less regularly spathulate outline, which is broken only by the projecting tip of the perithecium and the primary appendage. It is not nearly allied to other known acarine species, but is perhaps most nearly related to R. minuta. Rickia excavata nov. sp. General form roughly triangular, distally concave. Basal cell three or four times as long as broad, its distal fourth included in the angle between the two lower cells of the marginal series. Anterior series consisting of four cells, the lower three subequal, usually all appendiculate, the uppermost vertically elongated, externally convex, extending to the middle of the venter of the perithecium: posterior series consisting of usually seven cells, the four lower similar to those of the anterior series, usually all appendiculate, the subtending cells hardly intruded between adjacent members of the series, the three terminal cells successively smaller, flattened, their septa at right angles to the axis of the series which they form, and which is continu- ous with that of the primary appendage and its basal and subtending cells, which, together with the two terminal cells of the posterior series form a free subtriangular projection directed at an angle of 378 PROCEEDINGS OF THE AMERICAN ACADEMY. somewhat over 45° to the axis of the body of the perithecium: the median series consisting of usually five cells, the lowest larger, longer than broad and lying mostly below the base of the perithecium; the three upper successively narrower, extending to the base of the tip of the perithecium, forming a series almost symmetrical with that of the three terminal cells of the posterior series and the primary appendage, the axes of the two series nearly at right angles. Tip of the perithecium wholly free, bent strongly inward, the apex abruptly distinguished, the lip-cells rather prominent, the inner more so, rounded; the body nearly vertical or inclined very slightly outward, rather long and narrow and symmetrically rounded basally and distally. Spores 18 X ὅμ. Perithecium 80 10u. Appendages subceylindrical, small, about 6 X 2.5. Basal cell 20 X 6u. Total length to tip of perithecium 75 X 344, not including basal cell of primary appendage. On Celaenopsis sp. Trinidad, No. 2427. _ Clearly distinguished from other known species by its general form and excavated superior margin. Rickia Euzerconalis nov. sp. General form short-spathulate, hyaline. Basal cell very small and short, separating an appendiculate cell distally on the anterior side. Posterior marginal row consisting of usually eight, often nine cells, radially and obliquely but slightly elongated; all usually cutting off an appendiculate cell, except the distal one, which is small, tri- angular and subtends the large usually outcurved basal cell of the primary appendage which is free above it, two to three times as long as broad, and about the same diameter throughout: anterior marginal series consisting of usually five cells, more rarely four or six, the lowest separating an appendiculate cell below, which lies between it and the basal cell of the receptacle; the remaining cells large, each, except sometimes the lowest, separating an appendiculate cell distally; the uppermost extending to or beyond the middle of the perithecium with which its appendiculate cell with the appendage is in contact: median series consisting of almost invariably six, rarely five or seven, cells, not differing greatly in size, extending from just below the base of the perithecium nearly to its apex. Perithecium narrow, erect, its tip externally free, the inner lip-cell projecting as a short finger-like process. Appendages stout, yellowish-brown, 7 X 3.54. Spores 25 X 2.5. Perithecia 22-24 * Sy. Basal cell including foot, THAXTER.— RICKIA AND TRENOMYCES. 379 14-16 X 6-8 u. Total length to tip perithecium 50-70 Χ 24-32 μ. Basal and subtending cell of primary appendage 12-15 Χ 5 μ. On Euzercon spp. ‘Trinidad, Nos. 2432 and 2430; Kamerun, No. 2443. This species is most nearly related to R. Megisthani from which it differs in its more complicated receptacle, larger size and more or less evenly spathulate outline. In this, as well as in the following species (R. Megisthani) the lowest appendage on the anterior side is subtended by a cell which lies external and inferior in relation to the lowest cell of this series, instead of distal, and has the appearance often of having been separated, not from this cell, but from the basal cell of the re- ceptacle below and it is possible that this is its actual relation. Rickia Megisthani nov. sp. Hyaline. Basal cell rather short and stout, obliquely separated from the basal cell of the anterior series, which is angular, subiso- diametric and lies immediately below the base of the perithecium, cutting off an appendiculate cell which sometimes covers its whole outer margin, or more often lies external and inferior in relation to it; the series consisting of two other cells which are subequal, elongate; the base of the upper lying obliquely over the distal end of the lower, which may or may not cut off an appendiculate cell distally; the cell above it, sometimes lacking, with or without an appendiculate cell which lies in contact with the outer margin of the perithecium reach- ing to its upper third or half: the posterior series consisting of normally four cells, the lowest more often not appendiculate; the second and third equal and appendiculate; the fourth vertically elongated, its upper third or half quite free, straight or distally slightly geniculate and continued by the long free finger-like slightly curved basal cell of the primary appendage. Median series of three subequal cells, vertically placed and extending almost to the apex of the perithecium. Perithecium rather stout, its inner margin straight, its outer convex and one half to one third free; the tip very slightly bent inward; the outer lip-cell forming a small, short, finger-like projection. Appendages very short and small, 5 X 2.5. Spores 20 X 2u. Perithecia 30-32 8-117 μ. Basal cell, including foot, 16 X 7u. Total length to tip of perithecium 50-60 & 20-304. The free termination of the posterior series, including basal and subtend- ing cell of primary appendage 25-40 X 5 μ. On Megisthanus sp. Trinidad, No, 2435, 380 PROCEEDINGS OF THE AMERICAN ACADEMY. No antheridia have been seen in the numerous specimens examined. The species is very closely allied to R. Euzerconalis from which it differs in its smaller size, simpler structure and more irregular outline. Var. Trachyuropodae nov. var. Similar in general to the type. Somewhat smaller, the distal cell of the anterior series extending cushion-like usually to the tip of the perithecium; the posterior series consisting of five cells, the distal one wholly enclosed or hardly pro- truding, directed slightly inward, bearing the more slender base of the primary appendage which is erect or curved very slightly outward. Appendages stouter. On the thin anterior and lateral margins of Trachyuropoda spp. Ita- coatiara, Amazon, No. 2206, and Trinidad, No. 2429. Abundant material of both type and variety have been examined and the differences noted seem constant, though not sufficient for specific separation. Rickia Kameruna nov. sp. Hyaline asymmetrical. Basal cell small and short, abruptly dis- tinguished from the receptacle and from its broad pointed end, which is but slightly intruded between the two basal cells of the lateral series. Anterior series consisting of two cells without appendages, the upper partly overlapping the base of the perithecium which it subtends, and which is otherwise wholly free externally, rather long, its upper half bent slightly inward, the apex, only, free on the inner side, the outer and especially the inner lip-cell slightly prominent: the median series erect, consisting of five cells, the lowest not extend- ing to the base of the perithecium: posterior series consisting of seven to eight cells, all except the upper one or two cutting off a relatively large appendiculate cell, the two lower slightly elongated radially, the rest very similar to those of the median series beside which they lie; the terminal one bearing terminally and externally the basal cell of the primary appendage which projects outward obliquely, its axis parallel to that of the free upper oblique margin of the terminal cell of the median series. Appendages rather stout, 10 <3. Spores 18-20 X 2y. Perithecium 30-34 X 6-8 yu. Basal cell exclusive of foot 8 μ. Total length to tip of perithecium 40 X 20 μ. Basal cell of primary appendage, with subtending cell, 8 μ. On Euzercon sp. Kamerun, No. 2487. Although the posterior series in this species is not extended to form an appendage, it seems as nearly related to R. filifera as to any of the other species, owing to the small development of its posterior series, THAXTER.— RICKIA AND TRENOMYCES. 381 which leaves the external margin of the perithecium wholly free as in R. Celaenopsis. There appear to be two cells in the anterior series, the upper of which is almost concealed by the base of the perithecium. I have seen no indication of an antheridium in either of the three specimens from which the description has been drawn. Rickia filifera nov. sp. Small and slender. Basal cell obliquely separated from the lower cell of the anterior marginal series which consists of two subequal cells; the upper extending a short distance upward external to the base of the perithecium: posterior series consisting of a variable number of cells (eight to fifteen) the basal extending above the base of the perithecium, the subbasal lying opposite to it; the third extend- ing beyond its tip; the rest superposed to form a long, slender, erect, or slightly outcurved appendage, terminated by the undifferentiated basal cell of the primary appendage: the basal cell of the series, and many of the others, cutting off a small appendiculate cell distally and externally: median series consisting of two cells, the lower lying opposite the upper half or less of the perithecium, the upper in contact with the third and fourth cells of the posterior marginal series, its inner margin wholly free. Perithecium slender, the tip well dis- tinguished externally and bent slightly outward, the inner lip-cell forming a short projection. Appendages slender, cylindrical, hyaline, 10 X 2u. Spores 24 X 2.8 μ. Perithecia 35-45 X 8-12. Basal cell including foot 12 X 4-5y. Total length to tip of perithecium 35-45 X 8-12. Longest free flagellum, including primary append- age, 175 μ. On a very large mite allied to Megisthanus, on Passali. Kamerun, No. 2442. This species varies considerably in size and in the length of the extension of its posterior marginal row. No antheridia have been recognized, although material of various ages is available. It is perhaps most nearly related to R. Megisthani but resembles it only remotely, and cannot be confused with it on account of its free “ flagel- lum.” TRENOMYCES. This very curious genus was first discovered by Chatton in France on Mallophaga infesting domestic fowls, and had been received by me from Dr. Miiller who collected it at Elbing, Prussia, and from Dr. 382 PROCEEDINGS OF THE AMERICAN ACADEMY. Trinchieri who found it at Naples, before the appearance of the pre- liminary paper by Chatton & Picard in Comptes Rendus (CXLVI, p. 208, 1908) was published. It was thus discovered almost simul- taneously in Italy, Germany and France, and has since been found in New England and received from various other parts of North America. Having been interested to learn something further as to the distri- bution and characteristics of the species in this genus, I have made a special effort to accumulate material, and am especially indebted for an opportunity to do so to the kindness of Prof. V. L. Kellogg, who has allowed me to go over his very large accumulations of duplicates in alcohol, and of Mr. M. A. Carriker who put his valuable collection at my service. Mr. Kirkpatrick has also sent me Mallophaga from turkeys and pigeons collected for me at the Rhode Island Experiment Station, for which I am greatly indebted to him, and I have also obtained material from Guatemala collected by the late Professor W. A. Kellerman; from the Bahamas, (W. W. Worthington), as well as from other sources. The results of my examination of some thousands of Mallophaga have been somewhat disappointing, since their parasites are generally rare, and, if the data obtained may be assumed to indicate the actual conditions, have not found this aberrant group of insects a very favorable substratum for the development of numerous or character- . istic species. As will be seen the following enumeration includes only six additional forms, none of them, with the possible exception of 7. gibbus, departing very far from the characters of the type- species. In all a more or less complicated rhizoidal apparatus is developed, simple in one species, which penetrates the host. The receptacle consists of two cells terminated by a bicellular apiculate appendage resembling a spore of Puccinia, the upper giving rise to fertile branches which grow downward and corticate the lower, the corticating cells producing perithecia or antheridia according to the sex of the individual; although in some instances the corticating cells of the male are hardly developed, the antheridia arising directly from single cells obliquely separated from the lower margin of the subbasal cell of the receptacle. As in Dimeromyces and Dimorphomyces, to which the genus is most nearly related, the basal cells of the peri- thecium break down, and the cavity of the latter and that of the stalk- cell become continuous. a θα ν τ ναληδανηηι ae THAXTER.— RICKIA AND TRENOMYCES., 383 TRENOMYCES HISTOPHORUS Chat. & Picard. This species, which appears to vary greatly in size, has been re- ceived from Dr. Miiller, from Elbing, Prussia; from Prof. Trinchieri from Naples, Italy, and I have examined type material kindly sent me by Professor Chatton. In this country it has been obtained on species of Menopon and Goniocotes from Kittery Point, Maine, and from Newton, Mass. (on hosts kindly sent me by Mr. Walter Deane), on Menopon sp. from Gundlach’s mockingbird, Bahamas; on Meno- pon from hen, Jamaica, W. I., and Guatemala: in the Kellogg collec- tion on M. mesoleucum (crow), Palo Alto, California; M. tridens, Iowa; Menopon sp., No. 256b; on Goniocotes, Guatemala. A species has been examined from various species of Nirmus, N. punctatus (Calif.), N. maritimus (N. E. and Cal.), N. olivaceus (Elbing, Prussia, Dr. Miiller), which seems hardly separable from the many variations of 7’. histophorus. A variety, which may possibly prove a distinct species has also been found on Menopon numerosum (Kellogg, No. 24b), Menopon spp. (Kellogg, Nos. 80b, 256b, 74b), Docophorus sp. (Kellogg, No. 997). In this form the basal cell and the upper enlarged portion of the rhizoid are more or less conspicuously suffused with smoky brown in all cases. The ascogenic cell is usually near the base of the short stalk, and the distal cell of the appendage is somewhat more compressed than in the type but there are otherwise no distinctive characters. Trenomyces Lipeuri πον. sp. Male individual. Rhizoid more or less abruptly enlarged immedi- ately below the integument, the swollen portion variably elongated and passing below into a rather stout simple, cylindrical prolongation of variable length. Basal cell of the receptacle bent at right angles to the rhizoid, horizontally elongated and corticated on the upper surface by an irregularly double series of small cells, which give rise to a corresponding series of erect or slightly divergent antheridia, Stalk-cell of the antheridium very slender, broadened below the basal cells; the body rather short and stout, subfusiform, the efferent tube short and slender. Appendage lying horizontally; the distal cell twice as long as the basal. Length from tip of appendage to last corticating cell, largest specimen, 42 4. Appendage 15 X 9. ‘Total length of antheridium including stalk 35 μ; efferent tube 4 long; rest of body about 18 X 10 u. 384 PROCEEDINGS OF THE AMERICAN ACADEMY. Female individual. General structure like that of the male; the base of the rhizoid shorter and relatively broader with very thick walls, the rhizoid proper, simple. Corticating cells of the basal cell vertically elongated, closely associated in a double crest-like series, bearing two or three to fifteen perithecia. The latter yellowish more or less dis- tinctly tinged with brown, the stalk rather slender and clearly dis- tinguished, about one third as long as the body of the perithecium which is rather short and stout, subfusiform; the apex blunt and relatively broad, crowned by four more or less clearly defined promi- nences which surround the short rounded or slightly suleate apex. Perithecium, including stalk, 90-110 yu. The main body 60-80 X 20-28 μ. Total length of rhizoid about 90-100 μ the slender portion about 7 μ in diameter. On various parts of [ipeurus sp. on Buzzard, Los Amates, Guate- mala, No. 1547. On L. celer, Nos. 1564-67, California (Kellogg, Nos. 20a, 684c, 39a). This species is clearly distinguished by the horizontal arrangement of its perithecigerous cells and by its simphe rhizoid. It is somewhat variable in size, the specimens from Guatemala producing a greater number of smaller perithecia than those from California. The ap- pendage which also lies horizontally is usually quite hidden or broken off, and appears to be rather narrow, the distal cell larger. Trenomyces Laemobothrii nov. sp. Male individual. Corticating cells extending but slightly below the subbasal cell, the lower two thirds of the basal cell quite free, the latter thick-walled, somewhat broader distally, about as long as broad. Antheridia of the usual form suberect in a compact group, six or more in number, the stalk-cells rather long, broader distally and not abruptly distinguished from the body. Appendage relatively very large, the cells subequal, broadly rounded, the apiculus hardly distinguishable. Basal cell 18 Χ 18. Appendage 28 X 18m. An- theridia including stalk 45-50 Χ, the body 12 X 254, including efferent tube. Female individual. Basal cell rather large and rounded, more or less completely corticated, except at the base where the ends of the corticating branches may be clearly visible. Perithecia about six in number, rather slender, subfusiform, the stalk relatively short, not distinguished from the body, the tip large, its margins slightly convex, but otherwise not distinguished from the main body; the a 00 o THAXTER.— RICKIA AND TRENOMYCES. 385 rather prominent suleate apex subtended by four somewhat spreading bisuleate prominences. Appendage relatively very large, the subequal cells rounded as in the male. Perithecium, including stalk 140-160 20-25 μι Appendage 30 X 20 u. On Laemobothrium atrum from Coot, New England. M. C. Z., No. 1537. This species is most easily distinguished by its unusually large appendage, which resembles a stout spore of Puccinia. It seems most nearly related to 7. Lipeuri, the perithecia being very similar. The mode of growth is however, quite different. The rhizoids are entirely broken off in all the specimens. Trenomyces circinans nov. sp. Male individual. Corticating cells few and irregular, producing usually not more than two to four antheridia. Antheridia of the usual form, the body bent often at a right angle to the slender stalk- cell or sometimes recurved, the stalk 18 Χ 4 yu, the body 18 X 14 uz. Appendage relatively small, the cells about equal, 18 X 114, the .distal cell blunt pointed. Female individual. Swollen portion of the rhizoid bearing several horizontal or upcurved lobes from which arise usually furcate smaller lobes running to slender threads of no great length. Perithecia two to four, usually strongly circinate when young, at maturity typically bent or even recurved, rarely straight, the stalk relatively slender, the body often rather abruptly distinguished, broader distally below the tip, which may be subtended by a distinct elevation on one side and is well distinguished, its margin usually slightly convex, separated by a slight constriction from the crown formed by four symmetrically placed somewhat spreading lobes which surround the hardly promi- nent apex, the whole surface of the stalk and body more or less dis- tinctly roughened or granular, the walls much thickened. Appendage relatively small like that of the male. Perithecium including stalk 225-280 X 28-35 μ; the stalk 70-125 X 1ὸ or broader. Appendage 20 X 10-14 uw. On various parts, especially the head of Lipeurus sp., on pigeons, Kingston, R. I., No. 1549; on L. baculus, Elbing, Prussia (Dr. Miiller) ; on Docophorus Californicus, California, No. 1555 (Kellogg No. 666); on D. Montereyi, No. 1554 (Kellogg No. 264c). The Californian forms on Docophorus are not quite so well marked as those from Prussia and Rhode Island which, by their abruptly 386 PROCEEDINGS OF THE AMERICAN ACADEMY. curved habit, slender stalks, and roughened surface, are clearly dis- tinguished from other species of the genus. The tip of the perithe- cium in well developed specimens is not unlike that of Arthrorhynchus Eucampsipodae, but the conformation varies considerably and com- paratively few specimens have a well defined subterminal hunch. Several specimens on Docophorus colymbinus, Nos. 1556-7 (Kellogg, Nos. 14a and 12a), differ distinctly in that the tip is unmodified and hardly distinguished, the stalks stouter and less well distinguished. Further material may indicate that this form is distinct. Trenomyces gibbus nov. sp. Male individual unknown. Female individual. General structure like that of 7. histophorus. Swollen portion of the rhizoid producing several, horizontal lobes. Corticating cells very irregular, completely concealing the somewhat irregular basal cell, giving rise to numerous perithecia. Perithecia faintly tinged with yellowish, stout elongate, the stalk not distinguished from the body, the whole indistinctly roughened, and having the appearance of a goose’s neck and head owing to a subterminal protru-. sion which causes the tip and apex to be bent to one side at an angle 45° or more; the tip nearly symmetrical above the protrusion, broadly conical, the apex rather narrow, subtruncate, slightly indented. Total length of perithecium 300 μ, including stalk, which may be 30 u broad just above its origin; the tip above the hunch, 32 μ long, the base 28 to 30 u broad, the apex about 7 μ. Appendage 25 X 10 un. Described from a single female on Lipeurus longipilus. No. 1563 (Kellogg, No. 128d), California. This form is so peculiar that I have not hesitated to describe it from a single well developed female in good condition. There are a dozen or more perithecia on the specimen in various stages of develop- ment, the four which are mature suggesting the heads and necks of a flock of geese. The distal cell of the appendage is somewhat longer than the basal, tapering from base to apex. ppt νου. Proceedings of the American Academy of Arts and Sciences, Vor. XLVIII. No. 11.— Novemser, 1912. THE SPACE-TIME MANIFOLD OF RELATIVITY. THE NON-EUCLIDEAN GEOMETRY OF MECHANICS AND ELECTROMAGNETICS. By Epwin B. WILSON AND GILBERT N. LEwIs. THE SPACE-TIME MANIFOLD OF RELATIVITY. THE NON-EUCLIDEAN GEOMETRY OF MECHANICS AND ELECTROMAGNETICS. By Epwin B. WILSON AND GILBERT N. Lewis. Introduction. 1. The concept of space has different meanings to different persons according to their experience in abstract reasoning. On the one hand is the common space, which for the educated person has been formu- lated in the three dimensional geometry of Euclid. On the other hand the mathematician has become accustomed to extend the concept of space to any manifold of which the properties are completely de- termined, as in Euclidean geometry, by a system of self-consistent postulates. Most of these highly ingenious geometries cannot be expected to be of service in the discussion of physical phenomena. Until recently the physicist has found the three dimensional space of Euclid entirely adequate to his needs, and has therefore been in- clined to attribute to it a certain reality. It is, however, inconsistent with the philosophic spirit of our time to draw a sharp distinction between that which is real and that which is convenient,! and it would be dogmatic to assert that no discoveries of physics might render so convenient as to be almost imperative the modification or extension of our present system of geometry. Indeed it seemed to Minkowski that such a change was already necessitated by the facts which led to the formulation of the Principle of Relativity. 2. The possibility of associating three dimensional space and one dimensional time to form a four dimensional manifold has doubtless occurred to many; but as long as space and time were assumed to be wholly independent, such a union seemed purely artificial. The idea of abandoning once for all this assumption of independence, although fore-shadowed in Lorentz’s use of local time, was first clearly stated by 1 See, for example, H. Poincaré, La Science et |’ Hypothése. 390 PROCEEDINGS OF THE AMERICAN ACADEMY. Einstein. The theorems of the principle of relativity which correlate space and time appeared, however, far less bizarre and unnatural when Minkowski showed that they were merely theorems in a four dimensional geometry. Suppose that a student of ordinary space, habituated to the inter- pretation of geometry with the aid of a definite horizontal plane and vertical axis, should suddenly discover that all the essential geometri- cal properties of interest to him could be expressed by reference to a new plane, inclined to the horizontal, and a new axis inclined to the vertical. Whereas formerly he had attributed special significance to heights on the one hand and to horizontal extension on the other, he would now recognize that these were purely conventional and that the fundamental properties were those such as distance and angle, which remain invariant in the change to a new system of reference. Let us now consider a four dimensional manifold formed by ad- joining to the familiar ἃ, y, z axes of space a t axis of time. Any point in this manifold will represent a definite place at a definite time. Space then appears as a sort of cross section through this manifold, comprising all points of a given time. For convenience we may temporarily ignore one of the dimensions of space, say 5, and discuss the three dimensional manifold of x, y, t. This means that we will consider only positions and motions in a plane. The locus in time of a particle which does not change its position in space, that is, of a particle at rest, will be a straight line parallel to the ¢ axis. Uniform rectilinear motion of a particle will then be represented by a straight line inclined to the ἐ axis. 3. If we adopt the view that uniform motion is only relative, we may with equal right consider the second particle at rest and the first particle in motion. In this case the locus of the second particle must be taken as a new time axis. What corresponding change this will necessitate in our spacial system of reference will depend entirely upon the kind of geometry that we are led to adopt in order to make the geometrical invariants of the transformation correspond to the fundamental physical invariants whose occurrence in mechanics and electromagnetics has led to the principle of relativity. It is immediately evident that if uniform motion is to be repre- sented by straight lines, the statement that all motion is relative shows that the transformation must be of such a character as to carry straight lines into straight lines. In other words, the transformation must be linear. Further we must assume that the origin of our space and time axes is entirely arbitrary. WILSON AND LEWIS.— RELATIVITY. 391 The further characteristics of this transformation must be deter- mined by a study of the important physical invariants. Fundamental among these invariants is the velocity of light, which by the second postulate of the principle of relativity must be the same to all observ- ers. Hence any line in our four dimensional manifold which repre- sents motion with the velocity of light must bear the same relation to every set of reference axes. This is a condition which certainly cannot be fulfilled by any transformation of axes to which we are accustomed in real Euclidean space. It is indeed a condition sufficient to determine the properties of that non-Euclidean geometry which we are to investigate. Minkowski, in his two papers on relativity,? used two different methods. In his first and elaborate treatment of the subject he in- troduced the imaginary unit V— 1 in such a way that the lines which represent motion with the velocity of light become the imaginary invariant lines familiar to mathematicians who discuss the real and imaginary geometry of Euclidean space. In this way, however, the points of the manifold which represent a particle in position and time become imaginary; the transformations are imaginary; the whole method becomes chiefly analytical. In his second, a brief paper, Minkowski makes use of certain geometrical constructions which have their simplest interpretation only in a non-Euclidean geometry. 4. It is the purpose of the present work to develop the four dimen- sional non-Euclidean geometry which is demanded by the principle of relativity, and to show that the laws of electromagnetics and mechanics not only can be simply interpreted in this way but also are for the most part mere theorems in this geometry. In the first sections we shall develop in some detail the non-Eucli- dean geometry in two dimensions. For it is only by a thorough comprehension of this simpler case that it is possible to proceed into the more difficult domains involving three and four dimensions. This part of the paper will be continued by a discussion of vectors and the vector notation that will be employed. At this point it is possible in a few simple cases to show the applications of the non-Euclidean geometry to problems in kinematics and mechanics. The sections devoted to three dimensions will be occupied largely with numerous analytical developments of the vector algebra, many of which are directly applicable not only in space of higher dimensions 2 Gesammelte Abhandlungen von Hermann Minkowski, Vol. 2, pp. 352- 404 and pp. 431-444. 392 PROCEEDINGS OF THE AMERICAN ACADEMY. but also in Euclidean space. We are led further to a consideration of certain vectors of singular character. The study of the singular plane leads to the brief consideration of another interesting and im- portant non-Euclidean plane geometry. Passing to the general case of four dimensions we shall meet further new types of vectors, and shall attempt even here to facilitate as far as is possible the visualization of the geometrical results. We shall continue further the analytical development, and in particular con- sider the properties of the differential operator quad. In this con- nection a very general and important equation for the transformation of integrals is obtained. The idea of the geometric vector field will then be introduced, and the properties of these fields will be taken up in detail. The subject of electromagnetics and mechanics is prefaced with a short discussion of the possibility of replacing conceptually continuous and discontinuous distributions by one another, and we shall point out that in one important case such a transformation is impossible. The science of electromagnetics is treated both from the point of view of the point charge and from that of the continuous distribution. In both cases it is shown that the field of potential and the field of force are merely the geometrical fields previously mentioned, except for a constant multiplier. Particular attention is given to the field of an accelerated electron,® and in this field we find that the vectors of singular properties play an important rdle. With the aid of these vectors the problem of electromagnetic energy is discussed. The science of mechanics, which is treated in a fragmentary way in some preceding sections, is now given a more general treatment, and the conservation laws of momentum, mass and energy are shown to be special deductions from a single general law stating the constancy of a certain four dimensional vector, which we have called the vector of extended momentum. Finally it is pointed out that this last vector gives rise to geometric vector fields which can be identified with the 3 There seems to be a widespread impression that the principle of relativity is inadequate to deal with problems involving acceleration. But the essential idea of relativity can be expressed by the statement that there are certain vectors in the geometry of four dimensions which are independent of any arbitrary choice of the axes of space and time. Those problems which involve acceleration will be shown to possess no greater inherent difficulties than those that involve only uniform motion. It is, moreover, especially to be emphasized that the methods which are to be employed in this paper necessi- tate none of the approximations that are commonly employed in electro- magnetic theory. Such terms as “quasi-stationary,’’ for example, will not be used. WILSON AND LEWIS.— RELATIVITY. 393 fields of gravitational potential and gravitational force. Moreover, it is shown that these fields are identical in mathematical form with the electromagnetic fields, and that all the equations of the electro- magnetic field must be directly applicable to the gravitational. In an appendix a few rules for the use of Gibbs’s dyadies, which have occasionally been employed in the text, are stated. And a brief discussion of some of the mathematical aspects of our plane non- Euclidean geometry is given. Tue Non-EvuciipEaN GrEoMEtTRY IN Two DIMENSIONS. Translation or the Parallel Transformation. 5. In discussing a non-Euclidean geometry various methods of procedure are available; a set of postulates may be laid down, or the differential method of Riemann may be followed, or the theory of groups may be used as by Lie, or (if the geometry falls under the general projective type, as is here the case) the projective measure of length and angle may be made the basis. For our present purpose we need not restrict ourselves to any one of these; but since the first is familiar to all, we shall employ it as far as convenience permits. Some of the other methods will, however, be briefly discussed in the appendix, §§ 64, 65. With a view to simplicity we shall at first limit the discussion to the case of a plane. Points and lines will be taken as undefined, and most of the relations connecting them will be the same as in Euclidean plane geometry. Thus: * 1°. Through two points one and only one line can be drawn. 2°. Two lines intersect in one and only one point, except that 3°. Through any point not on a given line one and only one parallel (non-intersecting) line can be drawn. 4°. The line shall be regarded as a continuous array of points in open order. 6. In regard to congruence or “free mobility” it is important to proceed more circumspectly than did Euclid. The transformations of Euclidean geometry may be divided into translations and rotations, of which the former alone are the same for our geometry. It seems desirable, therefore, to discuss first and in some detail the postulates ’ 4 We make no claim of completeness or independence for these postulates, which are designed primarily to show the points of similarity or dissimilarity between our geometry and the Euclidean. A like remark may be made with respect to proofs of theorems. 394 PROCEEDINGS OF THE AMERICAN ACADEMY. and propositions relating to this type of transformation, and common to the two geometries. We therefore postulate for translation: 5°. Any point P can be carried into any point P’, and any two translations which carry P into P’ are identical. 6°. Any line is carried into a parallel line. 7°. Any line parallel to PP’ remains unchanged. 8°. The succession of two translations is a translation. These postulates determine the characteristics of a group of geome- tries of which the two most important are Euclidean geometry and that non-Euclidean geometry with which we are here concerned. Another non-Euclidean geometry belonging to this same group will be discussed briefly in §31. This group excludes such geometries as the Lobachewskian and the Riemannian in which a parallel to a given line at a given point is not uniquely defined. We shall first proceed to develop some of those general theorems which are true in this whole group of geometries. I. If two intersecting lines are parallel respectively to two other intersecting lines, the corresponding angles ° are congruent. For by translation the points of intersection may be made to coin- cide, and the lines of the first pair, remaining parallel with the lines of the other pair (6°), must come into coincidence with them, by postulate 3°. II. The opposite sides of a parallelogram are congruent. For if ABCD is a parallelogram and if A be translated to B, the line of DC remains unchanged, by 7°, and the line of AD falls along the line of BC by I. Hence D falls on C by 2°. Cor. If two points P, P’ are carried by a translation into Q, Q’, the figure PP’ Q’ Q is a parallelogram. 7. We may now set up a system of measurement along any line and hence along the whole set of parallel lines. Consider the segment PP’. By the translation which carries P into (δ΄, the point P’ is carried into a point P” of the same line. The measure of the separa- tion of P and P’ we will call the interval ® PP’. And since the segment PP’ is congruent to the segment P’ P”, the intervals PP’ and P’ P” are said to be equal. We may thus mark off any number of equal intervals along the line. We shall assume further the Archimedean postulate. 5 The word angle here refers to a geometrical figure only, and does not as yet imply any measure of angle. 6 We use the word interval to avoid all ambiguity. The notion of distance will be separately considered in Appendix, § 65. WILSON AND LEWIS.— RELATIVITY. 395 9°, Ifa sufficient number of equal intervals be laid off on a line, any point of the line may be surpassed. Now the whole theory of commensurability or incommensurability of two intervals along the same line or parallel lines may be treated by the usual methods. Thus the intervals along a line, starting from any origin upon the line, may be brought into one-to-one correspond- ence with the series of real numbers. It is, however, to be especially emphasized that we have not established, and cannot establish by the translation alone, any comparison between intervals on non-parallel lines. III. The diagonals of a parallelogram bisect each other.’ For let (Figure 1) the parallelogram ABCD, of which the diagonals intersect at 1, be translated into the position BB’ C’ C (by translating A to B), in which the diagonals intersect at Ε΄. Now BE’ is parallel to EC, and EL to CE’. Hence BE’ which is congruent to AF, is congruent to EC by II. Con- sequently 4H is congruent to EC by 8°. IV. If two triangles have the sides of one respectively parallel to the sides of the other, and if one side of one is congruent to one side of the other, then the remaining sides of the C,4A’ one are respectively congruent to the remain- ing sides of the other. For if the two congruent sides are brought into coincidence by translation, the two tri- , angles will either coin- cide throughout, or will together (Figure 2) form a parallelogram (II). ; Two triangles with the sides of one respectively parallel to the sides of the other will be called similar. VY. In two similar triangles the sides of the one are respectively proportional to the sides of the other. For if ABC and A’B’C’ are the triangles, the vertex A’ may be made to coincide with A by a translation (Figure 3). Suppose, now, FIGureE 1. B BD FIGURE 2. FIGURE 3. 7 Theorems like this and the preceding and some which are to follow are proved in elementary geometries by the aid of propositions (on congruence of triangles) not deducible from translations alone. 396 PROCEEDINGS OF THE AMERICAN ACADEMY. that AB’ falls along AB, and AC’ along AC. Assume that AC and AC’ are commensurable. Apply the common measure to the side AC, and through the points of division draw lines parallel to BC and to AB. In the small triangles thus formed the parallel sides will be equal by IV, and therefore the intervals cut off on AB must be equal by II. In case of incommensurability the method of limits may be applied.2 The case in which the two triangles fall on opposite sides of the common vertex may be treated in a similar manner by the aid of IV. 8. For our future needs, the conception and the measure of area are fundamental, and it is important to show that this subject may be satisfactorily treated with the aid of the parallel-transformation (that is, the translation) alone. Indeed, any arbitrarily chosen unit intervals along any selected pair of intersecting lines determine a parallelogram which may be taken as having a unit area. By ruling the parallelogram into equal parallelograms by lines parallel to its sides, an arbitrarily small element of area may be obtained. The area enclosed by any curve may be divided into like elements by similar rulings, and thus by the method of limits the enclosed area may be compared with the assumed unit area.’ In particular some simple propositions on areas will now be deduced. VI. Any parallelogram with sides parallel to those of the unit parallelogram has an area equal to the product of the intervals along two intersecting sides. 8 It may be observed at this point that if two intersecting lines be taken as axes of reference, if systems of measurement (as yet necessarily independent) be set up along the two lines with the point of intersection as common origin, and if to each point P of the plane are assigned coordinates (x, y) equal to the intercepts cut off from the axes by lines through P parallel to the axes, then straight lines are represented by linear equations, and conversely. For the deduction of the equation of a line depends merely upon the properties of triangles similar in our sense. The transformation from any such set of axis to any other such set will clearly be linear. 9 If axes be introduced as above, the area of a triangle and the area of any closed curve are expressed analytically by the usual formulas. πη | 41a yo 1! and ΤΩΣ = fudy ΞΞ —Lydte, [az ye 1| in terms of our assumed unit parallelogram. The theorems on areas could then be proved analytically, but the elementary geometric demonstrations seem preferable. It is important to observe further that in a transformation to new axes, such that x = ar’ ob by’ 55 ὯΣ y= a's’ 4+ by’ a cs WILSON AND LEWIS.— RELATIVITY. 397 VII. The diagonal of a parallelogram divides it into two equal areas. For if the sides of the parallel- Va ogram be divided by repeated bi- section into 2” parts, there will \/ be an equal number of equal τι parallelograms on each side of pes the diagonal (Figure 4), and in the limit the total area of these if parallelograms approaches the area of the triangles. Ficure 4. VIII. If from any point in the diagonal of a parallelogram lines be drawn parallel to the sides, the two parallelograms formed on either side of the diagonal are equal in area (Figure 5). γιὸς» IX. Two parallelograms between the same AWW YN / parallel lines and with congruent bases are equal Figure 5. in area. Cor. ‘Two triangles having congruent bases on one line and vertices on a parallel line have equal areas. Cor. The diagonals divide a parallelogram into four equal triangu- lar areas. Proofs may be given by obvious and familiar methods. X. Of all parallelograms having two sides common to two sides of a given triangle and a vertex on the third side of the triangle, that one has the greatest area whose vertex bisects that third side. For in the figure (Figure 6), where ABC is the triangle and F is the middle point of the third side, the difference of the two parallelograms is HBFE — IBGD = MGFE —IHMD = KMEL— IHMD = KMEL— KDNL = DMEN. Propositions IV and VIII are used in the proof. the value of the area, in terms of the area measured with reference to the new axes, 15 dx’ dy.’ dxdy = μη ᾿ Ια ὃ Hence if the measure of area is to be the same, that is, if the unit parallelogram on the new axes is to have a unit area referred to the old axes, the determinant of the transformation must be unity. This implies a relation between the choice of unit intervals on the new axes. Indeed when the unit interval on one of the new axes has been arbitrarily chosen, the unit interval on the other is determined. In other words the unit intervals on the new axes must each vary inversely as the other. 398 PROCEEDINGS OF THE AMERICAN ACADEMY. As an extension of the idea of similarity for triangles, we may say that any two polygons which have their corresponding sides parallel B G F σ Ficure 6. triangles ABF, CAE, BCD. and in proportion are similar. It fol- lows that if any two corresponding lines are drawn in the polygons, these lines must be parallel. XI. If on two sides of a triangle similar parallelograms be constructed, and on the third side a parallelogram with diagonals parallel to the diagonals of the other parallelograms, the area of this parallelogram will be equal to the difference of the areas of the other two. The areas (Figure 7) of the paral- lelograms on AB, CA, BC are respec- tively four times the areas of the If wetake the unit parallelogram with sides parallel to the diagonals, it will suffice to prove that FIGurReE 7. FBX AF = AE X EC— BD X CD, for each of these areas is twice the area of the corresponding triangle. In the similar triangles ACE and GCD, HO CD: vAL DEG: WILSON AND LEWIS.— RELATIVITY. 399 But by ΠῚ, BD is equal to DG. And writing AF = FB + BD, we have EC X BD = CD X FB + CD X BD. Add to each side the product FB & EC. Then EC(BD + FB) = CD X BD + FB(CD + EC). Hence ECO X:AE—CD X BD = FB X AP. Non-Euclidean Rotation. 9. The group of parallel geometries determined by Postulates 1°-9°, which, notwithstanding its generality, gives rise, as we have seen, to some interesting and important theorems, may be subdivided by adding a set of postulates belonging to a second transformation which by analogy may be called rotation. It is this set of postu- lates which will differentiate our non-Euclidean geometry from the Euclidean. The difference between our non-Euclidean rotation and the ordi- nary kind is that in addition to a fixed point, two real lines through the point remain unchanged. We may postulate for rotation: 10°. Any one point and only that one remains fixed. This point may be called the center of rotation. 11°. Two lines through this point remain unchanged. These lines may be called the fixed lines of the rotation. 12°. Any half-line (or ray) from the center, and lying in one of the angles determined by the fixed lines, may be turned into any other ray in the same angle, and this uniquely determines the rotation. 13°. The succession of two rotations about the same point is a rotation. 14°. The result of a rotation about O and a translation from O to O’ is independent of the order in which the rotation and transla- tion are carried out. It follows immediately from 14° that the fixed lines in a rotation about any point O are parallel to the fixed lines in a rotation about any other point Θ΄. All lines in the plane may now be divided into classes in such manner that neither translation nor rotation can change the classification. Namely, (a) lines parallel to one of the fixed directions, (8) lines parallel to the other of the fixed directions, 400 PROCEEDINGS OF THE AMERICAN ACADEMY. (y) lines which lie in one of the pairs of vertical angles determined by the fixed directions, (6) lines which lie in the other pair of vertical angles determined by the fixed directions. The lines of fixed direction, namely, the (a)-lines and (§)-lines, will be called singular lines. A system of measurement may be set up for angles between rays 19 which issue from a point into one of the angles determined by the fixed lines through the point. For a succession of rotations may be used (in the same manner as the succession of translations was used to establish the measure of interval along a line). Thus if a line a is carried into a line a’ and at the same time the line a’ is carried into the line α΄, the angles between a and a’ and between a’ and a” are congruent and the measures of the angles are said to be equal. Now as the rotation may be repeated any number of times without reaching the fixed line, it is possible to find an angle aa“ which shall be n times the angle aa’. We shall assume the postulate, analogous to the Archimedean: 15°. If a sufficient number of equal angles be laid off about a point from any initial ray, any ray of that class may be surpassed. It thus appears that the angles between any given line and other lines of the same class may be placed into one-to-one correspondence with all positive and negative real numbers, just as the intervals from a point on a line may be thus correlated.!! This constitutes a very great difference between our geometry and the Euclidean. It is impossible to show from the preceding statements that any given figure maintains a constant area during rotation.1? We shall therefore lay down the additional postulate: 10 The relations of order of all lines of a given class, (y) or (δ), are the same as those of points on a line, as in 4°. 11 The angle between two singular lines (α) and (8) can obviously not be measured. Such an angle, and also the angle between any line and a line of fixed direction, must be regarded as infinite. 12 This matter may readily be discussed analytically. As axes of reference choose the fixed lines, and let wu, v denote coordinates. As rotation is a linear transformation, the point P (u, v) and the transformed point P’ (μ΄, v’) are connected by the equations μ' =au+bv+e, υ' = du+ev+f. As the lines u = 0 and νυ = 0 are fixed, these equations reduce to τ΄ = au, υ' = ev; and as rotation depends on only one parameter, we may write e = d(a). The succession of two rotations is then expressed by (u’ = au {ὦ = bu’ ; u"”’ = abu lv Ξ-- φ(α), Lv” = φ()ν', = $(a)G(b)u, - | WILSON AND LEWIS.— RELATIVITY. 401 105. In rotation an area becomes an equal area.!% 10. We are now prepared to discuss in some detail the general characteristics of our rotation. Consider (Figure 8) a series of rota- tions about ὁ), whereby the point P assumes the positions P’, P”,.... Let the parallelograms on OP, OP’, OP”,.... as diagonals and with sides along the fixed lines be con- structed. Then by 16° the areas of these parallelograms are equal, and in terms of the intervals on the fixed lines OA X OB = OA’ X OB’ SOA 6 OR’. Ficure 8. The point P thus traces a curve which in ordinary geometry would be with the condition $(a)b(b) = (ab) necessitated by 13°. This is a functional equation of which the only (con- tinuous) solution is φί(α) = α΄. Hence rotation must be of the form a= au, w= αἴ. The unit parallelogram on the axes of τὸ and v is hereby transformed into a parallelogram on these same axes with intervals a and a” along u and v. By VI the area of the new parallelogram is therefore αὔτ]. If this is to be unity, r =-—l. The transformation equations for rotation are therefore oh S01, δ Soy Gp where a is necessarily positive because points do not change from one side of the axes to another. The intrinsic significance of these equations should not be overlooked. A rotation may be represented as a multiplication of all intervals along one of the fixed lines by a constant factor and a division of all intervals along the other fixed line by the same factor. Or, increasing the unit interval along one fixed line and decreasing it in the same ratio along the other is equivalent to a rotation. (This process effected along any other axes than the fixed lines would leave the area unchanged, but would not be a rotation). As the unit interval along one fixed line cannot be compared either by translation or by rotation with the unit along the other, and as one of these units is arbitrary, we have additional evidence that there is no natural zero of angle. 13 Such a postulate is unnecessary in Euclidean geometry owing to the cada nature of the Euclidean rotation. Postulate 16° could be replaced y one involving only the notion of symmetry between rotations in opposite directions. 402 PROCEEDINGS OF THE AMERICAN ACADEMY. considered a branch of a hyperbola.!* Since, however, this curve is here generated by the rotation of a line OP about its terminus Q, we shall call this locus (taken with the other branch Q Q’ Q” sym- metrically situated with respect to O) the pseudo-circle. By means of such a rotation we are able to compare intervals upon any line with intervals upon any other line of the same class. For the intervals of the congruent radii OP, OP’, OP” will be called equal. When we consider the fixed lines we observe that the effect of rotation is to carry the segment OA into OA’ or OA”. It is therefore evident that segments are congruent by rotation which are incongru- ent by translation. This source of ambiguity exists only in the case of singular lines, for in no other case is it possible to compare two segments both by rotation and by translation. We may remove this ambiguity at once by stating that intervals along singular lines, al- though metrically comparable with intervals on other singular lines of the same class by translation, are all of zero magnitude when compared with intervals on any non-singular line. This will become more evident later. Consider next (Figure 9) the inter- cept AB terminating on the fixed lines corresponding to a rotation with cen- ter at O. Let P be the middle point of the line, and C any other point. Through C draw a line parallel to OB, and on this line mark the point P’ such that the area OD P’G equals the area OF PH. The area OECG is less FIGURE 9. than each of these by X. Hence P’ lies on the further side of AB from Ὁ. But P’ is a point on the pseudo-circle through P concentric with O, as we have just seen. Since C was any point of AB, it follows that P’ may be any point of the pseudo-circle. Hence as the line AB meets the pseudo-circle at P and only at P, it is tangent to the curve. As a species of converse, we may state the theorem: 14 There is no special significance in the fact that a rectangular hyperbola is drawn in the figure and that the fixed lines a, 8 are perpendicular in the Euclidean sense; in subsequent figures the singular lines are often oblique. From the non-Euclidean viewpoint the question of perpendicularity or obliquity of the singular lines is of course meaningless. δ. (neler WILSON AND LEWIS.— RELATIVITY. 403 XII. The tangent to a pseudo-circle lies between the curve and its center, and the portion of the tangent intercepted between the two fixed lines is bisected at the point of tangency. 11. In a pseudo-circle the radius and the tangent at its extremity are said to be perpendicular. Or in virtue of XII we may say that the perpendicular from any point O to any non-singular line is the line from O to the middle point of that segment of the line which is inter- cepted by the fixed lines through ὦ. The construction of a perpendic- ular to any line of class (y) or (δ) at a point of the line is equally simple. By the aid of propositions concerning similar triangles, the follow- ing theorems concerning perpendiculars are readily proved. XIII. Ifa line ais perpendicular to a line b, then ὁ is perpendicular to a. XIV. Through any point one and only one perpendicular can be drawn to any line. XY. All lines perpendicular to the same line are parallel. XVI. The singular line of one class which is drawn through the intersection of any two perpendicular lines will bisect the segment intercepted by these lines upon any singular line of the other class (Figure 10).+° XVII. The perpendicular to a (y)-line Braun 10. is a (6)-line, and vice versa. Intervals along lines of class (6) cannot be compared by congruence with intervals along lines of the (y) class. We may, therefore, arbi- trarily define equality of intervals between the two classes. Jf two mutually perpendicular lines are drawn from any point and terminate on a singular line, the intervals of these lines will be said to be equal.'® The consistency of this definition is readily proved. The definition of perpendicularity is such that if two lines are per- pendicular they must remain perpendicular after a translation or rotation. The former case is obvious, and the latter becomes so when the lines are considered as radius and tangent in a pseudo-circle generated by the rotation; the more general case in which neither of the perpendicular lines passes through the center of rotation then follows with the aid of XV. It is important to observe one peculiar 15 In the figure BO and OC are equal, and AB and AC are perpendicular. 16 In Figure 10, the intervals AC and AB are therefore equal by this definition. 404 PROCEEDINGS OF THE AMERICAN ACADEMY. characteristic of our rotation, namely that two perpendicular lines approach each other and the fixed line between them scissor-wise, as may be seen, in Figure 11, where OC and OD become respectively OC’ and OD’, OC” and OD", ---- The pseudo-circles traced by OC and OD may be called conjugate pseudo- circles, since the interval OC equals the interval OD, the lines CD, C’D’, ----, being OS CSS ae singular, and bisected by a fixed line. Since two mutually perpendicular lines ap- proach, during rotation about their point of intersection, the same fixed line, we may extend our definition of perpendicularity by Figure 11. regarding every singular line as perpendicular to itself. This extension is also suggested by the fact that the fixed line may be considered an asymptote of a pseudo-circle. Special caution must be given against the idea that a singular line of one class is perpendicular to a singular line in the other class. The peculiarities of singular lines will become clearer in the work on vector analysis. 12. A triangle of which two sides are perpendicular will be called a right triangle, and the third side will be called the hypotenuse. A parallelogram of which the two adjacent sides are perpendicular and of equal interval will be called a square. The following theorem is obvious: XVIII. One diagonal of every square is a singular line and the other diagonal is a singular line of the other class. XIX. Pythagorean Theorem. The area of the square on the hypotenuse of a right triangle is equal to the difference of the areas of the squares on the other two sides. For by XVIII the diagonals of the squares are lines of fixed direction, and hence parallel each to each. The squares on the two legs are similar. And the proposition is evidently a special case of XI. (In Figure 7 if the dotted lines are singular lines, the lines AC and BC are so drawn as to be approximately perpendicular.) XX. Any two squares whose sides are of unit interval are equal in area. For by suitable translation and rotation one may be brought into coincidence with the other. The unit of area will henceforth be taken as the area of a square whose sides are of unit interval. Hence follows: WILSON AND LEWIS.— RELATIVITY. 405 Cor. The area of any rectangle is the product of the intervals of two adjoining sides. We may therefore obtain from XIX the theorem XXI. The square of the interval of the hypotenuse of a right triangle is equal to the difference in the squares of the intervals of the other two sides. Cor. The perpendicular from a point to a line has a greater interval than any other line of the same class drawn from the given point to the given line. Having now given a final definition of the measure of area, we may define the unit of angle. The radius of the pseudo-circle, in advancing by rotation over equal angles, necessarily sweeps out equal areas (by 16°). Hence by the familiar argument sectorial areas in any pseudo-circle are proportional to the angles at the center. The unit angle will be taken as that angle which, in a pseudo-circle of unit radius, encloses a sectorial area of one-half the unit area. Vectors and Vector Algebra. 13. ‘Translation or the parallel-transformation leads at once to the consideration of vectors. We have shown that when a translation carries A into B and A’ into B’ the directed segments AB and A’B’ are parallel and congruent (Cor. to 11). Hence a translation may be represented by a vector, that is, by any directed segment laid of from any origin and having the same interval and direction as AB. The succession of two translations is represented by the sum of their corresponding vectors. The addition and subtraction of vectors and their multiplication by scalars follows the usual laws (by δὲ 5-7). If two vectors a and Ὁ are laid off from a common origin, the paral- lelogram constructed on the vectors is called their outer product axb, and the magnitude of this product will be taken numerically equal to the area of the parallelogram.17 We must bear in mind that not this magnitude (nor yet a vector perpendicular to the plane), but the parallelogram itself is the outer product. We may, however, repre- sent the outer product by any other closed figure of equal area, pro- vided that it is taken with the same sign. The sign attributed to an 17 Our vector notation will be based upon that of Gibbs, and is identical with that employed by Lewis (Four dimensional Vector Analysis, These Proceedings, 46, 163-181) except in the designation of the inner product which we shall define asin that paper, but represent by a+b instead of ab; the latter form will be reserved to denote the dyad. The scalar magnitude of a vector will be represented by the same letter in italic type. 406 PROCEEDINGS OF THE AMERICAN ACADEMY. area does not arise from any positive or negative geometric charac- teristics of the area itself, but from an interpretation or convention concerning the way in which one area is considered as generated relative to another, and is required for analytic work. We shall make the convention that axb and (—a)xb or ax(—b) have opposite signs. The outer product of a vector by itself or by any parallel vector is zero, because the parallelogram determined by these vectors has zero area; thus axa = 0. The associative law for a scalar factor is valid, because multiplying one side of a parallelogram by a number multi- plies the area by that number; thus (na)xb = naxb = ax(nb). The distributive laws, ax(b + c) = axb + axe, (a+ b)xc = axc+ bxe, also hold; for inspection shows that the parallelogram ax(b + Ο) is equal to axb plus axc. The anti-commutative law, axb = — bxa, holds; for (a + b)x(a + b) = axa+ axb + bxa + bxb = 0. Hence axb = — bxa. 14. Thus far we have proceeded by means of the parallel-trans- formation alone. It is evident that this much of vector algebra is common to all geometries, including the Euclidean and our non- Euclidean geometry, in which there is such a parallel-transformation. The other type of product, the inner product, cannot be defined with- out some concept of rotation or perpendicularity, or its equivalent. We shall so define this inner product a:b that it obeys the associa- tive law for a scalar factor and the distributive and commutative laws, namely, (na)-b = na-b =a-(nb), a:(b + c) = a-b+ ac, a-b = bea, and furthermore remains invariant during rotation. As the fixed lines are fundamental in rotation it is sometimes ex- pedient to resolve vectors into components along these directions. Let p and q be definite vectors in the two fixed lines; any vector in WILSON AND LEWIS.— RELATIVITY. 407 the plane may be written as r = 0 - yq._ By the postulated formal laws, rr=2p-ep+ y2q-q + 2zy pq. We may now note that by rotation a vector along a fixed line is con- verted into a multiple of that vector. If p becomes np, and the inner product p-p remains invariant, then p-p = n*p+p; whence it is ob- vious that p-p = 0. In general: The inner product of any singular vector by itself is zero, and this suffices to characterize a singular vector. Hence r-r reduces to rer = 27ry p-q. Before proceeding further with the definition of the inner product, we may observe that the signs of xv and y are determined by that one of the four angles (made by the fixed lines) in which r lies. According, then, as x and y have the same sign or different signs, the vector r belongs to one or the other of the classes (γ) or (δ), and the product r-r will have one sign or the other. These considerations suffice to show that if r and r’ are two vectors, and if rer and r’-r’ have the same sign, the vectors are of the same class, but if rer and r’-r’ are of op- posite sign, rand r’ are of different classes. We have here a marked departure from Euclidean geometry, in which the inner product of a real vector by itself is always positive. We are now in a position to complete the definition of the inner product by stating that the product is a scalar, and that the product of a vector by itself is equal to the square of the interval of the vector, taken positively if the vector is of class (v), negatively if of class (δ). This does not imply any dissymmetry between the classes (γ) and (δ), but is only such a convention as is often made with respect to sign. The equation rer = 2xy p-q shows that the inner product of any singular vector and any singular vector of the other class is equal to one-half the inner product by itself of the diagonal of their parallelo- gram. The inner product of any vector and a perpendicular vector is zero. For by XVI it is evident that if p and q be the components along the fixed directions of any vector r, so that r= p+ q, then p—q is a perpendicular vector, and in general any perpendicular vector r’ has the form n(p — q). Hence 17 r-r = n(p — q)-(p+ q) = n(D-P + 6 — ap — a-d) = 0. 17 The fact that the inner product of a singul wr vector by itself vs anishes justifies our convention that a singular line is perpendicular to itself. 408 PROCEEDINGS OF THE AMERICAN ACADEMY. The inner product of any two vectors is equal to the inner product of either one by the projection of the other along it. For either vector may be resolved into two vectors one of which is parallel and the other perpendicular to the other vector. Thus Ὁ may be written as na + a’, where na is the projection of b on a, and a’ is perpendicu- lar toa. Therefore b-a = na-a+ a’-a = nasa, which was to be proved. Geometrically the only puzzling case is that in which the vectors are of different classes. Let OA (Figure 12) be a vector of class (vy) and OB of class (δ). The projections of OA on OB and of OB on OA are respectively OB’ and OA’. Note that whereas OB’ extends in the same direction as OB, the vector OA’ extends along the opposite direction to OA. Thus OB’ is a positive multiple of OB, whereas OA’ is a nega- tive multiple of OA. But the inner product of OB by itself is negative, since the vector is of class (6), while the inner product of OA by itself is positive, since the vector is of class (y). Hence the inner product of OA and OB has the same sign, whichever way the projection is taken. In obtaining the inner product of a singular and a non-singular vector by projecting one upon the other, it is necessary to project the singular vector upon the non-singular vector; for it is impossible to make a perpendicular projection upon a singular vector. In case both vectors are singular the method of perpendicular projection fails entirely, and we must use analytical methods (or have recourse to parallel projection). 15. It will often be convenient to select two mutually perpendicular lines as axes of reference. We will denote 18 by Κι and k, unit vectors along such axes, k, being the vector of the (7)-class, and Καὶ, of class (δ). For these vectors we have the rules of multiplication Figure 12. k, Κι = ily ky-ky i | k, ky = ky-k, = 0. 18 We reserve the symbols ky and ks for other unit vectors of class (7) in space of higher dimensions. WILSON AND LEWIS.— RELATIVITY, 409 Any two vectors ἃ and b’may be written in the form a= ak; + ayky, Ὁ = bik; + byky, and the inner product is then, by the distributive law, | arb = ab; — αὐι. In terms of these unit vectors we may also express outer products. If we write, for brevity, Κὰ = Κιχ Κι, the rules for outer multiplica- tion are Κι = —Ky, ki, = Ky = 0. The outer product of the vectors a and b is therefore ax) -Ξ (ayb4 == aby) ky. Since Κις represents a parallelogram of unit area, the question arises as to why we write k.xk, as ky, and not simply kxk, = 1. The answer is that the outer product axb possesses a certain dimension- ality, which, it is true, is not exhibited in a marked degree until we proceed into a space of higher dimensions, but which renders it un- desirable to regard the outer product as merely a scalar. We may call it a pseudo-scalar, and later extend this designation to n-dimensional figures in a manifold of m dimensions. Every vector in two dimensional space uniquely determines, except for sign, another vector, namely, the one equal in interval and per- pendicular to the first. This vector will be called the complement of the given vector. To specify this sign, the complement a* of the vector a may be defined as the inner product of a and the unit pseudo- scalar k,,, namely, a* = a+Ky, where the laws of this inner product are ki -kyy = — ky, ky-Kkyy = — kj. . Thus if a = ak, + ak,, then for the complement a* = (ayky + agky)* = (αἰκι + agky)+kyy = — agk, — ay ky. This type of multiplication, as will be seen later, obeys all the general laws of inner products (§§ 27, 29). Referred to a set of perpendicular unit vectors, the singular vectors take the form n(+ k, + k,). The complement of a singular vector is n(= ky + ky)*=n(+ ky + Κι) Κὶς = n(+ ky - Κι), that is, the complement of a singular vector is its own negative. 410 PROCEEDINGS OF THE AMERICAN ACADEMY. We may extend the idea of complements .to scalars and pseudo- scalars. The complement of the scalar n will be defined as the pseudo- scalar nk,,; the complement of the pseudo-scalar nk,, will be defined as the scalar — ἡ. This may be written (nky)* = nkyeky = — n, thus establishing the convention kKieki,= —1. It may readily be shown that, for any two singular vectors p and q of different class, the outer product is the complement of the inner product, that is, pxq = (Ῥ αὐ Κι. In other words the inner and outer products of singular vectors are numerically equal. Some Differential Relations. 16. As the inner product r-r of a vector by itself is numerically equal to the square of the interval of the vector r, the equation of the unit pseudo-circle of which the radii are all (y)-lines is rer = 1; and the equation of the conjugate unit pseudo-circle of which the radii are (6)-lines is rer = —1. As the tangents to a pseudo-circle are perpendicular to the radu, they must be of opposite class. A pseudo-circle of which any tangent is a (6)-line (the radii being (y)- lines) is called a (6)-pseudo-circle; and a pseudo-circle of which any tangent is a (y)-line (the radii being (6)-lines) is called a (y)-pseudo- circle. In general if a curve has tangents which are all of the same class (δ) or (vy), the curve may be designated as a (6)- or a (y)-curve; the normals to the curve will then be respectively of the opposite class (y) or (δ). The interval of the are of any such curve will be the limit of the sum of the intervals of the infinitesimal chords along the are. We shall not be obliged to consider any curve which is not altogether of one class as here defined. As dr is the infinitesimal chord as a vector quantity, the formula for the scalar arc is ee i sip de: sae ΟΝ Σὲ bp a Ἐπ ΤΠ according as the curve is a (γ)- or ἃ (6)-curve. The sectorial area in a unit pseudo-circle may be regarded as the sum of infinitesimal right triangles, of which the area is numerically equal to 4rxdr if r is drawn from the center. The numerical WILSON AND LEWIS.— RELATIVITY. 411 value of the area is therefore one-half the numerical value of dr, that is, one-half the infinitesimal interval of are. From our definition of unit angle (§ 12), it is evident that an angle is equal to the are sub- tended upon a unit pseudo-circle centered at the vertex of the angle. This might, in fact, have been made the definition of the measure of angle. It is evident from these considerations that a rotation turns all non-singular lines through the same angle. Angles may be classified according to the classes of their sides. If the two sides are (y)-lines, the angle will be designated as of class (yy); if they are (6)-lines, the angle is of class (66). Consideration of angles (y5), which have one side a (y)-line and the other a (6)-line, and which cannot be gener- ated by rotation, need not detain us here. (See Appendix.) If any line (Figure 13) through the center be taken from which to measure angle, posi- tion upon the unit pseudo-circle may be expressed parametrically in terms of the angle as follows. Let the given line be a line of class (y) (the pseudo-circle then being of class (6)), and construct the perpendicular Fievre 13. line of class (δ). These two lines may be taken respectively as axes of x, and x, with the unit vectors k, and Κι along them. The equation of the unit pseudo-circle is then rer = (ak, + agky)-(ayk; + ayky) = αἵ — af = 1. The differential of angle or arc is in this case d0=ds= V_dr.dr= V (kidx,+ k,dz:) . (k,d2,+ k,dx,) = Vde2—dx2 Whence, by differentiation of 2? — rf = 1, [« - [ὦ =) - dats ΕΞ . | dxy hy Nl ea Va? — 1 and x; = cosh 6, % = sinh, 6; where θ is the angle between the 2,-axis and the radius vector, and therefore of the class (yy). If the given line had been of class (δ) (the pseudo-circle of class (y)), and if the angle ¢ had been of class (65) measured from the a-axis to the radius vector, the results would have been 412 PROCEEDINGS OF THE AMERICAN ACADEMY. x, = sinh @¢, xs = cosh ¢, with 2°—a? = — 1 as the equation of the pseudo-circle. If now in general r be the radius of any pseudo-circle, the foregoing results may readily be generalized, and we obtain the following pair of equations. x; = r cosh 6, 2, = rsinh 0, Xs = 2, tanh 6; (1) x; = r sinh ¢, x4 = r cosh 9g, x; = x, tanh ¢. In the first case r is a (y)-vector and θ is a (yy)-angle; in the second, r is a (6)-vector and φ is a (66)-angle. We thus have equations which express the relations between the hypotenuse and the sides of any right triangle in terms of one angle. The inclination of the vector r to the axes k, or k, in the respective cases is the angle 6 = tanh! or oo tanh71 : 11 v4 and the slope of r relative to the axes is the hyperbolic tangent of the angle, not the trigonometric tangent. 17. Consider next any curve of class (δ). Let denote scalar arc along the curve, and let r be the radius vector from a fixed origin to any point of the curve. Then the derivative ἀντ dey, y divs Lillian ποτε Bi ΚΕ ΤΣ ds (2) is a unit vector tangent to the curve. If this vector makes the angle ¢ with the axis k;, so that the slope of the curve is ἢ = tanh ὦ = as (3) the components of the vector are day eh BC ay v dis Mm ἐν: 1 gs τ sinh Φ = ie: τὶ Ἐπ cosh Φ = Pipers (4) and Wa aoe (vk, + ky). (5) V1 — 7 * > ab WILSON AND LEWIS.— RELATIVITY. 413 If we had chosen a different set of perpendicular axes Κι', ky’, where k,’ makes an angle Ψ = tanh ''w with k,, so that the inclination of w to ky’ is φ' = φ — ψ, the new components of w would be dx’ : Σ : v’ ! = sinh φ' = cosh ¢coshy — sinh ¢ sinh y = ———— ds V1 — 0” ot Vi — # V1— wv dics! = cosh’ = cosh ¢ coshy — sinh¢ sinh y = —- : ds V1 —? Τ' 1 — vu τ Vo evil where 2 ᾿ a : ary! ; tanh φ — tanh i — al μ᾿ om τς 1 --- tanh ¢ tanh Ψ πὐ ΠΕ (6) It will be convenient to have a general equation for the components of a vector upon one set of axes in terms of its components on another set. Let Κι, ky be one set of perpendicular unit vectors, and ky’, k,’ another set. If the angle from Κι to Κι΄ be y, the angle from k, to Κι΄ is also ψ by ὃ 106. The products Κι Κι΄ = coshy, k,-k,’= — coshy, Κι Κι΄ = sinhy, k,’-k, = — sinhy, follow from (1). To obtain the transformation equations we write r= ak, + ayky = αἱ Κι + x Κῳ, and multiply by ky, ky, ky’, ky’; r-k, = 2; = x; coshy + ay’ sinhy, —reky = χὰ = 2 sinhy + ay’ coshy, (7) r-k,’ = 2’ = 2, cosh — ay sinhy, —r-k, = 2, = — 2x,sinhy + x coshy. Curvature in our non-Euclidean geometry is defined, as is ordinary geometry, as the rate of turning of the tangent relative to the are. As w is a unit tangent, dw is perpendicular to w and in magnitude is equal to the differential angle through which w turns. Hence 414 PROCEEDINGS OF THE AMERICAN ACADEMY. se 8) is the curvature, taken as a vector normal to the curve. Hence bee a eee ® In magnitude the curvature is dv Cx, APF dx. iy. dx ~~ — © τὸ es ιν. ΙΒ. -- yan 3 ὧν ἐν © δ ἢ i μ᾿ So | neal | oe Relative to axes k,’, k,’, the result is ‘ay k,’ v' ky’ dv’ a la —")? us (1 — ae dx4' _f d—w)k (v — μὴ ky’ ᾿ Ε —e2vi—w d—v? vli—w In complete analogy with the circle in Euclidean geometry the pseudo-circle in our non-Euclidean geometry has a curvature of con- stant magnitude throughout. The curvature of any other curve may always be represented as the curvature of the osculating pseudo-circle, and in magnitude is inversely proportional to the radius of that pseudo- cirele. Kinematics in a Single Straight Line. 18. Before proceeding to the discussion of the non-Euclidean geom- etry of more than two dimensions we may consider some simple but fundamental problems of physics which may be treated with the aid of the results which we have already obtained. The science of kinematics involves a four dimensional manifold, of which three of the dimensions are those of space, and one that of time. By neglecting two of the spacial dimensions, in other words by restricting our considerations to the motion of a particle 15. in a single straight line, kinematics becomes merely a two dimensional science. The theorems of kinematics, not in the classical form, but in the form given to them by the principle of relativity, are simply theorems in our non-Euclidean geometry. 19 By particle we do not as yet mean a material particle but merely an identifiable point in motion. ue lS WILSON AND LEWIS.— RELATIVITY. 415 The units of distance and time, namely the centimeter and second, were chosen without reference to each other. Retaining the centi- meter as the unit of distance, we may take as the unit of time one which had been frequently suggested as the rational unit long before the principle of relativity was enunciated, namely, the second divided by 3 X 10, or the time required by light in free space to travel one centimeter. The velocity of light then becomes unity. Let us consider in our geometry two perpendicular lines, and meas- ure along the (y)-line extension in space, along the (6)-line extension in time. Then any point in the plane will represent a given position at a given time. We are considering the motion of a particle along a specified straight line in space. If x denotes distance along the line from a chosen origin, then in terms of our previous nomenclature, we shall take x = αι andt = a; The k,- or f-axis, or any line in the at-plane parallel to this axis, represents the locus in time of a particle which does not change its position in space, in other words, of a sta- tionary particle. Any straight line of the (6)-class making a non- Euclidean angle Y with k,, represents the locus in space and time of a particle moving with a constant velocity dx Lanse δεν tanh y A singular line in our plane represents a velocity wu = 1, and is the locus of a particle moving with the velocity of light. We have seen that in our plane no pair of perpendicular lines is better suited to serve as coordinate axes than any other pair. If then we consider (Figure 14) two (6)-lines, marked ¢ and ?’, and the respectively perpendicular (y)-lines, marked «x and 2’, and if we regard the first (6)-line as the locus of a stationary particle and the second as the locus of a moving particle, we might expect to find that we could equally well regard the second (6)-line as the locus of a particle at rest and the first as the locus of a moving particle. And this is, in fact, the first postulate of the principle of relativity. The one relation between the two lines, which is independent of any assumption as to which line is the locus of a stationary point, 15 FIGURE 14. 416 PROCEEDINGS OF THE AMERICAN ACADEMY. the angle y whose hyperbolic tangent is the relative velocity which is the same by either of the assumptions. If now we have a third (6)-line t’’ making an angle ¢ with the first (6)-line, and ¢’ with the second, where ¢’ = ¢—y, and if we call the relative velocities corresponding to these angles v = tanh φ, v = tanh@’, u = tanhy, then it is not true that υ' = v—u, but since ¢’ = ¢—y, by (6). This is the theorem regarding the addition of velocities ob- tained by Einstein.?° The true significance of this result cannot be emphasized too strongly, namely, that the velocity as such can only be determined after a set of axes have been arbitrarily chosen; relative velocity, however, has a meaning independent of any co- ordinate system. Furthermore it is not the relative velocities, but the non-Euclidean angles, which are their hyperbolic anti-tangents, which are simply additive. If we were constructing a new system of kinematics uninfluenced by the historical development of the science, it might be preferable to make these angles fundamental rather than the velocities. Suppose that from a given (6)-line we lay off successively equal angles, so that each line determines with the preceding line the same relative velocity, then the angle measured from the given line increases without limit, but its hyperbolic tangent, which is the velocity relative to this line, approaches unity, that is, the velocity of light. The relative velocity, therefore, determined by any two (6)-lines whatever, is less than the velocity of light. The velocity of light itself appears the same regardless of the choice of coordinate axes. This is the sec- ond postulate of the principle of relativity. Indeed if angle, instead of relative velocity, had been made fundamental, the motion of light, as compared with all other motions, would have been characterized by an infinite value of the angle. 19. Let us return to our figure and consider once more the lines that have been marked ¢, t’, anda, α΄. If we take the ¢-line as the locus of a stationary particle, then all points along the line x or along any parallel line are said to be simultaneous, for along any line perpendicu- lar to the t-axis the value of ἐ is constant. In like manner if we con- 20 Hinstein, Jahrb. d. Radioak, 4, 423. WILSON AND LEWIS.— RELATIVITY. 417 sider the ?¢’-line as the locus of a particle at rest, then simultaneous points are those along x’ or along lines parallel to x’. Hence points which are simultaneous from one point of view, are not simultaneous from the other. In fact any two points through which a line of class (y) can be drawn may be regarded as simultaneous by choosing this (y)-line as the axis a, and the perpendicular line as the axis ἡ. Sim- ilarly any two points through which a (5)-line can be drawn may be regarded as having the same spacial position; in other words any point may be taken as a point at rest. It thus appears that the measurements of time and space are de- termined only relative to some selected set of axes. Further to exhibit this fact, and to determine the relations which exist between the measures of time and space when different sets of axes are chosen, let us consider (Fig- ure 15) two parallel (6)-lines in our non-Euclidean plane. These lines represent the loci of two particles which have no relative velocity. Let any set of axes of time and space be drawn. The constant intervals cut off by the two parallel (6)-lines from the x-axis and all lines parallel to this axis represent the constant distance, as Bicure 15. measured by these axes, between the two particles at any time. The constant intervals cut off by the two parallel (6)-lines on the f-axis and all lines parallel thereto repre-. sent the constant interval of time as measured by these axes, which must elapse between the instant when one of the particles has a certain position (upon the line in which we are considering rectilinear motion as taking place) and the instant when the other of the particles has this same position. One particular choice of axes is especially simple, namely, that in which the t-axis is parallel to the two (6)-lines, and the z-axis is perpendicular. Relative to this assumption of axes the particles are at rest. The distance between them is AB. If another set of axes is drawn, the particles appear to be in motion, and the distance be- tween them is taken as A’ B’. If y denotes the angle between the axes, the projection of A’B’ on AB is equal to AB, / / AB = A’B’ coshy = ἘΞ V1 — uw? 418 PROCEEDINGS OF THE AMERICAN ACADEMY. where w is the relative velocity determined by y. Or, A’'B' = AB sechy = AB V1 — w?. That is to say, the distance A’B’ between the particles when con- sidered in motion with the velocity wu is to the distance AB between the particles when considered at rest as V1 — u2:1. This statement embodies Lorentz’s theory of the shortening of distances in the direction of motion. Consider now (Figure 16) two intersecting (6)-lines along which equal (unit) intervals OT and OT" are marked. If OT is taken as the time-axis, the point 1], obtained by dropping vg oe from 7’ the perpendicular 7’M to OT, is as ΝΟ simultaneous with 7’. But the interval OM Ficure 16. is greater than OT in the ratio 1: V1 — wu where w= tanhy is the relative velocity determined by the two lines. Hence a unit time O7” as measured along OT’ appears greater with reference to OT than the unit OT itself. This is another statement of Einstein’s theorem that unit time, measured in a moving system, is longer than unit time measured in a stationary system. All of these special thorems follow directly from the general trans- formation equations (7). We have x = 2, cosh Ψ — ay sinh y, vy = —a,sinhy + a2 cosh y. Now substituting u/ V1 — wv, cosh y =1/ V1 — w, u = tanh y, sinh Ψ 1 σι ἘΞΞ SS (αι = U4), 1 4 = — = (a4 — Uni); Or, replacing a, by ἐ and 2: by x, we have the fundamental transfor- mation equations of Einstein for the change from stationary to moving coordinates. 20. Let us next consider instead of a (6)-line any (6)-curve. This will represent the space-time locus of a particle undergoing accelerated rectilinear motion. As the distinction between curved and straight ee WILSON AND LEWIS.— RELATIVITY. 419 lines is independent of any reference to axes, it follows that accel- erated motion must remain accelerated motion regardless of the axes chosen. Moreover, the curvature (§ 17) of a curve is also independent of any choice of axes. Hence, although it is impossible, as we have seen, to define absolute velocity (that is, all velocity is relative to some assumed set of axes), we may define absolute acceleration if we are willing to define it as the curvature or as any function of the curvature alone. If, however, we wish to use the ordinary measure of acceleration, we must consider the projection of the curvature upon a chosen z-axis, namely, 1 dw dw = —— - =— -- y2)2 ΓΝ ee Ok ape es ee Cry It is evident that curvature of constant magnitude does not mean uniform acceleration. Indeed if the numerical value of the curvature is constant the point in the vf-plane must move upon a pseudo-circle. Since the tangent to this curve approaches, but never reaches, the asymptotic fixed direction, it is clear that the velocity of the particle approaches as its limit the velocity of light. For such a motion, the relation between x and ἐ is easily seen to be (1 — v*) Be ΤΥ (ee) a pag) ξων; where /# is the radius of curvature, and ¢, 65 are constants of inte- gration depending on the choice of origin for x and ἡ. The interval of are along any (6)-curve is that which was called by Minkowski the Eigenzeit. This quantity is of course invariant in any change of axes. Thus Mechanics of a Material Particle and of Radiant Energy. 21. Hitherto we have not assigned to our moving particles any distinguishing characteristics. Let us now consider what follows if we attribute to each particle a mass. It is true, as we shall later see, that the phenomena which must be discussed in connection with the dynamics of a material particle, even in the case where that particle moves only in a straight line, cannot be adequately represented in our two dimensional diagram. Nevertheless those results which can 420 PROCEEDINGS OF THE AMERICAN ACADEMY. be discussed are so much more readily visualized in this simple case that we shall consider a few important theorems before entering upon the treatment of three and four dimensional manifolds. The meaning of the mass of a particle, when that mass is determined by a person at rest relative to the particle, will be taken as understood. We shall call that value of the mass mp. Let us consider a (6)-curve which represents the locus in time and space of this material particle, and at any point of the locus a tangent of unit interval (or unit tan- gent) w. By multiplying w by the scalar mo, we make a new vector which we shall call the extended momentum. Τῇ now we choose any pair of axes x and ft, the slope of the locus with respect to these axes, that is, the velocity of the particle, we have called v. The momentum vector may then be written, by (5), Mov Mw = ae Κι + 3 ky. (10) If the t-axis were chosen parallel to the tangent w, the coefficient of k,, that is, the component of the extended momentum mow along the time axis, would be simply mo, the stationary mass. If, as we have assumed, the particle is regarded as moving with the velocity v, we shall take the component of mow along the t-axis as the mass m. In other words, the mass of a body appears to increase with its velocity in the familiar ratio Mo m Fae (11) The component along the a-axis is then mv, the momentum. We may therefore write the vector of extended momentum as mw = mok, + mky. (42) 22. From our equation for the curvature we may write 1mow = d 1 d Ps dmow _ dmv ay a ae ΕΝ (Ξ ΠΝ a ae): (13) ds ΝῚ Sey? The vector moe we shall call the-extended force. Since our ordinary definition of force is time-rate of change of momentum, it is evident that the z-component of the extended force multiplied by V1 — v? is ordinary force. That is, dmv f= V1— ve me = Tie (14) WILSON AND LEWIS.— RELATIVITY. 421 By comparison with equation (9), or by substituting for m from (11) and differentiating, we obtain the results?! bm ea a ae dm moo dv dk — 5 ΞΞ 70 => ---- 10 ἀν (i— 2)! dt ‘ae: a where dE//dt represents the rate at which energy is acquired by the particle when acted upon by the force f. Since dE /dt and dm/dt are equal, we may, except possibly for a constant of integration, write E=m. This is a special statement which falls under the more general law, that the mass of a body, in the units which we employ, is equal to the energy of the body. We may therefore use the terms mass and energy interchangeably. The type of motion which, from the viewpoint of the principle of relativity, corresponds most closely to motion under uniform accelera- tion in Newtonian mechanics, is motion under a constant force f. The equation of motion may readily be integrated. . adm d v a asin Se MRC τς v K dx Kt tA); Fpl πὴ -- , v(1 — ἡ) {1 αἱ Ving? EKA Ey 2 2 and («2+ τὴ) — (t—t)? = = The representative point in the at-plane therefore describes a pseudo- circle of which the curvature is the constant force acting on the particle divided by m. The mass of the particle at any time is i EE 55 ( R) Sas a 7 aie t — x + στο which shows that the increase in mass is equal to the product of the force by the distance traversed, as it should be from the principle of energy above stated. 23. Let us consider the problem of the impact of two particles A and B of which the vectors of extended momentum (mW) are respec- 21 See later discussion ($36) of the so-called longitudinal mass. 422 PROCEEDINGS OF THE AMERICAN ACADEMY. tively a and Ὁ before collision, and a’ and b’ after collision. Several important laws are subsumed under a law which we may call the law of conservation of extended momentum, namely, atb=a'+Dd’. (17) Assume any set of space-time axes, and write a= ak, + ask, b = bik, + buku, a = aki+ ak, b’ = δι ky + δι. Then the law states that (a) + δι) Κι + (ας + δὼ) Κα = (ay! + by’) Κι + (αὐ + by’) ky, or a + by) = ay + by, (18) 4 -ἰ- by aS Qs + Da’. (19) Now (by ὃ 21) as and ὃς are the masses of the two particles before collision, a,’, b,’ the masses after collision, and equation (19) expresses the law of conservation of mass or energy. The components m, bi, ay’, δι΄, are the respective momenta (in the ordinary sense), and equa- tion (18) is the law of conservation of momentum. To assume that the impact is elastic is equivalent to assuming that the value of mp for each particle is unchanged by the collision; and since each value of mp is the magnitude of the corresponding vector of extended momentum, the assumption may be expressed in the equations b= Db’. The condition that the extended momentum / Ἢ ΞΞ ἢ" * λ », 1s unchanged gives Ν zo holly (@t b)s(at b) = (a + b)-(a’ + bY, Ν ΄ we or a:b = a+b’ “ὦν. by the above relations. Hence it follows 7 ‘ : = ios Dx (Figure 17) that . Ν ΄ ἣν Ry cosh @ = cosh ¢’, or Ὁ Ξ as is evident from the rules of projection previously deduced. It is thus seen that the relative velocity is the same before and after collision, and thereby a rule which has been found very useful in the discussion of simple Figure 17. Le WILSON AND LEWIS.— RELATIVITY. 423 problems in Newtonian mechanics proves equally applicable in the new mechanics. If the impact, instead of being perfectly elastic, were such that the particles remained together after the collision, the two vectors @ and b would merely be merged into a single vector ἃ - Ὁ. The sum of the mo’s would not in this case remain constant, but would be increased by the heat (or mass) produced by the impact and obtained from the “kinetic energy” of the relative motion. This is all equivalent to the simple geometrical theorem that the (5)-diagonal of a parallelo- gram whose sides are (6)-lines is greater than the sum of the two sides. 24. The concepts of momentum and energy (mass) are ordinarily extended from the primitive mechanical phenomena to those involving so-called radiant energy. We shall see that the ascription of mass and momentum to light or other radiation is in consonance with the geometrical representation which we have adopted. Let us consider a ray of light emitted in a single line for a definite interval of time. Such a ray alone can be considered in our two di- mensional system. If the interval of time is very short, so that the front and the rear of the ray are very near together, we may regard the ray as a particle of light. The motion of such a light-particle can only be represented in our geometry by a singular vector, and to any observer its velocity is unity. Although the interval of any singular vector is zero as compared with the interval of any (y)- or (5)-vector, intervals along a given singular vector are, as we have pointed out, comparable with one another." Supposing now that a given light-particle is represented by a definite singular vector, let us see whether such a vector can be regarded as an extended momentum. If so, its projection on any chosen space- axis must represent momentum, and its projection on the correspond- ing time-axis mass or energy. These two projections must, moreover, be of equal magnitude in this case, since the velocity of light is unity. It is immediately obvious that this latter condition is fulfilled, since the vector is singular (δ 11). If ἃ is the vector, then in terms of two sets of axes a= mk, + mk, = m ky’ + m ky’. If then a represents extended momentum, m must represent the mass of the light to an observer stationary with respect to the first system of axes, and m’ the mass as it appears to an observer stationary with respect to the other system. 424 PROCEEDINGS OF THE AMERICAN ACADEMY. If ¢ is the angle from Κι to k,’ or from k, to k,’, we have from (7) m' = mcosh ¢ — msinh¢ = m cpa. (20) where v = tanh ¢ is the relative velocity of the two sets of axes. But this is in fact the very relation between the energy of a given particle of light as measured by two different observers whose relative velocity is v. It is therefore, as far as the energy relations are con- cerned, proper to consider a as a vector of extended momentum. The final proof of the desirability of considering the vector a as extended momentum comes when we consider the interaction of a light-particle with a particle of the ordinary sort. We shall see that the law of the constancy of extended momen- A tum is true, and is only true, when we include Ξ / the momentum of radiant energy as well as Ἂς ay. that of so-called material particles. yy Let the vector a (Figure 18) be the vector 3 due to a light-particle, and Ὁ that due to ἃ a/\\s, material particle which has the power of absorb- a ing light. Then if our law of extended mo- 7 -ς mentum applies to ἃ and Ὁ, there will be a rs single vector after impact equal to a+ Ὁ which will represent the extended momentum of the material particle after it has absorbed the light. Let us choose any set of axes. Then FIGURE 18. a= aki+ ak, b = δι Κὶ + by ky, where ας = a, 15 the mass of the light-particle, and b, is the mass of the material particle before impact, while a and δι = b, v are the respec- tive momenta. The momentum after impact is ay -- by = ας + by v. Hence the change in momentum of the material particle is equal in our units to the energy of the light absorbed, which gives at once the well known formula of Maxwell and Boltzmann for the pressure of light. While it is evident, therefore, that such a vector a satisfies fully all the conditions of an extended momentum, it must as a singular vector have properties quite distinct from those of a momentum vector which can be written in the form of mow. Since a singular vector — ἥδ WILSON AND LEWIS.— RELATIVITY. 425 has zero magnitude we can ascribe to the light no finite value of mo or w. In this case, as in the case of inelastic impact between material particles, the total values of mo does not remain constant, but is larger after impact. In all cases we obtain the same results from the law of the constancy of extended momen- tum as those obtained by the appli- cation of the ordinary laws for the conservation of energy, mass, and mo- mentum, whatever axes be arbitrarily chosen. Another simple illustration of these laws is furnished (Figure 19) in the case where the material particle does not absorb the light, but acts as a perfect reflector, which corresponds closely to elastic impact between particles. Here a’ and b’ are the vectors of the light-particle and the Figure 19: material particle after impact; and these vectors are readily shown to be determined either by the condi- tion that the magnitude of b is equal to the magnitude of b’, that is that the value of mp for the material particle undergoes no change, or from the condition that the angle between Ὁ and ἃ - Ὁ is the same as the angle between b’ and a’+ b’. This latter condition may in fact be regarded as necessary ἃ priori, since it is the only construction which can be, in the nature of the case, uniquely determined. Let us now consider light traveling back and forth in a single line between two mirrors whose positions are fixed relative to one another. If the mirrors are very close to one another, \ we may as before consider the whole system as concentrated at a point. This gives us a new kind of particle, an infinitesimal one-dimensional Hohlrawm. Since how- ever the energy contained within the par- ticle is in part moving with the velocity of light in one direction and in part with the velocity of light in the other direction, Ficure 20. we may draw two singular vectors (Figure 20) to represent the extended momenta in the two directions. Now these vectors added together give a (6)-vector which will behave in every way like the extended momentum mow of 426 PROCEEDINGS OF THE AMERICAN ACADEMY. a material particle, and mp represents the mass or energy of the Hohi- raum as it appears to any observer at rest with respect to it. To such an observer the amount of energy traveling in one direction appears equal to that traveling in the opposite direction, and the resultant momentum is zero. To any observer moving with the velocity ὃ relative to the particle, the momentum is the difference between the momenta which he observes in the two directions, and the mass of the particle is increased in the ratio 1/¥1 —v?. These results are all evident geometrically, and follow analytically from (20). THe Non-EvucitipEAN GEOMETRY IN THREE DIMENSIONS. Geometry, Outer and Inner Products. 25. We shall now consider a three-dimensional space in which the meaning of points, lines, planes, parallelism, and parallel-transforma- tion or translation are precisely as in ordinary Euclidean geometry. In such a space, in addition to directed segments of lines or one-di- mensional vectors, we have directed portions of planes or two-dimen- sional vectors. Any two portions of the same or parallel planes having the same area and the same sign will be considered identical two-dimensional vectors, briefly designated as 2-vectors. The ordi- nary one-dimensional vectors may be called 1-vectors for definiteness. It is evident that the outer product axb of two 1-vectors in space is no longer a pseudo-scalar but a 2-vector lying in the plane determined by the two vectors and having a magnitude equal to the area of their parallelogram. The addition of two 2-vectors may be accomplished geometrically in the following way. Take a definite segment of the line of inter- section of the planes of the 2-vectors. In each plane construct on this segment as one side parallelograms equal respectively to the given 2-vectors. Complete the parallelepiped of which these two parallelo- grams are adjacent faces. The diagonal parallelogram of the paral- lelepiped, passing through the chosen segment, is the vector sum; the diagonal parallelogram parallel to the chosen segment is the vector difference. Let us consider the outer product of a l-vector and a 2-vector,?? axA. Let A be represented as a parallelogram, and a as a vector through one vertex; the product axA is the parallelepiped thus 22 In general 2-vectors will be designated by Clarendon capitals (except in the case of the unit coordinate 2-vectors). WILSON AND LEWIS.— RELATIVITY. 427 determined. This outer product axA, being three-dimensional in a three-dimensional space, is a pseudo-scalar; and different pseudo- scalars are distinguished only by magnitude and sign. If in axA we regard A as itself an outer product bxc, the parallel- epiped is written as ax(bx¢). This same parallelepiped can be re- garded, with the possible exception of sign, as (axb)xc. We shall in fact consider the sign as the same, and write ax(bxc) = (axb)xe = axbxce, so that the associative law holds for the three factors a, Ὁ, 6. As bxe = — exb, we shall write ax(bxc) = — ax(exb), in order that we may keep the law of association for the scalar factor. By succes- sive steps we may write axbxc = — bxaxc = bxexa; and hence the outer product of a 1-vector and a 2-vector is not anti- commutative but commutative, namely, axA = Axa. All of these statements are valid in any geometry of the group charac- terized by the parallel transformation. 26. In the three-dimensional non-Euclidean space, rotation about a fixed point is characterized by the existence of a fixed cone through the point, corresponding to the fixed lines in our plane geometry. An element of this cone always remains an element; points within the cone remain within, and points without remain outside. Besides the lines which are elements of this cone, or parallel to them, there are two classes, namely, (5)-lines through the vertex and lying within the cone, and all lines parallel to them, : (y)-lines through the vertex and lying outside the cone, and all lines parallel to them. In like manner planes may be separated into classes. Besides the planes of singular properties which are tangent to the cone along an element, or planes parallel to these, there are (5)-planes through the vertex cutting the cone in two elements, and all planes parallel thereto, (y)-planes through the vertex and not otherwise cutting the cone, and all parallel planes. The former set, the (6)-planes, contain (6)- 428 PROCEEDINGS OF THE AMERICAN ACADEMY. lines and also (y)-lines; the latter set, the (y)-planes, contain only (y)-lines. Any plane passed through a given (6)-line cuts the cone in two ele- ments and is therefore a (6)-plane. The geometry of such a plane is the non-Euclidean plane geometry above described, and the elements of the cone are the fixed directions. The-perpendicular in this plane to the given (6)-line is a (y)-line. The locus of the lines perpendicular to the given (6)-line in all the planes through the line is a (y)-plane. This (y)-plane will be called perpendicular to the (6)-line. Such a plane possesses no elements of the cone, that is, no lines which are fixed in rotation; hence the geometry of a (y)-plane is ordinary Euclidean geometry. In the plane any line may be rotated into any other line, and the locus of the extremity of a given segment issuing from the center of rotation is a closed curve which is the circle in that plane. Moreover, the idea of angle, and of perpendicularity between lines in the (y)-plane, being the same as in ordinary Euclidean geome- try, need not be further defined. A plane passed through a (y)-line may cut the cone in two elements and be a (6)-plane, or may fail to cut the cone and will then be a (y)- plane.?3 The perpendiculars to a (y)-line will therefore be in part (5)-lines and in part (y)-lines, and the plane perpendicular to a (7)- line will therefore be a (6)-plane. Thus a plane perpendicular to a (5)-line is a (y)-plane, and a plane perpendicular to a (y)-line is a (6)-plane. In any three dimensional rotation one line, the axis of rotation, remains fixed, and points in any plane perpendicular to the axis remain in that plane. If the axis is a (6)-line, the rotation is Euclidean; if a (y)-line, non-Euclidean. When all possible rotations, Euclidean and non-Euclidean, about axes through a given point are considered, the locus of the termini of a (y)-vector of fixed interval, and a (6)-vector of equal interval, issuing from the common center of the rotations, is a surface which from a completely Euclidean point of view appears to be the two conjugate hyperboloids of revolution asymptotic to the fixed cone, but which from our non-Euclidean viewpoint is really analogous to the sphere. The (6)-lines cuts the two-parted hyperboloid; the (y)- lines, the one-parted. 27. If we construct at a point three mutually perpendicular axes, two will be (v)-lines, and one a (6)-line. The unit vectors along these 23 Planes tangent to the cone will be discussed later. WILSON AND LEWIS.— RELATIVITY. 429 axes will be denoted respectively by Κι, Ko, and ky. The outer products Κιχκο, Kk, k.xk, will be denoted for brevity by Ky, Κι4, Ko4. In terms of these arbitrarily chosen axes a l-vector may be repre- sented as a= ak, + ak» + a4Ky. Similarly a 2-vector may be represented by the sum of its projections on the coordinate planes as A= Apky + Aki + AosKos. If we had chosen ky; in place of Kj. as one of our unit coordinate 2- vectors, we should have written A= Anko + Avkiy + Asko. Since A 12 Κιο ΞΞ Ay ko; and Κιο Ξε :- ko, we have A i — Ay. If we denote by Kjos the outer product k,xk»xk,, then Kin = — Kye = Kye = — Ky = ky, = — Koi, by the rules of outer products given above. In three-dimensional space these products are unit pseudo-scalars. In terms of their components we may now expand the two types of outer product which occur in three-dimensional space. In this expansion we employ the distributive law and the law of association for scalar factors. Then axb = (a,b aad αὐ.) kp + (ay, = ash) Ἐπ + (dob, — ας.) ko, axA = (a)Ao + Ag + ἀμ} ki. At this point we may discuss the general characteristics of inner and outer products of vectors of various geometric dimensionalities in an n-dimensional space. In such a space we have vectors of 0, 1, 2,..., n-1, n-dimensions, designated as OQ-vectors (or scalars), 1-vectors, 2-vectors, ..., (n—1)-vectors, and n-vectors (or pseudo-scalars). The outer product of a p-vector and a q-vector is a (p + q)-vector; the product vanishes if by translation the p-vector and g-vector can be made to lie in space of less than p + q dimensions. The inner product of a p-vector and a q-vector, where p = 4, will always be defined as a (p-q)-vector. Thus whereas the inner product of a l-vector by a 1-vector is a scalar, the inner product of a 1-vector and a 2-vector is a l-vector. Both the inner and outer products will obey the distributive law, and the associative law as far as regards multiplication by a scalar 430 PROCEEDINGS OF THE AMERICAN ACADEMY. factor. Furthermore the outer product will always obey the associa~- tive law, and the inner product the commutative law. 28. The inner product of any 1-vector into itself may, by an im- mediate generalization of the definition in plane geometry (§ 14), be defined as equal to the square of its interval, taken positively for (y)-vectors, negatively for (6)-vectors. The inner product of two 1-vectors is equal to the inner product of either one and the projection of the other upon it. The rules for the unit coordinate vectors are therefore Κι -k, = ky: ko = ἽΝ ky-k, — —— ik. k, «ky = Κι ky, = ky +k, he The product of two vectors @= mk, + mk, + asky, Ὁ = bk, + doko + bdiky, is arb = ab; + ab. — aybs. The inner product aA of a 1-vector and a 2-vector will be a 1-vector in the plane A and perpendicular to a (that is, perpendicular to the projection of a on A); its magnitude will be equal to the product of the magnitude of A and the magnitude of the projection of a on A; its sign is best determined analytically. If a and b are perpendicular l-vectors we may make the convention (axb)bi—= a(b-b), οἴ » (axb)-3\=> μμίλ:5.. (21) Thence follow the rules for the unit vectors, ky Ky =a Kp, κι Κα — ἔστ ky, Κι -Kos = 0, 0 Kyo — Κι, το" ἴα a 0, koe Kog ΞΘ - ky, Κι Κρ ἐπ 0, kyeky4 = — ky, Κι Koy = Ξ 5 kp. 24 Hence a-A= (ay. 1 τὶ 4A 14) kj, + (— a Aj. — a4Ao4) ky ++ (— aA, — az Aos) Ky. 24 We may show that these rules do give an inner product which in all cases agrees with the geometric definition above stated. The condition that a-A lies in the plane A is that the outer product of it and A shall vanish, that is, (a-A)xA = 0; the condition that it is perpen- dicular to @ is that the inner product of it and ἃ shall vanish, that is, (a-A)-a = 0. These two products are (a*A)xA = [(a2 Aw — ag Ay) 424 + (αι 415 + a4 4.4) Ata — (μά + a2Ao4) 4.15] King = 0, (8. Α).ἃ = αι (d2Ajq2 — a4Ay4) — a2 (Aq + a4Ao4) + ag (Arg + α5.4.4) = 0, as required. It is also necessary to show that the component of a perpendi- cular to A contributes nothing to the product aA, so that the component in WILSON AND LEWIS.— RELATIVITY. 431 The inner product of two 2-vectors is a scalar which is equal to the inner product of either vector by the projection of the other upon it. The inner product of two perpendicular 2-vectors is zero. The inner product of a 2-vector by itself is numerically equal to the square of its magnitude, and is positive in sign if the vector is of class (y), negative if of class (6). Hence we have as rules of inner multiplication for 2-vectors KK» = 1, KyeKiy = Κα = — 1, Κι Κα = Kyoko = Κι το, = 0, A-A= A)?” = Ay a Ao;’, A-B= “4.50 otk AysBiy —s 4..}.,. 29. Every 1-vector a, or 2-vector A in a three-dimensional space uniquely determines, except for sign, another vector (respectively a 2-vector or 1-vector) perpendicular to it and of equal magnitude. This vector will be called the complement of the given vector, and designated as a* or A* respectively. To specify the sign, the comple- ment may be defined as the inner product of the vector a or A and the unit 3-vector or pseudo-scalar Kj.4, where the laws of this inner product are τὸ τῷ Κι Κορ. = Kos, koeKyy = — Ky, kyekioy = — Ky, Κρ "Κι, = ky, Kysy+Kios = Kp, Koq*Kiog = — ky. Thus a* = (ak) + ake + agky) + Ky = — ak, — aokyy + ako, A* = (Apky + ArKiy + Aoskos) + Kies = — Aodks + Ayko + Akg. These complements satisfy the condition of perpendicularity pre- viously derived (footnote 24), and the inner products at-a* = α(" — a” — a;’, aca = ar+ a? — aZ, A*-A* = Ao? + Ay? — Ap’, A-A= Aj? — Ai? — Ao? the plane is alone of importance. We shall do this by deriving the expression for a vector perpendicular to the plane A. Let Cc=aki+ake+aky, n= 7 ki + mk. + τὰ ky be respectively any vector in the plane A and a vector perpendicular to the plane. Then the products oa (cyAo4 = c2Ay4 + C4A yo) Kj04 ΕΞ Cen = οι + ΟἿ — ON = 0 vanish. Hence it follows that the condition of perpendicularity for the vectors n and A is 71. 2. Ns = Ags! -- Aj: -- Aj, and that n must be some multiple of Agsk; — Ayko — Awky. By the rules, the inner product of this vector and A vanishes. 432 PROCEEDINGS OF THE AMERICAN ACADEMY. show that the magnitudes are equal. The reversal of sign is to be expected from the fact that the complement of a vector (whether 1- or 2-- of class (vy) is a (6)-vector (whether 2— or 1—), and vice versa. The use of the term complement in connection with scalars and pseudo-scalars is sometimes convenient. Since, by the rule of inner multiplication, we have Kj4*Ky2.4 = —1, the complement of any pseudo-scalar is a scalar of the same magnitude and of opposite sign. We may define the complement of a scalar a as the product of the scalar and the unit pseudo-scalar; thus αὖ = akjy,. All the special rules for the inner products of unit vectors (and pseudo-scalars) are comprised in the following general rule, which will also be applied in space of four dimensions: If either of two unit vectors has a subscript which the other lacks, the inner product is zero; in all other cases the inner product may be found by so trans- posing the subscripts that all the common subscripts occur in each factor at the end, and in the same order, by then canceling the com- mon subscripts, and by taking as the product the unit vector which has the remaining subscripts (in the order in which they stand), pro- vided that if the subscript 4 has been canceled, the sign is changed.?° Thus Κι Ks, = 0, Kjos? Kis = Kuo°Ky = ky, Κι». Κι == ky +k; a aa kp, Kjos Ky = a Κι», Kisae Kua ayes K314* Κα = Κ:. 80. Hitherto we have given little attention to the singular vectors of our geometry, namely, the lines which are elements of a singular cone and the planes which are tangent to a singular cone. We have seen (ὃ 14) that the inner product of a singular 1-vector by itself is zero, and have expressed that fact by stating that a singular line is perpendicular to itself. Analytically expressed, the condition that a l-vector a shall be singular is that aca = αι," Ἢ αο -- ας = 0. 25 Instead of regarding the common subscripts as canceled, it is possible to regard their corresponding unit l-vectors as multiplied by inner multiplica- tion,— and in this case the change of sign takes care of itself. Thus Kpgr* Ky ἐπ kp (ky° ky) (Κ, Κι). Indeed if a, b, ¢ are mutually perpendicular 1-vectors, then all the rules given above may be expressed in the equations (axb)+(axb) = (aa) (b-b), (axbxc) - (axbxc) = (8.8) (b-b) (66), (axb)*b = a(beb), (axbxc)*c = axb (c°c), (axbxc) + (bxc) = a (bb) (66). a= ak, + mk, + Va, + a’ky. The complement of a singular vector is A= a* = 8." Kjos τε ay Ko, — ky = Va; + a’? Kp. This 2-vector A must be itself a singular plane vector; for we have seen that the complement of any (6)-plane is a (y)-line and of any (y)-plane a (6)-line, and vice versa. The inner product of A by itself is obviously zero,?® for, A-A = — αι — a?’ + (a?+ αοὖ) = 0. Conversely if we consider any 2-vector A= ΑΚ ΞΕ Ayky a= AosKo4, such that A-A= 4." ἘΠῚ Ay? ar 4. = 0, its complement is a singular line, and it is itself a simgular 2-vector. The standard form may be taken as A= + VA24+ AoPKy + Auku + Adan. The outer product of a singular vector by its complement, whether a l-vector or a 2-vector, vanishes, as may be seen by multiplying out. Thus the singular vector and its complement lie in the same plane, that is, an element of the cone and the tangent plane through that element are mutually complementary. When we have to consider the inner product of any singular vector with any other vector, singular or not, the geometrical method de- pendent on projection often fails to be applicable; for it is impossible to project a vector upon a singular vector. We may in such cases employ the analytical method, which is universally applicable, or replace the inner product with an outer product by a method intro- duced in a following section (§ 32). We have seen that an element of the cone is complementary to the tangent plane to the cone through that element, that is, the element is perpendicular to the plane. Hence the element is perpendicular to every line in the plane (including itself). 26 A singular vector, or vector of zero magnitude, has, like any other vector, areal geometrical existence and is not to be confused with a zero vector, that 15, ἃ non-existent vector. 434 PROCEEDINGS OF THE AMERICAN ACADEMY. 31. We have seen that rotation in a ()-plane about the perpendicu- lar (6)-line is Euclidean, whereas rotation in a (6)-plane about the normal (y)-line is non-Euclidean. In this latter case not only do the (6)-planes normal to the axis remain fixed during the rotation, but the two singular planes through the axis and tangent to the cone also are fixed; for the axis remains fixed and the lines in which the planes are tangent to the cone are respectively the two fixed lines in the (6)- plane. As every point in the axis of rotation is fixed, the whole set of lines parallel to the elements of tangency is fixed. The effect in the two singular planes of a rotation is therefore to leave one line, the axis, fixed point for point, to leave a set of lines fixed, and to move the points on these lines either toward the axis or away from it by an amount which is proportional to the interval from the point to the axis. Since a rotation in a (6)-plane multiplies all intervals along one of the fixed directions in a certain ratio, and divides all intervals along the other fixed direction in the same ratio, the effect upon areas in the two singular planes is to multiply all areas in one of the planes in that same ratio, and to divide areas in the other in that ratio. This however is not inconsistent with our condition that areas should remain invariant; for it is evident that, when compared with areas in other planes, areas in singular planes are all of zero magnitude. This is also shown by the fact that the inner product of any singular vector by itself vanishes. That areas in a singular plane have a zero magnitude does not prevent our comparing two areas in the same singular plane or in parallel singular planes, just as the fact that intervals along singular lines had zero magnitude did not prevent our comparing intervals along any such line. A limiting case of rotation occurs when the axis of rotation is itself an element of the cone, that is, a singular line. Here the infinity of fixed planes perpendicular to the axis, and the two singular planes through it, have all coalesced into the one singular plane through this line and tangent to the cone. In this plane the rotation consists in a sort of shear. Every point moves along a straight line parallel to the axis. In this case areas are rotated into areas which are from every point of view equal. For if a parallelogram whose base is on the axis, which is fixed point for point, is subjected to this rotation, its base remains fixed and the parallelogram remains enclosed between the same two parallel lines (Theorem IX). The geometry in this plane, depending upon translation and upon such a rotation as has just been described, is interesting as affording a WILSON AND LEWIS.— RELATIVITY. 435 third geometry intermediate between the Euclidean and the non- Euclidean which we have discussed. In Euclidean plane geometry there is no line fixed in rotation, in our non-Euclidean plane geometry there are two fixed directions, in this new case there is just one. If we were to investigate this geometry, we should find one set of (parallel) singular lines and one set of non-singular lines. Every non-singular line may be rotated into any other. Angles about any point range from — οὐ τὸ - © on each side of the singular line through that point. The interval along any line intercepted between two singular lines is equal to the interval along any other line thus intercepted. Every non-singular line is perpendicular to the singular lines, as the singular line is complementary to the singular plane through it. Some Algebraic Rules. 32. We shall develop here a number of important relations be- tween outer products, inner products, and complements which will be of frequent use later. Many of these relations hold in any number of dimensions. We shall consider primarily a non-Euclidean space in which one of a set of mutually perpendicular lines is a (6)-line, the rest being (y)-lines. But except for occasional differences of sign, the results are valid in a Euclidean space. In a space of n dimensions, the complement of a vector of dimension- ality p is itself of dimensionality n — p. If a is a scalar and aisa vector of any dimensionality, then from the associative law for scalar factors, we have fan — eal Kg = ΜΙ; πα ΞΡ πὶ Ξ ao oa — ae... (24) Let a, 3, . . . be vectors of the respective dimensionalities p, icc! Then Bxa = (— 1)?%axf. (23) Owing to the availability of the distributive laws it is sufficient to prove such relations as this for the simpler case where the constituent vectors a, @ are unit vectors k,..., kj... of dimensionality p, 4. In the permutation of a and β, there are involved pq simple transposi- tions of subscripts; for each subscript in Κι... has to be carried past all the subscripts of k,... Hence there are pg changes of sign. Hence the outer product is commutative if either of the factors is even, but is anti-commutative if both factors are odd in dimensiona- lity. 436 PROCEEDINGS OF THE AMERICAN ACADEMY. We may next show that (axf)* = α.β". (24) Suppose again that a, β are unit vectors k,..., k;.... We have to show (ko. Ky...) Kye. = Ryn. (pe. Kr.) where kj... denotes the unit pseudo-sealar. Without changing this equation, it is possible on both sides to arrange at the end, the sub- scripts of the pseudo-scalar Κι... in the same order as in the factors k,..., Κι... Thus we have to show that (ΧΕ ΘΚ, (Karger ag anee: But now the products on the right are found by canceling succes- sively the common subscripts h... and g...; whereas the product on the left is found by canceling simultaneously the subscripts of k,...,..- The identity is therefore proved. As a corollary of the two preceding results we may write the formula (exo)? τὺ τ ξυ Ξε eB (25) All these rules are true for any space, Euclidean or non-Euclidean. The complement of the complement of a vector a is the vector itself, except for sign. [1 α is of dimensionality p in a space of ἢ dimensions, the exact relation is Ci) bee En (26) The complement of the complement of a vector will therefore be the negative of the vector except when p (n — p) is odd, that is, when the dimensionalities of the vector and of the space are respectively odd and even.?” For the proof, the consideration may be restricted to the case where a is a unit vector k,.... Then (a7) ake ik; ekg. =, (Kp. Kage ikea: ἘΞ (-- 1) PG Ὁ) ΕΠ: Here again the subscripts in the pseudo-sealar k;... have been re- arranged so as to bring g... to the end. Then as gq... denotes p subscripts and j ... denotes n — p, the permutation involves p (n— p) 27 In Euclidean space (a*)* = (— 1)?(—P). Some writers who have identi- fied vectors with their complements have perhaps overlooked this relation which would, upon their assumption, make a vector sometimes identical with its own negative. ων» WILSON AND LEWIS.— RELATIVITY. 437 changes of sign. In the final form thus found the subscripts g.. . and 7... have successively to be canceled. But one of these is necessarily the subscript 4 (corresponding to the (6)-vector), which requires a change of sign. Hence (ae) ee! ome ΞΞ -- (— 1) PP)... ’ and the desired result is proved. Consider the product a*+8*. We have by (24) either arbi (ax8) or B*sa* = (0*xa)*. (27) Now, although αἴθ and §*+a* are equal, the two expansions obtained are usually different. In fact, as the total dimensionality of an outer product cannot exceed n, the first formula holds only when p =p and the second only when g — p. Let us assume q=—p. Then α".β᾽ = peak = (β' κα)" = (— 1)ρίατῷ (axg)* ΕΞ (= 1) P(n—9) ἘΠ Ἐ (= 1)¢_ 9a) af. (28) As a corollary a*-a* = — area. (29) The complement of an inner product may likewise be proved to be (a -G)* — (= 1) Pie») oxo, (30) where it is assumed that the product αὐ has been so arranged that the second factor is of dimensionality q greater than the dimension- ality p of the first. We have furthermore a*xa =(aea)*; (31) and also if 9 is a pseudo-scalar (aeB)* = (— 1)2%?) B¥a = "Bea". (32) It is important to observe that by means of these rules it is possible to replace any outer product by an inner product, and vice versa. 33. Weare now able to obtain rules for the expansion of the vari- ous products in which three vectors occur. The simplest type, and one which needs no further comment, is (ax?)xy = αχ(βχγ), (33) which follows from the associative law. 438 PROCEEDINGS OF THE AMERICAN ACADEMY. Consider next the product a+(bxc) of three 1-vectors. Here a-(bxc) = (a-c) Ὁ — (a-b)e. (34) Perhaps the simplest proof is obtained from the relation 78 = (06) ὁ τ o- (bxe) C°C c°c b which states that a vector is equal to the sum of its components. By clearing and transposing, and by permuting the letters, we have c-(bxc) = (c-c) Ὁ — (e-b)c, b:(bxc) = (b-c) b — (b-b)c. If now ἃ is any vector perpendicular to Ὁ and 6, we have identically d-(bxc) = (d-c) Ὁ — (d-b)c = 0. If these equations be multiplied by 2, y, z and added, we have (xe + yb + 2d)-(bxe) = [(τὸ + yb + 2d)-c]b — [ἃ + yb + 2d)-ble, and any vector ἃ may be represented in the form ae + yb + 2d. From the rules (33), (84) combined with the rules (22)—(32) we may obtain a number of other reduction formulas by simply taking comple- ments of both sides of the equation. Thus (axb)-C = a-(b-C) = — b-(a-C). (35) 28 With the aid of inner and outer products we may write down expressions for the components of a 1-vector a along and perpendicular to another 1-vector b or a 2-vector A. The components of a along Ὁ and perpendicular to Ὁ are (a:b) b ἢ (axb)-b- b-b beb The components of a along A and perpendicular to A are . (a-A)-A ΓΝ (axA) +A, A-A A-A The component of the plane A on the plane B is (A+B) B B-B and a vector in the line of intersection of the two planes is A*-B or A-B*. WILSON AND LEWIS.— RELATIVITY. 439 For by (33) and (24), [(axb)xe]* = [ax(bxc)]*, (axb)-c* = a-(bxc)* = a-(b-c*). But since ¢ is any l-vector, its complement C is any 2-vector. Again, ax(b-C) = (axC)-b — (a-b)C. (36) For by (84), (22), and (30), [a+(bxe)]* = [(a-c¢) b]* — [(a-b) e]*, (— 1.18 ax(bxe)* = (— 116 (axc*)-b — (a+b) ο", ax(b-C) = (axC)-b — (a+b) C. Again, (pices οἰ: Αὐ (37) For from (35), (30), and (24), [C-(axb)]* = [(b-C)-a]*, (— 1)?8—2) Cx(axb)* — (= 1)16—D (b+C)xa*, — Cx(bxa)* = — Cx(b-A)=(b-C)xA. Again (b-C)-A = — b(C-A) + C:(bxA). (38) For from (36), (24), (32), (22), and (80), [(b-C)xa]* = — [b+(Cxa)]* + [C(b-a)]*, (b-C)-A = — b(Cxa)* + C-(b-a)*, — b(C-A) + (—1)!@—) C-(bxA). These rules (33) to (38) involve every possible combination of three vectors in three dimensional space. Since the formulas which we have used in deriving them, have the same form in Euclidean space, the rules will be true in Euclidean space. The particular use of the complement has implied a three dimensional space, and a similar use of the complenent in a four dimensional space would obtain analogous but different formulas; it should be observed, however, that the rules here obtained (with the exception of (87)) must hold in space of four dimensions, even when the three vectors in question do not lie wholly in a three dimensional space. For consider (36) as a typical case. Let Ὁ be a l-vector which does not lie in the space of a and C; we 440 PROCEEDINGS OF THE AMERICAN ACADEMY. may write Ὁ = b’ + b”, where b’ is in the space of a and C and b’” is perpendicular toa and C. Then by (36) ax(b’-C) = (axC)-b’ — (a-b’) C, and ax(b’’-C) = (axC)-b” — (a-b”)C holds identically, since each of its terms vanishes. Hence by addition (36) is seen also to hold in general. Some products involving more than three 1-vectors are of frequent occurrence. By (85) and (94) we may write immediately ih. ry ee Dee lacc aed) (axb)+(exd) = (a-c) (b-d) — (Ὁ 0) (a-d) = Ne bed (39) In a similar manner we may prove δι ἃ a-e a-f| (axbxc) + (dxexf) = Wed b-e bf), σ΄ c-e cf and (axb)+(exdxe) = (axb)-(dxe) ¢ + (axb)-(exe) ἃ + (axb)-(exd) e. These formulas are all valid in space of any dimensions. The Differentiating Operator VY. 34. In discussing the differential calculus of scalar and vector functions of position in space, the vector differentiating operator V/ is fundamental. The definition of this operator may be most simply obtained as follows. Consider a scalar function F of position in space. Let dr denote any infinitesimal vector change of position, and let dF denote the corresponding differential change in F. Then let V be defined by the equation dF = dr-V/F. Now VF is a vector. If dr is a vector in the tangent plane to the surface F = const., dF is 0, and as dr+\/F then vanishes, the vector dr and VF are perpendicular. Hence VF is a vector perpendicular to the surface F = const. Now V F may be a vector of the (6)- class or of the (y)-class, and the tangent plane is then respectively a (y)-plane or a (6)-plane.?9 29 In our non-Euclidean geometry VF’ will not be a vector in the line of the greatest change of F. If dr be written as Ὁ ds, where w is a unit vectorin the WILSON AND LEWIS.— RELATIVITY. 441 If we select three mutually perpendicular axes Κι, ky, ky, and denote by 21, 2, 24 the coordinates (intervals) along these axes, then dF = dx, oF + εχ. oF + dey oF = (dayk; + disks + dxyky)? VF. 02, ΠΕΣ O24 From this V may be determined as V= I ob ea go — κε σο: (40) Thus V appears formally as a 1-vector, and may be treated formally as such.3° direction of dr and where ds is the interval or magnitude of dr, we may write dF = dsu-vVF or u-VF = ἐπ, 8 Hence the component of V F along the direction dr is the directional derivative of F in that direction. Consider now two neighboring surfaces of constant F’. Suppose first that the (approximately parallel) tangent planes to the surfaces are of class (7), so that the perpendicular VF is a (4)-vector. Then, in our geometry, the perpendicular from a point on one surface to a point of the other is, of all lines of its class, the line of greatest interval ds (§12). The directional derivative along the normal is therefore numerically a minimum (instead of a maximum) relative to neighboring directions. In fact, the derivative along a line of fixed direction would be infinite, because along the fixed cone ds = 0. Along the (y)-lines the directional derivative varies between Ὁ and ». Suppose next that the tangent planes are of class (δ), so that the perpendicular VF is a (y)-line. Then the interval ds along the perpendicular from a point on one surface to a point on the other is neither a maximum nor a minimum, but a minimax. For it is less than along any neighboring direction (of the same class) which with the perpendicular determines a (y)-plane, but greater than along any neighboring direction (of the same class) which with the perpendicular determines a (5)-plane. 30 The above definition of VF depends on inner multiplication, and hence upon the notion of perpendicularity or rotation. It is, however, interesting to observe that we may define a differential operator y’ dependent upon the outer product, and hence upon the idea of translation alone. The definition would then read axbxcdF = drxv’/F = (adr, + bdx: + ¢dxs)xv’F, where a, b, ¢ are any three independent vectors, and where αὶ, 22, Ys are co- ordinates referred to a set of axes along a, Ὁ, 6. Then = bxe τς, : τ: oa τς ΒΗ axb Ὁ ΕΝ (41) Now V’ may be regarded as ἃ "Ov ector ope pais in Site same sense as V 18 regarded as a l-vector. To show the relation of τ΄ to Vv, when the ideas of perpendicularity are assumed, we may take a, Ὁ, ¢ as Κι, Κα, Ky and ἃς as zy Then - 6 a] a) fe) 0 \" = Καὶ -- 4 a πε ὶ | Re St τ k ΛΎΣΙΣ . ὃ 7 = " 0x2 OX2 δος ; Ons (x Ox, τῇ - OX2 : Ons Thus V’ is the sheen νον In fact if (dF)* = drxv’F and dF =dr-VF, our rule of operation (30) shows that ν΄ = v*. 442 PROCEEDINGS OF THE AMERICAN ACADEMY. If we consider a field of 1-vectors, that is, a 1-vector function f of position in space, we are naturally led to enquire what meaning, if any, should be associated with the formal combinations ΧΕ and ΧΕ obtained by operating with the l-vector V. Let f (αι, 2, v4) = Aiki + foko + fake. Of Oe , Ofe 02, τ OX» Ἢ Oxy Of. δῇ Of πὸ ἢ ὁ. δῇ, ἡὐητς- (2 - ral Eel aa vy το uae - ει: Ox. *) ks Then ἊΣ ef — Of these the first, Vf, is a scalar function of position, and the second, V ΧΕ, is a 2-vector function of position. They correspond respectively to the divergence and curl in Euclidean three dimensional space. The first, V +f, has indeed the same formasusual. And this was to be expected: for physically or geometrically the idea of divergence depends on translation alone and not on rotation, and it would also have appeared analytically evident if we had used in the definition of divergence the operator Δ ἢ instead of V. The second, V*xf, differs from the ordinary curl not only in that we have retained it as a 2-vector (instead of replacing it by the 1-vector, its complement, as is usually done in Euclidean geometry of three dimensions), but also in that it represents non-Euclidean rotation in the vector field in the same sense that the curl represents ordinary rotation. If F is a scalar function of position, then V F is a 1-vector function. We may then form αν πο υ". Of these the second, VxV/ F, vanishes identically, as may be seen by its expansions or by regarding it as an outer product in which one vector is repeated. The first, V+ V F, may be expanded as fe Oe OLE sooner and her tere er τι» δὰ 0..." Ong and VY’: V corresponds to ee operator in Euclidean seometry. If fis a 1-vector function, there are four different expressions which involve the operator V twice, namely N7 Waa: VeVi V+ Vxé, VxVxe. WILSON AND LEWIS.— RELATIVITY. 443 Of these the last is a 3-vector function, which clearly vanishes identi- cally. The first three are 1-vector functions, and are connected by the relation ViVx'¥= V(V-f)—VVeE, as may be seen by expansion or by the application of (34). Kinematics and Dynamics in a Plane. 35. The three dimensional non-Euclidean geometry which we have developed is adapted to the discussion of the kinematics and dynamics of a particle constrained to move in a plane. The two dimensions of space and the one of time constitute the three dimen- sions of our manifold. Any (y)-plane in this manifold may be called space, and extension along the complementary (6)-line may be called time. As in the simpler case, any (6)-line represents the locus in time and space of an unaccelerated particle, and any (6)-curve the locus of an accelerated particle. If we choose an¥ two perpendicular axes 21, 2 of space, and the perpendicular time axis x4, then if the locus of any particle is inclined at the non-Euclidean angle ¢ to the chosen time axis, the particle is said to be in motion with the velocity v of which the magnitude is » = tanh φ. For the locus of a particle let a { Ἢ ay PCM be the arc measured along the (6)-curve, and let r be the radius vector from any origin to a point of the curve. Then the derivative of r by s is the unit tangent w to the curve. We have dx, day W= Κι ἢ ἀξ ke = a πὶ If the velocity vis | v= μάν τ oe, ° dis om γὴν tt then since iin cosh ¢ = WF ae we write 3! 1 dx, dato ) v+k, v=) ky κι) = . 42 Ἧι Ξ (i ΓΝ ΤΣ Ξ na ἢ ( ) 31 By a transformation to a new set of axes we may derive at once the ge Hen form of Einstein’s equation for the addition of velocities. 444 PROCEEDINGS OF THE AMERICAN ACADEMY. To obtain the vector curvature of the locus we write dw ἀκ ΑΝ ἂν ν ἘΚ, dv ds ἀῶ 1—vwvdxy ( --- Yr)? ee Cc = or Att av v+tk, do iene a a {1 — ee” dt (43) If v be written as v = vu, where Ὁ is a unit vector, the resolution of ¢ into three mutually perpendicular components along u, ky, and du follows immediately: a yuu ok dv dt dt * dt 2S aes ar = oh (44) The magnitude of ¢ is dv\? ,du du]? dt "dt dt ν τ: = 7 τ a—* Go] (45) a 2 1 ΝΣ 1 Σ κῶν alert r= i) To γδ νὸς νϑλ οἰ οτννς In case the acceleration is along the line of motion, these expressions reduce to those previously found; the additional term is due to the acceleration normal to the line of motion. 36. Mass may now be introduced just as in the simpler case already discussed, and here likewise we are led to the equation The extended momentum in this case is also mow, that is, mw = mv + mk. (46) We may speak of w as the extended velocity, of ¢ as the extended acceleration, and of mp¢ as the extended force. It is to be noted that while ordinary momentum is the space component of extended momen- tum, ordinary velocity, acceleration, and force are not the space com- WILSON AND LEWIS.— RELATIVITY. 445 ponents of the corresponding extended vectors. Indeed the space component of the extended velocity is v/¥1 — vw. The ordinary force, measured as rate of change of momentum, is mo — Mov mM dmv dv dm oT alt *" at ᾿Ξ =m—+v = —; + =, (47 dt dt dt (i .Ὦ (1 —2)3 ) which is the space component of mp¢ multiplied by V1 — υ", It is evident that in our mechanics the equations dmv ee where a = dv/dt, are not equivalent, and it is the first of these which we have chosen as fundamental. This makes the mass a definite scalar property of the system. Those who have used the second of the equations have been led to the idea of a mass which is different in different directions, and indeed have introduced as the “longitudi- nal’’ and the “transverse” mass the coefficients and fi ΠΝ Mo Mo Gist... ae of the components of acceleration along the path and perpendicular to it, that is, of the longitudinal and! transverse accelerations, which are respectively ait dt’ dt The disadvantages of this latter system are obvious. An interesting case of planar motion is that under a force constant in magnitude and in direction, say f, = 0, f, = —k. The momen- tum in the z-direction is constant, that in the y-direction is equal to its initial value less kt. From these two equations the integration may be completed. Or, in place of the second, the fact that the increase in mass (that is, energy) is equal to the work done by the force, may be used to give a second equation. The trajectory of the particle is not a parabola, but a curve of the form y + a = —b cosh (ca — d), resembling a catenary. The space-time locus of uniform circular motion is a helix r = a(k, cos nt + ky sin nt) + ky. 446 PROCEEDINGS OF THE AMERICAN ACADEMY. Then mv = man(— k,sin nt + 0 cos nt) + my, lmv ᾿ ἘΝ ΞΞΞ rae = — man*(k cos nt + ke sin nt) = — mn’r,, where r, is the component of r on the two-dimensional “space.” The force is directed toward the center, as usual. It may be observed that if in general the force is central, the moment of momentum is constant. For if d d d ii (ny) =f, To (mv) = 7 ὐπν =" rb): That the rate of change of moment of momentum is equal to the mo- ment of the force is therefore a principle which holds in non-Newtonian as in ordinary mechanics. Tue Non-EvucLuipEAN GEOMETRY IN Four DIMENSIONS. Geometry and Vector Algebra. 37. Consider now a space of four dimensions in which the elements are points, lines, planes, flat 3-spaces or planoids, and which is sub- ject to the same rules of translation or parallel-transformation as two or three dimensional space. If a and Ὁ are two 1-vectors, the product axb is a 2-vector, that is, the parallelogram determined by the vectors. The parallelograms axb and bxa will be taken as of opposite sign, but otherwise equal. The equation axb = 0 ex- presses the condition that a and Ὁ are parallel. If ¢ is any third 1- vector, not lying in the plane of a and b, the product axbxe, which is now itself a vector will represent the parallelepiped deter- mined by the three vectors. The condition axbxe = Ὁ there- fore states that the three 1-vectors lie ina plane. If now dis a fourth l1-vector, not lying in the 3-space or planoid determined by a, b, 6, the product axbxexd will represent the four dimensional parallel figure determined by the vectors. This product is a pseudo-scalar of which the magnitude is the four dimensional content of the parallel figure. The condition axbxexd = 0 shows that the four vectors lie in some planoid. In all these outer products the sign is’ changed by the interchange of two adjacent factors, as in the case of lower dimensions. Moreover, the associative law, the distributive law, and the law of association for scalar factors will also hold, as is evident from their geometrical interpretation. WILSON AND LEWIS.— RELATIVITY. 447 Two l-vectors are added in the ordinary way by the parallelogram law. The same is true of two 2-vectors if they intersect in a line, that is, if they lie in the same 3-space (ἢ 25). It is, however, clear that in four dimensional space it is possible to have two parallelograms which have a common vertex but which do not lie in any planoid, that is, do not intersect in a line. For two such 2-vectors the construction previously given for the sum is not applicable, and it is indeed impossi- ble to replace the sum of the two 2-vectors by a single plane vector. The sum may, however, be replaced in an infinite variety of ways by the sum of two other 2-vectors. For if A and B are any two 2-vectors, and if a and Ὁ be two 1-vectors drawn respectively in the planes of A and B, then the 2-vector axb = C may be added or subtracted from A and B so that A+ B= (A+ C)+ (B—C)= A'+B. The sum of more than two 2-vectors can, however, always be reduced to a sum of two. For if three planes in four dimensional space pass through a point, at least two must intersect in a line. A sum of 2-vectors, which is not reducible to a single uniplanar or simple 2- vector will be called a biplanar or double 2-vector whenever it is important to emphasize the difference. Since the analytical treatment of these two kinds of 2-vectors is not materially different, they will be designated by the same type of letters (clarendon capitals). A vector of the type axbxe will be called a3-vector. As two planoids which have a point in common, intersect in a plane, a geometric construction for the sum of two 3-vectors may be given in a manner which is the immediate extension of the rule for 2-vectors in three dimensional space. The sum of two 3-vectors is always a simple 3-vector. In respect to rotation and to the classification of lines, planes, and planoids, our four dimensional geometry will be non-Euclidean in such a manner as to be the natural extension of the non-Euclidean geometries of two and three dimensions which have been already discussed. As in two dimensions there were two fixed lines through a point, and in three dimensions a fixed cone, so in four dimensions there will be a fixed conical spread of three dimensions, or hypercone, which separates lines within the hypercone and called (6)-lines, from lines outside the hypercone, which are called (y)-lines. Besides the singular planes which are tangent to the hypercone, there are two classes of planes, namely, (6)-planes which contain a (6)-line, and (γ)- planes which contain no (6)-line. Besides the singular planoids which 448 PROCEEDINGS OF THE AMERICAN ACADEMY. are tangent to the hypercone, there are two classes of planoids, namely, (6)-planoids which contain a (6)-line, and (y)-planoids which contain no (6)-line. In the (vy)-planoids the geometry is the ordinary three dimensional Euclidean geometry; in the (6)-planoids the geometry is that three dimensional non-Euclidean geometry which we have discussed at length. Every (6)-line determines a perpendicular planoid of class (vy), and every (y)-line determines a perpendicular planoid of class (δ). Thus if we construct four mutually perpendicular lines, one will be a (6)-line, and three will be (y)-lines. A plane determined by one pair of these four mutually perpendicular lines is completely perpendicular to the plane determined by the other pair, in the sense that every line of one plane is perpendicular to every line of the other, and the planes therefore have no line in common. Jn general every plane determines uniquely a completely perpendicular plane. One of the planes is a (y)-plane and the other is a (6)-plane. As in our previous geometries, perpendiculars remain perpendicular during rotation. If then in a rotation any plane remains fixed, its completely perpendicular plane will also remain fixed; and a general rotation may be regarded as the combination of a certain ordinary Euclidean rotation in a certain (y)-plane, combined with a certain non-Euclidean rotation in the completely perpendicular (6)-plane. 38. Let ki, ke, ks, Κὰ be four mutually perpendicular unit vectors of which the last is a (6)-vector. The six coordinate 2-vectors may then be designated 53 as ky:, Kos, Kgs, Kos, κοι, Kix. There are furthermore four coordinate unit 3-vectors Ko3i, K3i4, Kiog, Kie3; and a unit pseudo- scalar Κρ. We may represent 1-vectors, 2-vectors and 3-vectors, as the sum of their projections on the coordinate axes, coordinate planes, and coordinate planoids. Thus ak, + doko + aj3k3 + ας, A= Ayaky4 == AoiKos He ΑΚ ar Aoskos == Agiksi =F Apkis, Zl == ΡΝ ὍΝ ΞιΞ Moisksis == Yoko -Ξ ϑ[γ031Κ|53. a The outer product of any two vectors is defined geometrically and expressed analytically in a manner entirely analogous to that of the simpler cases already discussed. We thus obtain the following equa- tions for the different types of products. 32 The particular order of subscripts is chosen for convenience only. WILSON AND LEWIS.— RELATIVITY. 449 axb = -- bxa = (aby — ayb,)Ki4 ΒΕ (anbs — aybo) Koy ΒΒ (agb, — abs) Kg4 + (a2b3 — ash) ko; ++ (αὶ — ayb3)Ks31 - (aib, — arb) Ky, axA = (a2 A3x — a3Aoy + a4Ao3) Koss ie (ag3Ai4 — aA + a;A31) Κρ ΕΙΣ (αι.1:. - a@Ay+ αι 12) Κι» + (a, Ao3 + azA3) + ἀμ.) Kus, ax = — Ζῖχα = (ales, + a2lsig + asXliex — ayes) Ki2s4, AxB = (AyBx3 + AuBs: + AsBr2 + AgsBiy + Asi Bo + As2Bas) Kiss. The outer product of two vectors the sum of whose dimensions is greater than four vanishes. The outer product of a vector by itself vanishes except in the case of the biplanar or double 2-vector where the product becomes AxA = 2(Aj4Ao ΞΕ Ao4A31 ΞῈ 4:4.) Kj231. If the biplanar vector be written as A = B+ C, where B and C are two simple plane vectors, the product may be written AxA = (B+ C)x(B+ C) = 2BxC. It thus appears that AxA is twice the four dimensional parallele- piped constructed upon any pair of planes into which the double vector may be resolved. The vanishing of the outer product, AxA = 0), is the necessary and sufficient condition that A be uniplanar. The general rule for all cases of inner product has been stated (§ 29). We may tabulate the following cases. aeb = ab, + abo + ash3 — asda, a-A = (a2 Ayo — a3A3; — a4A 14) |i (— aAy. + a3Ao3 — a4Ao4) kp + (a,A31 — d2Ao3 — a4A34) k3 + (— αι — 2A — α3. 14) Κι, a- Al = (a3 Asia a αὐϑί.»4) ky + (alos = a3 234) Koy ΞΞ (ayYlozs Ξ-- a 2314) k34 + (αιϑί 0. = a42lo34) ko; a (alias — ayrlsi4) ἵκει + (aslo — ἀμί 4) Kip, A-B = — AyBy — AuBoy — AgsBo, + Avg Bo3 + Agi Bs: + AwBry, A-A = (— AosQliog + Azalgiy + Aos2ia3) ki (δέ μος — Ags Mos, + Agi 223) & Γ (— Aylsis Ξε ΓΌΝΥ + 4.) 153) ks ΞΕ (Ao3%oss ai Ag lois Ξε Ayo) k,, A-3B Se ΡΥ = Us Bsis st ΟΝ ΞΕ Wros Bros. The geometrical interpretation of these inner products follows the same lines as before. The inner product of a vector into a vector 450 PROCEEDINGS OF THE AMERICAN ACADEMY. of equal dimensions is a scalar, and is the product of either into the projection of the other upon it. In the case where a biplanar 2-vector is projected, or is projected upon, each simple plane has to be treated, and the results compounded. That this may be done follows at once from the distributive law. The product of two vectors of dif- ferent dimensionality is a vector of which the dimension is the differ- ence of the dimensions of the factors; this vector lies in the factor of larger dimensions and is perpendicular to the factor of smaller dimensions. ‘However, the product a:A, if A is biplanar, is com- pounded of two 1-vectors lying in the two component planes. The complement of a vector is again defined as its inner product with the unit pseudo-scalar ky31. ‘The complement of a 1-vector is a perpendicular 3-vector, and vice-versa; that of a simple 2-vector is the completely perpendicular 2-vector. We may tabulate the results for the unit vectors. Κι = - kos, k.* = — Kgu, k;* = — Kj, k,* = — kps, ky," = — ky, Κο = — βι, k34* = — Kp, k3* = Ku, κοΐ = Ku, Κιῦ = Ky, ko34* a ee ἘΠ ks14* = —k, Κι = — k;, Κι = — ky. With the aid of complements a unique resolution of a given 2-vector into two completely perpendicular parts may be accomplished. Sup- pose the resolution effected as A= mM-+ aN where M is a unit vector of class (y) and N one of class (6) so chosen that MXN is a positive unit pseudo-scalar. Then A* = —nM + mN, mA — πὰ n& + mA* d M = = "--τττ--.-.-.--ὄο sie m? + nz n m? + n? nA — mnA* nA + nmA* Η αν τὸ =i ea ici: m= ue n ἢ m+ n? Let p= A-A= m — n’, gq = A-A* = — 2mn. The quantities m, n may then be expressed in terms of p, 4, that is, in terms of A-A, Δ. A*. The result is Pe i a artes al a a a Vp + ᾧ Ve + ᾧ WILSON AND LEWIS.— RELATIVITY. 451 The general relationships between products of vectors and _ their complements have been developed in a previous section for aspace of any dimensions. It was there shown that (except 37) formulas (34)-(39) for the expansion of all types of products involving 1-vectors and 2-vectors would be true in higher dimensions, and this is true even if the 2-vectors involved happen to be biplanar, because any such vectors is the sum of two uniplanar vectors and the equations are linear or bilinear in the vectors. Similar equations may, if occasion requires, be developed for products involving 3-vectors. 39. We have not yet considered those vectors whose inner products with themselves are zero. The case of the 1-vector, which is an ele- ment of the hypercone, need not be treated again in detail. For such a vector aca = ay + ly” + a3” == ag? =i (i) A uniplanar 2-vector such that A+A = 0 satisfies the conditions AxA = 2 (4...4. εἰν AgsA31 ΞΕ AA 12) Kyo34 = 0, A-A = — Ai? — Ao? — Az? + 452 + An? + Ax? = 0. Such a vector is obviously a plane tangent to the hypercone; for it can be neither a (y)- nor a (6)-plane. The singular plane has the same properties as in three dimensional space. The element of tangency may be found as follows. If a is any vector, ἃ" Α is a line in the plane A, and (a*A)-A is a perpendicular line of the plane. But the only line which is perpendicular to another line in this peculiar two dimensional space is the singular line, that is, the element of tangency with the hypercone. If ky be taken as a, the element may be written as (Κ..4}.4 = ky, (Ag143: — AvtA we) + ke (μά. — 4.4.3) ΞΞ ks (Ao Ao ik A\4A31) =F ky (Ay? ΞΕ 40. ΞΞ 44), an equation which we shall find serviceable. The complement of a uniplanar singular 2-vector is itself such a vector, and it may readily be shown to pass through the same element of tangency. Indeed through every element of the hypercone is a whole single infinity of tangent planes which are mutually comple- mentary In pairs. If a 2-vector be biplanar, that is, if AxA is not zero, the condition A-A = 0 is satisfied when, if the vector be resolved into the two complementary (γ)- and (6)-vectors, these have the same magnitude. For if A= mM + oN, A-A = πηι — n’. Such a vector is singular only in an analytical sense. 452 PROCEEDINGS OF THE AMERICAN ACADEMY. The complement of a singular 1-vector is a 3-vector which itself is evidently singular. It is the planoid tangent to the hypercone through the given element.3? It contains, besides the pencil of singu- lar planes through the element of tangency, only (y)-planes. We may take this opportunity of summarizing the properties of singular vectors in general. The inner product of any singular vector by itself is 0. Every singular vector is perpendicular to itself and to every singular vector lying within it. The magnitude of a singular vector is zero. This does not imply that such a vector is not a definite geometric object, but only that the interval of a singular 1-vector, the area of a singular 2-vector, and the volume of a singular 3-vector are zero when compared with non-singular intervals, areas, and volumes. The visualization of the geometrical properties of a four dimensional and especially of a non-Euclidean four dimensional geometry is extremely difficult. It is of course possible to rely wholly on the analytic relations, and thus avoid the difficulty. But we believe that it is of the greatest importance to realize that we are dealing with perfectly definite geometrical objects which are independent of any arbitrary axes of reference, and that it is therefore advisable to make every effort toward the visualization. It seems probable that Min- kowski, although he employed chiefly the analytical point of view in his great memoir, must himself have largely employed the geo- metrical method in his thinking. The Differentiating Operator }. 40. By analogy we may in four dimensions define the operator ©, called quad, by the equation “Ξε (48) When referred to ἃ set of perpendicular axes, quad takes the form fe} fe) 6 fe} O= kbs the ths ka (49) Me o and like V7 it may be regarded formally as a 1-vector. 38 The geometry in a singular planoid is analogous to that in a singular plane (§31). In this3-space there are two classes of lines, singular lines, all of which are parallel to each other, and non-singular lines, (7)-lines, all of which are perpendicular to the singular lines. Similarly there are two classes of planes, singular planes, all of which are parallel to the singular lines, and non-singular (y)-planes, which are perpendicular to every singular plane. Volumes are comparable with one another but are all of zero magnitude as compared with a volume in any non-singular planoid. WILSON AND LEWIS. — RELATIVITY. 453 We may therefore write the following equations. The result of applying © to a scalar function F is a 1-vector OF, which might be called the gradient of F. oF oF oF OF = nS Ὁ + iS + Ks, — ky On: 7. v3 Oxy The application of © to a 1-vector function f by inner multiplication is a scalar, which might be called the divergence of f. Of. 9 U2 ai. On, Of of = = 9 π ἰἀψ μεν Ὁ τ τ The application of ©, by outer multiplication, to the 1-vector f is a 2-vector function, which might be called the curl of f. Ont = (Fe ὀπὴν, + (et Fe) tas + (+P Oxy 0x4 Ox 87: af) df, as ) (38 δ ate bs vp 0x3 Kos τε (2 aa Ox, ἴοι Ἂν Ox, ary OX, Ke. The expression °F is a 1-vector. _ (Af — fs iu) (ὦ: Of “ἢ ° og (2 ἊΝ 0X3 Ox Κι +r Ox O24 ὡ Ofsi ὁ [5 δ = (ὦ: ΟΝ at + (ge ὅπ ὅθι, ὃ Tae as τὰ The product *xF is ἃ 3-vector. _ (Of _ ofes — 22) Of Afss ue OF i (4 — O23 O24 Koss & a θαι e O24 ὼς Ofer δίμ — ve) Ofes , fn δ΄ *) a5 (: Oa. ΤΟΣ Kio + fe τὸ Ox as ὃ. Kis. We might likewise expand ©-Jf and Oxf. The rules (30) and (24) for operation with the complement enable us to write (O-a)* = — Oxa*, (xa)* = τὰ" when a is a vector function of any dimensionality in four dimensional space. It is important to note in all these equations that while quad operates as a l-vector, it is not a l-vector in any geometrical sense. 454 PROCEEDINGS OF THE AMERICAN ACADEMY. Thus we find, for example, that xf is not always a plane passing through f, and in fact will usually be a biplanar vector. Also ΟΕ is not necessarily in the plane of F. We have used the same symbol <> for our differential operator as was used by Lewis in his discussion of the vector analysis of four dimensional Euclidean space, and which corresponded to the “lor” of Minkowski. There seems no danger of confusion, since it will never be desirable to work simultaneously in Euclidean and non- Euclidean geometry. Sommerfeld?* has also developed a vector analysis of essentially Euclidean four dimensional space, and his notation is an extension of that current in Germany for the three dimensional case. For the sake of reference we will compare the two notations, as far as the differential operator is concerned, in the follow- ing table. OF ὦ Grad F, Of w Div f, Oxf ὦ Rot f, O:Fo Div F. Operations involving © twice are of frequent use in a number of important equations. These may be obtained by rules already given if } be regarded as a 1-vector. OXOF) = 0, (50) Oxn(Or€) = 0, (51) Oil Ei, (52) Ox(OrxF) = 0, (53) Ca τι (54) DOD) = OOD AIOE (55) QPOs DN NOT ΕΟ ΕΣ (56) CAO) ΞΟ ΚΟΥ. (57) The important operator ©: or <* has sometimes been called the D’Alembertian. In the expanded form it is ὁ: ὁ: 6: ὍΝ 4 ὁ: ios = Die eyes ; O23" Ox Oxy (58) where V now denotes the Euclidean differentiating operator in the Κιο5 space. 34 Sommerfeld, Ann. d. Physik [4] 33, 649. 35 Kraft (Bull. Acad. Cracovie A, 1911, p. 538) devotes a paper to the proof and application of this formula. WILSON AND LEWIS.— RELATIVITY. 455 41. In the ordinary integral calculus of vectors the theorems due to Gauss and Stokes play an important réle. In our notation we may express these laws with great simplicity and generalize them to a space of any dimensions. Let us consider first the form of these theorems in the case of two dimensions, beginning with the more familiar Euclidean case. Stokes’s theorem states that the line integral of a vector function f around a closed path is equal to the integral of the curl of f over the area bounded by the curve. The analytic statement is J dset = J fascut, where ds is the vector element of arc, and dS the scalar element of area. In our notation 35 this becomes ist = [ [aso where d§$ is now the 2-vector element of area (a pseudo-sealar) and Vf is a pseudo-sealar (the complement of curl f, which itself is a scalar in the two dimensional case). Transforming by (35), we may also write Fre ἐὸν eae te Gauss’s theorem states that the integral of the flux of a vector through a closed curve is equal to the integral of the divergence of the vector f over the area bounded by the curve. The analytic statement 1S ‘where f,, is the component of f normal to the curve. In our notation this becomes -- [ὦ - [ fasyt= f fastv-t, or, by taking the complement of both sides, — [ avet -- [[ὠν- 36 One of the advantages of our system of notation is that if one term in an equation is a vector of p dimensions, every other term is a vector of p dimen- sions. This furnishes at once a check on the correctness of any equation. 456 PROCEEDINGS OF THE AMERICAN ACADEMY. and transforming by (36), where in two dimensions fxd§S vanishes, we obtain the form fcc ON oa he «0 Equations (59) and (60) can be combined into the operational equation ; F fao=—f favo, (61) where the operators may be applied to f in either inner or outer multiplication. In three dimensions Stokes’s theorem states that the line integral of a vector around a curve is equal to the surface integral of the normal component of the curl of the vector over any surface spanning the curve, with proper regard to sign. The ordinary statement is fost - [{{{ (curl ἢ),, which in our notation becomes ΤΠ f face: and may be transformed by (35) into fis-t= ἘΠ (62) In like manner Gauss’s theorem states that the integral of the flux of a vector through a closed surface is equal to the integral of the divergence of the vector over the volume inclosed by the surface. Thus, if dS is the scalar element of volume, ff fras= ff faveras: In our notation, if dS denotes vector element of volume, this becomes fuss [J frevse ff fase which, by transformation by (24) and (82), becomes [fase = ff fase os) WILSON AND LEWIS.— RELATIVITY. 457 As an example of a similar formula involving a scalar function f, we may take the familiar theorem of hydrodynamics that the surface integral of the pressure is equal to the volume integral of the gradient of the pressure f. This is usually written as J sas = [ff sraasaz, but in our notation becomes J fas- Jf fes-con - f fas-vv. 42. All these formulas lead us to suspect the existence of a single operational equation which is valid when applied to scalar functions and to any vector functions whether with the symbol (+) or (x). This would have the form J dono ai (— Def (dain) Ch (64) where do, is the p-vector element of a closed spread bounding a spread of p+ 1 dimensions. We may extend this equation to four (or more) dimensions, and demonstrate its validity as follows. It will perhaps be sufficient to give the proof of the formula in case the (p+ 1)-spread is a rectangular parallelepiped with p+ 1 pairs of opposite faces. For let do (p41) == K193.. pil da,dxodx3 δἰ Ὁ dts 11. Then, by the rules for multiplication, ὯΝ 6 d S = P οὐ τη, « 2 0 y 2 a ἥν. ἀν τὰ Saat ΟΣ dips οι, oi Ox, 0 dxydxs . . . dp Κι ρει dx δε ᾿ The partial integrations may now be effected upon the right, and leave J doon> = (—1 fda, JY (p+1) (p) if it be remembered that Ko3,p,1, — Kuis,.ps1, - - - are the positive faces perpendicular to ki, ko,... It will be evident from this mode of proof that (64) is valid both 458 PROCEEDINGS OF THE AMERICAN ACADEMY. for Euclidean and for our non-Euclidean geometry. The equation may be put in another form by the aid of rules previously given.?7 J mate “OS i Beda τς (a: (65) In four dimensions a large number of special formulas may be obtained by applying our operational equation to scalars and to vectors of any denomination with either symbol of multiplication. As examples we may write the formulas corresponding to Stokes’s and Gauss’s theorems. Let p = 1 and apply the operator by inner multiplication to a 1-vector function. Then [st = — f { @s-o)-t = ff [ as-#). This is the extended Stokes’s theorem. Again let p = 3 and apply the operator by outer multiplication to a 1-vector function. Then {{{{Φ4- -Γ{{{|{.0»«- Ὡ[7 ἘΠ This is the extended Gauss’s theorem, where d= represents a differ- ential (pseudo-scalar) element of four dimensional volume. In these cases also the same equations apply in Euclidean and in our non-Euclidean space. If, however, we write these two equations in non-vectorial form, they become in the non-Euclidean case Jide + fodar oa fsdas = fadvs) = of: 3 ch ΓΕ: of: Of ) = Ἵ τς pean daodi3 + i Tas dx3dx, ag (Ξ = ΕῚ αατάχ. -- (ee == ah daydatg Ox, O25 OX) θα, Ofs | ont fs ch) Bet se ( δὲ Ἐ ae) deades — ( πε ΕΒ davde | 37 This equation embraces both of the operational equations given by Gibbs in δὲ 164-5 of his pamphlet Vector Analysis (1884) reprinted in his Scientific Papers, 2. In case p + 1 is equal to n, the number of dimensions of space, then do(p41)* is ascalar and the equation has no meaning unless we adopt the convention ™xa = ma, where mis a scalar and ἃ any vector. This convention would lead to no contradiction, and might occasionally be useful. WILSON AND LEWIS.— RELATIVITY. 459 and | Υ yi a (fidaed. xg -- fe dusdayd v4 + fadaid xed v4 fid xd xodars) ὅπ Ὁ ὁ: ἢ {ΜΕ}: Ox, + oh hyo τ dajdiodagdiry. The theorems may be used to demonstrate in a vectorial manner such an equation as (52), O+(:F) = 0. For Jf ffeocon-- ff fon τ ff [aso - f fase. As the final integral extends over the bowndary of the closed three dimensional spread which bounds the given region of four dimensions, the final integral vanishes, since the closed spread has no boundary. Geometric Vector Fields. 43. The idea of a vector field is ordinarily associated with concepts such as those of force or momentum, which are not wholly geometri- cal in character; but it is per- fectly possible to construct ra vector fields which are purely R n= geometrical. Thus in ordinary er geometry we may derive a R} 4 vector field, when a single k 4 point is given, by constructing (δ) “ae at every other point the vector “ ΄ from that point to the given eS) point, or that vector multiplied pg = by any function of the dis- “2 ὟΣ tance. In our non-Euclidean four dimensional space we may as- sociate with any (6)-curve a vector field derived from that curve in the following way. At each point of the (6)-curve construct the forward unit tangent w, and the forward hypercone.3® At each point Q . these hypercones construct the vector w/R, parallel to the vector FIGURE 21. 38 Tv hat half of the hypercone lying above the origin, ee ue w Hak will represent later times than the time of the origin, will be called the forward hypercone. 460 PROCEEDINGS OF THE AMERICAN ACADEMY. w at the vertex, and equal in magnitude to the reciprocal of the interval A along the perpendicular drawn from the point Q to that tangent produced (Figure 21). On account of analogies which will soon become apparent we shall call this vector function the extended vector potential of the given (6)-curve.29 We shall write Ww ig = R (66) We shall next consider the 2-vector field il il P= Op = (Opt κίον). (67) We shall consider the evaluation of xp in two steps. First we shall assume that the original (6)-curve is a straight line. In this case w is constant and ©xw = 0. If we arbitrarily take ky along w, we may write 1 1 Ὁ ἢ 1 Vie νῶν for it is clear that a displacement parallel to w does not change R. It is evident that R becomes a radius vector in the 3-space perpendicu- lar to w. Τῇ ἢ represents a unit vector from the point Q normal to w, that is, in the direction in which R was measured, then by the well known formula, VR! = ἢ ΠΣ, Hence And hence P= Op = ΞΕ (68) The determination of +p follows in precisely the same way; in each of the above formulas the symbol of inner multiplication will replace that of outer multiplication, and we find that O-p= Ξὺ (69) for n is perpendicular to w. Of all the geometrical vector fields which might have been con- structed from a given (6)-curve, we shall show later that those which we have just derived are the most fundamental (footnote § 44). The 39 The vector fields produced at a point by two or more (s)-curves may be regarded as additive. The locus of all possible singular lines 1 drawn (as in Fig. 21) from (s)-curves to a given point is the backward hypercone of which that point is the apex. WILSON AND LEWIS.— RELATIVITY. 461 2-vector xp is a simple plane vector in the plane of the point Q and of w. The 1-vector p has everywhere the direction of the funda- mental vector w; if 1 be the singular vector from the vertex of the cone to the point Q, the scalar product 1+p is constant. In fact Ww lxw τ (1-w)* are the expressions for the fields in terms of 1 and w. Let us now choose arbitrarily a time-axis ky, and then the perpen- dicular planoid is our three dimensional space. We may resolve our l-vector and 2-vector fields as follows. aie (70) lew’ p — «τὸ δὰ πον cod _ La ΞΕ κι τ SS l-w (1; + Usky)+(v + ky) (71) = — Vv ky eee ie where 1, and p, are the space components of land p._ As 1 isa singular vector, J; is equal to the magnitude of ],. ἘΠ τῆνος 2) (Is ++ Liks)x(v + ky) a ame ἐν, ees oa pee 2s (1 — v) xv ᾳ — υἢ) (eS Lw)xKy, i. (Ls Ἐς τε 1.. νυ)" (Ly =e 1,-v)8 Of these two planes into which P is now resolved, the first lies in “space”? and the second passes through the time axis and is perpen- dicular to “space.” We shall attempt to show with the aid of a diagram (Figure 22) the geometrical significance of the various terms which we have employed in the above formulas. The origin, that is, the vertex of the hyper- cone, is any chosen point O on the given (6)-line w. A point upon the forward hypercone is Q, and 1 is the element OQ. The unit vector ἢ is drawn along QJ from @ towards and perpendicular to the vector w. The intervals OJ and QJ are equal, and equal to R = —1-w. The vector p drawn at Q parallel to w and of magnitude 1/R is the extended vector potential at Q due tow. The 2-vector P lies in the plane 0/Q, and is equal in magnitude to 1/R?. The arbitrarily cho- sen time-axis is ky, and on the planoid perpendicular to ky (that is, on “‘space”’) the vector 1 projects into 1, = O’Q. The intersection of the line of w with the planoid is G (the point of the line w which is simultaneous with Q). Similarly O’ is the intersection of ky with the 462 PROCEEDINGS OF THE AMERICAN ACADEMY. planoid. The line OO’ = I; represents the lapse of time between O and Θ΄; and this is equal in magnitude to 0’Q or 1,, the space compo- nent of 1. The interval OG = 1, ¥1 — υ and the interval O’G = lw = 1,v. The direction w projects into the direction v. Hence as a vector, O’G is equal to /,v. The quantity 1,-v = O’F may be obtained by Figure 22. dropping a perpendicular from G to 0’Q. The interval FQ is then 1, — 1,*v or 1; —1,*v, the expression which occurs in the denomina- tors. The vector GQ = ris clearly 1, — lv or 1, — lav. 44. We shall now remove the restriction that the (6)-curve which gives rise to the potential p = w/R = — w/(l-w) is rectilinear, and consider the general case of any (6)-curve. For the sake of simplic- ity in this complex problem we shall use dyadic notation (see appen- dix § 61, ff.). The results, however, might all be obtained by means of the more elementary geometric and vector methods. We may write w Op= OF = (Opt pOW= — POR Wt ZOw. Now ΟΝ is defined so as to satisfy the relation dr-Ow = dw. A displacement (Figure 23) dr = w ds parallel to w, makes a change dw = cds. A displacement dr along the vector 1 (Figure 24) intro- WILSON AND LEWIS.— RELATIVITY. 463 duces no change in w, and in like manner a displacement dr in the plane perpendicular to that of w and 1 does not affect w. Hence we may write 1 1 w= c= ——le. 73 NS pe R (73) a 4 n τὰ coe Ca ae ΄ Pi cs adr=d1 4a ie - wl” wl Ie, Ag (8) ψ ὃν \ds ds” wx x FIGURE 23. FIGuRE 24. To compute OR = — © (1-w), we may write O(l-w) = (O)-w+ (Ὁ ν}:}. Here ὧν is already known. To find ΟἹ observe that dl = dr-©1 is equal to dr when dr is along 1 (Figure 24). Further if dr is elsewhere in the hypercone, for instance, in the plane perpendicular to that of 1 and w then also dl = dr. But when dr = wds 1s along w the differ- ential dl vanishes. Hence we may write 1 1 ΦΙΞΙ- τ ἘΞ ῚΈ εν, (74) where I is the idemfactor. Thus we have il ] > (1-w) = (1 { R Iw)-w— le+l, or, performing the multiplication by w, OR= —O(1-w) = —wt From this it follows at once that 1 1 eM 1+ le (76) = ae (tc + R lw — ww ) i+ le alt (75) 464 PROCEEDINGS OF THE AMERICAN ACADEMY. The two expressions xp and +p may now be obtained by inserting the cross and dot in ©p. Hence nD R AC e+ ee bw) (77) 1 1 Ξ Se ey 77) (1-¢ Ἢ a “ew + 1) = 0. (78) Here also <>+p vanishes, since l-w = — R. As 1 varies with R, the parts of

+p = 0, and therefore O-OrP = Op = 0. (86) The existence of this extended Laplacian equation justifies the use of the term potential *° for p. 40 It isinteresting to enquire what form the potential p might be given other thanw/Rk. Suppose that p should be independent of the curvature of the (6)- curve. The only vectors then entering into the determination of p at any point Q would be w and 1. The only possible form of a 1-vector potential would therefore be P= ο( πο - (hl, where R = —l-w. ‘The expression for Op becomes Op = 0 (R)( - wx. 1) =% (R) ple + f’ (R) [- wt ! 1) +5@) (1 Ἔ ΩΣ 466 PROCEEDINGS OF THE AMERICAN ACADEMY. ELECTROMAGNETICS AND MECHANICS. The Continuous and Discontinuous in Physics. 45. It has been customary in physics to regard a fluid as composed of discrete particles (as in the kinetic theory) or as a continuum (as in hydrodynamics) according to the nature of the problem under investi- gation; it has been assumed that even if a fluid were made up of discrete particles, it could be treated as a continuum for the sake of convenience in applying the laws of mathematical analysis. For example we introduce the concept of density which may have no real exact physical significance, but which by the method of averages yields apparently correct results. Provided that the particles in a discontinuous assemblage are sufficiently small, numerous, and regu- larly distributed, it is assumed that any assemblage of discrete particles can be replaced without loss of mathematical rigor by a continuum. However, when we investigate problems of this character in the light of our four dimensional geometry, we are led to the striking conclusion that in some cases it is impossible, except by methods which are unwarrantably arbitrary, to replace a discontinuous by a continuous distribution and vice versa. Especially we shall see that this is the case with radiant energy, a conclusion which 15. particularly Hence Op ΞΞ --ἰ Ἰτρ [- (R) + ze ®)| ΞΕ (pcr) ΞΕ 3f(R) )- If ep is to vanish eek of the curvature of the (s)-curve, then ¢’ (R) nee Rn?) Rf'(R) + 3f(R The integration of ae equations determines ¢ and f as A B Chi R’ ii = R® where A and B are constants. The expression for ©>xp is A 1+ 1c 2B xp pe [το - τας ἢ bw] =a Ixw. The calculation of -+Oxp = — +p gives O-Op = 2 Β [Ἐπ +3 ει ) It therefore appears impossible to satisfy <>+p = Oand +>p = ὁ with δὴν other form of potential, dependent only on 1 and w, than the one chosen. WILSON AND LEWIS.— RELATIVITY. 467 notable when taken in connection with the recent theories regarding the constitution of light, embodied in the quantum hypothesis. Let us for simplicity first consider such cases as arise in our two dimensional geometry. Consider a material rod of infinitesimal cross section moving uniformly in its own direction. Suppose now that we regard this rod as made up of discrete particles. Then in our geometrical representation each particle will give rise to a vector of extended momentum mow, and these vectors will all be parallel. The whole space-time locus of the rod will be a set of parallel (6)-lines. The rod as a spacial object possessing length has no meaning until a definite set of space-time axes have been chosen, and this choice is arbitrary. There is, however, one such choice which is unique, and that is the selection of the time-axis along w, and the space-axis per- pendicular thereto. In this system the mass of each particle is its mo, and the sum of the m’s of any segment of the rod divided by the length of the segment is the average density. If the particles are sufficiently numerous, we may regard the rod as continuous and re- place conceptually the locus of the rod as a set of discrete (6)-lines by a vector field continuous between the two (6)-lines which mark the termini of the rod, and represented at each point by a vector parallel to w and equal in magnitude to the density at that point. This is the density as it appears to an observer at rest with respect to the rod, and may be called up. The vector uw has therefore a defi- nite four dimensional significance. Its projections on any arbitrarily chosen space and time axes are, however, not respectively the density of momentum and mass in that system. For Haw = Ξ- ΕἸ τοῦ a (87) Vi—? But μ, the density in this system, is not equal to μη. V1 — v, but Mo = (88 1 ---οἴ' (88) μ as the units of mass and length both change with a change of axes. Conversely we may replace a continuous by a discrete distribution. ' Let us consider a continuous vector field f of (6)-lines. Then any region of this field, embraced between two (6)-lines sufficiently near together, may be replaced by one or several parallel (6)-vectors, of which the sum is equal to f multiplied by the length of the line drawn between and perpendicular to the boundary (6)-lines. We may also 468 PROCEEDINGS OF THE AMERICAN ACADEMY. use another construction which is essentially identical with this. Let dr be any vector drawn from one boundary line to the other. Then (drxf)*f/f is the same vector as the one just obtained. Although the method of obtaining this vector may seem somewhat artificial, the vector is, however, a definite vector obtainable from the field without any choice of axes. 46. These methods fail completely when the vector field is com- posed of singular vectors. Let us consider instead of a material rod, a segment of a uniform ray of light. If this can be represented by a continuous vector field bounded by two lines representing the loci of the termini of the segment then all these vectors must be singular. Let 1 be (Figure 25) the value of the vector through- out the field. It is evident that we cannot, as in the former case, draw any line across Fiqure 25. the field perpendicular to 1. The second method likewise fails because it would involve the magnitude of 1 which is zero. Moreover it can be stated that there is no method whatever, independent of any choice of axes, which will enable us to change from this continuous distribution of the light to a set of light particles. Conversely it is equally true that given a system of light particles moving in a single ray it is quite impossible to replace them by means of any continuous distribution, and this is true no matter how small and numerous and close to- gether these particles are. This statement regarding singular vectors will be seen to hold also in space of higher dimensions,*! and is of fundamental importance. While it is impossible, therefore, to find continuous and discontinu- ous distributions of singular vectors which are equivalent to one another, it is possible to obtain by four dimensional methods out of a specified region of a singular vector field a single vector or group of discrete vectors uniquely determined by that vector field but quadratic instead of linear in the vectors of the field. Consider any portion of the field bounded by two singular vectors sufficiently near together. Let 1 be the vector of the field, and then if dr is any vector drawn from — 41 In the case of the peculiar geometry of a singular plane (§ 31), the interval dr from one singular line to another is independent of the direction of αὐ. It is therefore possible to replace the field 1 between two boundary lines by the single vector ldr linear in 1. Thus there are exceptional singular fields in higher dimensions for which the passage from continuous to discrete and vice versa may be accomplished. WILSON AND LEWIS.— RELATIVITY. 469 one boundary to the other (Figure 25), the 2-vector dr«1 is independ- ent of the way in which dr was drawn and the 1-vector (drxl)*1 is determined, and is in a certain sense representative of the region of the field chosen. It may be of interest to obtain the projection of 1 and (dr«1)*1 upon two sets of axes Κι, ky and k,’, ky’ where the angle from ky, to Κι' is φ = tanh !v. Let the vector 1 be written as l= a (κι + κω) — a’ (k,’ + k,’). Now by the transformation equations (7) we have a’ = a(coshy — sinhy) = a erst τὴν ΕΝ « = 2! VI — 2 L+u Hence the ratio of the components of 1 along the new axes to the components along the old axes is V1 — v/ V1 +. But (drx1)* is a member independent of any system of axis. Hence the ratio for (drx1)* 1 is the same as that for 1. Now while it is impossible by any four dimensional methods to redistribute the vector (drxl)*1] as a continuous vector field, it is always possible after arbitrary axes of space and time have been chosen to make such a distribution. Thus if between the two bound- ary lines dr be taken parallel to Κι and dr’ parallel to k’:, then as before drxl = dr’*1. By taking the complement of both sides and ap- plying (24), then, since 1 is its own complement, we find αὐ] Ξε σ΄]. But dr-l is equal to adr-k, = adr, and dr’-1=/’dr’. Hence dr/dr’ = a’/a. Thus the density of the components of the vector (drx1)*1 in the one case is to the density of the components in the other case as a” is to a”, equal to (1 — v)/(1+). Thus while we have seen that the energy and momentum of a light-particle (§ 24) appear different in the ratio V1 — v/ V1 + » to two observers, if the energy and momentum are regarded as distributed their densities will appear different to the two observers in the ratio (1—v)/(1 + 2). Let us proceed at once to the discussion of similar problems arising in space of four dimensions. Here also it is possible to pass at will from a consideration of continuous 1l-vector fields to a consideration of equivalent discontinuous distributions of 1-vectors in the case of all non-singular vectors, by an extension of either of the methods which we have used in two dimensional space. Thus if a region of the field is cut out by a (hyper-) tube of lines parallel to the vector of the field, then the original vector multiplied by the volume of inter- 470 PROCEEDINGS OF THE AMERICAN ACADEMY. section of a perpendicular planoid is a single vector (or the sum of a group of vectors) which may replace the original field within the tube. Or if f represents the vector field and d the 3-vector cut off on any planoid by the tube, then the same result as before may be obtained by the operation (dSxf)*f /f. In the case of singular vectors we encounter the same difficulties as in two dimensions. Let us consider a field of singular 1-vectors 1, and a portion of this field cut off by a small tube of lines parallel to 1. A little consideration shows that it is impossible by any means what- ever to replace this portion of the field by a single equivalent vector along 1. It is possible, however, as before to obtain a single vector quadratic in 1 and determined by the given portion of the field. Let d& be the 3-vector volume cut off on any planoid by the tube. Then (d§}~1) is independent of the planoid chosen, and (d»1)* 1= dg is the vector thus determined. 47. Now it is impossible to distribute the vector just obtained over that portion of the four dimensional spread which has given rise to it. But there is, nevertheless, in one case another kind of dis- tribution which is possible and which possesses considerable interest. In order to introduce the somewhat difficult construction which is necessary in this case let us investigate first a particular type of singular vector field in three dimensions. Let ds be a small vector segment of a (6)-curve. Each point of this segment determines a forward cone. The field which we wish to consider is such that at each point the vector 1 is along an element of the cone and of any interval which is a continuous function of position. This construc- tion gives a limited field bounded by the two forward cones from the termini of the segment ds. Let a plane cut across the two cones. The region of this plane intercepted between the two boundary cones is the surface lying between two nearly concentric circles. Let dS be an element of this surface. Now just as before the vector (d§x1)*1 = dg may be formed and is different for each element dS. The singular lines drawn from all the points bounding dS to the corre- sponding points of the segment ds determine a sort of tube of nearly parallel singular lines. The value of dg for each tube is at each point independent of the particular position of the plane through that point whose intersection with the tube is dS. If therefore the whole field is divided up into an infinite number of such tubes, the infinitesimal vectors of the second order in 1 obtained for the several tubes are at each point independent of the plane which was used in constructing them. WILSON AND LEWIS.— RELATIVITY. 471 Now it is impossible to redistribute the discrete vectors dg over the three dimensional field from which they were derived, but it is possible to replace them by a continuous distribution over a two dimensional spread in one of the cones. Let us assume that the infinitesimal tubes are so chosen that the elements of surface dS = dqxdr are four-sided figures approximately rectangular and that the outer cone is divided into small regions lying between the elements of the cone, a, a’, a’, ... (Figure 26). In each of these small two dimensional regions we may place the corresponding vector dg. Now any two neighboring lines drawn from a to a’ are of equal interval because they lie in a singular plane between two singular lines (see preceding footnote and ὃ 31). The vec- tor dg/dr is therefore determined at each point of the cone independent of the direc- tion of dr. It is a vector representing a Ficure 26. kind of density and when all the vectors dg - are similarly treated, it is continuously distributed over the whole cone. The vector dg /dr is a function of the interval ds. Let us determine this relation analytically. Since dS= dqxdr we may write dg = (dqxdrx1)*1 = [(dqxdr)*+1]1 = 11 dqdr, where /; is the component of 1 perpendicular to dqxdr; for since dq is perpendicular to dr, (dqxdr)* is a l-vector perpendicular to dqxdr and of magnitude dqdr._ We therefore find dg/dr = Ildqg. It remains to determine dq in terms of ds. The plane of intersection having been chosen, the two circles are in general eccentric and the distance de between their centers is the pro- jection of the segment ds upon their plane (Figure 27). If the normal to this plane makes an angle with ds whose hyperbolic tangent is ὃ, then de = rds/ V1 — x. The two segments cut off by the two circles on de produced are found as follows. Pass a plane through de and ds. Then AB isreadily shown to be ds V1 — »/ V1 + », and CD = dsV1 + v/ V1 — 2: Then the value of dq is readily proved by Euclidean methods to be 472 PROCEEDINGS OF THE AMERICAN ACADEMY. (1—v cos Φ) ds/ V1 — v2, where ¢ is the angle between dg and AD. Hence dg ii 1 — vecos¢ Ap 4 ae, lds. (89) We have gone through this somewhat complicated calculation for the three dimensional case because of the greater ease of visualisation FIGURE 27. and because the results obtained are applicable without essential change to four dimensions. Again let ds be a segment of any (6)- curve each point of which determines a forward hypercone. Let us consider the four dimensional vector field 1 bounded by the two limiting forward hypercones, 1 at every point lying along an element of one of the hypercones whose apex is on ds. Any (y)-planoid will intersect the limited vector field in a three dimensional volume bounded by the intersections of the two limiting hypercones with the planoid; these surfaces of intersection appear in the planoid as two nearly concentric spherical surfaces. If as before the vector field is divided into infinitesimal portions, so that the volume of intersection is divided into the infinitesimal vol- umes d, each of which is approximately a rectangular parallelepiped, and one of the surfaces of intersection is thus divided into the infi- nitesimal portions dS such that dqxd$ = d%, then for each infinitesimal portion of the field we may at any point obtain as above the vector dg = (d%»1)*1. Then precisely as in the previous case 52 42 In the peculiar three dimensional geometry of a tangent (singular) planoid there is one set of parallel singular lines, and every plane in the planoid is erpendicular to these lines. Every cross-section of a given tube of singular ΤῊΝ has the same area. WILSON AND LEWIS.— RELATIVITY. 473 dg = (dqxdSx1)*1 = Ildq dS, and dg/dS = [4] ἀη. This vector is distributed uniformly over one of the hypercones and is independent of the particular planoid used in obtaining it. Then also just as before dg 1 — veos¢d το Ξε ἰς ——=— lds (90 dS V1 — yy ; ) where ¢ is the angle between v, which passes through the centers of the two spheres, and the line, from either center, to the chosen point upon the surface. The Field of a Point Charge. 48. Much of recent progress in the science of electricity has been due to the introduction of the electron theory, in which electricity is regarded not as a continuum but as an assemblage of discrete particles. In Lorentz’s development of this theory he has deemed it necessary, however, to regard the electron itself as distributed over a minute region of space known as the volume of the electron. This deprives the theory of some of that simplicity which it would possess if the charge of an electron could be regarded as in fact concentrated at a single point. Whether the theory of the point charge can be brought into accord with observed facts and with the laws of energy cannot at present be decided. It seems, however, highly desirable to develop this theory as far as possible. In our application of our four dimensional geometry to electricity we shall therefore consider first an electric charge as a collection of discrete charges or electrons, each of which is concentrated at a single point. The locus of a point electron in time and space must be a (6)-curve. If w is a unit tangent to such a curve, then we may consider at every point the vector ew, where εἰ is the magnitude of the charge, negative for a negative electron, and positive for a positive electron (if such there be). It is explicitly assumed that εἰ is a constant. We shall show that the geometric fields obtained from this vector by the methods of § 48 give precisely the equations which are of importance in electromagnetic theory. The vector w determines at every point of our time-space manifold the vector p = w/R. Similarly the vector ew determines the vector field ew ΕΥ̓͂ ek, (91) τ εν τὴς ΣΎΝ, 474 PROCEEDINGS OF THE AMERICAN ACADEMY. The last equality is obtained when any ky, axis has been arbitrarily chosen. Then v is the velocity of the electron and /,;—1,°v is the distance FQ in Figure 22, that is, the projection of the distance from the point of observation to the contemporaneous position of the electron (if assumed to be moving uniformly) upon the line 1, joining the “retarded”’ position of the electron to the point of observation. We may call m the extended electromagnetic vector potential. Its projections on space and on the time-axis are respectively the vector potential a and the scalar potential φ, eV € "τ ΘΕ στο δ ἐὰν l —— lev (92) precisely in the form first obtained by Liénard.*® From (69) we have ὃ Om = (ν a κι}. + ok) = 0. Hence Vato = 0. We see therefore that the Liénard potentials are connected by the same familiar equation as connects the ordinary vector and scalar potentials. Assuming that vector fields produced by two or more electrons are additive, these equations are true for the general case. The 2-vector field produced by an electron, whether in uniform or accelerated motion, is obtained immediately from (81)—(83). Mm << — ες ΡΞ -: a [wxc — nxc+ n-c nxw| + <“nxw. i ra QE} Or ( € : € tes € 7 < € M = — RB Ix[1-(wxe)| — R kw = — RB (Ixwxe)-1 — pw: (94) The first term in this expression vanishes when the curvature is zero. The fact that this term is a singular vector has already been pointed out, and the great importance of this fact in electromagnetic theory will be pointed out later. In the second term nxw is the unit 2-vector determined by the line wand the point Q where the field is being dis- cussed. 49. In case the electron is unaccelerated the equation assumes the simple form € ΝΞ pow. (95) 43 Eclairage électrique, 16, 5 (1898). WILSON AND LEWIS.— RELATIVITY. 475 This may be expanded according to (72) when an axis of time has been chosen. Then, noting that lx«v = (1, — lv)xv, M = —e—]— mv — ε- τ κα, (96) Where r is the vector r = 1, — lv from the contemporaneous position of the charge to the point Q in the field, and γ΄ = ἰῷ — 1,-v. The 2-vector M is thus split automatically into two 2-vectors, of which one passes through the time-axis ky, and the other lies in the planoid Κις which constitutes ordinary space. These will be designated respectively by the letters Eand H. Thus M= H+ E. ᾿ (97) This separation may in all cases be made whether the field is caused by one or more electrons in constant or accelerated motion. We shall thus see that the 2-vector M is precisely the “ Vektor zweiter Art” which Minkowski introduced to express the electric and magnetic forces. Out of H and E spacial 1-vectors h and e may be obtained by the equations hi H:-kps, e = E-ky. (98) Then h is the three-dimensional complement of H, and e the inter- section of E with three-dimensional space. Evidently hy = Π5., hy = Hz, hs = Hy, a= — ἔνι, e = — En, 68 = — Ey. (99) Referring now to (96) we see that in the case of a uniformly moving electron 1—?” rae rr r3 Vv τ “h= -- ae (rxv) Kis, (100) or Ὁ ΘΞΞ ΠΕ. — Ho. Noting that (rxv)+kw3 is that which in ordinary vector analysis is known as the vector product of r and v, we see that these equations are precisely the equations for the electric and magnetic forces.** It may seem surprising to one who is not fully convinced of the very fundamental relationship between the four dimensional geometry of relativity and the science of mechanics that we should thus be led 44 See Abraham, Theorie der Elektrizitat, 2, p. 88. 476 PROCEEDINGS OF THE AMERICAN ACADEMY. from simple geometrical premises to conclusions of so purely physical a character. Of course it is to be noted that while our values of e and h are identical in mathematical form with equations for electric and magnetic force, we should need some additional assumptions before actually identifying these quantities. 50. Our next step will be to show that the values of e and h derived from the 2-vector xm = M are identical with the expressions for electric and magnetic force in the general case in which the electron is no longer restricted to uniform motion. We have from (94) M =