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 stalkcell 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 stalkcell; 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
stalkcells, while the distinction between venter, neck and tip is not
well marked. The apex, in all three species, is rather characteristi
eally shaped, flatconical, 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
amberbrown, paler distally; the base inflated, tapering thence
gradually to the blunt conical apex; the stalkcell 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 stalkcell 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 stalkcell 1627 μ (distally). Appendage,
including stalkcell, 4454 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 twocelled 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 innerside, 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 twocelled
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 wallcells
greatly elongated, usually distinctly suffused with purplebrown,
and more or less inflated; or the whole perithecium of nearly the same
diameter to the tip; which is well distinguished, bluntconical, the
apex flat, papillate, subtended by a slight elevation: the basal cell
region relatively short and compact, concolorous with the part above,
the stalkcell 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 4055 μ, ascigerous part 165270, stalkcell 4060 2030 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. 17034.
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 stalkcell relatively small, narrow, hyaline distally,
but otherwise rich redbrown, 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 redbrown
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 crestlike series of branchlets
(usually broken) arising from its broad distal surface, the most external
opaque or basally suffused. Perithecium 175 X28 yu including basal
cellregion (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 amberbrown; 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 stalkcell
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 fanlike 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 bristlelike 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, threecelled, 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 cellsof the receptacle hyaline, small, subequal,
or the subbasal larger. Perithecium, including basal cellregion,
118125X3440 yp, the sporiferous part 751004; the stalkcell
4060X 1220 4. Spores 24X2.5y. Greatest length of whole ap
pendage 150360 yu. Receptacle, including foot, 4020p. Total
length to tip of perithecium 85235 μ.
var. gracilis nov. var. Perithecium and its stalkcell 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 100156 X 2035 μ, including basal cellregion;
stalkcell 175X20 yu. Greatest length of appendage 150430 μ. Total
length to tip of perithecium 180385 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 bristlelike 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 stalkcell 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 crestlike 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 6575: ascigerous portion 5570; the stalkcell 2812 yu.
Receptacle 20X16 u. Total length of appendage including branch
lets 6080 u. Total length to tip of perithecium 95120 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 rockerlike basal cell of the receptacle,
which is usually quite hyaline. The species bears more resemblance
to the Stilicusinhabiting 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 cellregion clearly distinguished by a distinct
constriction from the stalkcell, which may be nearly straight, or
strongly curved, distally broader or slightly inflated, about twice as
long as broad; the stalkcell 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 fanlike 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 5862204y; asci
gerous portion 5458y; the stalkcell 233012. 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 Stilicusinhabiting 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
amberbrown, asymmetrical; the basal cellregion 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: stalkcell 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 5866 X 2428 μ: stalkcell 406020 μ. Spores 26X
2.5 w (measured in perithecium). Receptacle 20X 12 μ, its outgrowth
20305 uw. Total length of appendage 3040 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 amberbrown; the basal cellregion
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: stalkcell but faintly suffused, broader distally, and
distinguished from the basal cellregion 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 stalkcell 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 7085 2327 μ: stalk
cell 60X18 μ. Receptacle including foot 40 u. Total length to tip
of perithecium 135170 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 cellregion not distinguished from the asci
gerous portion, which tapers distally to its peculiarly modified tip,
the blackish suffusions of which extend to an opaque, hooklike pro
longation which, bending at right angles, forms a lid immediately above
and often partly concealing the hyaline apex: the stalkcell nearly
hyaline, variously, often greatly, elongated, curved, or often straight
and erect; distally broader than the basal cellregion, 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 7085X2026 u;
its stalkcell 50125 15 μ, distally, 20 u broad. Appendages, we
75 wu. Receptacle, including foot, 3040 μ, its outgrowth 3060 μ
Total length to tip of perithecium, 150250 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 amberbrown, the basal
cellregion 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 stalkcell 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 stalkcell 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 stalkcell in length.
Spores 32X3 u. Perithecium 6070X 2023 μ, its terminal projection,
upper margin 28 μ, lower 40 μ; stalkcell 3045X1218 μ. Recep
tacle 3040 u. Longest appendage 175 yu. Total length to tip of
perithecium 120150 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 wallcell on the inner side; subclavate
in form, the gradual distal enlargement extending to the subterminal
wallcell; distally curved outward to the subhyaline apex which is
slightly cleft, and subtended on the inner side by a long, straight,
rather slender unicellular spinelike process which tapers slightly to a
blunt apex and projects at right angles; basal cellregion well devel
oped, concolorous, not distinguished from the ascigerous part, nar
rower below where it connects with the rather slender free, subcylin
drical stalkcell. 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 cellregion, 240250
46 w: the subterminal spine 8090 uX810 uw near base; the stalkcell
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 cellregion
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 wallcells deeply suffused with brown, and forming a free
truncate projection immediately beside the flatconical, hyaline,
slightly geniculate apex: stalkcell 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, 100110X25 μβ. Receptacle,
including foot, 55X18 u. Appendage 5055X810 yu. Total length
to tip of perithecium 1501804. 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 cellregion distinct from the more or
less inflated basal ascigerous part; the midregion sometimes rather
abruptly narrower and elongate; the tip not distinguished, symmet
rical; the apex narrow subsymmetrical, hyaline, abruptly papillate:
stalkcell 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, 120135X2022 μ:
the stalkcell 6XSu. Receptacle 5558 u. Spores 30X3 yu. Append
age 95100 1214 μ its longest branches 155 uw. Total length to tip
of perithecium 200230 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 cellregion 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 stalkcell 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
5060 X 1520 μ. Receptacle, including foot, 3055X912 uw. Main
axis of appendage 3035 12 μ; total length to tip of longest branch
lets, 75 uw. Total length to tip of perithecium, 90125 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 insertioncell,
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 wallcells strongly spiral and marked
by fine irregularly parallel lines; the tip deeply suffused, the lipedges
hyaline, subequal, the apex suleate and turned strongly inward.
Spores 75X8 uw. Perithecium 150175X4550 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 wallcells.
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
95100 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 olivebrown 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 (stalkcell) 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 stalkcell. 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 662023 uw. Spores 222.6 (in perithecia). Receptacle
85X23 uw. Appendages to tips of longest branchlets, 105 yu. Total
length to tip of perithecium 100120 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 insertioncell 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 lipcells, 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 125130X24 u. Receptacle 235200 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
insertioncell 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 insertioncell, 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 lipcells large rounded and bent slightly inward. Spores 453.5 μ.
Perithecium 90100 X 2428 μ. Receptacle 80 15155 X22. Longer
appendages 135150 u. Total length to tip of perithecium 175280 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 insertioncell, 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 cellregion
forming an external rounded prominence, the junction of the basal
and subbasal wallcells 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 62702022 μ. Receptacle
80100 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 olivebrown, deeper externally below the very thick
dark insertioncell; 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 olivebrown, 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 120135X4045 μ. Receptacle 150225. Longest ap
pendage 80110 μ. Total length to tip of perithecium 250325 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 insertioncell 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 bellshaped 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 156X4855 uw. Receptacle 20075 μ. Total
length to tip of perithecium 330350 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 granularpunctate, 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. Insertioncell 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 lipcell
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 insertioncell, 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 wallcells
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 olivebrown extends hori
zontally across the perithecium, which above it is pale amberbrown;
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 175185 4550 uw. Receptacle 310415 X 6278 μ; larg
est subbasal cell 187X75y. Appendages 200 μ, longest 215 μ.
Total length to tip of perithecium 350585 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 insertioncell,
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 235335 5070 uw. Longest appendage 200yu. Total
length to tip of perithecium 300500 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 insertioncell. Insertioncell 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 lipedges hya
line and evenly rounded; if the twist is absent, oblique, or sometimes
fourlobed if the twist is one eighth. Spores 78X7u, Perithecium
125145 3540 uw. Receptacle 100135 4. Total length to tip of
perithecium 225275 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 amberyellow. Receptacle
normal, the subbasal cell variably elongated, rarely minutely corru
gated; cell V parallel to cell IV and slightly longer. Appendages
hyaline, the insertioncell 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 wallcell 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 100235 yp. Total length to
tip of perithecium, 150350 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 wallcell of the perithecium.
Laboulbenia australis nov. sp.
Receptacle indistinctly punctate, cells I and II becoming dirty
yellowish, often contrasting with the frequently deeply suffused
yellowbrown distal portion which often becomes somewhat olivace
ous. Insertioncell 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 cellof 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 125275 μ. Total
length to tip of perithecium average 250275 μ (150300 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
insertioncell 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 insertioncell; its tip abruptly
distinguished, narrow, prominent, opaque, contrasting abruptly with
the hyaline symmetrically rounded apex. Perithecium 155200
4855 wu. Receptacle 275390 uw. Outer appendage 135155x40 μ
at base, longest 20050 uw: inner appendage 506012 μ; longest
branchlets 120140 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
hostgenus 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 insertioncell. 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 stalkcells
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 olivebrown, 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 110128X3538 μ, the stalk
8X1520 uw. Receptacle 984045 μ, its basal cell 455020 μ.
Main appendages 20 μ, their branches 5075 uw. Antheridia 20 μ,
their stalkcells 1012 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. Insertioncell 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 250275 Χ 52 uw exclusive of the stalk (58X30 μ).
Receptacle 19020090 μ. Appendages to tips of branches about
175 μ; the antheridia 24 μ, their basal cell 204. Total length to tip
of perithecium average 500510 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 insertioncell 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. Insertioncell 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, 215235X45~47 μ, the’ stalk 2731X27 uy. Receptacle 215X
66 μ. Appendages, to tips of branches, longest, 1604. Antheridia
2327 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 strawyellow, 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. Insertioncell
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
sertioncell 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 snoutlike 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
110120X35+40 uw. Receptacle 156235 uw. Appendages 5060 μ.
Total length to tip of perithecium 220340 wu.
Growing in various positions on species of Heteroceros sent from
La Plata by P. Spegazzini in 1907, Nos. 167980. 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 insertioncell 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 toothlike 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 110155 35
40 »; the longer terminal appendage (longest) 20 uw. Total length
to tip of perithecium 235350 μ; greatest width 3866 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 120404383 yw. Longest
appendages about 95 uw. Total length to tip of perithecium 310425 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 stalkcell;
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 amberbrown 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 stalkcell 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 fewcelled 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
stalkcells 1518 u. Appendages to tips of branchlets 704. Total
length to tip of primary perithecium 150310 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
stalkcells 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 severalcelled 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 amberyellow, concolorous, becoming dirty amber
brown with age. Perithecium subsymmetrical; main body distin
guished from the slightly broader basal cellregion; 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, 95110 3040 yu,
the perithecial branch 25120 μ, total length, including branch, 130
250 μ; basal cellregion 2040 2530 u. Total length to tip of long
est branchlets (largest) 400 4. Supporting outgrowths 100275 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 spermcells 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 longconical with straight margins,
subtruncate or rounded, the apex symmetrical and subtended ex
ternally by a distinct prominence; the basal cellregion not distin
guished, its cells flattened around the ascogenic cells; borne on a
distinct short stalkcell. 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) 3035X3 yu. Perithecium 70X14 uw; the stalkcell
8X10 uw. Receptacle 230275X79 μ. Total length 290340 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 hyphalike 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 footcell
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 neckportion, 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 wallcell; 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 stalkcells 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 120130X 2327 “μ.
Receptacle 5565 yw. Footappendage 18 uw. Appendage 3550 μ,
its branches 7590 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 flaskshaped, 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
sixpapillate; the outer, lower wallcell slightly prominent below the
neck; the two upper basal cells extending upward beside the venter,
the stalkcell short and subtriangular. Perithecium 140200X38
44 μ, smallest 100X 25 μ, stalkcells and lower basal cells 20 u. Spores
in peritheclum 35X3.5 uw. Receptacle average 200 μι Appendage
60100 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
wallcells 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 95110X
THAXTER.— ARGENTINE LABOULBENIALEFS. 215
4045 μ, its longer appendage 60200 μ, the shorter 7078 μ. Ap
pendage 1385. Receptacle 80X35 4. Total length to tip of peri
thecium 175190 uw; to tip of inner appendage 310370 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 wallcells 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 reddishamber tinge. Peri
216 PROCEEDINGS OF THE AMERICAN ACADEMY.
thecium relatively very large and long, about fortyfive cells in each
row of wallcells; more or less evenly curved away from the append
age, deeply rich red amberbrown, except at its pale narrower base,
of the lower half characterized by a bellylike enlargement; the upper
half of nearly the same diameter throughout; the tip subtended
externally by a vesicular enlargement of one of the wallcells, 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 550700X 100110 uw (lower half) and 6575 uw (upper half),
the appendage 250350 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 potbellied perithecium.
Ceratomyces marginalis nov. sp.
Uniform dirty translucent amberbrown. 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 wallcells 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 90110 3545 μ, the longest appendage 100 μ.
The receptacle, including foot, 556030. Appendage 60X7 μ.
Total length to tip of perithecium 135150 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 amberbrown, 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 amberbrown, 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
amberbrown, 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 wallcells 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 31039080105 yu, the base
5060 4; the appendage 105170yu. Receptacle 7482X7578 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
wallcells 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 spermcells 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 X80390105 μ; its appendage without terminal branchlets 110
120 μ. Receptacle 25027 7080 μ distally. Appendage (broken)
160 1518 μ. Total length to tip of perithecium 700750 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 Aspergilluslike 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 flaskshaped. These conidia were also
borne individually on the sides of ordinary hyphae, being abstricted
in chains from flaskshaped 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 acremoniumlike 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, brownishred 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, brownishyellow 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 bottleshaped 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) goldenyellow bulbils,
which resembled those described and figured by Mattirolo in Melano
spora Gibelliana and which seem to be shortlived 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 400500 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 clubshaped with deep smokecolored 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 “sporeballs”’
(bulbils) and definite perithecia.
The sporeballs, 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
sporeballs gave sporeballs and small reddishbrown, hardwalled
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 reddishbrown 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 “ bulbillike”’ 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 culturestudies 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 reddishbrown or chocolatecolored clusters of cells, more or less
globose in shape, and usually 6580 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 Oidiumtype.
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
sporeform to sporeform. 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 dissectingneedles under a dissecting microscope, and after thor
ough washing in sterilized water on a flamed slide, was transferred to
a testtube 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 1520° 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 2530° C.
In Cambridge many were grown in an oven kept at various constant
temperatures, 2025° 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 preservejars with
cotton plugs. These were filled about onethird 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 sporepicking apparatus (’07) was employed.
Platecultures 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 lifecycle.
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 formgenus, 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 formgenus 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 128.
“Plants single or gregarious, often crowded, sessile or narrowed to a
stemlike base, turbinate, 310 mm. in diameter. Disk cupulate or
saucershaped, the hymenium pale fawncolor, 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 laceratedentate. Consistency subgelatinous, excipulum
pseudoparenchymatous throughout, of nearly rounded cells, 2025 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, 75100 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 tubecultures 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 dogdung 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 darkcolored, almost black, bulbils. The best sub
stratum for producing apothecia is bran, or rat or dogdung, although
they developed quite readily on sweetpotato 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 broomlike structure.
The bulbils— Often within fortyeight 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 1114, 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,
75100 μ in diameter, the cells in the center, usually considerably
larger, but all filled with protoplasm, without any definite differentia
tion of cellcontents 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 23,
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 56, 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, woollike 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 910, 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 310 mm. in diameter and 35 mm. in height; often in
groups and occasionally with a short stemlike base.
246 PROCEEDINGS OF THE AMERICAN ACADEMY.
When a portion of the hymenium containing some of the large
cells below the subhymenium 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 2025 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 coverglass. 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 (714 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 twentyfour 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 bulbillike 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 bulbillike 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, 160220 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 brownishred 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 “bulbillike” 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 ‘sporebulbils’ 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 3040 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, screwlike.” 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 twentyfour to
thirtysix 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 bottledshaped sterigmata. ‘The
ascospores are yellowish brown, becoming black or smokecolored,
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, tendrillike filament, something like
the paint as it is squeezed out of an artist’s painttube. These cirrose
structures may measure from 1018 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 halffilled 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 sporeforms 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 Melanosporalike 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 126.
Perithecia scattered or gregarious, superficial, membranous, semi
translucent, strawcolored to light brown, globose to pyriform,
350450 uw X 400500 yp, papilla surmounted by erect, somewhat
divergent, continuous setae, 100170 ww in length; primordium a
group of one or more intercalary cells; ascospores asymmetrical,
somewhat crescentshaped 10 X 25 yp, yellowish brown becoming
black; conidia abundant, hyaline, spherical to ovoid, on flaskshaped
sterigmata; bulbils yellowish brown, irregular in outline, 5060 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
agaragar, or even on a medium well supplied with food material if
made very hard (about 4050 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 47 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 strawcolored 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 testtubes, 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 25, 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 5060 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 2425), 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 tubecultures, 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 strawcolored, globose or slightly pyri
form, measuring 400500 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, 100170 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 1820, Plate 2).
Not infrequently a series of these swollen cells appears terminating a
branch and these become spherical and form a bulbillike 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 bottleshaped 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 sclerotiumlike. In each case, however,
the cells of the original bulbil retain their deep tancolor, 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 branagar 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 1624.
Perithecia scattered or gregarious, superficial, membranous, semi
transparent, strawcolored to brownish, globose to pyriform, 350
450 Χ 450550 μ, with a definite neck 85140y in length, terminal
setae 100170 uw in length, erect, somewhat divergent, continuous,
sharp, subulate; lateral setae on the neck and upper part of the peri
thecium; ascospores asymmetrical, somewhat crescentshaped 10 Χ
25 μ, yellowish brown becoming black; conidia common in tube
cultures, hyaline, spherical to ovoid, on flaskshaped sterigmata;
bulbils yellowish brown, irregular, normally 5060 mw in diameter,
sometimes 100 yu, primordium one or more intercalary cells. This
form is also said to produce conidia on secondary “ Harzialike”’
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 400500 μι in diameter, has a
definite neck 85140 yw in length, while MW. papillata which is slightly
smaller, seldom reaching 500 in diameter, has no neck but often
a papillalike 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 57 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 “ Harziatype”’ 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
bottleshaped 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 “ Harzialike”’ 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 2730; Puate 3, Figures 115.
Perithecia scattered or gregarious, superficial membranous, straw
colored or light brown, globose or subglobose, 250350 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 crescentshaped 14 Χ 28 μ, yellowish brown
becoming brownish black; conidia, hyaline, spherical to ovoid, on
flaskshaped sterigmata: bulbils yellowish brown, variable in size
70140 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 nutrientagar 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 2.57 μ.
The bulbils.— Scattered over the aerial hyphae and on the substra
tum are seen numerous small yellowishbrown 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
agaragar, 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 300400 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 yellowishbrown 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 910, 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 bottleshaped 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
(300400 yw in diameter) with a papilla about the ostiole, while the
latter is 400500 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 twocelled 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 Stysanuslike 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 bottleshaped
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 brownishblack 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,
250280 μι in diameter and 360450 uw (rarely 500 μὴ long. The neck
HOTSON.— CULTURE STUDIES OF FUNGI. 201
is well developed, 110200 μ in length. The ascospores differ from
those, already described, in being symmetrical. The other forms
have asymmetrical ascospores which are somewhat crescentshaped.
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 subglobose, 400500 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 brownishblack and asymmetrical.
The bulbils are yellowish, nearly spherical, 80150 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 bottleshaped sterigmata resembling
those of the melanosporous forms already referred to.
The chlamydospores, which measure 3240 X 2425 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 sclerotiumlike 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 147.
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 testtube 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 2937,
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 crosierlike 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 socalled
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 112 and 1527, 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 twentyfour 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 1218 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 1823.
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 socalled
“chlamydospores”’ which at maturity consist usually of one large,
thickwalled, chocolatebrown, 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 oneandahalf 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 2530 uw in diameter. Fig
ures 1823, 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 clampconnections;
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 110.
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 clampconnections,
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 strawcolored 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 growthactivity 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, ropelike 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 clampconnec
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 49, 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 testtube cultures of branagar 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, chalkwhite 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, 400500 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 Oidiumtype 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 1119.
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 clampconnections, 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 1315, 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 1115, 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
clearcut, 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 clampconnections 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 papillalike 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 basidialike 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 FORMGENUS
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, brownishyellow, central cells 2855 » in diameter.
P. immersa τι. sp.
2. Bulbils strawcolor, central cells 1020 μ in diameter.
P. irregularis τι. sp.
Bulbils dark psoray hyphae with clampconnections.
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, 26 central cells.
(a) Bulbils a dark brown......... P. coprophila.
(b) ΠΕΡΙ ΘΕ. P. rubidan. sp.
b. Cells of bulbil homogenous.
i. Bulbils brown 2136 win diam... P.sporotrichoides τι. sp.
il. ‘‘ steel gray 2136 » in diam... ..P. cinerea τι. sp.
2. Primordium not a spiral.
a. Bulbils large, 1007504 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
formgenera 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
formgenus.. 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 formgenus 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 sporelike 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 sporelike.
Papulospora immersa n. sp.
PuaTE 10, Figures 1725.
Mycelium white, septate, scanty, procumbent, growing in or on
the substratum; bulbils, light brownishyellow, irregular, 88150 μ
in diameter, but very variable, sometimes the long axis exceeding
260 μ, often immersed; central cells large 2855 w in diameter,
angular, with conspicuous oil globules; 5070 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 tubecultures 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 1922 are further stages in its develop
ment. In the latter case a more or less spherical bulbil is the result
(110148 » 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, 2835 μι in diameter,
“and more angular than those nearer the periphery.
Papulospora pannosa n. sp.
Puate 6, Figures 2025; Piate 8, Figures 2831; Pate 9,
Figures 1820.
Mycelium white at first, becoming dark smokecolored, 810 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, 200300 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 corncobs
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 tubecultures, appears somewhat shaggy, is white at first,
gradually becoming dark smokecolored, with prominent cross walls
which remain rigid when the cells collapse (Figure 31, Plate 8). The
hyphae which are 34 » in diameter when young and hyaline, gradually
increase in size until they are 810 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 2024,
Plate 6, and Figures 1819, 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 2830, 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 1117.
Mycelium white, more or less procumbent; bulbils hyaline, be
coming light strawcolor, somewhat spherical (140170 μ᾽ in diam.)
to irregular in outline (250300 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 strawcolor. They are usually
somewhat spherical in form, measuring 140170 » in diameter, but
HOTSON.— CULTURE STUDIES OF FUNGI. 277
frequently run into irregular sclerotiumlike bodies, 250300 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 1117, Plate 9, illus
trate the mode of development of this bulbil.
Papulospora spinulosa, n. sp.
PLATE 9, Figures 110.
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
latebrown, somewhat spherical, 5588 » in diameter, 5060 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 chocolatebrown
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 14, 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 onehalf turns, divides into cells from which short
lateral branches are produced, usually growing towards the center,
rarely outward (Figures 57, 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 910,
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 5588 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 116.
Mycelium white, septate, flocculent, abundant, persistent; bulbils,
dark brown, more or less spherical, 3040 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 brickred,
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 1315,
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 16, 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 1011, 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 12, 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 56, 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 ‘“ Aspergilluslike” 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 1227.
Mycelium white, procumbent or slightly aerial on some media;
bulbils more or less spherical, 3040 yw in diameter, with 25 large
central cells surrounded by a layer of empty cells which usually
retain their yellowish red color, at maturity the whole culture has a
brickred aspect; primordium a spiral, with many modifications;
conidia on bottleshaped 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 brickred 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 testtubes 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 314 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 1215, 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, 3040 w in diameter
with 15 (usually 2 or 3) large central cells each of which varies from
1014 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 bottleshaped 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 141.
Mycelium white, procumbent, usually scanty; bulbils dark choco
late colored, somewhat spherical or flattened, 2136 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 19, 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 1214, 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 1925, 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 bulbiltype 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 3536, 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 3841,
Plate 12) was continuously followed in Van Tieghem cells until
bulbils were produced on the mycelium derived from them.
The conidiophores (Figures 3536, 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 3941, Plate 12).
Papulospora cinerea n. sp.
PLATE 8, Figures 111.
Mycelium white, septate, procumbent, forming a felted mass over
the substratum; bulbils steelgray or slatecolored, somewhat spheri
cal and flattened, 2136 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 2136 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 slatecolored bulbils
being scattered among the white hyphal filaments, finally giving the
whole culture a bluish gray or steelgray 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
78, 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 117.
Mycelium septate, white, flocculent; bulbils light brown, nearly
spherical, 1521 μι in diameter, with a single large central cell sur
rounded by a single layer of empty colorless cells; primordium a
spiral, coiled crosierfashion.
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 crosierfashion 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
1014 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 1521 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 2436 hours and
send out one or two germ tubes, as shown in Figures 1516, 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.
Conidialike 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 120.
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, sclerotiumlike
HOTSON.— CULTURE STUDIES OF FUNGI. 289
bulbils. Pure cultures from a halfdozen 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 Aspergilluslike 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 1718, 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 brownishred 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 23, 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 24, Plate 7), then lightyellow, 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, sclerotiumlike 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.
Aspergilluslike fructification. Conidia are frequently produced
both on Aspergilluslike heads and also laterally, on the sides of the
hyphae (Figures 1011, 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).
Chlamydosporelike bodies occur quite frequently. They are
mostly intercalary but sometimes terminal (Figures 1316, 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
sporelike 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 Aspergilluslike 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 “Aspergilluslike” 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 sclerotiumlike 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 sclerotiumlike. 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
gilluslike 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 sclerotiumlike 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 Acremoniellalike 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
platecultures under various conditions. In no instance, however,
have bulbils or Aspergilluslike 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 shortstalked, 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 sporeforms 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 Acremoniellalike 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 socalled
“Harzia type” of fructification, as seen in M. cervicula and the
“ Aspergilluslike” 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 twocelled, warty, chlamydospores, this species also
produces conidia on “Aspergilluslike” 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 113.
Hyphae septate, hyaline, scanty, procumbent, 57 μι in diameter
(sometimes as much as 9 μὴ); bulbils dark redbrown 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,
119122 μ in diameter, composed of closely compact angular cells,
150200 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 sporeform 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 tancolored, 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, 119122 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 “stalklike” 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 tancolor, 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 strawcolored, 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 (1722 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 “sporemasses” was followed
carefully in Van Tieghem cells — some crushed, others not — and it
was found that nearly all the spores germinated in twentyfour hours,
some producing one, others two germ tubes, which were hyaline and
septate, becoming much branched (Figures 910, Plate 11). When
allowed to remain adherent, the sporemass 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 9b13, 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 6090 μ in
diameter, consist of an “aggregation of conidialike, inflated, spherical,
cells, 1218 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, brownishred 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 2434, 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 bulbillike 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 “sporeballs”’ of some of the Ustilaginales,
and the structures under consideration; in fact it would be quite
impossible to differentiate the sporeballs 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 sporelike 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
bulbillike 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 hymeniumlike 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
thickwalled 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 sporeforms might be mentioned which bear more or
less resemblance to bulbils, but those above enumerated are sufficient
for purposes of illustration. Before leaving bulbillike 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, bulbillike bodies which consist of
loosely coherent sporemasses 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 sporeballs 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
lifecycle 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 bulbilprimordium 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 nonsexual 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 nonsexual 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 smokecolored.
1. Bulbils 75100 in diam. margin even........ Cubonia bulbifera.
2. ~ 2003002." — * ‘irregular. . Papulospora pannosa.
Il. Bulbils yellowish red to dark brown.
1. Hyphae showing clampconnections.
1. Bulbils dark brown or chocolate colored.
i. Bulbils 6580. in diam. clamps conspicuous.
Corticium alutaceum.
11. “1251754 “ “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 clampconnections.
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 brownishyellow, dente cells 2855 up
ae {{ “ec
SLOW RL OT: Vat να ν Papulospora immersa.
(1). Bulbils strawcolored, central cells 1020% 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, 26 central
cells.
a Bulbils dark brown.
Papulospora coprophila.
8 Bulbils brick red.
Papulospora rubida.
§§ Cells homogenous, bulbils 2136 yu in
diam. brownish producing sporo
trichum spores.
Papulospora sporotrichoides.
HOTSON.— CULTURE STUDIES OF FUNGI. 303
b. Primordium not a spiral.
§ Bulbils large, 100750 wu, irregular.
Papulospora aspergilliformis.
70150 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, 3035 » in diam.
Papulospora candida.
IV. «steel gray, 2186 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 Microorganisms. 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 sporeformation 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. 251256. 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:
377483; pl. 1928. 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 Kryptogamenflora. 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. 405411. 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. 7683. 1865.
’80 Helicosporangium Karst. Deutschen Flora, p. 123. 1880.
’88. Bary’s “ Zweifelhafte Ascomyceten.” Hedwigia, Vol.
XXVII, pp. 182144. 2 Figs. 1888.
Kirchner, O.
’88. Ueber einen im Mohndél lebenden Pilz. Ber. deutsch.
bot. Gesell. GeneralVersammlung, 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.
125209, plates 1826. 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 TextBook 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. 121154, 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. 644647. 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 : 4552. 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. 2126, Taf.
2, Figs. 14. 1885.
’86. Untersuchungen iiber den biologischen und morpholo
gischen Werth der Pilzbulbillen. Verh. k. k. Zool. bot.
Ges. Wien, Vol. XXXVI, pp. 123135, plate 4. 1886.
EXPLANATION OF PLATES.
The figures of Plates 112 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 threequarters.
PLATE 1.
CUBONIA BULBIFERA.
Ficures 16. 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 1116. 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 2224. 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 16. Stages in the development of the bulbil.
Figure 7. A group of Chlamydosporelike intercalary cells.
Fiaures 810. 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 1320. Germinating ascospores.
Figures 21, 22. Forms produced in Van Tiegham cell cultures.
Fiaure 23. Conidia on flaskshaped 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 2730. Stages in the development of the bulbil.
PLATE 3.
MELANOSPORA ANOMALA.
Fiaures 112. Stages in the development of the perithecium.
Figure 12. Mature perithecium.
Figure 13. (a) Germinating ascospore showing a bottleshaped sterigma.
; (0) Bottleshaped 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 flaskshaped
sterigmata.
Figure 20. Mature perithecium.
Figure 21. Abnormal forms common among the hyphae.
Figure 22. Chlamydospores of the Acremoniella type.
Figures 23, 24. “Harzialike”’ fructification.
Hotson. — Cucture Stupies oF Funei. Piate 3.
Sse : SSE
Proc. Amer. Acav. Arts ano Sciences. Vor. XLVIII.
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PLATE 4.
PAPULOSPORA CANDIDA.
Ficures 1,2. Variation in the size of the conidia.
Ficures 312, and 1527. Stages in the germination of the conidia and the
development of the bulbil from them.
Ficures 2841. 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.
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PLATE δ.
PAPULOSPORA PARASITICA.
Ficures 114. 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. Conidialike bodies connected with the bulbil.
Ficures 35b, 36. Swollen intercalary cells.
ACROSPEIRA MIRABILIS.
Fiaures 1823. Stages in the development of the bulbil.
Figure 20. The endcell 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 2434. 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.
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PLATE 6.
GRANDINIA CRUSTOSA.
Ficure 1. Pustulate habit of the fructification.
Figure 2. Hymenium with basidiospores.
Figure 3. Basidiospore.
Ficures 410. Stages in the development of the bulbil.
Fieure 10. Mature bulbil with the same magnification as all the other mature
bulbils.
PAPULOSPORA ANOMALA.
FicurE 1117. 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 2024. 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.
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PLATE 7.
PAPULOSPORA ASPERGILLIFORMIS.
Ficures 14, & 6. Stages in the development of the bulbil.
Figure 5. A group of Chlamydosporelike bodies.
Ficure 7. A primordium that produces a very irregular bulbil.
Ficure 8. ‘Aspergilluslike’’ heads produced directly from the bulbil.
Ficures 912. Different forms of the “ Aspergilluslike” fructification.
Fraure 12. Abnormal conditions.
Ficures 1316. 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 110. 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 1216. Stages in the development of the bulbil.
Fiaures 25a27a, 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 2830. 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 17. 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 1117. Stages in the development of the bulbil.
Figure 17. A mature bulbil.
PAPULOSPORA PANNOSA.
Ficures 1820. 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 18. 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 1315. 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 1925. Stages in the development of the bulbil.
10.
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PLATE 11.
PAPULOSPORA POLYSPORA.
Ficures 17. 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, 1113. 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 19. 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 1216. 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 1925. Modifications in the development of the bulbils which are
hyaline.
Figures 2628. Semidiagrammatie 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 3234. Germinating bulbils.
Fiaures 2526. 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.
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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
pvt surface over the domain in question: Both the first and second
1 Bridgman, These Proceedings, 47, 439558 (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, 281288 (1909).
3 Parsons and Cook, Proc. Roy. Soe. A, 85, 332349 (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 temperaturepressure 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, 201217 (1909).
5 Bridgman, These Proceedings, 47, 319343 (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
chromevanadium 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 “pinchingoff 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, 6379 (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 pressureresistance 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 pv 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 aftereffects
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 pressureswe 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
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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.
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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. Such a general
treatment must be reserved for another paper, when the data for
more liquids are in hand.
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PROCEEDINGS OF THE AMERICAN ACADEMY.
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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 .  τας = ++ —
= 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 socalled 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, 199251 (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 636010,000 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 cellseries 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 cellseries, 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 inthis 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 hostgenera.
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 lipcell
protruding as a fingerlike 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 welldeveloped appressed
appendages or antheridia (?). Appendages subcylindrical, 816 X
2.5u. Perithecium 3040 X 8104, including terminal projection
THAXTER.— RICKIA AND TRENOMYCES. 367
(2.53 μ). Spores about 25 X 2.54. Total length to tip of perithe
cium 4070 μ, to tip of posterior branch 50175 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, 718 X 4μ.
Spores 30 X 3y. Perithecia 70 X 1820. Diameter of branches
810 μ, greatest length 460520 μ, in largest specimens. Basal and
subtending cell of primary appendage 1820 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 cellseries 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 cellseries 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, bladderlike, 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 lipcell forming a fingerlike straight
free process. Spores about 25 X 3 yu (in perithecium). Perithecia 30
35 X 810.54. Basal cell 1418 Χ 56.5y. Basal and subtending
cell of primary appendage 1617 X 7μ. Appendages 9X 4.5
7X 6u. Total length 4856 Χ 1416 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
cellseries 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 4852 X 1825. Total length 250350 Χ 18
32 μ, measured below the perithecium. Appendages 710 X 34 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
twentyfive 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 6585 X
20 yu. Appendages about 15 X 4y. Total length 200220 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 thirtysix 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 6075 X 4852 u. Appendages 1625 X
4, becoming brownish and subtended by the usual dark cuplike
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 cellseries. 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 spinelike 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 celldivisions
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 3540 X 15y. Basal cell 1850 X 615. Appendages
3560 Χ 46. Total length exclusive of stalk 4856 Χ 3538 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 90100 X 7885 μ. Basal cell including
foot 2835 X 1518 yu. Appendages 2435 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 fingerlike upgrowth, geniculate at its base; body of the
perithecium rather long and narrow, subsymmetrical. Spores (in
perithecium) 22 Χ 2.5. Perithecium 3040 Χ 1012 μ, not includ
374 PROCEEDINGS OF THE AMERICAN ACADEMY.
ing the projection which is 7 X 2u. Basal cell, including foot,
816 u. Total length of body 5066 Χ 3540 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 wallcell.
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 lipcell
projecting very slightly and somewhat irregularly. Spores 25 X 3y
(in perithecium). Perithecium 3840 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 domeshaped,
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 Peyritschiellatype, 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; 1525 X 3.5 y, the
primary one 3045 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 5565 X 1822 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
crestlike 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 4046 X 1620 yu. Basal cell, including
foot, 2833 X 9llyu. Total length, not including primary ap
pendage base, 1216 X 6Su. 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 lipcells 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 shortspathulate, 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 lipcell projecting as a short fingerlike
process. Appendages stout, yellowishbrown, 7 X 3.54. Spores
25 X 2.5. Perithecia 2224 * Sy. Basal cell including foot,
THAXTER.— RICKIA AND TRENOMYCES. 379
1416 X 68 u. Total length to tip perithecium 5070 Χ 2432 μ.
Basal and subtending cell of primary appendage 1215 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 fingerlike 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 lipcell forming a small, short, fingerlike
projection. Appendages very short and small, 5 X 2.5. Spores
20 X 2u. Perithecia 3032 8117 μ. Basal cell, including foot,
16 X 7u. Total length to tip of perithecium 5060 & 20304. The
free termination of the posterior series, including basal and subtend
ing cell of primary appendage 2540 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
cushionlike 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 lipcell 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
1820 X 2y. Perithecium 3034 X 68 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 lipcell
forming a short projection. Appendages slender, cylindrical, hyaline,
10 X 2u. Spores 24 X 2.8 μ. Perithecia 3545 X 812. Basal
cell including foot 12 X 45y. Total length to tip of perithecium
3545 X 812. 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,
Stalkcell 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 crestlike 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, 90110 yu. The main body 6080 X
2028 μ. Total length of rhizoid about 90100 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. 156467, 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 thickwalled, somewhat broader distally, about as long as
broad. Antheridia of the usual form suberect in a compact group,
six or more in number, the stalkcells 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 4550 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 140160
2025 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
225280 X 2835 μ; the stalk 70125 X 1ὸ or broader. Appendage
20 X 1014 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. 15567 (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 SPACETIME MANIFOLD OF RELATIVITY. THE
NONEUCLIDEAN GEOMETRY OF MECHANICS
AND ELECTROMAGNETICS.
By Epwin B. WILSON AND GILBERT N. LEwIs.
THE SPACETIME MANIFOLD OF RELATIVITY.
THE NONEUCLIDEAN 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 selfconsistent
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
foreshadowed 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 nonEuclidean 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 nonEuclidean geometry.
4. It is the purpose of the present work to develop the four dimen
sional nonEuclidean 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 nonEucli
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 nonEuclidean
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. 431444.
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 nonEuclidean 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 “quasistationary,’’ 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 NonEvuciipEaN GrEoMEtTRY IN Two DIMENSIONS.
Translation or the Parallel Transformation.
5. In discussing a nonEuclidean 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 (nonintersecting) 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 nonEuclidean geometry with which we are here concerned.
Another nonEuclidean 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 onetoone 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 nonparallel
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 paralleltransformation
(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.
NonEuclidean 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 nonEuclidean geometry from the
Euclidean.
The difference between our nonEuclidean 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 halfline (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 onetoone 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 pseudocircle.
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 nonsingular
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 pseudocircle 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 pseudocircle. Hence as the line
AB meets the pseudocircle 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 nonEuclidean 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 pseudocircle 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 pseudocircle 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 nonsingular 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 pseudocircle
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 scissorwise,
as may be seen, in Figure 11, where OC and
OD become respectively OC’ and OD’, OC”
and OD",  The pseudocircles 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
pseudocircle. 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 pseudocircle, in advancing
by rotation over equal angles, necessarily sweeps out equal areas
(by 16°). Hence by the familiar argument sectorial areas in any
pseudocircle are proportional to the angles at the center. The unit
angle will be taken as that angle which, in a pseudocircle of unit
radius, encloses a sectorial area of onehalf the unit area.
Vectors and Vector Algebra.
13. ‘Translation or the paralleltransformation 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 57).
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, 163181) 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 anticommutative 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 paralleltrans
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 paralleltransformation.
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 = nab =a(nb),
a:(b + c) = ab+ ac,
ab = 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=2pep+ y2qq + 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 pp remains invariant, then pp = n*p+p; whence it is ob
vious that pp = 0. In general: The inner product of any singular
vector by itself is zero, and this suffices to characterize a singular
vector. Hence rr reduces to
rer = 27ry pq.
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
rr 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 pq shows that the inner product of any
singular vector and any singular vector of the other class is equal to
onehalf 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
rr = n(p — q)(p+ q) = n(DP + 6 — ap — ad) = 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
ba = naa+ 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 nonsingular
vector by projecting one upon the other, it is necessary to project the
singular vector upon the nonsingular 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 kyky i  k, ky = kyk, = 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 pseudoscalar, and later extend this designation to ndimensional
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, kyKkyy = — 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 pseudoscalar 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 rr of a vector by itself is numerically
equal to the square of the interval of the vector r, the equation of
the unit pseudocircle of which the radii are all (y)lines is rer = 1;
and the equation of the conjugate unit pseudocircle of which the
radii are (6)lines is rer = —1. As the tangents to a pseudocircle
are perpendicular to the radu, they must be of opposite class. A
pseudocircle of which any tangent is a (6)line (the radii being (y)
lines) is called a (6)pseudocircle; and a pseudocircle 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 pseudocircle 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 onehalf the numerical value of dr, that
is, onehalf 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 pseudocircle 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 nonsingular 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 pseudocircle may be
expressed parametrically in terms of the
angle as follows. Let the given line be a
line of class (y) (the pseudocircle 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 pseudocircle 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 pseudocircle of class (y)), and if the angle ¢ had been of class
(65) measured from the aaxis 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 pseudocircle.
If now in general r be the radius of any pseudocircle, 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’;
rk, = 2; = x; coshy + ay’ sinhy,
—reky = χὰ = 2 sinhy + ay’ coshy, (7)
rk,’ = 2’ = 2, cosh — ay sinhy,
—rk, = 2, = — 2x,sinhy + x coshy.
Curvature in our nonEuclidean 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
pseudocircle in our nonEuclidean 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 pseudocircle,
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 nonEuclidean 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 nonEuclidean 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 faxis, or any line in the
atplane 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 nonEuclidean angles, which are their hyperbolic antitangents,
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 taxis 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
nonEuclidean 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
xaxis 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 faxis 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 taxis is parallel to the two (6)lines, and the zaxis 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
timeaxis, 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 spacetime 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 zaxis, 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 vfplane must move upon a pseudocircle.
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 taxis 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 taxis 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 aaxis 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 theextended force. Since our ordinary
definition of force is timerate of change of momentum, it is evident
that the zcomponent 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 atplane 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 socalled 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 spacetime 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
socalled 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 lightparticle
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 lightparticle 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 timeaxis 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
lightparticle 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 socalled material particles.
yy Let the vector a (Figure 18) be the vector
3 due to a lightparticle, 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 lightparticle, 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 lightparticle 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
onedimensional 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 NonEvucitipEAN GEOMETRY IN THREE DIMENSIONS.
Geometry, Outer and Inner Products.
25. We shall now consider a threedimensional space in which the
meaning of points, lines, planes, parallelism, and paralleltransforma
tion or translation are precisely as in ordinary Euclidean geometry.
In such a space, in addition to directed segments of lines or onedi
mensional vectors, we have directed portions of planes or twodimen
sional vectors. Any two portions of the same or parallel planes
having the same area and the same sign will be considered identical
twodimensional vectors, briefly designated as 2vectors. The ordi
nary onedimensional vectors may be called 1vectors for definiteness.
It is evident that the outer product axb of two 1vectors in space is no
longer a pseudoscalar but a 2vector lying in the plane determined
by the two vectors and having a magnitude equal to the area of their
parallelogram.
The addition of two 2vectors may be accomplished geometrically
in the following way. Take a definite segment of the line of inter
section of the planes of the 2vectors. In each plane construct on
this segment as one side parallelograms equal respectively to the given
2vectors. 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 lvector and a 2vector,??
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 2vectors will be designated by Clarendon capitals (except in
the case of the unit coordinate 2vectors).
WILSON AND LEWIS.— RELATIVITY. 427
determined. This outer product axA, being threedimensional in a
threedimensional space, is a pseudoscalar; 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 1vector and a 2vector 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 threedimensional nonEuclidean 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 nonEuclidean plane geometry above described, and the elements
of the cone are the fixed directions. Theperpendicular 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, nonEuclidean.
When all possible rotations, Euclidean and nonEuclidean, 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 nonEuclidean viewpoint is really analogous to
the sphere. The (6)lines cuts the twoparted hyperboloid; the (y)
lines, the oneparted.
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 lvector may be repre
sented as
a= ak, + ak» + a4Ky.
Similarly a 2vector 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 threedimensional
space these products are unit pseudoscalars.
In terms of their components we may now expand the two types
of outer product which occur in threedimensional 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
ndimensional space. In such a space we have vectors of 0, 1, 2,...,
n1, ndimensions, designated as OQvectors (or scalars), 1vectors,
2vectors, ..., (n—1)vectors, and nvectors (or pseudoscalars). The
outer product of a pvector and a qvector is a (p + q)vector; the
product vanishes if by translation the pvector and gvector can be
made to lie in space of less than p + q dimensions. The inner product
of a pvector and a qvector, where p = 4, will always be defined as a
(pq)vector. Thus whereas the inner product of a lvector by a
1vector is a scalar, the inner product of a 1vector and a 2vector is
a lvector.
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 1vector 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
1vectors 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 kyk, — —— 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 1vector and a 2vector will be a 1vector
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
lvectors we may make the convention
(axb)bi—= a(bb), 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
aA= (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 aA lies in the plane A is that the outer product of it
and A shall vanish, that is, (aA)xA = 0; the condition that it is perpen
dicular to @ is that the inner product of it and ἃ shall vanish, that is,
(aA)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 2vectors 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 2vectors is zero. The inner
product of a 2vector 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 2vectors
KK» = 1, KyeKiy = Κα = — 1,
Κι Κα = Kyoko = Κι το, = 0,
AA= A)?” = Ay a Ao;’, AB= “4.50 otk AysBiy —s 4..}.,.
29. Every 1vector a, or 2vector A in a threedimensional space
uniquely determines, except for sign, another vector (respectively
a 2vector or 1vector) 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 3vector or pseudoscalar 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
ata* = α(" — a” — a;’, aca = ar+ a? — aZ,
A*A* = Ao? + Ay? — Ap’, AA= 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
pseudoscalars is sometimes convenient. Since, by the rule of inner
multiplication, we have Kj4*Ky2.4 = —1, the complement of any
pseudoscalar 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 pseudoscalar; thus αὖ = akjy,.
All the special rules for the inner products of unit vectors (and
pseudoscalars) 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 1vector 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 lvector 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 lvectors 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 1vectors, then all the rules given
above may be expressed in the equations
(axb)+(axb) = (aa) (bb), (axbxc)  (axbxc) = (8.8) (bb) (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 2vector 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,
AA = — αι — a?’ + (a?+ αοὖ) = 0.
Conversely if we consider any 2vector
A= ΑΚ ΞΕ Ayky a= AosKo4,
such that
AA= 4." ἘΠῚ Ay? ar 4. = 0,
its complement is a singular line, and it is itself a simgular 2vector.
The standard form may be taken as
A= + VA24+ AoPKy + Auku + Adan.
The outer product of a singular vector by its complement, whether a
lvector or a 2vector, 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 nonexistent 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 nonEuclidean. 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 nonEuclidean 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 nonsingular lines. Every nonsingular
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
nonsingular 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 nonEuclidean 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 anticommutative 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 pseudosealar. Without changing this
equation, it is possible on both sides to arrange at the end, the sub
scripts of the pseudoscalar 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 nonEuclidean.
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 pseudosealar 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 pseudoscalar
(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 1vectors. Here
a(bxc) = (ac) Ὁ — (ab)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) = (cc) Ὁ — (eb)c,
b:(bxc) = (bc) b — (bb)c.
If now ἃ is any vector perpendicular to Ὁ and 6, we have identically
d(bxc) = (dc) Ὁ — (db)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(bC) = — b(aC). (35)
28 With the aid of inner and outer products we may write down expressions
for the components of a 1vector a along and perpendicular to another 1vector
b or a 2vector A. The components of a along Ὁ and perpendicular to Ὁ are
(a:b) b ἢ (axb)b
bb beb
The components of a along A and perpendicular to A are
. (aA)A ΓΝ (axA) +A,
AA AA
The component of the plane A on the plane B is
(A+B) B
BB
and a vector in the line of intersection of the two planes is
A*B or AB*.
WILSON AND LEWIS.— RELATIVITY. 439
For by (33) and (24),
[(axb)xe]* = [ax(bxc)]*,
(axb)c* = a(bxc)* = a(bc*).
But since ¢ is any lvector, its complement C is any 2vector.
Again,
ax(bC) = (axC)b — (ab)C. (36)
For by (84), (22), and (30),
[a+(bxe)]* = [(ac¢) b]* — [(ab) e]*,
(— 1.18 ax(bxe)* = (— 116 (axc*)b — (a+b) ο",
ax(bC) = (axC)b — (a+b) C.
Again,
(pices οἰ: Αὐ (37)
For from (35), (30), and (24),
[C(axb)]* = [(bC)a]*,
(— 1)?8—2) Cx(axb)* — (= 1)16—D (b+C)xa*,
— Cx(bxa)* = — Cx(bA)=(bC)xA.
Again
(bC)A = — b(CA) + C:(bxA). (38)
For from (36), (24), (32), (22), and (80),
[(bC)xa]* = — [b+(Cxa)]* + [C(ba)]*,
(bC)A = — b(Cxa)* + C(ba)*,
— b(CA) + (—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 lvector 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’ — (ab’) C,
and ax(b’’C) = (axC)b” — (ab”)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 1vectors are of frequent
occurrence. By (85) and (94) we may write immediately
ih. ry ee Dee lacc aed)
(axb)+(exd) = (ac) (bd) — (Ὁ 0) (ad) = Ne bed (39)
In a similar manner we may prove
δι ἃ ae af
(axbxc) + (dxexf) = Wed be bf),
σ΄ ce 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 = drV/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 nonEuclidean 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 1vector, 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 = dsuvVF or uVF = 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 lvector. 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 =drVF,
our rule of operation (30) shows that ν΄ = v*.
442 PROCEEDINGS OF THE AMERICAN ACADEMY.
If we consider a field of 1vectors, that is, a 1vector 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 lvector 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 2vector 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 2vector (instead of replacing it by the 1vector, its complement,
as is usually done in Euclidean geometry of three dimensions), but
also in that it represents nonEuclidean 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 1vector 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 1vector 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 3vector function, which clearly vanishes identi
cally. The first three are 1vector functions, and are connected by
the relation
ViVx'¥= V(Vf)—VVeE,
as may be seen by expansion or by the application of (34).
Kinematics and Dynamics in a Plane.
35. The three dimensional nonEuclidean 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 nonEuclidean 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 zdirection is constant, that in the ydirection 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 spacetime 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 twodimensional “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 nonNewtonian
as in ordinary mechanics.
Tue NonEvucLuipEAN 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 3spaces or planoids, and which is sub
ject to the same rules of translation or paralleltransformation as two
or three dimensional space. If a and Ὁ are two 1vectors, the product
axb is a 2vector, 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 1vectors lie ina plane. If now dis a fourth
l1vector, not lying in the 3space 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 pseudoscalar
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 lvectors are added in the ordinary way by the parallelogram
law. The same is true of two 2vectors if they intersect in a line, that
is, if they lie in the same 3space (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 2vectors the construction
previously given for the sum is not applicable, and it is indeed impossi
ble to replace the sum of the two 2vectors by a single plane vector.
The sum may, however, be replaced in an infinite variety of ways by
the sum of two other 2vectors. For if A and B are any two 2vectors,
and if a and Ὁ be two 1vectors drawn respectively in the planes of A
and B, then the 2vector 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 2vectors 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
2vectors, which is not reducible to a single uniplanar or simple 2
vector will be called a biplanar or double 2vector whenever it is
important to emphasize the difference. Since the analytical treatment
of these two kinds of 2vectors 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 a3vector. As two planoids
which have a point in common, intersect in a plane, a geometric
construction for the sum of two 3vectors may be given in a manner
which is the immediate extension of the rule for 2vectors in three
dimensional space. The sum of two 3vectors is always a simple
3vector.
In respect to rotation and to the classification of lines, planes, and
planoids, our four dimensional geometry will be nonEuclidean in
such a manner as to be the natural extension of the nonEuclidean
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 nonEuclidean 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
nonEuclidean 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 2vectors may
then be designated 53 as ky:, Kos, Kgs, Kos, κοι, Kix. There are furthermore
four coordinate unit 3vectors Ko3i, K3i4, Kiog, Kie3; and a unit pseudo
scalar Κρ. We may represent 1vectors, 2vectors and 3vectors,
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[γ031Κ53.
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 2vector 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,
aA = (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,
AB = — AyBy — AuBoy — AgsBo, + Avg Bo3 + Agi Bs: + AwBry,
AA = (— 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,,
A3B 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 2vector
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 1vectors lying in the two component planes.
The complement of a vector is again defined as its inner product
with the unit pseudoscalar ky31. ‘The complement of a 1vector is a
perpendicular 3vector, and viceversa; that of a simple 2vector is
the completely perpendicular 2vector. 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 2vector
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 pseudoscalar. 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= AA= m — n’, gq = AA* = — 2mn.
The quantities m, n may then be expressed in terms of p, 4, that is,
in terms of AA, 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 1vectors
and 2vectors would be true in higher dimensions, and this is true
even if the 2vectors 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 3vectors.
39. We have not yet considered those vectors whose inner products
with themselves are zero. The case of the 1vector, 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 2vector such that A+A = 0 satisfies the conditions
AxA = 2 (4...4. εἰν AgsA31 ΞΕ AA 12) Kyo34 = 0,
AA = — 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 2vector 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 2vector be biplanar, that is, if AxA is not zero, the condition
AA = 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, AA = πηι — n’.
Such a vector is singular only in an analytical sense.
452 PROCEEDINGS OF THE AMERICAN ACADEMY.
The complement of a singular 1vector is a 3vector 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
1vector, the area of a singular 2vector, and the volume of a singular
3vector are zero when compared with nonsingular intervals, areas,
and volumes.
The visualization of the geometrical properties of a four dimensional
and especially of a nonEuclidean 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 1vector.
38 The geometry in a singular planoid is analogous to that in a singular plane
(§31). In this3space there are two classes of lines, singular lines, all of which
are parallel to each other, and nonsingular 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 nonsingular
(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 nonsingular planoid.
WILSON AND LEWIS. — RELATIVITY. 453
We may therefore write the following equations. The result of
applying © to a scalar function F is a 1vector 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 1vector 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 1vector f is a
2vector 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 1vector.
_ (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 3vector.
_ (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
(Oa)* = — 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 lvector, it is not a lvector 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 1vector.
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 2vector element of area (a pseudosealar) and
Vf is a pseudosealar (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 fastvt,
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
fist= 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 fescon  f fasvv.
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 pvector 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 nonEuclidean 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 1vector function. Then
[st = — f { @so)t = ff [ as#).
This is the extended Stokes’s theorem. Again let p = 3 and apply
the operator by outer multiplication to a 1vector function. Then
{{{{Φ4 Γ{{{{.0»« Ὡ[7 ἘΠ
This is the extended Gauss’s theorem, where d= represents a differ
ential (pseudoscalar) element of four dimensional volume.
In these cases also the same equations apply in Euclidean and in
our nonEuclidean space. If, however, we write these two equations
in nonvectorial form, they become in the nonEuclidean 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 1645 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 nonEuclidean 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 2vector 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 3space 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
Op= 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
2vector xp is a simple plane vector in the plane of the point Q
and of w. The 1vector 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 (1w)*
are the expressions for the fields in terms of 1 and w.
Let us now choose arbitrarily a timeaxis ky, and then the perpen
dicular planoid is our three dimensional space. We may resolve our
lvector and 2vector fields as follows.
aie (70)
lew’
p — «τὸ δὰ πον cod _ La ΞΕ κι τ SS
lw (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 = —1w.
The vector p drawn at Q parallel to w and of magnitude 1/R is the
extended vector potential at Q due tow. The 2vector P lies in the
plane 0/Q, and is equal in magnitude to 1/R?. The arbitrarily cho
sen timeaxis 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/(lw) 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 drOw = 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 = — © (1w), we may write
O(lw) = (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 ]
> (1w) = (1 { R Iw)w— le+l,
or, performing the multiplication by w,
OR= —O(1w) = —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 lw = — R.
As 1 varies with R, the parts of +p = 0, and therefore
OOrP = 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 1vector potential
would therefore be
P= ο( πο  (hl,
where R = —lw. ‘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
OOp = 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 spacetime 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 spacetime 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 timeaxis along w, and the spaceaxis 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 2vector dr«1 is independ
ent of the way in which dr was drawn and the 1vector (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 drl is equal to adrk, = 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 lightparticle (§ 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 1lvector fields to a consideration
of equivalent discontinuous distributions of 1vectors in the case of
all nonsingular 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 3vector 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 1vectors 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 3vector 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
foursided 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 lvector 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 crosssection 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 timespace 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 timeaxis 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 2vector field produced by an electron, whether in uniform or
accelerated motion, is obtained immediately from (81)—(83).
Mm << — ες ΡΞ : a [wxc — nxc+ nc 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 2vector
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
2vector M is thus split automatically into two 2vectors, of which
one passes through the timeaxis 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 2vector M is precisely the “ Vektor zweiter Art”
which Minkowski introduced to express the electric and magnetic
forces.
Out of H and E spacial 1vectors h and e may be obtained by the
equations
hi H:kps, e = Eky. (98)
Then h is the threedimensional complement of H, and e the inter
section of E with threedimensional 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 2vector 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 =