The growth and development of an Actinomyxidian have been recently

worked out by Caullery and Mesnil in the case of _Sphaeractinomyxon

stolci_. A noteworthy point is the differentiation of an external

(covering) cellular layer, which affords, perhaps, the nearest approach

to distinct tissue-formation known among Protozoa. This envelope is

formed soon after nuclear multiplication of the young trophozoite has

begun, and is constituted by two nuclei and a thin, peripheral layer of

cytoplasm. It remains binuclear throughout the entire period of

development, and serves as a delicate cyst-membrane. The multiplication

of the internal nuclei is accompanied by a corresponding division of the

cytoplasm; so that instead of a multinucleate or plasmodial condition,

distinct uninucleate cellules are formed, up to sixteen in number. These

cellules, as a matter of fact, are sexual elements or gametes; and eight

of them can be distinguished from the other eight by slight differences

in the nuclei. The gametes unite in couples, each couple being most

probably composed of dissimilar members: in other words, conjugation is

slightly anisogamous. Each of these eight copulae gives rise to a spore.

As the name of the order implies, there are always eight spores formed.

These differ from other Endosporan spores in having invariably a ternary

symmetry and constitution (fig. 9). The wall of the spore is composed of

three valves, each formed from an enveloping cell, and three capsular

cells, placed at the upper or anterior pole, and containing each a

polar-capsule, visible in the fresh condition. The valves are usually

prolonged into processes or appendages, whose form and arrangement

characterize the genus; but in _Sphaeractinomyxon_ the spore is

spherical and lacks processes. The sporoplasm may be either a plasmodial

mass, with numerous nuclei, or may form a certain number of uninuclear

sporozoites. A remarkable feature in the development of the spore is

that the germinal tissue (sporoplasm) arises separate from and outside

the cellules which give rise to the spore-wall; later, when the

envelopes are nearly developed, the sporoplasm penetrates into the

spore.

Four genera have been made known. (1) _Hexactinomyxon_, Stolc. Spores

having the form of an anchor with six arms; sporoplasm plasmodial,

situate near the anterior pole of the spore. One sp. _H.

psammoryctis_, from _Psammoryctes_. (2) _Triactinomyxon_, St. Spores

having the form of an anchor with three arms; distinct sporozoites,

disposed near the anterior pole. _T. ignotum_, with eight spores, from

_Tubifex tubifex_, and also from an unspecified Tubificid; another

sp., unnamed, with 32 sporozoites, also from T. t. (3)

_Synactinomyxon_, St. Spores united to one another, each having two

aliform appendages; sporoplasm plasmodial. One sp., _S. tubificis_,

from _T. rivulorum_. (4) _Sphaeractinomyxon_, C. and M. Spores

spherical, without aliform prolongations; sporoplasm gives rise to

very many sporozoites, occupying the whole spore. One sp., _S.

st_olci, from _Clitellio_ and _Hemitubifex_.

[Illustration: From Lankester's _Treatise on Zoology_, vol. Protozoa.

FIG. 9.--Spores of Actinomyxidia (after Stolc).

a, _Hexactinomyxon psammoryctis_ (par. _Psammoryctes barbatus_).

b, _Synactinomyxon tubificis_ (par. _Tubifex rivulorum_); the mass

of united spores.

c, _Triactinomyxon ignotum_ (par. _Clitellio_, sp.).

d, Upper portion of _Hexactinomyxon_, showing two of the three polar

capsules, one with filament discharged.]

[Illustration: From Wasielewski's _Sporozoenkunde_.

FIG. 10.--A. Sarcosporidia in the ox; a transverse section of the

oesophagus, natural size, showing the parasites in the outer (a, b, c,

d, e) and inner (f, g, h) muscular coats.

B. Longitudinal section of a muscle-fibre containing a Sarcosporidian

parasite, X 60.]

3. Order--Sarcosporidia. With the exception of one or two forms

occurring in reptiles, these parasites are always found in warm-blooded

Vertebrates, usually Mammals. They are of common occurrence in domestic

animals, such as pigs, sheep, horses and (sometimes) cattle. A

Sarcosporidian has also been described from man. The characteristic

habitat is the striped muscle, generally of the oesophagus (fig. 10, A)

and heart, but in acute cases the parasites overrun the general

musculature. When this occurs, as often happens in mice, the result is

usually fatal. Unless, however, the organisms thus spread throughout the

body, the host does not appear to suffer any serious consequences. In

addition to the effects produced by the general disturbance to the

tissues, the attacked animals have apparently to contend--at any rate in

the case of _Sarcocystis tenella_ in the sheep--with a poison secreted

by the parasite. For Laveran and Mesnil (27) have isolated a toxine from

this form, which they have termed sarcocystin.

In the early stages of growth, a Sarcosporidian appears as an elongated

whitish body lodged in the substance of a muscle-fibre; this phase has

long been known as a "Miescher's tube," or _Miescheria_. The youngest

trophozoites that have been yet observed (by Bertram, 1) were

multinucleate (fig. 11, A), but there is no reason to doubt that they

begin life in a uninuclear condition. The protoplasm is limited by a

delicate cuticle. With growth, organellae corresponding to the

Myxosporidian pansporoblasts are formed by the segregation internally of

little uninuclear spheres of protoplasm. At the same time, a thick

striated envelope is developed around the parasite, which later comes to

look like a fur of fine filaments. The probable explanation of this

feature (given by Vuillemin, 44) is that it is due to the partial

breaking down of a stiff, vertically (or radially) striated external

layer (fig. 12, A), such as is seen in _Myxidium lieberkuhnii_.

Immediately internal to this is a thin, homogeneous membrane, which

sends numerous partitions or septa inwards; these divide up the

endoplasm into somewhat angular chambers or alveoli (fig. 12). In each

chamber is a pansporoblast, which divides up to produce many spores;

hence the spores formed from different pansporoblasts are kept more or

less separate. The pansporoblasts originate, in a growing

Sarcosporidian, at the two poles of the body, where the peripheral

endoplasm with its nuclei is chiefly aggregated. More internally,

spore-formation is in progress; and in the centre, pansporoblasts full

of ripe spores are found.

By this time the parasite has greatly distended the muscle-fibre in

which it has hitherto lain, absorbing, with its growth, practically all

the contractile-substance, until it is surrounded only by the sarcolemma

and sarcoplasm. It next passes into the adjacent connective-tissue, and

in this phase has been distinguished from _Miescheria_ as _Balbiania_,

under the impression that the two forms were quite distinct. In the

later stages, the parasite may become more rounded, and a cyst may be

secreted around it by the host's tissue. In these older forms, the most

centrally placed spores degenerate and die, having become over-ripe and

stale.

[Illustration: After Bertram, from Wasielewski's _Sporozoenkunde_.

FIG. 11.--Stages in the growth of _Sarcocystis tenella_ of the sheep. A,

Youngest observed stage in which the radially striated outer coat has

not appeared; the body of the trophozoite is already divided into a

number of cells or pansporoblasts (k). B and C, Older stages with

numerous pansporoblasts and two envelopes, an inner membrane and an

outer radially striated layer.]

With regard to the spores themselves and what becomes of them, our

knowledge is defective. Two kinds of reproductive germ have been

described, termed respectively _gymnospores_ (so-called sporozoites,

"Rainey's corpuscles") and _chlamydospores_, or simply spores. It seems

probable that the former serve for endogenous or auto-infection, and the

latter for infecting fresh hosts. Unfortunately, however, both kinds of

germ are not yet known in the case of any one species. The gymnospores,

which are the more commonly found (e.g. in _S. muris_, _S. miescheriana_

of the pig, and other forms), are small sickle-shaped or reniform

bodies which are more or less amoeboid, and capable of active movement

at certain temperatures. They appear to be naked, and consist of finely

granular protoplasm, containing a single nucleus and one or two

vacuoles. The chlamydospores, or true spores, occur in _S. tenella_ of

sheep (fig. 13), and have been described by Laveran and Mesnil (26).

They also are falciform, but one extremity is rounded, the other

pointed. There is a very thin, delicate membrane, most unlike a typical,

resistant spore-wall; and the spores themselves are extremely fragile

and easily acted upon and deformed by reagents, even by distilled water.

The rounded end of the spore contains a large nucleus, while at the

other end is an oval, clear space, which, in the fresh condition, shows

a distinct spiral striation. The exact significance of this structure

has been much debated. In position and appearance it recalls the

polar-capsule of a Myxosporidian spore. The proof of this interpretation

would be the expulsion of a filament on suitably stimulating the spore;

while, however, some investigators have asserted that such a filament is

extruded, this cannot be regarded as at all certain. Hence it is still

doubtful whether this striated body really corresponds to a

polar-capsule.

[Illustration: From Wasielewski's _Sporozoenkunde_.

_Fig._ 12.--A, _Sarcocystis miescheriana_ (Kuhn) from the pig: late

stage in which the body has become divided up into numerous chambers or

alveoli, each containing a number of germs.

B, _Sarcocystis_ of the ox: section of a stage similar to fig. 12. a,

Substance of muscle-fibre; b, envelope of parasite; c, nuclei of the

muscle; d, parasitic germs (gymnospores); e, walls of the alveoli. In

the peripheral alveoli are seen immature germs.]

[Illustration: (After Laveran and Mesnil, from Lankester's _Treatise on

Zoology_, vol. Protozoa.)

FIG. 13.--Spores of _Sarcocystis tenella_, Raill., from the sheep.

a, Spore in the fresh condition, showing a clear nucleus (n) and a

striated body or capsule (c).

b, Stained spore; the nucleus (n) shows a central karyosome; the

striations of the polar capsule (c) are not visible.]

Nothing whatever is known as to the natural means by which infection

with Sarcosporidia is brought about. Smith (39) showed that mice can be

infected with _Sarcocystis muris_ by simply feeding them on the flesh of

infected mice. It is not very likely, however, that this represents the

natural mode, even in the case of mice; and it certainly cannot do so in

the case of Herbivora. The difficulty in the way is the delicacy of the

spores, which seem totally unfitted to withstand external conditions. It

may be that some alternative (intermediate) host is concerned in

dispersal; but this has yet to be ascertained.

All known Sarcosporidia are included in a single genus _Sarcocystis_,

Lank. (= _Miescheria_ + _Balbiania_, Blanchard.) Some of the principal

species are: _S. miescheriana_, from pigs; _S. tenella_, from sheep;

_S. bertrami_, from horses; _S. blanchardi_, from Bovines; _S. muris_,

from mice; _S. platydactyli_, from the gecko; and lastly, _S.

lindemanni_, described from man.

4. Order--Haplosporidia. The Sporozoa included in this order are

characterized by the general simplicity of their development, and by the

undifferentiated character of their spores. The order includes a good

many forms, whose arrangement and classification have been recently

undertaken by Caullery and Mesnil (15), to whom, indeed, most of our

knowledge relating to the Haplosporidia is due. The habitat of the

parasites is sufficiently varied; Rotifers, Crustacea, Annelids and

fishes furnishing most of the hosts. A recent addition to the list of

Protozoa causing injury to man, a Haplosporidian, has been described by

Minchin and Fantham (29d), who have termed the parasite

_Rhinosporidium_, from its habitat in the nasal septum, where it

produces pedunculate tumours.

[Illustration: From Minchin, in Lankester's _Treatise on Zoology_, vol.

Protozoa.

FIG. 14.--_Bertramia Asperospora_ (Fritsch) from the body-cavity of

_Brachionus_. X 1040.

a, Young form with opaque, evenly-granulated protoplasm and few

refringent granules; the nuclei (n) are small, and appear to be

surrounded each by a clear space.

b and c, Full-grown specimens with large nuclei and clearer

protoplasm, containing numerous refringent granules (r. gr.).

d and e, Morula stages, derived from b and c by division of the body

into segments centred round the nuclei, each cell so formed being a

spore. Between the spores a certain amount of intercellular substance

or residual protoplasm is left, in which the refringent granules seem

to be embedded. The morula may break up forthwith and scatter the

spores, or may first round itself off and form a spherical cyst with

a tough, fairly thick wall.

f, Empty, slightly shrunken cyst, from which the spores have escaped.

g, Free spore or youngest unicellular trophozoite.

h, i, j, Commencing growth of the trophozoite, with multiplication of

the nuclei, which results ultimately in forms such as a and b.]

_Bertramia_, a well-known parasite of the body-cavity of Rotifers, will

serve very well to give a general idea of the life-cycle so far as it

has yet been made out (fig. 14). The trophozoite begins life as a small,

rounded uninucleate corpuscle, which as it grows, becomes multinucleate.

The multinuclear body generally assumes a definite shape, often that of

a sausage. Later, the protoplasm becomes segregated around each of the

nuclei, giving the parasite a mulberry-like aspect; hence this stage is

frequently known as a morula. The uninuclear cellules thus formed are

the spores, which are ultimately liberated by the break-up of the parent

body. Each is of quite simple, undifferentiated structure, possesses a

large, easily-visible nucleus, and gives rise in due course to another

young trophozoite. In some instances, as described by Minchin, the

sporulating parasite becomes rounded off and forms a protective cyst,

doubtless for the protection of the spores during dissemination.

In some forms (e.g. _Haplosporidium_ and _Rhinosporidium_) the

spore-mother-cells, instead of becoming each a single spore, as in

_Bertramia_, give rise to several, four in the first case, many in the

latter. Sometimes, again, the spore, while preserving the essentially

simple character of the sporoplasm, may be enclosed in a spore-case;

this may have the form of a little box with a lid or operculum, as in

some species of _Haplosporidium_, or may possess a long process or tail,

as in _Urosporidium_ (fig. 15).

[Illustration: From Caullery and Mesnil, _Archives de zoologie

experimentale_, vol. 4, 1905, by permission of Schleicher Freres et

Cie, Paris.

FIG. 15.--Spores of various Haplosporidia.

1. _Haplosporidium heterocirri_:

a, on liberation;

b, after being in sea-water.

2, _H. scolopli_.

3, _H. vejdovskii_.

4, _Urosporidium fuliginosum_:

a, surface-view;

b, side-view. X 1000.]

The _Haplosporidia_ are divided by Caullery and Mesnil into three

families, _Haplosporidiidae_, _Bertramiidae_ and _Coelosporidiidae_;

one or two genera are also included whose exact position is doubtful.

(a) _Haplosporidiidae_: 3 genera, _Haplosporidium_, type-species _H.

heterocirri_; _Urosporidium_, with one sp., _U. fuliginosum_; all

parasitic in various Annelids; and _Anurosporidium_, with the species

_A. pelseneeri_, from the sporocysts of a Trematode, parasitic on

_Donax_.

(b) _Bertramiidae_: 2 genera, _Bertramia_, with _B. capitellae_ from

an Annelid and _B. asperospora_, the Rotiferan parasite above

described; and _Ichthyosporidium_, with _I. gasterophilum_ and _I.

phymogenes_, parasitic in various fish.

(c) _Coelosporidiiae_: genera _Coelosporidium_, type-species _C.

chydoriclola_; and _Polycaryum_, type-species _P. branchiopodianum_.

These forms are parasitic in small Crustacea. The genus _Blastulidium_

is referred, doubtfully, by Caullery and Mesnil to this family; but

certain phases of this organism seem to indicate rather a vegetable

nature.

The genus _Rhinosporidium_ should probably be placed in a distinct

family. The only species so far described is _R. kinealyi_ from the

nasal septum of man, to which reference has above been made. Another

form, _Neurosporidium cephalodisci_, agreeing in some respects with

_Rhinosporidium_, has been described by Ridewood and Fantham (37a)

from the nervous system of _Cephalodiscus_.

A parasite whose affinities are doubtful, but which is regarded by

Caullery and Mesnil as allied to the Haplosporidia, is the curious

parasite originally described by Schewiakoff as "endoparasitic tubes"

of _Cyclops_; it has been named by Caullery and Mesnil,

_Scheviakovella_. This organism is remarkable in one or two ways: it

possesses a contractile vacuole; the amoeboid trophozoites tend to

form plasmodia; and the spores, of the usual simple type, may

apparently divide by binary fission.

5. There remain, lastly, certain forms, which are conveniently grouped

together as "Sporozoa _incertae sedis_," either for the reason that it

is impossible to place them in any of the well-defined orders, or

because their life-cycle is at present too insufficiently known.

Serosporidia is the name given by Pfeiffer to certain minute parasites

of the body-cavity of Crustacea; they include _Serosporidium_,

_Blanchardina_ and _Botellus_. _Lymphosporidium_, a form with

distributed nucleus, causing virulent epidemics among brook-trout, is

considered by Calkins(3) to be suitably placed here. Another parasite of

lymphatic spaces and channels is the remarkable _Lymphocystis_,

described by Woodcock (46), from plaice and flounders, which in some

respects rather recalls a Gregarine. The group Exosporidia was founded

by Perrier to include a peculiar organism, ectoparasitic on Arthropods,

to which the name of _Amoebidium_ had been given by Cienkowsky. It has

recently been shown, however, that this organism is most probably an

Alga. Another genus, _Exosporidium_, described by Sand (38), is placed

at present in this group. For details of the structure of these forms

and others like _Siedleckia_, _Toxosporidium_, _Chitonicium_

_Joyeuxella_ and _Metschnikovella_, a comprehensive treatise on the

Sporozoa, such as that of Minchin, should be consulted.

To complete this article, it will be sufficient to mention various

enigmatical bodies, associated with different diseases, which are

regarded by their describers as Protozoa. Among such is the

"_Histosporidium carcinomatosum_" of Feinberg, which he finds in

cancerous growths. _Cytoryctes_, the name given to "Guarnieri's bodies"

in small-pox and vaccinia, has been recently investigated by Calkins

(3a), who has described a complex life-cycle for the alleged parasite.

Other workers, however, such as Siegel, give a quite different account

of these bodies, and, moreover, find similar ones in scarlet-fever,

syphilis, &c.; while yet others (e.g. Prowazek) deny that they are

parasitic organisms at all.

BIBLIOGRAPHY.--(For general works see under SPOROZOA.) (1) Bertram,

"Beitrage zur Kenntnis der Sarcosporidien," _Zool. Jahrb. Anat._ 5,

1902; (2) L. Brasil, "Joyeuxella toxoides," (n.g., n.sp.), _Arch.

zool. exp._ N. et R. (3) 10, p. 5, 7 figs., 1902; (3) G.N. Calkins,

"Lymphosporidium truttae," (n.g., n.sp.), _Zool. Anz_. 23, p. 513, 6

figs., 1903; (3a) ib. _The Life-History of Cytoryctes Variolae_;

Guarnieri, "Studies path. etiol. variola," _J. Med. Research_ (Boston,

1904), p. 136, 4 pls.; (3b) M. Caullery and A. Chappellier,

"Anurosporidium pelseneeri, (n.g., n.sp.), Haplosporidie," &c., _C. R.

soc. biol._ 60, p. 325, 1906; (4) M. Caullery and F. Mesnil, "Sur un

type nouveau" (_Metchnikovella_, n.g.), _C. R. ac. sci._ 125, p. 787,

10 figs., 1897; (5) ib. "Sur trois Sporozoaires parasites de la

Capitella," _C. R. soc. biol_. 49, p. 1005, 1877; (6) ib. "Sur un

Sporozoaire aberrant" (_Siedleckia_, n.g.), op. cit. 50, p. 1093, 7

figs., 1898; (7) ib. "Sur le genre Aplosporidium" (nov.), op. cit. 51,

p. 789, 1899; (8) ib. "Sur les Aplosporidies," _C. R. ac. sci._ 129,

p. 616, 1899; (9) ib. "Sur les parasites intimes des Annelides"

(_Siedleckia_, _Toxosporidium_), C. R. ass. franc., 1899, p. 491,

1900; (10) ib. "Sur un type nouveau (_Sphaeractinomyxon_, n.g.)

d'Actinomyxidies," _C. R. soc. biol._ 56, p. 408, 1904; (11) ib.

"Phenomenes de sexualite dans le developpement des Actinomyxidies,"

op. cit. 58, p. 889, 1905; (12) ib. "Recherches sur les

Actinomyxidies," _Arch. Protistenk._ 6, p. 272, pl. 15, 1905; (13) ib.

"Sur quelques nouvelles Haplosporidies d'Annelides," C. R. soc. biol.

58, p. 580, 6 figs., 1905; (14) ib. "Sur des Haplosporidies parasites

de poissons marins," ib. p. 640, 1905; (15) ib. "Recherches sur les

Haplosporidies," _Arch. zool. exp._ (4) 4, p. 101, pls. 11-13, 1905;

(16) L. Cohn, "Uber die Myxosporidien von Esox lucius," _Zool. Jahr.

Anat_. 9, p. 227, 2 pls., 1896; (17) ib. "Zur Kenntniss der

Myxosporidien," Centrbl. Bakt. 1, Orig. 32, p. 628, 3 figs., 1902;

(18) ib. "Protozoen als Parasiten in Rotatorien," Zool. Anz. 25, p.

497, 1902; (19) F. Doflein, "Uber Myxosporidien," Zool. Jahr. Anat.

11, p. 281, 6 pls., 1898; (20) ib. "Fortschritte auf dem Gebiete der

Myxosporidienkunde," _Zool. Centrbl_. 7, p. 361, 1899; (21) R. Gurley,

"The Myxosporidia," _Bull. U.S. Fish. Comm., 1892_, p. 65, 47 pls.,

1894; (22) E. Hesse, "Sur une nouvelle Microsporidie tetrasporee du

genre Gurleya," _C. R. soc. biol._ 55, p. 495, 1903; (23) ib.

"Thelohania legeri" (n.sp.), op. cit. 57, pp. 570-572, 10 figs., 1904;

(24) ib. "Sur Myxocystis Mrazeki Hesse," &c., op. cit. 58, p. 12, 9

figs., 1905; (25) A. Laveran and F. Mesnil, "Sur la multiplication

endogene des Myxosporidies," op. cit. 54, p. 469, 5 figs., 1902; (26)

ib. "Sur la morphologie des Sarcosporidies," op. cit. 51, p. 245,

1899; (27) ib. "De la Sarcocystin," op. cit. p. 311, 1899; (28) L.

Leger, "Sur la sporulation du Triactinomyxon," op. cit. 56, p. 844, 4

figs., 1904; (29) ib. "Considerations sur ... les Actinomyxidies," op.

cit. p. 846, 1904; (29a) L. Leger and E. Hesse, "Sur une nouvelle

Myxosporidie, Coccomyxa, n.g.," _C. R. ac. sci._, 1st July 1907; (29b)

ib. "Sur la structure de la paroisporale des Myxosporidies," op. cit.

142, p. 720, 1906; (29c) A. Lutz and A. Splendore, "Uber 'Pebrine' and

verwandte Mikrosporidien," _Centrbl. Bakt_. 1, 33, Orig. p. 150, 1903,

and 36, Orig. p. 645, 2 pls., 1904; (29d) E.A. Minchin and H.B.

Fantham, "Rhinosporidium kinealyi" (n.g., n.sp.), _Q. J. Micr. Sci._

49, p. 521, 2 pls., 1905; (30) A. Mrazek, "Uber eine neue

Sporozoenform" (_Myxocystis_), _S. B. Bohm. Ges._ 8, 5 pp., 9 figs.,

1897; (31) ib. "Glugea lophii," Doflein, op. cit. 10, 8 pp., 1 pl.,

1899; (32) C. Perez, "Sur un organisme nouveau, Blastulidium," _C. R.

soc. biol._ 55, p. 715, 5 figs., 1903; (33) ib. "Sur nouvelles

Glugeidees," op. cit. 58, pp. 146-151, 1905; (34) ib. "Microsporidies

parasites des crabes," _Bull. sta. biol. d'Arcachon_, 8, 22 pp., 14

figs., 1905; (35) W.S. Perrin, "Pleistophora periplanetae," _Q. J.

Micr. Sci._ 49, p. 615, 2 pls., 1906; (36) L. Plate, "Uber einen

einzelligen Zellparasiten" (_Chitonicium_), _Fauna Chilensis_, 2, pp.

601, pls., 1901; (37) M. Plehn, "Uber die Drehkrankheit der

Salmoniden" (_Lentospora_, n.g.), _Arch. Protistenk_. 5, p. 145, pl.

5, 1904; (37a) W.J. Ridewood and H.B. Fantham, "Neurosporidium

cephalodisci, n.g., n.sp.," _Q. J. Micr. Sci._ 51, p. 81, pl. 7, 1907;

(38) R. Sand, "Exosporidium marinum" (n.g., n.sp.), _Bull. soc. micr.

belge_, 24, p. 116, 1898; (39) T. Smith, "The production of

sarcosporidiosis in the mouse," &c., _J. Exp. Med._ 6, p. 1, 4 pls.,

1901; (40) W. Stempell, "Uber Thelohania mulleri," _Zool. Jahr. Anat._

16, p. 235, pl. 25, 1902; (41) ib. "Uber Polycaryum branchiopodianum"

(n.g., n.sp.), _Zool. Jahrb. Syst._ 15, p. 591, pl. 31, 1902; (42) ib.

"Uber Nosema anomalum," _Arch. Protistenk_, 4, p. 1, pls. 1-3, 1904;

(43) P. Thelohan, "Recherches sur les Myxosporidies," _Bull. sci.

France belg._ 26, p. 100, 3 pls., 1895; (44) P. Vuillemin, "Le

Sarcocystis tenella, parasite de l'homme," _C. R. ac. sci._ 134, p.

1152, 1902; (45) H.M. Woodcock, "On Myxosporidia in flat fish," _Proc.

Liverp. Biol. Soc._ 18, p. 126, pl. 2, 1904; (46) ib. "On a remarkable

parasite" (_Lymphocystis_), op. cit. p. 143, pl. 3, 1904.

(H. M. Wo.)

ENDYMION, in Greek mythology, son of Aethlius and king of Elis. He was

loved by Selene, goddess of the moon, by whom he had fifty daughters,

supposed to represent the fifty moons of the Olympian festal cycle. In

other versions, Endymion was a beautiful youth, a shepherd or hunter

whom Selene visited every night while he lay asleep in a cave on Mount

Latmus in Caria (Pausanias v. 1; Ovid, _Ars am._ iii. 83). Zeus left him

free to choose anything he might desire, and he chose an everlasting

sleep, in which he might remain youthful for ever (Apollodorus i. 7).

According to others, Endymion's eternal sleep was a punishment inflicted

by Zeus upon him because he ventured to fall in love with Hera, when he

was admitted to the society of the Olympian gods (Schol. Theocritus iii.

49). The usual form of the legend, however, represents Endymion as

having been put to sleep by Selene herself in order that she might enjoy

his society undisturbed (Cicero, _Tusc. disp._ i. 38). Some see in

Endymion the sun, setting opposite to the rising moon, the Latmian cave

being the cave of forgetfulness, into which the sun plunges beneath the

sea; others regard him as the personification of sleep or death (see

Mayor on Juvenal x. 318).

ENERGETICS. The most fundamental result attained by the progress of

physical science in the 19th century was the definite enunciation and

development of the doctrine of energy, which is now paramount both in

mechanics and in thermodynamics. For a discussion of the elementary

ideas underlying this conception see the separate heading ENERGY.

Ever since physical speculation began in the atomic theories of the

Greeks, its main problem has been that of unravelling the nature of the

underlying correlation which binds together the various natural

agencies. But it is only in recent times that scientific investigation

has definitely established that there is a quantitative relation of

simple equivalence between them, whereby each is expressible in terms of

heat or mechanical power; that there is a certain measurable quantity

associated with each type of physical activity which is always

numerically identical with a corresponding quantity belonging to the new

type into which it is transformed, so that the energy, as it is called,

is conserved in unaltered amount. The main obstacle in the way of an

earlier recognition and development of this principle had been the

doctrine of caloric, which was suggested by the principles and practice

of calorimetry, and taught that heat is a substance that can be

transferred from one body to another, but cannot be created or

destroyed, though it may become latent. So long as this idea maintained

itself, there was no possible compensation for the destruction of

mechanical power by friction; it appeared that mechanical effect had

there definitely been lost. The idea that heat is itself convertible

into power, and is in fact energy of motion of the minute invisible

parts of bodies, had been held by Newton and in a vaguer sense by Bacon,

and indeed long before their time; but it dropped out of the ordinary

creed of science in the following century. It held a place, like many

other anticipations of subsequent discovery, in the system of Natural

Philosophy of Thomas Young (1804); and the discrepancies attending

current explanations on the caloric theory were insisted on, about the

same time, by Count Rumford and Sir H. Davy. But it was not till the

actual experiments of Joule verified the same exact equivalence between

heat produced and mechanical energy destroyed, by whatever process that

was accomplished, that the idea of caloric had to be definitely

abandoned. Some time previously R. Mayer, physician, of Heilbronn, had

founded a weighty theoretical argument on the production of mechanical

power in the animal system from the food consumed; he had, moreover,

even calculated the value of a unit of heat, in terms of its equivalent

in power, from the data afforded by Regnault's determinations of the

specific heats of air at constant pressure and at constant volume, the

former being the greater on Mayer's hypothesis (of which his calculation

in fact constituted the verification) solely on account of the power

required for the work of expansion of the gas against the surrounding

constant pressure. About the same time Helmholtz, in his early memoir on

the Conservation of Energy, constructed a cumulative argument by tracing

the ramifications of the principle of conservation of energy throughout

the whole range of physical science.

_Mechanical and Thermal Energy._--The amount of energy, defined in this

sense by convertibility with mechanical work, which is contained in a

material system, must be a function of its physical state and chemical

constitution and of its temperature. The change in this amount, arising

from a given transformation in the system, is usually measured by

degrading the energy that leaves the system into heat; for it is always

possible to do this, while the conversion of heat back again into other

forms of energy is impossible without assistance, taking the form of

compensating degradation elsewhere. We may adopt the provisional view

which is the basis of abstract physics, that all these other forms of

energy are in their essence mechanical, that is, arise from the motion

or strain of material or ethereal media; then their distinction from

heat will lie in the fact that these motions or strains are simply

co-ordinated, so that they can be traced and controlled or manipulated

in detail, while the thermal energy subsists in irregular motions of the

molecules or smallest portions of matter, which we cannot trace on

account of the bluntness of our sensual perceptions, but can only

measure as regards total amount.

_Historical: Abstract Dynamics._--Even in the case of a purely

mechanical system, capable only of a finite number of definite types of

disturbance, the principle of the conservation of energy is very far

from giving a complete account of its motions; it forms only one among

the equations that are required to determine their course. In its

application to the kinetics of invariable systems, after the time of

Newton, the principle was emphasized as fundamental by Leibnitz, was

then improved and generalized by the Bernoullis and by Euler, and was

ultimately expressed in its widest form by Lagrange. It is recorded by

Helmholtz that it was largely his acquaintance in early years with the

works of those mathematical physicists of the previous century, who had

formulated and generalized the principle as a help towards the

theoretical dynamics of complex systems of masses, that started him on

the track of extending the principle throughout the whole range of

natural phenomena. On the other hand, the ascertained validity of this

extension to new types of phenomena, such as those of electrodynamics,

now forms a main foundation of our belief in a mechanical basis for

these sciences.

In the hands of Lagrange the mathematical expression for the manner in

which the energy is connected with the geometrical constitution of the

material system became a sufficient basis for a complete knowledge of

its dynamical phenomena. So far as statics was concerned, this doctrine

took its rise as far back as Galileo, who recognized in the simpler

cases that the work expended in the steady driving of a frictionless

mechanical system is equal to its output. The expression of this fact

was generalized in a brief statement by Newton in the _Principia_, and

more in detail by the Bernoullis, until, in the analytical guise of the

so-called principle of "virtual velocities" or virtual work, it finally

became the basis of Lagrange's general formulation of dynamics. In its

application to kinetics a purely physical principle, also indicated by

Newton, but developed long after with masterly applications by

d'Alembert, that the reactions of the infinitesimal parts of the system

against the accelerations of their motions statically equilibrate the

forces applied to the system as a whole, was required in order to form a

sufficient basis, and one which Lagrange soon afterwards condensed into

the single relation of Least Action. As a matter of history, however,

the complete formulation of the subject of abstract dynamics actually

arose (in 1758) from Lagrange's precise demonstration of the principle

of Least Action for a particle, and its immediate extension, on the

basis of his new Calculus of Variations, to a system of connected

particles such as might be taken as a representation of any material

system; but here too the same physical as distinct from mechanical

considerations come into play as in d'Alembert's principle. (See

DYNAMICS: _Analytical_.)

It is in the cases of systems whose state is changing so slowly that

reactions arising from changing motions can be neglected, that the

conditions are by far the simplest. In such systems, whether stationary

or in a state of steady motion, the energy depends on the configuration

alone, and its mathematical expression can be determined from

measurement of the work required for a sufficient number of simple

transformations; once it is thus found, all the statical relations of

the system are implicitly determined along with it, and the results of

all other transformations can be predicted. The general development of

such relations is conveniently classed as a separate branch of physics

under the name _Energetics_, first invented by W.J.M. Rankine; but the

essential limitations of this method have not always been observed. As

regards statical change, the complete specification of a mechanical

system is involved in its geometrical configuration and the function

expressing its mechanical energy in terms thereof. Systems which have

statical energy-functions of the same analytical form behave in

corresponding ways, and can serve as models or representations of one

another.

_Extension to Thermal and Chemical Systems._--This dominant position of

the principle of energy, in ordinary statical problems, has in recent

times been extended to transformations involving change of physical

state or chemical constitution as well as change of geometrical

configuration. In this wider field we cannot assert that mechanical (or

available) energy is never lost, for it may be degraded into thermal

energy; but we can use the principle that on the other hand it can never

spontaneously increase. If this were not so, cyclic processes might

theoretically be arranged which would continue to supply mechanical

power so long as energy of any kind remained in the system; whereas the

irregular and uncontrollable character of the molecular motions and

strains which constitute thermal energy, in combination with the vast

number of the molecules, must place an effectual bar on their unlimited

co-ordination. To establish a doctrine of _energetics_ that shall form a

sufficient foundation for a theory of the trend of chemical and physical

change, we have, therefore, to impart precision to this motion of

available energy.

_Carnot's Principle: Entropy._--The whole subject is involved in the new

principle contributed to theoretical physics by Sadi Carnot in 1824, in

which the far-reaching modern conception of cyclic processes was first

scientifically developed. It was shown by Carnot, on the basis of certain

axioms, whose theoretical foundations were subsequently corrected and

strengthened by Clausius and Lord Kelvin, that a reversible mechanical

process, working in a cycle by means of thermal transfers, which takes

heat, say H1, into the material system at a given temperature T1, and

delivers the part of it not utilized, say H2, at a lower given

temperature T2, is more efficient, considered as a working engine, than

any other such process, operating between the same two temperatures but

not reversible, could be. This relation of inequality involves a definite

law of equality, that the mechanical efficiencies of all reversible

cyclic processes are the same, whatever be the nature of their operation

or the material substances involved in them; that in fact the efficiency

is a function solely of the two temperatures at which the cyclically

working system takes in and gives out heat. These considerations

constitute a fundamental general principle to which all possible slow

reversible processes, so far as they concern matter in bulk, must conform

in all their stages; its application is almost coextensive with the scope

of general physics, the special kinetic theories in which inertia is

involved, being excepted. (See THERMODYNAMICS.) If the working system is

an ideal gas-engine, in which a perfect gas (known from experience to be

a possible state of matter) is passed through the cycle, and if

temperature is measured from the absolute zero by the expansion of this

gas, then simple direct calculation on the basis of the laws of ideal

gases shows that H1/T1 = H2/T2; and as by the conservation of energy the

work done is H1 - H2, it follows that the efficiency, measured as the

ratio of the work done to the supply of heat, is 1 - T2/T1. If we change

the sign of H1 and thus consider heat as positive when it is restored to

the system as is H2, the fundamental equation becomes H1/T1 + H2/T2 = 0;

and as any complex reversible working system may be considered as

compounded in various ways of chains of elementary systems of this type,

_whose effects are additive_, the general proposition follows, that in

any reversible complete cyclic change which involves the taking in of

heat by the system of which the amount is [delta]H, when its temperature

ranges between T_r and T_r + [delta]T, the equation [Sigma][delta]H_r/T_r

- 0 holds good. Moreover, if the changes are not reversible, the

proportion of the heat supply that is utilized for mechanical work will

be smaller, so that more heat will be restored to the system, and

[Sigma][delta]H_r/T_r or, as it may be expressed, [int]dH/T, must have a

larger value, and must thus be positive. The first statement involves

further, that for all reversible paths of change of the system from one

state C to another state D, the value of [int]dH/T must be the same,

because any one of these paths and any other one reversed would form a

cycle; whereas for any irreversible path of change between the same

states this integral must have a greater value (and so exceed the

difference of entropies at the ends of the path). The definite quantity

represented by this integral for a reversible path was introduced by

Clausius in 1854 (also adumbrated by Kelvin's investigations about the

same time), and was named afterwards by him the increase of the _entropy_

of the system in passing from the state C to the state D. This increase,

being thus the same for the unlimited number of possible reversible paths

involving independent variation of all its finite co-ordinates, along

which the system can pass, can depend only on the terminal states. The

entropy belonging to a given state is therefore a function of that state

alone, irrespective of the manner in which it has been reached; and this

is the justification of the assignment to it of a special name, connoting

a property of the system depending on its actual condition and not on its

previous history. Every reversible change in an isolated system thus

maintains the entropy of that system unaltered; no possible spontaneous

change can involve decrease of the entropy; while any defect of

reversibility, arising from diffusion of matter or motion in the system,

necessarily leads to increase of entropy. For a physical or chemical

system only those changes are spontaneously possible which would lead to

increase of the entropy; if the entropy is already a maximum for the

given total energy, and so incapable of further continuous increase under

the conditions imposed upon the system, there must be stable equilibrium.

This definite quantity belonging to a material system, its entropy

[phi], is thus concomitant with its energy E, which is also a definite

function of its actual state by the law of conservation of energy;

these, along with its temperature T, and the various co-ordinates

expressing its geometrical configuration and its physical and chemical

constitution, are the quantities with which the thermodynamics of the

system deals. That branch of science develops the consequences involved

in just two principles: (i.) that the energy of every isolated system is

constant, and (ii.) that its entropy can never diminish; any

complication that may be involved arises from complexity in the systems

to which these two laws have to be applied.

_The General Thermodynamic Equation._--When any physical or chemical

system undergoes an infinitesimal change of state, we have [delta]E =

[delta]H + [delta]U, where [delta]H is the energy that has been acquired

_as heat_ from sources extraneous to the system during the change, and

[delta]U is the energy that has been imparted by reversible agencies

such as mechanical or electric work. It is, however, not usually

possible to discriminate permanently between heat acquired and work

imparted, for (unless for isothermal transformations) neither [delta]H

nor [delta]U is the exact differential of a function of the constitution

of the system and so independent of its previous history, although their

sum [delta]E is such; but we can utilize the fact that [delta]H is equal

to T[delta][phi] where [delta][phi] is such, as has just been seen. Thus

E and [phi] represent properties of the system which, along with

temperature, pressure and other independent data specifying its

constitution, must form the variables of an analytical exposition. We

have, therefore, to substitute T[delta][phi] for [delta]H; also the

_change_ of internal energy is determined by the change of constitution,

involving a differential relation of type

[delta]U = -p[delta]v + [delta]W + [mu]1[delta]m1 + [mu]2[delta]m2

+ ... + [mu]_n[delta]m_n,

when the system consists of an intimate mixture (solution) of masses m1,

m2, ... m_n of given constituents, which differ physically or chemically

but may be partially transformable into each other by chemical or

physical action during the changes under consideration, the whole being

of volume v and under extraneous pressure p, while W is potential energy

arising from physical forces such as those of gravity, capillarity, &c.

The variables m1, m2, ... m_n may not be all independent; for example,

if the system were chloride of ammonium gas existing along with its

gaseous products of dissociation, hydrochloric acid and ammonia, only

one of the three masses would be independently variable. The sufficient

number of these variables (independent components) together with two

other variables, which may be v and T, or v and [phi], specifies and

determines the state of the system, considered as matter in bulk, at

each instant. It is usual to include [delta]W in [mu]1[delta]m1 + ...;

in all cases where this is possible the single equation

[delta]E = T[delta][phi] - p[delta]v + [mu]1[delta]m1 + [mu]2[delta]m2

+ ... + [mu]_n[delta]m_n (1)

thus expresses the complete variation of the energy-function E arising

from change of state; and when the part involving the n constitutive

differentials has been expressed in terms of the number of them that are

really independent, this equation by itself becomes the unique

expression of _all_ the thermodynamic relations of the system. These are

in fact the various relations ensuring that the right-hand side is an

exact differential, and are of the type of reciprocal relations such as

d[mu]_r/d[phi] = dT/dm_r.

The condition that the state of the system be one of stable equilibrium

is that [delta][phi], the variation of entropy, be negative for all

formally imaginable infinitesimal transformations which make [delta]E

vanish; for as [delta][phi] cannot actually be negative for any

spontaneous variation, none of these transformations can then occur.

From the form of the equation, this condition is the same as that

[delta]E - T[delta][phi] must be _positive for all possible_ variations

of state of the system as above defined in terms of co-ordinates

representing its constitution in bulk, without restriction.

We can change one of the independent variables expressing the state of

the system from [phi] to T by subtracting [delta]([phi]T) from both

sides of the equation of variation: then

[delta](E - T[phi]) = - [phi][delta]T - p[delta]v + [mu]1[delta]m1

+ ... + [mu]_n[delta]m_n.

It follows that for _isothermal_ changes, i.e. those for which [delta]T

is maintained null by an environment at constant temperature, the

condition of stable equilibrium is that the function E - T[phi] shall be

a minimum. If the system is subject to an external pressure p, which as

well as the temperature is imposed constant from without and thus

incapable of variation through internal changes, the condition of stable

equilibrium is similarly that E - T[phi] + pv shall be a minimum.

A chemical system maintained at constant temperature by communication of

heat from its environment may thus have several states of stable

equilibrium corresponding to different minima of the function here

considered, just as there may be several minima of elevation on a

landscape, one at the bottom of each depression; in fact, this analogy,

when extended to space of n dimensions, exactly fits the case. If the

system is sufficiently disturbed, for example, by electric shock, it may

pass over (explosively) from a higher to a lower minimum, but never

(without compensation from outside) in the opposite direction. The

former passage, moreover, is often effected by introducing a new

substance into the system; sometimes that substance is recovered

unaltered at the end of the process, and then its action is said to be

purely _catalytic_; its presence modifies the form of the function E -

T[phi] so as to obliterate the ridge between the two equilibrium states

in the graphical representation.

There are systems in which the equilibrium states are but very slightly

dependent on temperature and pressure within wide limits, outside which

reaction takes place. Thus while there are cases in which a state of

mobile dissociation exists in the system which changes continuously as a

function of these variables, there are others in which change does not

sensibly occur at all until a certain _temperature of reaction_ is

attained, after which it proceeds very rapidly owing to the heat

developed, and the system soon becomes sensibly permanent in a

transformed phase by completion of the reaction. In some cases of this

latter type the cause of the delay in starting lies possibly in passive

resistance to change, of the nature of viscosity or friction, which is

competent to convert an unstable mechanical equilibrium into a

moderately stable one; but in most such reactions there seems to be no

exact equilibrium at any temperature, short of the ultimate state of

dissipated energy in which the reaction is completed, although the

velocity of reaction is found to diminish exponentially with change of

temperature, and thus becomes insignificant at a small interval from the

temperature of pronounced activity.

_Free Energy._--The quantity E - T[phi] thus plays the same fundamental

part in the thermal statics of general chemical systems at uniform

temperature that the potential energy plays in the statics of mechanical

systems of unchanging constitution. It is a function of the geometrical

co-ordinates, the physical and chemical constitution, and the

temperature of the system, which determines the conditions of stable

equilibrium _at each temperature_; it is, in fact, the potential energy

generalized so as to include temperature, and thus be a single function

relating to each temperature but at the same time affording a basis of

connexion between the properties of the system at different

temperatures. It has been called the _free energy_ of the system by

Helmholtz, for it is the part of the energy whose variation is connected

with changes in the bodily structure of the system represented by the

variables m1, m2, ... m_n, and not with the irregular molecular motions

represented by heat, so that it can take part freely in physical

transformations. Yet this holds good only subject to the condition that

the temperature is not varied; it has been seen above that for the more

general variation neither [delta]H nor [delta]U is an exact

differential, and no line of separation can be drawn between thermal and

mechanical energies.

The study of the evolution of ideas in this, the most abstract branch of

modern mathematical physics, is rendered difficult in the manner of most

purely philosophical subjects by the variety of terminology, much of it

only partially appropriate, that has been employed to express the

fundamental principles by different investigators and at different

stages of the development. Attentive examination will show, what is

indeed hardly surprising, that the principles of the theory of free

energy of Gibbs and Helmholtz had been already grasped and exemplified

by Lord Kelvin in the very early days of the subject (see the paper "On

the Thermoelastic and Thermomagnetic Properties of Matter, Part I."

_Quarterly Journal of Mathematics_, No. 1, April 1855; reprinted in

Phil. Mag., January 1878, and in _Math. and Phys. Papers_, vol. i. pp.

291, seq.). Thus the striking new advance contained in the more modern

work of J. Willard Gibbs (1875-1877) and of Helmholtz (1882) was rather

the sustained general application of these ideas to chemical systems,

such as the galvanic cell and dissociating gaseous systems, and in

general fashion to heterogeneous concomitant phases. The fundamental

paper of Kelvin connecting the electromotive force of the cell with the

energy of chemical transformation is of date 1851, some years before the

distinction between free energy and total energy had definitely

crystallized out; and, possibly satisfied with the approximate exactness

of his imperfect formula when applied to a Daniell's cell (_infra_), and

deterred by absence of experimental data, he did not return to the

subject. In 1852 he briefly announced (_Proc. Roy. Soc. Edin._) the

principle of the dissipation of mechanical (or available) energy,

including the necessity of compensation elsewhere when restoration

occurs, in the form that "any restoration of mechanical energy, without

more than an equivalent of dissipation, is impossible"--probably even in

vital activity; but a sufficient specification of available energy (cf.

_infra_) was not then developed. In the paper above referred to, where

this was done, and illustrated by full application to solid elastic

systems, the total energy is represented by c and is named "the

intrinsic energy," the energy taken in during an isothermal

transformation is represented by e, of which H is taken in as heat,

while the remainder, the change of free (or mechanical or available)

energy of the system is the unnamed quantity denoted by the symbol w,

which is "the work done by the applied forces" at uniform temperature.

It is pointed out that it is w and not e that is the potential

energy-function for isothermal change, of which the form can be

determined directly by dynamical and physical experiment, and from which

alone the criteria of equilibrium and stress are to be derived--simply

for the reason that for all _reversible_ paths at constant temperature

between the same terminal configurations, there must, by Carnot's

principle, be the same gain or loss of heat. And a system of formulae

are given (5) to (11)--_Ex. gr._ e = w - t(dw/dt) + J [int]s dt for

finding the total energy e for any temperature t when w and the thermal

capacity s of the system, in a standard state, have thus been

ascertained, and another for establishing connexion between the form of

w for one temperature and its form for adjacent temperatures--which are

identical with those developed by Helmholtz long afterwards, in 1882,

except that the entropy appears only as an unnamed integral. The

progress of physical science is formally identified with the exploration

of this function w for physical systems, with continually increasing

exactness and range--except where pure kinetic considerations prevail,

in which cases the wider Hamiltonian dynamical formulation is

fundamental. Another aspect of the matter will be developed below.

A somewhat different procedure, in terms of entropy as fundamental, has

been adopted and developed by Planck. In an isolated system the trend of

change must be in the direction which increases the entropy [phi], by

Clausius' form of the principle. But in experiment it is a system at

constant temperature rather than an adiabatic one that usually is

involved; this can be attained formally by including in the isolated

system (cf. _infra_) a source of heat at that temperature and of

unlimited capacity, when the energy of the original system increases by

[delta]E this source must give up heat of amount [delta]E, and its

entropy therefore diminishes [delta]E/T. Thus for the original system

maintained at constant temperature T it is [delta][phi] - [delta]E/T

that must always be positive in spontaneous change, which is the same

criterion as was reached above. Reference may also be made to H.A.

Lorentz's _Collected Scientific Papers_, part i.

A striking anticipation, almost contemporaneous, of Gibbs's

thermodynamic potential theory (_infra_) was made by Clerk Maxwell in

connexion with the discussion of Andrews's experiments on the critical

temperature of mixed gases, in a letter published in Sir G.G. Stokes's

_Scientific Correspondence_ (vol. ii. p. 34).

_Available Energy._--The same quantity [phi], which Clausius named the

entropy, arose in various ways in the early development of the subject,

in the train of ideas of Rankine and Kelvin relating to the expression

of the _available energy_ A of the material system. Suppose there were

accessible an auxiliary system containing an _unlimited_ quantity of

heat at absolute temperature T0, forming a condenser into which heat can

be discharged from the working system, or from which it may be recovered

at that temperature: we proceed to find how much of the heat of our

system is available for transformation into mechanical work, in a

process which reduces the whole system to the temperature of this

condenser. Provided the process of reduction is performed reversibly, it

is immaterial, by Carnot's principle, in what manner it is effected:

thus in following it out in detail we can consider each elementary

quantity of heat [delta]H removed from the system as set aside at its

actual temperature between T and T + [delta]T for the production of

mechanical work [delta]W and the residue of it [delta]H0 as directly

discharged into the condenser at T0. The principle of Carnot gives

[delta]H/T = [delta]H0/T0, so that the portion of the heat [delta]H that

is not available for work is [delta]H0, equal to T0[delta]H/T. In the

whole process the part not available in connexion with the condenser at

T0 is therefore T0[int][delta]H/T. This quantity must be the same

whatever reversible process is employed: thus, for example, we may first

transform the system reversibly from the state C to the state D, and

then from the state D to the final state of uniform temperature T0. It

follows that the value of T0[int]dH/T, representing the heat degraded,

is the same along all reversible paths of transformation from the state

C to the state D; so that the function [int]dH/T is the excess of a

definite quantity [phi] connected with the system in the former state as

compared with the latter.

It is usual to change the law of sign of [delta]H so that gain of heat

by the system is reckoned positive; then, relative to a condenser of

unlimited capacity at T0, the state C contains more mechanically

_available energy_ than the state D by the amount E_C - E_D + T0

[int]dH/T, that is, by E_C - E_D - T0([phi]_C - [phi]_D). In this way

the existence of an entropy function with a definite value for each

state of the system is again seen to be the direct analytical equivalent

of Carnot's axiom that no process can be more efficient than a

reversible process between the same initial and final states. The name

_motivity_ of a system was proposed by Lord Kelvin in 1879 for this

conception of available energy. It is here specified as relative to a

condenser of unlimited capacity at an assigned temperature T0: some such

specification is necessary to the definition; in fact, if T0 were the

absolute zero, all the energy would be mechanically available.

But we can obtain an intrinsically different and self-contained

comparison of the available energies in a system in two different states

at different temperatures, by ascertaining how much energy would be

dissipated in each in a reduction to the _same_ standard state of the

system itself, at a standard temperature T0. We have only to reverse the

operation, and change back this standard state to each of the others in

turn. This will involve abstractions of heat [delta]H0 from the various

portions of the system in the standard state, and returns of [delta]H to

the state at T0; if this return were [delta]H0T/T0 instead of [delta]H,

there would be no loss of availability in the direct process; hence

there is actual dissipation [delta]H - [delta]H0T/T0, that is

T([delta][phi] - [delta][phi]0). On passing from state 1 to state 2

through this standard state 0 the difference of these dissipations will

represent the energy of the system that has become unavailable. Thus in

this sense E - T[phi] + T[phi]0 + const. represents for each state the

amount of energy that is available; but instead of implying an unlimited

source of heat at the standard temperature T0, it implies that there is

no extraneous source. The available energy thus defined differs from E -

T[phi], the _free energy_ of Helmholtz, or the _work function of the

applied forces_ of Kelvin, which involves no reference to any standard

state, by a simple linear function of the temperature alone which is

immaterial as regards its applications.

The determination of the available mechanical energy arising from

differences of temperature between the parts of the same system is a

more complex problem, because it involves a determination of the common

temperature to which reversible processes will ultimately reduce them;

for the simple case in which no changes of state occur the solution was

given by Lord Kelvin in 1853, in connexion with the above train of ideas

(cf. Tait's _Thermodynamics_, S179). In the present exposition the

system is sensibly in equilibrium at each stage, so that its temperature

T is always uniform throughout; isolated portions at different

temperatures would be treated as different systems.

_Thermodynamic Potentials._--We have now to develop the relations

involved in the general equation (1) of thermodynamics. Suppose the

material system includes two coexistent states or phases, with

opportunity for free interchange of constituents--for example, a salt

solution and the aqueous vapour in equilibrium with it. Then in

equilibrium a slight transfer [delta]m of the water-substance of mass

m_r constituting the vapour, into the water-substance of mass m_s,

existing in the solution, should not produce any alteration of the first

order in [delta]E - T[delta][phi]; therefore [mu]_r must be equal to

[mu]_s. The quantity [mu]_r is called by Willard Gibbs the potential of

the corresponding substance of mass m_r; it may be defined as its

marginal available energy per unit mass at the given temperature. If

then a system involves in this way coexistent phases which remain

permanently separate, the potentials of any constituent must be the same

in all of them in which that constituent exists, for otherwise it would

tend to pass from the phases in which its potential is higher to those

in which it is lower. If the constituent is non-existent in any phase,

its potential when in that phase would have to be higher than in the

others in which it is actually present; but as the potential increases

logarithmically when the density of the constituent is indefinitely

diminished, this condition is automatically satisfied--or, more

strictly, the constitutent cannot be entirely absent, but the presence

of the merest trace will suffice to satisfy the condition of equality of

potential. When the action of the force of gravity is taken into

account, the potential of each constituent must include the

gravitational potential _gh_; in the equilibrium state the total

potential of each constituent, including this part, must be the same

throughout all parts of the system into which it is freely mobile. An

example is Dalton's law of the independent distributions of the gases in

the atmosphere, if it were in a state of rest. A similar statement

applies to other forms of mechanical potential energy arising from

actions at a distance.

When a slight constitutive change occurs in a galvanic element at given

temperature, producing available energy of electric current, in a

reversible manner and isothermally, at the expense of chemical energy,

it is the free energy of the system E - T[phi], not its total intrinsic

energy, whose value must be conserved during the process. Thus the

electromotive force is equal to the change of this free energy per

electrochemical equivalent of reaction in the cell. This proposition,

developed by Gibbs and later by Helmholtz, modifies the earlier one of

Kelvin--which tacitly assumed all the energy of reaction to be

available--except in the cases such as that of a Daniell's cell, in

which the magnitude of the electromotive force does not depend sensibly

on the temperature.

The effects produced on electromotive forces by difference of

concentrations in dilute solutions can thus be accounted for and traced

out, from the knowledge of the form of the free energy for such cases;

as also the effects of pressure in the case of gas batteries. The free

energy does not sensibly depend on whether the substance is solid or

fused--for the two states are in equilibrium at the temperature of

fusion--though the total energy differs in these two cases by the heat

of fusion; for this reason, as Gibbs pointed out, voltaic

potential-differences are the same for the fused as for the solid state

of the substances concerned.

_Relations involving Constitution only._--The potential of a component

in a given solution can depend only on the temperature and pressure of

the solution, and the densities of the various components, including

itself; as no distance-actions are usually involved in chemical physics,

it will not depend on the aggregate masses present. The example above

mentioned, of two coexistent phases liquid and vapour, indicates that

there may thus be relations between the constitutions of the phases

present in a chemical system which do not involve their total masses.

These are developed in a very direct manner in Willard Gibbs's original

procedure. In so far as attractions at a distance (a uniform force such

as gravity being excepted) and capillary actions at the interfaces

between the phases are inoperative, the fundamental equation (1) can be

integrated. Increasing the volume k times, and all the masses to the

same extent--in fact, placing alongside each other k identical systems

at the same temperature and pressure--will increase [phi] and E in the

same ratio k; thus E must be a homogeneous function of the first degree

of the independent variables [phi], v, m1, ..., m_n, and therefore by

Euler's theorem relating to such functions

E = T[phi] - pv + [mu]1m1 + ... + [mu]_nm_n.

This integral equation merely expresses the additive character of the

energies and entropies of adjacent portions of the system at uniform

temperature, and thus depends only on the absence of sensible physical

action directly across finite distances. If we form from it the

expression for the complete differential [delta]E, and subtract (1),

there remains the relation

0 = [phi][delta]T - v[delta]p + m1[delta][mu]1

+ ... + m_n[delta][mu]_n. (2)

This implies that in each phase the change of pressure depends on and is

determined by the changes in T, [mu]1, ... [mu]_n alone; as we know

beforehand that a physical property like pressure is an analytical

function of the state of the system, it is therefore a function of these

n + 1 quantities. When they are all independently variable, the

densities of the various constituents and of the entropy in the phase

are expressed by the partial fluxions of p with respect to them: thus

[phi] dp m_r dp

----- = --, --- = -------.

v dT v d[mu]_r

But when, as in the case above referred to of chloride of ammonium gas

existing partially dissociated along with its constituents, the masses

are not independent, necessary linear relations, furnished by the laws

of definite combining proportions, subsist between the partial fluxions,

and the form of the function which expresses p is thus restricted, in a

manner which is easily expressible in each special case.

This proposition that the pressure in any phase is a function of the

temperature and of the potentials of the independent constituents, thus

appears as a consequence of Carnot's axiom combined with the energy

principle and the absence of effective actions at a distance. It shows

that at a given temperature and pressure the potentials are not all

independent, that there is a necessary relation connecting them. This is

the _equation of state_ or constitution of the phase, whose existence

forms one mode of expression of Carnot's principle, and in which all the

properties of the phase are involved and can thence be derived by simple

differentiation.

_The Phase Rule._--When the material system contains only a single

phase, the number of independent variations, in addition to change of

temperature and pressure, that can spontaneously occur in its

constitution is thus one less than the number of its independent

constituents. But where several phases coexist in contact in the same

system, the number of possible independent variations may be much

smaller. The present independent variables [mu]1, ..., [mu]_n are

specially appropriate in this problem, because each of them has the same

value in all the phases. Now each phase has its own characteristic

equation, giving a relation between [delta]p, [delta]T, and

[delta][mu]1, ... [delta][mu]_n, or such of the latter as are

independent; if r phases coexist, there are r such relations; hence the

number of possible independent variations, including those of v and T,

is reduced to m - r + 2, where m is the number of independently variable

chemical constituents which the system contains. This number of degrees

of constitutive freedom cannot be negative; therefore the number of

possible phases that can coexist alongside each other cannot exceed m +

2. If m + 2 phases actually coexist, there is no variable quantity in

the system, thus the temperature and pressure and constitutions of the

phases are all determined; such is the triple point at which ice, water

and vapour exist in presence of each other. If there are m + 1

coexistent phases, the system can vary in one respect only; for example,

at any temperature of water-substance different from the triple point

two phases only, say liquid and vapour, or liquid and solid, coexist,

and the pressure is definite, as also are the densities and potentials

of the components. Finally, when but one phase, say water, is present,

both pressure and temperature can vary independently. The first example

illustrates the case of systems, physical or chemical, in which there is

only one possible state of equilibrium, forming a point of transition

between different constitutions; in the second type each temperature has

its own completely determined state of equilibrium; in other cases the

constitution in the equilibrium state is indeterminate as regards the

corresponding number of degrees of freedom. By aid of this phase rule of

Gibbs the number of different chemical substances actually interacting

in a given complex system can be determined from observation of the

degree of spontaneous variation which it exhibits; the rule thus lies at

the foundation of the modern subject of chemical equilibrium and

continuous chemical change in mixtures or alloys, and in this connexion

it has been widely applied and developed in the experimental

investigations of Roozeboom and van 't Hoff and other physical chemists,

mainly of the Dutch school.

_Extent to which the Theory can be practically developed._--It is only

in systems in which the number of independent variables is small that

the forms of the various potentials,--or the form of the fundamental

characteristic equation expressing the energy of the system in terms of

its entropy and constitution, or the pressure in terms of the

temperature and the potentials, which includes them all,--can be readily

approximated to by experimental determinations. Even in the case of the

simple system water-vapour, which is fundamental for the theory of the

steam-engine, this has not yet been completely accomplished. The general

theory is thus largely confined, as above, to defining the restrictions

on the degree of variability of a complex chemical system which the

principle of Carnot imposes. The tracing out of these general relations

of continuity of state is much facilitated by geometrical diagrams, such

as James Thomson first introduced in order to exhibit and explain

Andrews' results as to the range of coexistent phases in carbonic acid.

Gibbs's earliest thermodynamic surface had for its co-ordinates volume,

entropy and energy; it was constructed to scale by Maxwell for

water-substance, and is fully explained in later editions of the _Theory

of Heat_ (1875); it forms a relief map which, by simple inspection,

reveals the course of the transformations of water, with the

corresponding mechanical and thermal changes, in its three coexistent

states of solid, liquid and gas. In the general case, when the substance

has more than one independently variable constituent, there are more

than three variables to be represented; but Gibbs has shown the utility

of surfaces representing, for instance, the entropy in terms of the

constitutive variables when temperature and pressure are maintained

constant. Such graphical methods are now of fundamental importance in

connexion with the phase rule, for the experimental exploration of the

trend of the changes of constitution of complex mixtures with

interacting components, which arise as the physical conditions are

altered, as, for example in modern metallurgy, in the theory of alloys.

The study of the phenomena of condensation in a mixture of two gases or

vapours, initiated by Andrews and developed in this manner by van der

Waals and his pupils, forms a case in point (see CONDENSATION OF GASES).

_Dilute Components: Perfect Gases and Dilute Solutions._--There are,

however, two simple limiting cases, in which the theory can be completed

by a determination of the functions involved in it, which throw much

light on the phenomena of actual systems not far removed from these

ideal limits. They are the cases of mixtures of perfect gases, and of

very dilute solutions.

If, following Gibbs, we apply his equation (2) expressing the pressure

in terms of the temperature and the potentials, to a very dilute

solution of substances m2, m3, ... m_n in a solvent substance m1, and

vary the co-ordinate m_r alone, p and T remaining unvaried, we have in

the equilibrium state

d[mu]_r d[mu]1 d[mu]_n

m_r------- + m1------ + ... + m_n------- = 0,

dm_r dm_r dm_r

in which every m except m1 is very small, while d[mu]1/dm_r is

presumably finite. As the second term is thus finite, this requires

that the total potential of each component m_r, which is m_r

d[mu]_r/dm_r, shall be finite, say k_r, in the limit when m_r is null.

Thus for very small concentrations the potential [mu]_r of a dilute

component must be of the form k_r log m_r/v, being proportional to the

logarithm of the density of that component; it thus tends

logarithmically to an infinite value at evanescent concentrations,

showing that removal of the last traces of any impurity would demand

infinite proportionate expenditure of available energy, and is

therefore practically impossible with finite intensities of force. It

should be noted, however, that this argument applies only to fluid

phases, for in the case of deposition of a solid m_r is not uniformly

distributed throughout the phase; thus it remains possible for the

growth of a crystal at its surface in aqueous solution to extrude all

the water except such as is in some form of chemical combination.

The precise value of this logarithmic expression for the potential can

be readily determined for the case of a perfect gas from its

characteristic properties, and can be thence extended to other dilute

forms of matter. We have pv = R/m.T for unit mass of the gas, where m

is the molecular weight, being 2 for hydrogen, and R is a constant

equal to 82 X 10^6 in C.G.S. dynamical units, or 2 calories

approximately in thermal energy units, which is the same for all gases

because they have all the same number of molecules per unit volume.

The increment of heat received by the unit mass of the gas is [delta]H

= p[delta]v + [kappa][delta]T, [kappa] being thus the specific heat at

constant volume, which can be a function only of the temperature. Thus

[phi] = [int]dH/T = R/m.log v + [int][kappa]T^(-1)dT;

and the available energy A per unit mass is E - T[phi] + T[phi]0 where

E = [epsilon] + [int][kappa]dT, the integral being for a standard

state, and [epsilon] being intrinsic energy of chemical constitution;

so that

A = [epsilon] + [phi]0T + [int][kappa]dT - T [int][kappa]T^(-1)dT -

R/m.T log v.

If there are [nu] molecules in the unit mass, and N per unit volume,

we have m[nu] = Nmv, each being 2 [nu]', where [nu]' is the number of

molecules per unit mass in hydrogen; thus the free energy per molecule

is a' + R'T log bN, where b = m/2[nu]', R' = R/2[nu]', and a' is a

function of T alone. It is customary to avoid introducing the unknown

molecular constant [nu]' by working with the available energy per

"gramme-molecule," that is, for a number of grammes expressed by the

molecular weight of the substance; this is a constant multiple of the

available energy per molecule, and is a + RT log[rho], [rho] being the

density equal to bN where b = m/2[nu]'. This formula may now be

extended by simple summation to a mixture of gases, on the ground of

Dalton's experimental principle that each of the components behaves in

presence of the others as it would do in a vacuum. The components are,

in fact, actually separable wholly or partially in reversible ways

which may be combined into cycles, for example, either (i.) by

diffusion through a porous partition, taking account of the work of

the pressures, or (ii.) by utilizing the modified constitution towards

the top of a long column of the mixture arising from the action of

gravity, or (iii.) by reversible absorption of a single component.

If we employ in place of available energy the form of characteristic

equation which gives the pressure in terms of the temperature and

potentials, the pressure of the mixture is expressed as the sum of

those belonging to its components: this equation was made by Gibbs the

basis of his analytical theory of gas mixtures, which he tested by its

application to the only data then available, those of the equilibrium

of dissociation of nitrogen peroxide (2NO2 <--> N2O4) vapour.

_Van 't Hoff's Osmotic Principle: Theoretical Explanation._--We proceed

to examine how far the same formulae as hold for gases apply to the

available energy of matter in solution which is so dilute that each

molecule of the dissolved substance, though possibly the centre of a

complex of molecules of the solvent, is for nearly all the time beyond

the sphere of direct influence of the other molecules of the dissolved

substance. The available energy is a function only of the co-ordinates

of the matter in bulk and the temperature; its change on further

dilution, with which alone we are concerned in the transformations of

dilute solutions, can depend only on the further separation of these

molecular complexes in space that is thereby produced, as no one of them

is in itself altered. The change is therefore a function only of the

number N of the dissolved molecules per unit volume, and of the

temperature, and is, per molecule, expressible in a form entirely

independent of their constitution and of that of the medium in which

they are dissolved. This suggests that the expression for the change on

dilution is the same as the known one for a gas, in which the same

molecules would exist free and in the main outside each other's spheres

of influence; which confirms and is verified by the experimental

principle of van 't Hoff, that osmotic pressure obeys the laws of

gaseous pressure with identically the same physical constants as those

of gases. It can be held, in fact, that this suggestion does not fall

short of a demonstration, on the basis of Carnot's principle, and

independent of special molecular theory, that in all cases where the

molecules of a component, whether it be of a gas or of a solution, are

outside each other's spheres of influence, the available energy, so far

as regards dilution, must have a common form, and the physical constants

must therefore be the known gas-constants. The customary exposition

derives this principle, by an argument involving cycles, from Henry's

law of solution of gases; it is sensibly restricted to such solutes as

appear concomitantly in the free gaseous state, but theoretically it

becomes general when it is remembered that no solute can be absolutely

non-volatile.

_Source of the Idea of Temperature._--The single new element that

thermodynamics introduces into the ordinary dynamical specification of a

material system is temperature. This conception is akin to that of

potential, except that it is given to us directly by our sense of heat.

But if that were not so, we could still demonstrate, on the basis of

Carnot's principle, that there is a definite function of the state of a

body which must be the same for all of a series of connected bodies,

when thermal equilibrium has become established so that there is no

tendency for heat to flow from one to another. For we can by mere

geometrical displacement change the order of the bodies so as to bring

different ones into direct contact. If this disturbed the thermal

equilibrium, we could construct cyclic processes to take advantage of

the resulting flow of heat to do mechanical work, and such processes

might be carried on without limit. Thus it is proved that if a body A

is in temperature-equilibrium with B, and B with C, then A must be in

equilibrium with C directly. This argument can be applied, by aid of

adiabatic partitions, even when the bodies are in a field of force so

that mechanical work is required to change their geometrical

arrangement; it was in fact employed by Maxwell to extend from the case

of a gas to that of any other system the proposition that the

temperature is the same all along a vertical column in equilibrium under

gravity.

It had been shown from the kinetic theory by Maxwell that in a

gas-column the mean kinetic energy of the molecules is the same at all

heights. If the only test of equality of temperature consisted in

bringing the bodies into contact, this would be rather a proof that

thermal temperature is of the same physical nature in all parts of the

field of force; but temperature can also be equalized across a distance

by radiation, so that this law for gases is itself already necessitated

by Carnot's general principle, and merely confirmed or verified by the

special gas-theory. But without introducing into the argument the

existence of radiation, the uniformity of temperature throughout all

phases in equilibrium is necessitated by the doctrine of energetics

alone, as otherwise, for example, the raising of a quantity of gas to

the top of the gravitational column in an adiabatic enclosure together

with the lowering of an equal mass to the bottom would be a source of

power, capable of unlimited repetition.

_Laws of Chemical Equilibrium based on Available Energy._--The complete

theory of chemical and physical equilibrium in gaseous mixtures and in

very dilute solutions may readily be developed in terms of available

energy (cf. _Phil. Trans_., 1897, A, pp. 266-280), which forms perhaps

the most vivid and most direct procedure. The available energy per

molecule of any kind, in a mixture of perfect gases in which there are N

molecules of that kind per unit volume, has been found to be a' + R'T

logbN where R' is the universal physical constant connected with R

above. This expression represents the marginal increase of available

energy due to the introduction of one more molecule of that kind into

the system as actually constituted. The same formula also applies, by

what has already been stated, to substances in dilute solution in any

given solvent. In any isolated system in a mobile state of reaction or

of internal dissociation, the condition of chemical equilibrium is that

the available energy at constant temperature is a minimum, therefore

that it is stationary, and slight change arising from fresh reaction

would not sensibly alter it. Suppose that this reaction, per molecule

affected by it, is equivalent to introducing n1 molecules of type N1, n2

of type N2, &c., into the system, n1, n2, ... being the numbers of

molecules of the different types that take part in the reaction, as

shown by its chemical equation, reckoned positive when they appear,

negative when they disappear. Then in the state of equilibrium

n1(a'1 + R'T log b1N1) + n2(a'2 + R'T log b2N2) + ...

must vanish. Therefore N1^(n1) N2^(n2) ... must be equal to K, a

function of the temperature alone. This law, originally based by

Guldberg and Waage on direct statistics of molecular interaction,

expresses for each temperature the relation connecting the densities of

the interacting substances, in dilution comparable as regards density

with the perfect gaseous state, when the reaction has come to the state

of mobile equilibrium.

All properties of any system, including the heat of reaction, are

expressible in terms of its available energy A, equal to E - T[phi] +

[phi]0T. Thus as the constitution of the system changes with the

temperature, we have

dA dE d[phi]

-- = -- - T------ - ([phi] - [phi]0)

dT dT dT

where

[delta]E = [delta]H + [delta]W, [delta]H = T[delta][phi],

[delta]H being heat and [delta]W mechanical and chemical energy imparted

to the system at constant temperature; hence

d(A - W) d(A - W)

-------- = -([phi] - [phi]0), so that A = E + T--------,

dT dT

which is equivalent to

d /A - W\

E - W = -T^2 -- (-------).

dT \ T /

This general formula, applied differentially, expresses the heat

[delta]E - [delta]W absorbed by a reaction in terms of [delta]A, the

change produced by it in the available energy of the system, and of

[delta]W, the mechanical and electrical work done on the system during

its progress.

In the problem of reaction in gaseous systems or in very dilute

solution, the change of available energy per molecule of reaction has

just been found to be

[delta]A = [delta]A0 + R'T log K', where K' = b1^(n1) b2^(n2) ... K;

thus, when the reaction is spontaneous without requiring external work,

the heat absorbed per molecule of reaction is

d [delta]A0 d

-T^2 -- ---------, or -R'T^2 -- log K.

dT T dT

This formula has been utilized by van 't Hoff to determine, in terms of

the heat of reaction, the displacement of equilibrium in various systems

arising from change of temperature; for K, equal to N1^(n1) N2^(n2) ...,

is the reaction-parameter through which alone the temperature affects

the law of chemical equilibrium in dilute systems.

_Interfacial Phenomena: Liquid Films._--The characteristic equation

hitherto developed refers to the state of an element of mass in the

interior of a homogeneous substance: it does not apply to matter in the

neighbourhood of the transition between two adjacent phases. A

remarkable analysis has been developed by J.W. Gibbs in which the

present methods concerning matter in bulk are extended to the phenomena

at such an interface, without the introduction of any molecular theory;

it forms the thermodynamic completion of Gauss's mechanical theory of

capillarity, based on the early form of the principle of total energy.

The validity of the fundamental doctrine of available energy, so far as

regards all mechanical actions in bulk such as surface tensions, is

postulated, even when applied to interfacial layers so thin as to be

beyond our means of measurement; the argument from perpetual motions

being available here also, as soon as we have experimentally ascertained

that the said tensions are definite physical properties of the state of

the interface and not merely accidental phenomena. The procedure will

then consist in assuming a definite excess of energy, of entropy, and of

the masses of the various components, each per unit surface, at the

interface, the potential of each component being of necessity, in

equilibrium, the same as it is in the adjacent masses. The interfacial

transition layer thus provides in a sense a new surface-phase coexistent

with those on each side of it, and having its own characteristic

equation. It is only the extent of the interface and not its curvatures

that need enter into this relation, because any slight influence of the

latter can be eliminated from the equation by slightly displacing the

position of the surface which is taken to represent the interface

geometrically. By an argument similar to one given above, it is shown

that one of the forms of the characteristic equation is a relation

expressing the surface tension as a function of the temperature and the

potentials of the various components present on the two sides of the

interface; and from the differentiation of this the surface densities of

the superficial distributions of these components (as above defined) can

be obtained. The conditions that a specified new phase may become

developed when two other given ones are brought into contact, i.e. that

a chemical reaction may start at the interface, are thence formally

expressed in terms of the surface tensions of the three transition

layers and the pressures in the three phases. In the case of a thin

soap-film, sudden extension of any part reduces the interfacial density

of each component at each surface of the film, and so alters the surface

tension, which requires time to recover by the very slow diffusion of

dissolved material from other parts of the thin film; the system being

stable, this change must be an increase of tension, and constitutes a

species of elasticity in the film. Thus in a vertical film the surface

tension must be greater in the higher parts, as they have to sustain the

weight of the lower parts; the upper parts, in fact, stretch until the

superficial densities of the components there situated are reduced to

the amounts that correspond to the tension required for this purpose.

Such a film could not therefore consist of pure water. But there is a

limit to these processes: if the film becomes so thin that there is no

water in bulk between its surfaces, the tensions cannot adjust

themselves in this slow way by migration of components from one part of

the film to another; if the film can survive at all after it has become

of molecular thickness, it must be as a definite molecular structure all

across its thickness. Of such type are the black spots that break out in

soap-films (suggested by Gibbs and proved by the measures of Reinold and

Rucker): the spots increase in size because their tension is less than

that of the surrounding film, but their indefinite increase is

presumably stopped in practice by some clogging or viscous agency at

their boundary.

_Transition to Molecular Theory._--The subject of energetics, based on

the doctrine of available energy, deals with matter in bulk and is not

concerned with its molecular constitution, which it is expressly

designed to eliminate from the problem. This analysis of the phenomena

of surface tension shows how far the principle of negation of perpetual

motions can carry us, into regions which at first sight might be classed

as molecular. But, as in other cases, it is limited to pointing out the

general scheme of relations within which the phenomena can have their

play. There is now a considerable body of knowledge correlating surface

tension with chemical constitution, especially to a certain extent with

the numerical density of the distribution of molecules; thus R. Eotvos

has shown that a law of proportionality exists for wide classes of

substances between the temperature-gradient of the surface tension and

the density of the molecules over the surface layer, which varies as the

two-thirds power of the number per unit volume (see CHEMISTRY:

_Physical_). This takes us into the sphere of molecular science, where

at present we have only such indications largely derived from

experiment, if we except the mere notion of inter-atomic forces of

unknown character on which the older theories of capillarity, those of

Laplace and Poisson, were constructed.