462 
NATURE 
[Marcu 18, 1897 
second cylinder, and have the pressure #” at the diaphragm ; 
and so on, the connections being so made, and the quantities of 
the several kinds of molecules so regulated, that the pressures 
at all the diaphragms shall have the same two values. p 
It is evident that the vertical distance between successive 
connections must be everywhere the same, say 7; also, that at 
all the diaphragms, on the side of the greater pressure, the 
proportion of molecules which can and which cannot pass the 
diaphragm must be the same. Let the ratiobe 1:7. If we 
write ya, yp, &c., for the densities of the several kinds of mole- 
cules, and + for total density, we have for the second cylinder 
EAE ES a5 HF. 
Ya 
For the third cylinder we have this equation, and also 
ge A lta 
GEN e907} 
which gives c 
UMC Aaah r= (1 + 2). 
Ya 
In this way, we have for the 7th cylinder 
pan 
Y= (1 + 2) 
Ya 
Now the vertical distance between equal pressures in the first 
and 7th cylinders, is 
(7 — DL 
Now the equilibrium will not be destroyed if we connect all 
the cylinders with the first through diaphragms impermeable 
to all except A-molecules. And the last equation shows that as 
y/ya increases geometrically, the vertical distance between any 
pressure in the column when this ratio of densities is found, 
and the same pressure in the first cylinder increases arith- 
metically. This distance, therefore, may be represented by 
log (y/y.) multiplied by a constant. This is identical with our 
result for a volatile liquid, except that for that case we found 
the value of the constant to be a¢/e. 
The following demonstration of van *t Hoffs law, which is 
intended to apply to existing substances, requires only that the 
solutum, 7.¢. dissolved substance, should be capable of the ideal 
gaseous state, and that its molecules, as they occur in the gas, 
should not be broken up in the solution, nor united to one 
another in more complex molecules. 
It will be convenient to use certain quantities which may be 
called the fotentzals of the solvent and of the solutum, the term 
being thus defined :—In any sensibly homogeneous mass, the 
Potential of any independently variable component substance is 
the differential coefficient of the thermodynamic energy of the 
mass taken with respect to that component, the entropy and 
volume of the mass and the quantities of its other components 
remaining constant. . The advantage of using such fofentéals in 
the theory of semi-permeable diaphragms consists partly in the 
convenient form of the conditions of equilibrium, the potential 
for any substance to which a diaphragm is freely permeable 
having the same value on both sides of the diaphragm, and 
partly in our ability to express yan ’t Hoff’s law as a relation 
between the quantities characterising the state of the solution, 
without reference to any experimental arrangement (see 7yavsac- 
‘tons of the Connecticut Acadenry, vol. iii. pp. 116, 138, 148, 
94). 
Let there be three reservoirs, R’, R”, R’’, of which the first 
contains the solvent alone, maintained in a constant state of 
temperature and pressure, the second the solution, and the third 
the solutum alone. Let R’ and R” be connected through a 
diaphragm freely permeable to the solvent, but impermeable to 
the solutum, and let R” and R’” be connected through a 
diaphragm impermeable to the solvent, but freely permeable to 
the solutum. We have then, if we write a, and p, for the 
potentials of the solvent and the solutum, and distinguished by 
accents, quantities relating to the several reservoirs, 
Hol” = fa'”. 
Now if the quantity of the solutum in the apparatus be varied, 
the ratio in which it is divided in equilibrium between the reser- 
voirs R” and R” will be constant, so long as its densities in the 
two reservoirs, o'", yo". aresmall. For let us suppose that there 
is only a single molecule of the solutum. It will wander through 
R’ and R’”, and in a time sufficiently long the parts of the time 
<”, which for convenience we may 
spent respectively in R” and R 
NO. 1429, VOL. 55] 
, 
aa 9 "const. 
suppose of equal volume, will approach a constant ratio, say 
1:B. Now if we put in the apparatus a considerable number 
of molecules, they will divide themselves between R’ and R” 
sensibly in the ratio 1: B, so long as they do not sensibly inter- 
fere with one another, z.e. so long as the number of molecules 
of the solutum which are within the spheres of action of other 
molecules of the solutum is a negligible part of the whole, both 
in R’ and R’’. With this limitation we have, therefore, 
yait = By. 
Now in R’” let the solutum have the properties oi an ideal gas, 
which give for any constant temperature (zz. p. 212) 
” 
He’ = ast log yo’ + C, 
where a, is the constant of the law of Boyle and Charles, and C 
another constant. Therefore, 
Mg’ = agt log (Bys”) + C. 
This equation, in which a single constant may evidently take 
the place of B and C, may be regarded as expressing the 
property of the solution implied in van t’ Hoff’s law. For we 
have the general thermodynamic relation (zézd. p. 143) 
vdp = ndt + mdi, + Mido, 
where 7 and 7 denote the volume and entropy of the mass con- 
sidered, and #2, and m, the quantities of its components. 
Applied to this case, since ¢and my, are constant, this becomes 
dip" = -yx!'dys! 
Substituting the value of dy,’ derived from the last finite 
equation, we have 
dp” = a,tdy," 
whence, integrating from 7.” = 0 and 7” = 7’, we get 
p’- 7 
which evidently expresses van ’t Hoff’s law. 
We may extend this proof to cases in which the solutum is 
not volatile by supposing that we give to its molecules mutually 
repulsive molecular forces, which, however, are entirely inopera- 
tive with respect to any other kind of molecules. In this way 
we may make the solutum capable of the ideal gaseous state. 
But the relations pertaining to the contents of R” will not be 
affected by these new forces, since we suppose that only a 
negligible part of the molecules of the solutum are within the 
range of such forces. Therefore these relations cannot depend 
on the new forces, and must exist without them. 
To give up the condition that the molecules of the solutum 
shall not be broken up in the solution, nor united to one another 
in more complex molecules, would involve the consideration of 
a good many cases, which it would be difficult to unite in a 
brief demonstration. The result, however, seems to be that 
the increase of pressure is to be estimated by Avogadro’s law 
from the number of molecules in the solution which contain 
any part of the solutum, without reference to the quantity in 
each. J. WILLARD GIBBS. 
New Ilaven, Connecticut, February 18. 
= agtys’’s 
Changes in Faunze due to Man’s Agency. 
Pror. Happon’s interesting article in NATURE of 
January 28 certainly deserves serious attention. Probably few 
naturalists realise the rapid changes which are being brought, 
avout through the agency of man. The way in which the 
Coccidee (scale-insects) are carried from country to country is 
amazing ; and with them go their hymenopterous parasites in 
many instances. Thus, in 1894, Mr. L. O. Howard described 
the chalcidid Homalopoda cristata from St. Vincent, W.I. ; in 
1896 I described Aspidiotus secretus—a coccid—from Japan. 
Later, in 1896, Mr. Howard was able to report that Mr. Greem 
had bred his St. Vincent chalcidid from my Japanese coccid— 
inCeylon! Mr. Howard, in 1896, described another chalcidid, 
parasitic on scale-insects, from Ceylon; and the day he received 
the separate copies of his paper, he received the chalcidid from 
the Southern U.S. Aurivillius, in 1888, described a remarkable 
parasite of Coccidz bred by him in Sweden ; already it is known 
also from two localities in the United States, and from Ceylon. 
As early as 1863, Prof. A. Costa described from Italy a re- 
markable genus of Chalcididz, after taking great pains to learn 
that it was unknown in Europe; but, as Mr. Howard has lately 
shown, it had been described (the very same species) in 1859 by 
Motschulsky from Ceylon, whence it had undoubtedly been 
