194 MATHEMATICS: F. L. HITCHCOCK Proc. N. A. S. 
hence ionization as calculated from freezing-point by the ordinary formula 
should be exact. 
12. Generalization of the mass law. — ^The case is quite otherwise, however, 
when we consider equilibrium between the ions and un-ionized salt. For 
if we let /i and denote the chemical potentials of the ions, /a that of the 
un-ionized salt, the general condition (4) of equilibrium gives 
/l+/2=/3 (38) 
as the equation which determines the ionization. Now by definition 
Jo = - — ,/i ,/2 = — and /a = — - 
owo (pnii 0W2 om4 
and if T be regarded as constant we may expand P{c,T) in the form 
P{c) = i [aiml -h ai2WiW2 + etc. ] 4- [anml -}- amm\m2 -f etc. ], (39) 
Wo Wo 
while, from (17), /o can be written 
/o = (^o - RTXn into + wi H- W2 + Ws) + RTlnmo + P{c) (40) 
Taking the derivative with respect to Wi, 
— = h — [2anmi + ai2W2 + aiaWa] 
C)Wi Wo + Wi + W2 + W3+ Wo 
+ -ij [SaiiiWi + 2aii2WiW2 + 2aii3WiW3 + a^w^ + amm^m^ + aisaw]] 
Wo 
Now we have identically 
dwi dwodwi dwo 
whence /i = f — (iwo + terms not involving Wo . Therefore, 
J dwi 
h = <P, + RT\J ^ 
nto + Wi + W2 + W3J 
- -i-[2aiiWi + ai2W2 + aiaWa] ^ [SamWi^, etc.] 
Wo 2Wo 
where (pi involves temperature (and pressure) only. The presence of the 
term RThinti is determined from our hypothesis that the solution acts 
like a perfect gas to a first approximation.^^ It is noteworthy that terms 
in fo involving squares of concentrations lead, in /i, to terms of the first 
degree. Expressions for /2 and fz are similarly found. If we put 
mi 
W^o + W^l + W2 + W3 
with similar notation for X2 and Xz, and substitute in (38) the expressions 
for /i, /2 and fz we shall have as the condition of equilibrium 
