Vol. 6, 1920 
MATHEMATICS: F. L. HITCHCOCK 
which holds good when the solution is "ideal." It is evident that (15) 
is a limiting case of (16). If, as is the main object of the present paper, 
we wish to investigate solutions which are not ideal, we may write the per- 
fectly general equation 
fo = cpo - RTln(l + Zc) + P{c,T) (17) 
where Sc means the sum of the concentrations of the various solutes, 
and P{c,T) is a power series in concentrations and temperature. This 
series will, for the most general expression, contain the squares, products, 
and higher powers of the concentrations. It will not contain the first 
power, for (15) must be the limiting form of the equation as the solution 
becomes very dilute. 
Instead of the power series P{c,T) explicit algebraic functions might be 
introduced, based either on theoretical considerations or on empirical 
data.^ The main point is that some form of function in concentration 
and temperature must be added to the right side of (16) to obtain sufficient 
generality. The object of this paper is to indicate how the entrance of 
such a function affects the ordinary theories of freezing-point, heat of dilu- 
tion, and ionization, and for this purpose the power series appears the most 
convenient, and can be made as general as we please by taking terms 
enough. 
It is important to notice that equations (15), (16) and (17) all assume 
that we know the molecular weights of the dissolved substances, that is, 
2c depends on the actual number of dissolved molecules. This is fully 
in accord with the usual theory of very dilute solutions. But, equally 
important, nothing whatever is assumed as to possible hydration of these 
molecules, nor as to molecular complexes occurring in the solvent. As 
stated above, the number of mols Wo of solvent means merely formula- 
weight. It is precisely such matters as hydration, polymerization of 
solute, and departure from gas laws which are to be taken care of by the 
series P{c,T). 
It will be most convenient to expand this series first in powers of the 
absolute temperature T, thus 
P{cJ) = Po + PiT + P2r2 + etc. (18) 
It is doubtful whether we possess any data warranting the use of powers 
of T higher than the square. That the square, at least, is necessary will 
appear. The coefficients Po, Pi and P2 will be power series in the con- 
centrations only, it being assumed that the pressure is constant. (If we 
vary the pressure the coefficients in these new series will vary with pres- 
sure.) The expansion (18) is, however, general enough for our present 
purpose. We shall, therefore, have the potential fo of the solvent in the 
form 
/o = - i^nn(l + Xc) + Po + PiP + P2P (19) 
where Po, Pi and P2 are functions of the concentrations, which, if ex- 
