Vol. 6, 1920 MATHEMATICS: F. L. HITCHCOCK 191 
It is more convenient to expand Qo in powers of t, where / = 0 is the freez- 
ing-point of pure solvent, 
Qo = q + q't (27) 
so that q is the latent heat of freezing per mol in contact with pure solvent, 
while q', to a good approximation, is the difference of specific heats of 
solvent in liquid and solid state. It is not likely that a term in will be 
needed with our present data, but can be introduced if wanted. It is 
important to notice that q and q' are not functions of concentration, by 
definition. Inserting values of H and Qo on the right side of the identity 
(20) and performing the integration we have (putting t = T - To), 
^(^o+Po -f)+Pi - Mn(l + 20 -{-P,T = ^~ - ^ + 
H.lnT + U2T + Pi - i^ln(l -f Sc) + K - q'lnT; (28) 
Comparing like terms on the two sides of this identity, we see that on the 
left the only term involving and varying with concentration is Po/T 
while on the right the only like term is -Ho/T, hence Ho = -Po- 
Again, on the left there is no term in InT varying with concentration, 
hence i/i = 0. Terms in T give H2 = P2- Thus 
H = -Po + P2T2 
which checks with (24). Collecting results, and substituting in (22) 
we have the equation of the melting-point curve as 
^ ~ ^ ° + Y + P2P - i?ln(l + i:c)+K + Pi- q'\nT = 0 (29) 
where the constant K is yet undetermined. When T = To the quanti- 
ties Po, Pi and P2 are zero since the concentration is zero. This gives 
for 
K = ^+q'-^ q'hiT (30) 
whence the melting-point equation by a little arrangement is 
i?rin(l + 2c) - P{c,T) = g[l - f] - q'(T - T.) - q'T\a~m) 
10. Discussion of the melting point equation. — Equation (31) departs from 
strict generality only in so far as we have failed to expand to higher powers 
of T. In principle it is entirely rigorous, and whenever we have data to 
warrant it we may expand as far as we please. It shows that a rigorous 
theory differs from the ordinary theory only through the function P{c,T) 
which, if expanded in powers of the concentrations, begins with terms 
of the second degree, but which, if expanded in powers of T, contains a 
term PiT playing no part in heat of dilution. To omit P{c,T) altogether 
is, therefore, equivalent to something more than assuming heat of dilu- 
tion to be zero. 
