MATHEMATICS: F. L. HITCHCOCK Proc. N. A. S. 
panded in series, will all begin with terms of the second degree. In other 
words, terms in the first powers of the concentrations occur nowhere ex- 
cept in the logarithmic term; if it were otherwise, the solution would fail 
to obey the perfectly well established laws for very dilute solutions. 
8. Determination of the function F{c). — Returning now to equation (12), 
writing y = fo — /, and putting for /o its value we have 
]-{<Po + Po -f)+Pi-Rln{l + i:c)+P2T=- (^dT+F{c)+K (20) 
where the only terms on the left which are functions of the concentrations 
alone are Pi — R\n{l + He). Hence, by equating, 
F{c) = Pi - RXnil + 2c) (21) 
and by using this result the equation of the melting-point curve (13) 
becomes 
- {^dT + Pi - R\n{l + 2c) + /I = 0 (22) 
J 7"2 
where, as already stated, Pi is a function of concentration (or of pressure 
if that is not constant), but not of temperature. 
9. Heat of dilution. — Before developing (22) further, it will be well to 
consider in a brief way the heat of dilution emitted when 1 mol of water 
is mixed with a large amount of solution under the same pressure. Take 
the complete system as consisting of a mass of pure water M together 
with the solution. The potential of pure water is ipo . By reasoning pre- 
cisely as in Art. 5, and letting H stand for the heat emitted when 1 mol 
of water is mixed with a large amount of solution, 
H =T^^-fo -f"^^ (23) 
dP dP 
Introducing the value of /o from (19) this becomes 
H = Po - + P2P2 (24) 
The absence of the term in the first power of P is noticeable. 
// the heat of dilution at constant pressure he expanded in powers of P, 
the first power is rigorously absent. 
9, The equation of the melting-point curve. — Returning now to (22), 
we may think of the heat of melting Q as composed of two parts, first the 
heat of melting at that temperature in contact with pure water, second 
the heat of dilution. The first we may call Qo, the second is —H (since 
Q was taken as heat emitted on freezing). Thus 
Q = Qo - H (25) 
Without assuming the results of the last article, we may expand H 
in powers of P, 
H = Ho -\- HiT + H2T' (26) 
