Vol. 6, 1920 
PHYSICS: A. G. WEBSTER 
303 
clear and inasmuch as in the five-foot shelf of treatises on thermody- 
namics in my library I cannot find the answer to the question that I have 
in mind, I think it worth while to take the matter up ab initio. To be 
sure my colleague Professor Charles A. Klraus has pointed out to me that 
van der Waals, in his excellent Lehrbuch der Thermodynamik, Kap. 
II, has partially treated the subject, and Professor Gilbert N. Lewis, 
whom I met yesterday on the Avenue, referred me to the same "well- 
known formula," and intimated that there was nothing more to be added. 
I am not quite convinced. After teaching thermodynamics for thirty 
years I find it is only now that my ideas on some points are becoming 
clarified. 
Let us suppose any system to be specified by two coordinates, x and y, 
in terms of which all thermal data may be expressed, neither being the 
temperature. If we deal with unit of mass of a substance, the specific 
heat at constant x is defined by the equation 
where JT^ is the change of temperature made when x is constant and dQ 
is the amount of heat required for a change of state. Similarly for the 
specific heat at constant y: 
dOy = CydTy, 
Now we have in any change 
and consequently: 
dT — dx + dy, 
dx by 
dQx = Cx dOy = C^-^dx. 
dy dx 
Since for an infinitesimal change the total amount of heat required is the 
sum of the two preceding, we have for any change whatever 
dQ = Cy^dx + dy. 
dx dy 
But, by the first law of thermodynamics, when we are dealing with a 
system under a uniform normal pressure p we have 
dQ = dU -\- pdv 
where U is the intrinsic energy of the substance. Writing out the perfect 
differentials dU and dv in terms of dx and dy, which are independent, 
we have the two partial differential equations : 
dJJ^ C — - — 
dx ^ dx dx' 
dU ^ dT dv 
dy dy dy 
Vrom these, eliminating U by cross differentiation, we get : 
