24 
Proceedings of the Royal Society of Edinburgh. [Sess. 
fix a, V, V a , v 0 , v a , A and or, but not g, v, s. The system will, in general, 
have its state completely defined by fixing any three of p, T, v, g , s. (It 
can, of course, be defined by other combinations of three variables.) Let 
us select as our controllable variables p, T, and s. 
We have the thermodynamical relations 
dE = dH-dW 
and 
dli = T d<f> 
for the reversible change considered, where dE and 
differentials. 
/0E N 
and 
+ 
\ 0S Jp, T 
ds 
are complete 
• ■ (3) 
d<f> = (<£E + dW)/T 
— (dE +pdv - c rds)/I 
1/0E dv 
= TV0T +J °df 
1 /0E 0r 
1/0E dv 
dT + + %r + tU +i % - v ds 
e> 
If we employ as variables p, T, and g, equation (4) takes a more 
symmetrical form, while if we use v, T, and s, it assumes a simpler form. 
Evidently 
<ty\ = \( d 3 , 
0T Jp,s TV0T Z 0T/ 
df\ _ I/0E dv 
dp )t, s TV dp dp 
dcf> 
ds 
Since 
we have from (6) and (7) 
1/0E dv 
p ,\ T T\0s ”^0S 
d^u d‘ z u 
dydx dxdy ’ 
<T 
(5) 
(6) 
(7) 
1 / 3' 2 E d 2 v \ _ 1/ 0^E dv 3 2 « SoA 
T\dpds ^^dpds) T\0y0s ds ^ dpds dp) 
or 
/dv\ 
( 8 ) 
^ = • - 
Now 
\ ds/p, T 
/0cr N 
dcr \ 
dp/s, 
da\ 
