18G 
DR. E. P. PERMAN", DR. W. RAMSAY, AND MR. J. ROSE-INNES, 
area ABCD = increase of pressure when we pass from lower to higher adiabatic 
at constant volume. 
X increase of volume when we pass from higher to lower isothermal 
along adiabatic. 
Now, whatever the variations of pressure volume and temperature, they are subject, 
if small, to the relation 
Let us denote differentiation with regard to v by accents, then we may put this 
dj) = bdT -{-{h'T - a')dv. 
Also any adiabatic alteration will be subject to the relation 
dp — (^kxp' + y') di\ 
Hence, 
h dT + (//T - a') dv = {kx}j' + x') dv 
ch 
dT ;,- kf' + ^' + a' -¥T , .,P + 
kf + X + (I -b —p- 
W + x' + ft' \ + a) k lir' - + X + (x + ft) 
Therefore 
- xIj{v) 
dv 
\ftT/^ 
\ 
V 
;.■ ( -vlr I - ^ (x + ft) - X - a' 
Now this quantity is to be independent of T so long as we keep k unaltered, hence 
we have 
^b' <r'\ . V 
,1b' f \ / . N X + ^' n • r. 7 1 
(v “ 4+) + SV (X + “) - "ST " ^ 
' ^ ft' 
This can only be the case if the expressions y, — — , (y + «) “ ^ d. 
■’ ^ b^ byjr ■ b-yp- ^ byp 
are 
functions of k only. But both these expressions, from their form, can be seen to be 
functions of v only ; hence they must be numerical constants. Let 
h'/lr — \p'/h\p = C, say. 
