438 MR W. J. M. RANKINE ON THE CENTRIFUGAL THEORY OF ELASTICITY, 
(¥) will have, in those two operations, equal arithmetical values, of opposite signs. 
Each of the quantities ¥ consists partly of heat and partly of expansive power, 
the proportion depending on the mode of intermediate variation of the volume 
and temperature, which is arbitrary. Ifthe mode of variation be different in the 
two operations, the effeet of the double operation will be to transform a portion 
of heat into expansive power, or vice versd. 
Let (a) denote the first operation: (0) the reverse of the second. Then 
Y=Y, 
The terms of ¥ which involve functions of + only, or of V only, are not affected 
by the mode of intermediate variation of those quantities. The term on which the 
mutual conversion of heat and expansive power depends, is therefore 
Nae dV (8) =f[o-9% Pav (a) 
or, S (G-2) vo= ele p) ava) 
Hence, 
[VO -~fRavo =foavea— foavn 
which last quantity is the amount of the heat transformed into expansive power, 
or the total latent heat of expansion in the double operation. 
Let dp re ra 
¥ec dV = = [os - aV=F 
Then because 

TE Ge yak 

aV 
we have 
Mi V4 By ay 
paV (a)—f paV(b)=}/ (t—k) da F (a)—] (1t—k) d F (6) 
ferro fpervafe-vare finer 
=) \(n— 7) ab aidaten 
=f¢ err ON) NN ae Oe Se 
In which 7. and 7; are the pair of absolute temperatures, in the two operations 
respectively, corresponding to equal values of F. 
This equation gives a relation between the heat transformed into expansive 
power by a given pair of operations on a body, the latent heat of expansion in 
the first operation, and the mode of variation of temperature in the two opera- 
tions. It shews that the proportion of the original latent heat of expansion finally 
transformed into expansive power, is a function of the temperatures alone, and 
is therefore independent of the nature of the body employed. 
Equation (28) includes Carnot’s law as a particular case. Let the limits of 
vt 
ee 
ssid 
