162 MR W. J. M. RANKINE ON THE 
Suppose a portion of any substance, of the weight wnity, to pass through a 
variety of changes of temperature and volume, and at length to be brought back 
to its primitive volume and temperature. Then the absolute quantity of heat in 
the substance, and the molecular arrangement and distribution, being the same 
as at first, the effect of their changes is eliminated ; and the algebraical sum of the 
vis viva expended and produced, whether in the shape of expansion and compression, 
or in that of heat, must be equal to zero :—that is to say, if, on the whole, any 
mechanical power has appeared, and been given out from the body, in the form 
of expansion, an equal amount must have been communicated to the body, and 
must have disappeared in the form of heat; and if any mechanical power has 
appeared and been given out from the body in the form of heat, an equal 
amount must have been communicated to the body, and must have disappeared 
in the form of compression. This principle expressed symbolically is 
STE G10 eee a (2) 
Where u, when positive, represents expansive power given out, when negative, 
compressive power absorbed ; and Q’ represents, when positive, heat given out, 
when negative, heat absorbed. 
To take the simplest case possible, let the changes of temperature and of 
volume be supposed to be indefinitely small, and to occur during distinct intervals 
of time, so that 7 and V are independent variables. Let the initial absolute tem- 
perature be 7, the initial volume V, and the initial total elasticity P; and let the 
substance go through the following four changes. 
First, Let its temperature be raised from 7 to 7+07, the volume remaining 
unchanged. Then the quantity of heat absorbed is 
dQ 7t-Kk dU 
0 aaa) 
and there is no expansion nor compression. 
Secondly, Let the body expand, without change of temperature, from the 
volume V to the volume V+0V. Then the quantity of heat absorbed is 
—dV. 

T+O7T-K/1 d GU: «tk 
Cn M (y-a¥ : SO a b7)) 
while the power given out by expansion is 
= dP 
ONCE Ot) 
Thirdly, Let the temperature fall from ++067 to its original value 7, the 
volume V+6V continuing unchanged; then the heat given out is 
dQ t—-K d dU gy. 
+80(73 ~CaM +z) 
and there is no expansion nor compression. 

