(" 



Mr. W. J. M. Rankine on the Mechanical Action of Heat, 19 



the initial volume V, and the initial total elasticity P ; and let 

 the substance go througli the following four changes : — 



First. Let its temperature be raised from t to t + 8t, the 

 volume remaining unchanged. Then the quantity of heat 

 absorbed is 



J^ CnM. Ji^)' 



and there is no expansion nor compression. 



Secondly. Let the body expand, without change of temperature, 

 from the volume V to the volume V + SV. Then the quantity 

 of heat absorbed is 



while the power given out by expansion is 



8V(P+^8t). 



Thirdly. Let the temperature fall from t + St to its original 

 value r, the volume V + SV continuing unchanged; then the 

 heat given out is 



and there is no expansion nor compression. 



Fourthly. Let the body be compressed, without change of 

 temperature, to its original volume V ; then the heat given out is 



-8V^' 



mVv d\)' 



CnM 



while the power absorbed in compression is 



-SV.P. 



The body being now restored in all respects to its primitive 

 state, the sum of the two portions of power connected with 

 change of volume, must, in virtue of the principle of vis viva, be 

 equal to the sum of the four quantities of heat. Those additions 

 being made, and the sums divided by the common factor 8V8t, 

 the following equation is obtained : — 



dr ~ CnMVV d\) ^ ' 



The integral of this partial differential equation is 



U = 0.T+/^v(l-CnM^). . . (10) 



C2 



