The integral being so taken as to be = for the state of per- 

 fect gas. 



Those two equations comprehend the whole theory of the mecha- 

 nical action of heat, and agree with those given in the author's pre- 

 vious paper on that subject. In that paper the assistance of Joule^s 

 Law was used in investigating the second equation ; in the present 

 paper it is deduced directly from the hypothesis. The following are 

 some of its consequences. 



Let P 5 V be the expansive power given out by the body while 

 the variations b r and h V take place. Then 



aQ + aQi-PaV=5Y(V,r) 



is the exact variation of a function of r and V. This is the ma- 

 thematical expression of Joule's Lavj. 



Let unity of weight of a substance be brought from the volume 

 Vq, and absolute temperature r^, by the process (a), to the volume 

 Vj^ and absolute temperature r^, and restored to the original volume 

 and temperature by the reverse of the process (6). Let r„ and r^ 

 be a pair of temperatures in the two processes, corresponding to the 



same value of / -— d V. The result of the pair of processes will be 



the transformation of a certain quantity of heat into expansive power, 

 whose value is as follows : — 



r VdY(a)- I 'TdV{b)=r \r^-T,)%^ dY. 



This equation comprehends as a particular case, Carnot^s Law 

 of the effect of machines working by expansion. 



The following equation, hitherto known for perfect gases only, is 

 shewn to be true for all fluids. Let a denote the velocity of sound 

 in a fluid ; Ky ^^^ ^^p the specific heats at constant volume and 

 constant pressure ; then, 



•=^{'■^4;) 



"■v 



VOL. III. 



