or Centrifugal Theory of Elasticity. 415 



tegral is the following function of the volume and temperature : — 

 ^ = fe(T-«) + |^(log.T+^)+/{(T-«)|-;;}^V 



-Jf{y)dy (27) 



Accordingly, the total amount of power which must be ex- 

 ercised upon unity of weight of a substance, to make it pass 

 from the absolute temperature Tq and volume Vq to the absolute 

 temperature Tj and volume V,, is 



^(V„T,)-^F(Vo,To). 



This quantity consists partly of expansive or compressive 

 power, and partly of heat, in proportions depending on the mode 

 in which the intermediate changes of temperature and volume 

 take place ; but the total amount is independent of these changes. 



Hence, if a body be made to pass throiic/h a variety of changes 

 of temperature and volume, and at length be brought back to its 

 primitive volume and temperature, the algebraical sum of the por- 

 tions of poioer applied to and evolved from the body, whether in the 

 form of expansion and compression, or in that of heat, is equal to 

 zero-. 



This is one form of the law proved experimentally by ]\Ii\ 

 Joule, of the equivalence of heat and mechanical power. In my 

 original paper on the jNIechanical Action of Heat, I used this 

 law as an axiom, to assist in the investigation of the Equation of 

 Latent Heat. I have now deduced it from the hypothesis on 

 which my researches are based ; not in order to prove the law, 

 but to verify the correctness of the mode of investigation which 

 I have followed. 



Equations (26) and (27), like equation (23), are made appli- 

 cable to unity of weight of a mixture, by putting 2«k for fe, and 



^ hfi „ hjM 

 Snj^for^j. 



The train of reasoning in this article is the converse of that 

 followed by Professor William Thomson of Glasgow, in article 20 

 of his paper on the Dynamical Theory of Heat, where he proves 

 from Joule's law, that the quantity corresponding to h^ is an 

 exact differential. 



(11.) Mutual Conversion of Heat and Expansive Power. Car- 

 not's Law of the Action of Expansive Machines. — If a body be 

 made to pass from the volume Vg and absolute temperature Tq 

 to the volume V, and absolute temperature t,, and be then 

 brought back to the original volume and temperature, the total 

 power exerted (^) will have, in those two operations, equal arith- 

 metical values, of opposite signs. Each of the quantities SP' 



