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



T—ic rfU 

 — Ts-^rv St -t- expresses the variation of heat due to change of 

 KjUm. dr 



molecular distributiou dependent on change of temperature. 



7. The function U is one depending on molecular forces, the 



nature of wJiich is yet unknown. The only case in which it can 



be calculated directly is that of a perfect gas. Without giving 



the details of the integration, it may be sufficient to state, that 



in this case 



U=-, 



T 



and therefore that 



dr ~ T^' dN 



(7) 



In all other cases, however, the value of this function can be 

 determined indirectly, by introducing into the investigation the 

 principle of the conservation of vis viva*. 



Suppose a portion of any substance, of the weight uniti/, to 

 pass through a variety of changes of temperature and volume, 

 and at length to be brought back to its primitive volume and 

 temperatm-e. 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 jjroduced, whether in 

 the shape of eapansion 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 

 foj'tn of expansion, an equal amount must have been communi- 

 cated 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 symboli- 

 cally, is . 



An-AQ' = 0; (8) 



where IT, when positive, represents expansive power given out, 

 when negative, compressive power absorbed ; and Q' represents, 

 when negative, heat given out, when positive, heat absorbed. 



To take the simplest case possible, let the changes of tempe- 

 rature and of volume be supposed to be indefinitely small, and to 

 occur during distinct mtervals of time, so that t and V are in- 

 dependent variables. Let the initial absolute temperature be t, 



* The function U is determined by direct integration in a paper already- 

 referred to on the Centrifugal Theory of Elasticity (Trans. Roy. Soc. Edinb. 

 vol. XX. part 3). 



