416 Mr. Rankine on the Hypothesis of Molecular Vortices , 



consists partly of heat and partly of expansive power, the pro- 

 portion depending on the mode of intermediate variation of the 

 volume and temperature, which is arbitrary. If the mode of 

 variation be different in the two operations, the effect of the 

 double operation will be to transform a portion of heat into ex- 

 pansive ]30wer, or vice versd. 



Let {a) denote the first operation, {b) the reverse of the second. 

 Then 



O It 



The terms of ^ which involve functions of t only, or of V 

 only, are not affected by the mode of intermediate variation of 

 those quantities. The term on which the mutual conversion of 

 heat and expansive power depends is therefore 



/{(T-«)|-;'}-;vw=/{(T-«)|-;,)iVW, 



or 



Hence 



y^dY{a)-/^dY{b)=fpdY{a)-fpdY{b) ; 



which last quantity is the amount of the heat transformed into 

 expansive power, or the total latent heat of expansion in the 

 double operation. 

 Let 



r±dY=r-^-'^^dY=^. 



■J dr i/t — K d\ 

 Then, because 



^dY={T-K)d^, 



we have 



f%dY{a)-f'^'pdY(b)=f \T-K)d¥{a)-r \r-K)dY{b) 



. =x! '-^^i«=f:r^^^- ■ • • • p«) 



in which t„ and tj are the pair of absolute temperatures, in the 

 two operations respectively, corresponding to eqitai values ofY*. 



This equation gives a relation between the heat transformed 

 into expansive power by a given pair of operations on a body, 

 the latent heat of expansion in the first operation, and the mode 



* F is what has siuce been called a " Ileat-poteutial," and with the ad- 

 dition of the term /( k z^J^ j dr, a " Thernio-dvnamic runctiou." 



(1855.) 



