20 Ml'. W. J. M. Rankiue on the Mechanical Action of Heat. 

 Now . T being tlie same for all densities, is the value of U for 

 the perfectly gaseous state, or -; for in that state the integral = 0. 

 The values of the partial differential coefficients are accordingly 





(11) 



and they can, therefore, be deteiinined in all cases in which the 

 quantity /f=CnjK,i, and the law of variation of the total elasticity 

 with the volume and temperature are known, so as to complete 

 the data required in order to apply equation 6 of this section to 

 the calculation of the mechanical vahie of the variations of heat 

 due to changes of volume and molecular arrangement. 



The total elasticity of an imperfect gas, according to equations 

 VI. and XII. of the introduction, being 



P = 



CnMV 



,(i_f(d.i))+/(D), 



its first and second partial differential coefficients with respect to 

 the temperature are 



dr '' 



CnMV 



0-( 



1 + T 



d_ 



dr. 

 d^ 



Consequently, for the imperfectly gaseous state. 



>■ <12) 



dV K /. d f/2 X n 



'(-^^.) 



(8.) It is to be observed, that the process followed in ascer- 

 taining the nature of the function U is analogous to that em- 



* The substitution of these values in equation (J reduces it to the follow- 

 ing form, which, being the more convenient, hiis been employed in most 

 subsequent investigations : — 



