OF THE ENERGIES OF A SYSTEM. 89 



In like manner, let us denote by V q the extension-in-configu- 

 ration below a certain limit of potential energy which we may 

 call e g . That is, let 



JV 



(2T1) 



the integration being extended (with constant values of the 

 external coordinates) over all configurations for which the 

 potential energy is less than e g . V q will be a function of e q 

 with the external coordinates, an increasing function of e 3 , 

 which does not become infinite (in such cases as we shall con- 

 sider *) for any finite value of e q . It vanishes for the least 

 possible value of e ? , or for e q = oo , if e q can be diminished 

 without limit. It is not always a continuous function of e g . 

 In fact, if there is a finite extension-in-configuration of con- 

 stant potential energy, the corresponding value of V q will 

 not include that extension-in-configuration, but if e q be in- 

 creased infinitesimally, the corresponding value of V q will be 

 increased by that finite extension-in-configuration. 

 Let us also set 



(272) 



The extension-in-configuration between any two limits of 

 potential energy e q and e q f may be represented by the integral 



(273) 



whenever there is no discontinuity in the value of V q as 

 function of e q between or at those limits, that is, when- 

 ever there is no finite extension-in-configuration of constant 

 potential energy between or at the limits. And hi general, 

 with the restriction mentioned, we may substitute e^ q de q for 

 Aj dq 1 . . . dq n in an w-fold integral, reducing it to a simple 

 integral, when the limits are expressed by the potential energy, 

 and the other factor under the integral sign is a function of 



* If V q were infinite^ for finite values of e,, V would evidently be infinite 

 for finite values of e. 



