OF THE ENERGIES OF A SYSTEM. 91 



the integration being extended, with constant values of the 

 coordinates, both internal and external, over all values of the 

 momenta for which the kinetic energy is less than the limit e p . 

 V p will evidently be a continuous increasing function of e p 

 which vanishes and becomes infinite with e. Let us set 



The extension-in-velocity between any two limits of kinetic 

 energy e p and e p " may be represented by the integral 



f 



e* p de p . (278) 



And in general, we may substitute e^ p de p for A,* dp l . . . dp n 

 or Ag* dq l . . . dq n in an w-fold integral in which the coordi- 

 nates are constant, reducing it to a simple integral, when the 

 limits are expressed by the kinetic energy, and the other factor 

 under the integral sign is a function of the kinetic energy, 

 either alone or with quantities which are constant in the 

 integration. 



It is easy to express V p and $ p in terms of e p . Since A^ is 

 function of the coordinates alone, we have by definition 



1 ...dp n (279) 



the limits of the integral being given by e p . That is, if 



e p = F( Pl ,... Pa ), (280) 



the limits of the integral for e p = 1, are given by the equation 

 F( Pl ,... Pa ) = \, (281) 



and the limits of the integral for e p a 2 , are given by the 

 equation 



='. (282) 



But since F represents a quadratic function, this equation 

 may be written 



1 (283) 



