182 



Theory of a Non-Conservative Gas [CH. vni 



216. Let <l> denote a maximum value of <I>, and suppose that this value 

 occurs at the point p^, p 2 ' ... q n ' ; then in the immediate neighbourhood of 

 this point we can expand <l> in the form 



.(4*3), 



'^ 



where the differential coefficients are all evaluated at the point at which 

 <f> = <I> . Since <E> is, by hypothesis, a maximum so long as E is kept 

 constant, it follows that over the surface E = E^ 



9*3? 9^ 

 ( p l PI) + . . . + (QTI Qn) ^0 (444). 



Hence the region for which E = E l and for which <l> lies between <3> and 

 an adjacent value <J\ is bounded by 



When p l pi, p 2 p 2 ', . . . are small enough, terms of degree higher than the 

 second in these quantities may be neglected, so that the equation is that of 

 a small closed boundary, an "ellipsoid" of 2w dimensions. The volume of 

 this ellipsoid is a Dirichlet integral of which the value is found to be 



F(H-n) 



.(446), 



where 



A = 



8 2 <|) 



8 2 <l> 



.(447). 



The equation of the plane E = E t is equation (444), and the intersection 

 of this plane with the 2n dimensional ellipsoid (445) determines a 2?i 1 

 dimensional ellipsoid. This latter ellipsoid contains all points for which 

 E = E l and ^ < <I> < <f> . To find the volume of this 2n 1 dimensional 

 ellipsoid, we must first find the lengths of its principal axes. 



A 2w dimensional sphere of radius r intersects the 2n dimensional 

 ellipsoid (445) in a cone of which the equation is 



