186 Theory of a Non-Conservative Gas [CH. ym 



In the most general case it is obvious that the quadratic expression on the 

 left hand is finite. The expression is, however, by hypothesis, positive for all 

 values of (PI PI) - and as it contains the 2n variables^ p^, ... q n qn, it 

 may be expressed as a sum of 2n squares of linear functions of these variables. 

 When n is infinite, therefore, each square will be vanishingly small, so that 

 Pi Pi> Pz pz> will be infinitesimal. We may say, then, that inside any 

 cluster the coordinates pi , p 2 . . . of any point are indistinguishable from the 

 coordinates p^, p z ' ... of the corresponding point on the central line of the 

 cluster. 



221. The analysis has supposed the system to have started from an 

 initial value of the energy E = E Q , and has, moreover, supposed all configura- 

 tions for which E = E to have been equally probable as the starting point. 

 In nature it is not easy to find any value of E which shall represent the con- 

 ditions of the problem. When the system under consideration is a gas, we 

 may probably assert with confidence that the gas has cooled from some higher 

 temperature at which its energy has some value E , but we cannot say that 

 at this temperature all configurations for which E = E were equally likely. 

 For obviously if it had previously cooled from some still higher temperature 

 this statement would not be true. Again, the value E = oo does not supply 

 a solution of the difficulty, for in taking this value we should not only have 

 to suppose that the gas has cooled from a temperature in which it probably 

 could not exist as a gas, but we should also have to suppose that it had cooled 

 undisturbed throughout infinite time. 



222. An escape from this difficulty is found as soon as we examine the 

 way in which our work has depended on E . It will be seen that E has 

 entered only as a limit of integration in the definition of <i>, namely (from 

 equation (441)), 



=pdE 



L 



.- (455). 



In this equation the integral has to be taken tentatively along all stream 

 lines starting from E = E , and the course of the cluster or family is deter- 

 mined by the maximum values of <E>. The integrand contains a factor F 

 in the denominator. From its nature F cannot be negative, and in general 

 will not vanish. If at a special point F does vanish, then the integrand in 

 the first integral becomes infinite, but the value of the integral taken over 

 any part of the path which includes this point remains finite, because so long 

 as F = we have also dE = 0. The same result is of course obvious from the 

 second form of the integral. If, however, F remains very small over a finite 

 range of values of E, the contribution to the integral from this range of 

 values will be very great. In this case the definition of <I>, for values of E 



