358 On Maxwell's Investigation of Boltzniann's Theorem. 



considered as absolutely fixed throughout the discussion. 

 The following, dependent upon the substitution for the 

 " action" A of Hamilton's "principal function" S, seems to 

 meet the requirements of the case. By definition, 



and, as in Thomson and Tait's ' Natural Philosophy/ § 319, 

 8S = i8A- PsVdt 



= \{Zm(xhx+ . . .)}-\\tm(xhx+ . . .)] 



+ iVdt[8T+8Y-2SV] ; 



so that 



8S = {Zm(itSa+ . . .)}-\$m{x8x+ . . .)], 



or in generalized coordinates 



8S=2/?^-2p r S/ (1) 



In this equation all the motions contemplated are uncon- 

 strained, and occupy the fixed time t — t'. The total energy 

 E is variable from one motion to another, and S is to be 

 regarded as a function of the g's and g' 3 s. 



The initial and final momenta are thus expressed by means 

 of S in the form 



,_ d$ _ d& . 9 v 



Pr ~ dq r " Pr ~d^'' W 



so that 



dpr __ __ d 2 8 _ __ dps _ 



dq 8 ' dq r f dq s dg/' ' J 



Thus, using S with t—t' constant, instead of (as in Maxwell's 

 investigation) A with E constant, we get 



* As an example the motion of a particle in two dimensions a"bout a 

 centre of force may be considered. q r , qs are then the rectangular co- 

 ordinates of the particle at a fixed time t ; q r ', q s ' the coordinates at the 

 fixed time t', while p r ,ps and p r ' p s ' are the component velocities at the 

 same moments. 



In equation (3) r and s may be identical. 



