Law of Distribution of Energy. 149 



If, further, 



// „ „ \ _ ^^ -— C«i*i 2 +&ia«i«a+ • • •) 



(2tt)3 

 we see that 



f fe.. . <g = /(c x . . . cj. 



Now let us write 



, #n + #2i+ • • • +%m 



* = 71 — — 



I — ^12 + ^22+ • • • + #N2 ^ 



*» - - - 4/N 

 Also 



Sj_ = a^n 2 + &i2^n^i2 + • • • 



S 2 = a^i + ^i 2 a?2i^22 + • » • 

 Ac. = Ac. 



S' = ajc-l 2 + btfpc-lxj'. 

 Therefore expanding a?/ 2 &c, 



S'= — %* — - L - : + terms containing products of 



the form x x 2 , that is products of two x 9 8 taken from different 



associations, which terms are not generally zero. 



(6) The truth of the foregoing propositions will not be 

 affected if for a 1? b 12 , &c. we write koi, fcb 12) &c, where k is 

 any positive constant. In this case D becomes K n T> and 

 S becomes «S. It will be convenient if at this point we 

 introduce a factor corresponding to Boltzmann's h, namely, 



rn 



wm, so that henceforth we will write 



S = a 1 x x 2 + b 12 x 1 x 2 + &c, 



where 



and 



F(#! . . . #J = Ce 

 " "V2T/ 



_nS 

 2T 



D = S± 2ai2a 8 &c. 

 In this case also 



2T D u — _ 2T D 



®x = - -jy , x x x 2 = — jy > &c«, 



