68 



The Law of Distribution 



[OH. v 



The quantities rj l , ?/ 2 ... will be referred to as momentoids, and the form 

 of equation (153) shews that the kinetic energy of the molecule may be 

 regarded as the sum of contributions from various momentoids. 



There are an infinite number of possible substitutions of the kind re- 

 quired, for if in (152) we replace 6 n , 6 12 ... by Kb u , Kb lz ..., and r} 1 , t]^... by 

 r]i/K, ijvIK , the substitution is still a possible one and reduces *2L to a sum 

 of squares as before. The modulus of transformation for this new substitution 

 is the Jacobian 



( n+i > s n+2 



and is therefore K n times the modulus of the original substitution. By a 

 suitable choice of K, the right-hand member of the foregoing equation can 

 always be made equal to unity. We shall suppose such a value of K to have 

 already been absorbed in the quantities which occur in the transformation 

 (152), so that 



9(|n. +1 , gn +2 ..j = 1 ...(154). 



9 (171,%...) 



If we now transform variables according to scheme (152), so that the new 

 variables are 



1, &' %n, i)i > fli T]n (155), 



we see from expression (137) that the number of molecules for which lies 

 between ^ and ^ + d^, ^ between t) l and 771 + d^, etc., will be 



1) ** '' - } ^ ... d* B> 

 , % ) 



and this, by equation (154), may be written 



^a/a^i, | 2 n, t)i fjn)d^d^ ... dgndlji ... dr) n (156), 



while from equation (146) we have 



f a = A a er* E * = A a er* hv - h i e nt + "- +e * t > (157). 



If, then, we denote by c^ 2 the value of dV averaged over all molecules 

 of type a, we shall have 



f [IT. . . 



. . . dfc^ . . . d rj n 



JJJJ ". 



...... (158). 



The integrals in both numerator and denominator are of the type 

 JJJJ . . . le-^ 



