100 The Law of Distribution [CH. v 



The remaining averages are required only when r = 0. We may therefore 

 write U for V, and u' u for a, etc. 



It is obvious at once that when r = 0, {ZjZ/} = 0, and hence that as the 

 averaged value of a further term in equation (238) we have 



As regards the remaining terms, on putting r we obtain 



{D} = 2 |[>1X 2 ] ctu} + 2 {[A^} (0w + /v)} ............ (241) 



by equation (236) ; and since the various systems of quantities may now be 

 supposed to vary independently 



{[A\*] cm} = {A} {A, 2 } (u' -u)u 



by equation (234), and 



{[A/j,v](l3w + <yv)} = {(v v')w + (w' w)v}- ^ --{-4}, etc., e 

 and from equation (228) 



Substituting these values in equation (241), we find that 



m\ l v / ' x 



\D\ = JTT-JJ 2M (U U), 



and in a similar way we have, for the remaining term in equation (238), 

 {A F 2 cos 2 i/r} = {A } F 2 {cos 2 ^} 



2 . 'ft 



2 r 2 



= ^- F 2 I (cos 2 i/r) 2 sin -^ cos 



OK' J 



since we are now neglecting r altogether. 



Hence, averaging equation (238), we obtain on substituting the averages 

 which have been found for the separate terms, 



