58 C. V. L. Charlier 



Inversely we have 



(125*) f'i = y^iv^+ v^), V^^y^ (- v, + I'J , 



so that 



(126) (O = 1/2 + IV + W',/ = 1/2 fi^ . 



The lacobiana being =1, we have 



zl£, Jsg = cZ?/, (Z r, d W\ dU^ dV^dW^. 



As furthermore 



2 



2 



so is 



2a^ 2a' 



and we get 



(127) V.oo = \Uffll i-^U,' + ^2' + W/) If,f, dU, dV, dW, dU, dV, dW„ 



ivhere I depends only on . 



For performing the integration pohir coordinates are introduced. Let us liowever 

 first consider the development of fif^- 



44. Development of • 



We have supposed that may be represented as an ^.-series having the 

 generating function 



^ + v^^ + 



a='(2;ry 



so that 



(129) f, = IA,.,'tO^^. 



where 



^ ^ 8'+^'+^^cc(i) _ 8'>(?<J d'''i[w ^) 

 ~ du^' dv^' du'i'' ~ 8?Vi'' div^'' 



In like manner we have 

 where 



