50 C. V. L. Charlier 



(79) AJ:,y) = — -— e W ; f,{y)=-—-=e '2,/ 



Oj [/ 27r 1/ 2jt 



then we get from (78) according to (51) and (52) 



(80) a{m) = 77^T7=^ ^'^ 2(0/^+0/) 



and 



(80*) M„(r) = e ^1'+^^' . e ^(a,^ + o,^) _ 



Equating the right members of these expressions with the formulée (76) 

 and (77) for a{m) and Mm{r) we get the relations 



(81) 



p - 





N = 





mo = 



— ; 













miOj^ + OTj 

 _ 2 1 ^ 2 



2 "1 -"2 



lOe"*'"' =e =i' + '2^ + 



Solving these relations with respect to o^, Og, , and C we get 



C = N; 



(82) a^2 = X,Ä;2; 



o,^:=(l-XJF; 



>«2 — = m^, 



X, + m,{l - X J = 5 + 5X, + ^ è X,(l - X J F 



The two last eqvations for and m.^ may be written 



1 



(82**) 



= -5 + (5-mo) X, + ^ ^ ^1 " ^i) 

 = -f 



The equations (82) and (82**) solve the problem. It remains however to 

 determine from A^f«/), according to the equation 



D(r) = fl(.-'») = iÄ,W- 

 Substituting here the formula (79) for ^^{y) we get 



by] ^0 



(83) D{r) Die = e 2c,' , where 



1/2;: 



mg = ni^ -f 3fc Oj^, and 

 " a, (üb 



