1209 



Tliis result leads also to a new tbrtn for all the functions L,i (.r), ïov 





Vjx 







and, according to (12) 



Ln = HnL,-^~:é>^'Tn-X ...... (27) 



where 



T„_i = ^„_i - 2(n-2) F„_3 + 2Xn— 3) (n— 4) //„_5 - . . . (28) 



11. Applying the preceding expansion, the problem of the momenta 

 may be solved. 

 Let 



CO 



«n = j f{y) y"dy 







the question is to determine the function /(y) when a,, is given for 

 all i)Ositive integral values of ?i. 

 Putting 



fiy) = e-"' VKH,{y) + h,H,{y) +h,H^{y) :f . . .] 



we have to determine the coefficients h from 







h U-y'y''Hp{y)dy 



Here p-\-n is an even number, for the integral vanishes for /> -f- '^ 

 odd. Moreover the integral vanishes if p'^n therefore 



iX 



an = !k bpje-I/'y"Hj,{y)dy 

 — « 

 or, according to the expansion I Art. 8 





an=n.'[/? ^ '- 



n — p 

 2"-/^ -f 



which may be written 



""^ = A„ = é '" 



^'^\^jr n — » 



*^ ^ 2»-p - I 



2 



Solving these linear equations, we get immediately 



