194 PROCEEDINGS OF THE AMERICAN ACADEMY. 



for k ^ m, 

 (90) 



s 



for k ^ m, and 

 (91) 



s 



Also, from (83), (87), and (88) follow 



n'S 



ds 

 so that 



c's 



^ (S-Zm+l ^2,,,) = ~ (^ '" + 1) ' '^2o,+l Rim+1 + '^Sm ^im ) 



(92) CS,,„ R,,„ ■ ds = [S,,, i?,„._Ol + 2 ;/i • J^„„_i i?2„. 1 • ds, 



ds. 



(93) J^„„^,/?.,.+i • ^s = - 2^ [>S.,.+i i?.,.f + 2;;^/'^^"' ^-^ 



By alternate applications of (92) and (93) any integral of the type 

 I S,„ B,„ • ds can be reduced to a linear function of expressions of the type 



[aS^. ^i-i]! for values of k descending from m to 1 and of the integral 



(94) Cs,R,-ds= CeT^ 



ds. 



X. Case in avhich all Values of s are possible. 



For s = — oc and s =- -\- yz (that is, if all values of the residual are 

 possible), the integrals in question have simple numerical expressions. 

 Namely, whatever integer k may be, 



_«' 

 s* e ^ = f or s = ± X ; 



