Harmonics of the Second Type. 5 



as against 



p*h(o)-o, p 2 „(0) = (-)» 1 2 3 ; 4 ;; 2 ^~ 1 . . (9) 



The Q's admit a recurrence formula 



(?i + l)Q«+i-(27i + l) / u,Q /i + nQ re _ 1 = 0, 

 so that Q n+1 (0) / Q n _x (0) = - n J (72 + 1) 



or . Q„ +2 (O)/Q„(0) = -0i + l)/( W + 2). 



We thus find without difficulty, that * 



2.4. ...2n 



H2nV 



j,-v, W«i-ri '3.5....2n + r 



From another recurrence formula, 





(2n.+ l)/i Q„' = (n + 1) Q'„_! + n Q'„ +1) 



we have 



Q'» + i(0)/Q'»- 1 (0) = -(»+l)/n 



or 



Q'„ + 2(0)/Q„'(0; = -(n + 2)/(» + l). 



But 



Q '( M )=- 1 - 2 , Q '(0)=1, 



whence 



Qi'(0)=o, 



(10) 



QWi(0)=0, Q' a ,(0) = (-)» 1 2 3 4 ;- 2w 2 " 1 . (11) 

 Returning to the formula 



(m-n)(m4-n + l)rP n Q 7 ^ y a=(l- / . 2 )(Q w P ? /-P n Q m / ), 



which is true between any two limits, we can evaluate the 

 whole set of integrals between limits zero and unity. Thus 



(2ro- 2n)(2m + 2n + 1) ( F 2n Q 2m dfjL 



Jo 



= -P 2 ,(l)P2.(l)-{Q 2Hl (0)P' 2 ,c0)-P 2/t (0)Q' 2m (0)} 



- 1-1 .(-VM-n 1 - 3 - --- 2 "- 1 2.4. ...2m a2 . 



" i + ( ^ ' 2.4.~2^"'1.3....2m-l' ' (1J) 



(277i-2?i-l)(2m + 2n + 2) ( 1 p 2 n+iQ2m^/* 



Jo 



= -l-{Q 2 »(0)P' 2 „,i(0)-P 2K+1 (0)Q' 2ra (0)} = -l. (13) 



