Products of Legendre Functions. 769 



The operation 



may be conveniently called D, so that 



(D + M + N)?/ = 2P'Q'(1-/* 2 ). 

 Accordingly 



= _4 / ,(l_^ ) p/Q/ + 2 (l- / , 2 ) 2 (P"Q'i-Q"P'j 

 = -^(l-^p/Q/ + 2(i-/x 2 )Q'(2/xP'-NP) 



+ 2(1- / . 2 )F(2/,Q'-MQ) 

 = 4 / x(l-^)P / Q'-2(MFQ + NPQ0(l-/ A 2 ). 

 Substituting for P'Q', this may be reduced to 



(1 - ^) £ . (D 4- M + N V - 2fi(D + M + N)y 



= -2(l-V)(MFQ + NPQ') 

 Differentiating again with respect to /-t, 



D(D + M + N)sf -2~.fx(D + M 4- N)y 



= 4 / a(MP'Q-fNPQ / )-2(l- A 6 2 )(M+N)P , Q' 



-2(1-/* 2 )(MP"Q + NPQ") 

 = 4//(MP'Q + NPQ / )-2(l- A 6 2 )(M + N)P'Q' 



-2MQ(2 At P / -NP)-2NP(2 A tQ / -MQ) 

 = 4MN J /-2(l- A 6 2 )(M-tN)P / Q / 

 on reduction. Again substituting for P'Q', 



D(D + M+N)y~2^-.A*(D + M + N)y-4-MNy 



+ (M + N)(D + M + N)ya*0, 

 which we may write in the form 



(D-FM + X) 2 ^2^ t/i (Di-M + :% k4M%. 

 Pfo'Z 3%. S. 6. Vol. 43. No. 256. 4/m7 1922. 3 JD 



