Products o, Legendre Functions, 773 



and the corresponding solution 



y ««. { P„W + <^- 2 P^O) + *& . ^P. + iG0 +-...), 



where a a is arbitrary. 



For the first series, write a = m + n + l, and we find the 

 solution 



1 2f?i + 2 .2n + 2 2m + 2n + 3 2m + 2n+l 

 y ~~ n,+n+1+ 2' 2-/n + 3. 2n + 3' 2 wi + 2» + 4 # 2m + 2n + 3 



X -L m + n + 3 



1.3 2m + 2.2m + 4.2/ ? H-2.2/i + 4 

 + 2 . 4 ' 2m + 3 . 2m + 5 . 2n+ 3 . 2rc + 5 



2»i + 2tt + 3.2m + 2tt+5 2m + 2n+7 . 2m + 2/i + ll 



* 2??2 + 2?2 -f 4 . 2m + 2^ + 6 ' 2/?i + 2/i + 3 . 2;?i -}- 2?i + 7 



multiplying by a constant, we may take instead the form 



y 



' § 2 _ 2 „ T(m + r+l)r(n + r + l)F(m + n + r + +) 

 r % ' Tim + r + f) T(n + r + f) F(m + n+r +2) 



,j^j^(2m + 2n+4:r + 3)F m+n+1+ 



2r, 



which may be called Series A, and to which we shall return 

 later. 



For the second ascending series, we suppose m > n. Then 

 a = m — n, and the series becomes 



If 2m 4-2 2/i 2m — 2ft + r f 2m — 2n + 5 

 y-JV-« + 2 ^^+3 ' 2n- 1 ' 2m-2n + 2 J 2m-2n + 1 



X "»l— Tl+2 



1.3 / 2m + 2.2m + 4 2n.2n — 2 

 + 2 . 4\2m + 3 . 2m-h5*2/i-1.2/i-~3 



2m -2/1 + 1 . 2m — 2n + 3 ") 2m-2/i + 9 

 • 2m-2n + 2 . 2m-'Zn + 4:j 2m-2/i + 1 m ~ n+i *"" 



We notice that it terminates with the term in P m + n , so 

 that it must be a multiple of Adams' series, and in fact 

 can be arranged equally as a descending series. We may 

 call it Series B. 



