OF TWO LEGENDRE'S COEFFICIENTS. 359 



In the same way it may be shewn that 



in n " \ / / . _ _ 



1 . 3 ._5_... (m-2) 1 .3 . 5 ... n 

 * if 4 . 6 ... (m+1) 2 . 4 . 6 ... (- 1) ' 



p 

 which may also be found from ' S n P m d^ by interchanging m and ?i. 



Hence 



P (S m P )t + S..PJ c^ - k^T - - [<S',,A] M= o 

 Jo L Jo 



- yv" 1-3.5 ...(m-2) 1.3.5 ...(n-2) 

 ' 2.4. 6. ..(m+1) 2.4.67.. (w+l) 



m(m+l)[m(m+l) w(n+ 1) 2] n (+ 1) [m (m+ 1) n(n+ l) + 2] 

 (m nl)(m n+l)(m + n) (m + n + 2) 



If m + I = n or if n + 1 = m the numerator of the last fraction vanishes, 

 hence (m n+1) and (m n 1) are factors of it, also (m + n) is a factor 

 and the remaining factor is (m + n + 2); hence the fraction = 1, 



and f ' S m P n dp. + f 1 S n P m dp. = \S m S,^ 

 Jo Jo Jo 



xT 1.3.5 ...(m-2). 1.3.5 ...(n- 2) 

 ' 2. 4.6 ... (m+1). 2. 4.6 ... (n+l)' 



1 4. We will take an example of the application of these last formulae. 

 Let then m = 3 and n I . 



Then by formulae just found 



^ = - -L and 



o o 



f 



Jo 



By actual formation of SiP 3 and S 3 P, and integration, 



