OF TWO LEGENDRE'S COEFFICIENTS. 351 



Following out the law of formation of the terms, we see that the 

 terms are alternately positive and negative, the numerical coefficients are 

 those of a binomial raised to the power indicated by the number of 

 integrations, the denominators are products of factors 



and of those factors all diminished by 2, 4, 6 &c. with the omission in 

 the case of any term involving P n+r of the factor 2n + 2r+l. 



Thus if we have m integrations the factors for the first denominator 

 are 



(2n+l)(2n + 3) ... (2n + 2m-l), 



and the factors for the (r + 1 )th denominator would be 



(2n-2r+ 1) (2n-2r + 3) ... (2n + 2m-2r + l), 

 of which the factor (2n + 2m 4r+ 1) is omitted. 



Hence taking r from to in, the general term of the expression of 

 n <1p m is 



/ . y m ! _ 2n + 2m-4r+l _ p , , 



' r\ (m-r)!(2n-2r+l)(2n-2r + 3) ... (2n + 2m-2r+ I) n+ 



Hence we can find 



for if p = n + m 2r, where r is not greater than m or less than 0, and 



fm 



if S 1 ^ be written for shortness instead of I P n dp. m , we shall have 



Jj 



m - 2n + 2m-4r+l 



r! (m-r)\(2n-2r+l)(2n-2r + 3). 



^ ; _________J_ ___... ( 6 ). 



Hence we see that n + m p must be an even number not greater 

 than 2m in order that 



n 



may not vanish. 



