496 THE PRODUCT OF ANY TWO LEGENDRE'S COEFFICIENTS, ETC. [60 

 Hence we can find at once the value of the definite integral 



P 



J -i 



for if p = n + m 2r we have 



(m+p-n\ . (n + m-p\ . /n+p-m 

 



pp 2 



m " 



n+m+p+l 



+ &c. 

 Hence 



P 



/ m+ p-n\ A t n + m-p\ . /n+p-m\ 

 A (--2T-) A (--T) A (--2.) 



n + m + p + 1 





. n + m + p 



or if ^ - = s, 



Zi 



P p p p j.. 2 ^ (* - m ) A ( s - n } A(s^P) 



m n p 



2s + 1 A (s) 



where as above 



I 3 5 / 1 



ioc /OAVI 1 \ 9 9 9 \ 9 



. , . i . 6 . ..\Lm- i) _ e)m * * * \ * 



A( ^ = 1.2.3...m 1.2.3...W 



It is clear that, in order that this integral may be finite, no one of 

 the quantities m, n, and p must be greater than the sum of the other 

 two, and that m + n+p must be an even integer. 



I learn from Mr Ferrers that, in the course of the year 1874, he 

 likewise obtained the expression for the product of two Legendre's co- 

 efficients, by a method very similar to mine. In his work on " Spherical 

 Harmonics," recently published, he gives, without proof, the above result for 



the value of the definite integral I P m P n P p dp.. 



