60. 



ON THE EXPRESSION OF THE PRODUCT OF ANY TWO LEGENDRE'S 

 COEFFICIENTS BY MEANS OF A SERIES OF LEGENDRE'S COEFFICIENTS. 



[From the Proceedings of the Royal Society, No. 185, 1878.] 



THE expression for the product of two Legendre's coefficients which 

 is the subject of the present paper, was found by induction on the 13th 

 of February, 1873, and on the following day I succeeded in proving that 

 the observed law of formation of this product held good generally. Having 

 considerably simplified this proof, I now venture to offer it to the Royal 

 Society ; and, for the sake of completeness, I have prefixed to it the whole 

 of the inductive process by which the theorem was originally arrived at, 

 although for the proof itself only the first two steps of this process are 

 required. The theorem seems to deserve attention, both on account of its 

 elegance, and because it appears to be capable of useful applications. 



As usual let Legendre's nth coefficient be denoted by P n , then P n 

 may be defined by the equation 



1 d n 

 P - (u:-\Y 



n ~2 n [n' dX 



It is well known that the following relation holds good between three 

 consecutive values of the functions P, viz. 



(n + 1) P n+1 = (2n +l)p.P n - 



