1878.] Prof. J. C. Adams on Legendre's Coefficients. G3 



III. " On the Expression of the Product of any two Legendre's 

 Coefficients by means of a Series of Legendre's Co- 

 efficients." By Professor J. C. Adams, M.A., F.R.S. Re- 

 ceived November 22, 1877. 



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„ 

 may be defined by the equation 



1 rl n 



It is well known that the following relation holds good between 

 three consecutive values of the functions P, viz. : 



Now Pj=/* 



( W + l)P n+l =(2 W + l>P 11 -wP.-i 



* * rirM -2n + l r " +1 + 2^ + l rB - 1 



3 1 



Again , we have P 2 — - jliP 1 — - 



_3 n + 1 3 n 1 



~22^+I^ +1 + 2 **- l ~2 * 



Substitute for yuP n+1 and fiP n . l their equivalents obtained by writ- 

 ing n-\-l and n — 1 successively for n in the above formula 



. 3 (n + l)(n + 2) 



* ' 2 n 2(2»+l) (2« + 8) r?t+2 



,J3 (n + iy 13 n* l p 



L2(2» + l) (2n + 3) 2 + 2 (2»-l) (2rc + l) J n 



+ 3 (n — l)n p 



2(2w— 1) (2rc+l) 



