382 Mr. I. Todhunter on Legendre's Coefficients. [May 16, 



1.3.5. .(2,-1) 

 1 . 2 . 6 . . r 



and in all other cases A(V) is to be considered zero, except when r=0, 

 and then it is to be considered =1. 



2?* + l 



It will be observed that A(r + 1) = A(r), and this is the essen- 



r+1 J 



tial property of the function for onr purpose. The demonstration 

 consists of three parts. 



I. The proposed formula holds when m=0, whatever n and jp may 



be. This is seen to be true in virtue of the known theorem that 



f 1 .2 



I Y n Ypdn is zero when n and p are unequal, and is equal to 



J-i 2n + l 



when p = n. 



II. The proposed formula holds when m=l, whatever n andp may 

 be. For by a known theorem we have 



p p H + 1 p I n p 



therefore 



J- 1 J_x\2»+1 2?^ + l 7 



and this vanishes unless p = n-\-\ or n — 1, and then its value is 

 2s 



.where 2s=?i + p + l. And if we examine the formula 



(2 s +l)(2s-l)' 



to be established we shall find if m = l the product A(s— n)A(s— p) 

 vanishes, except when jp = n + \ or n — 1, and then it becomes unity. 



Also Y^ = 2~~ 1 ' anC ^ ^ nS ^ lie re 9. n * re ^ resu -lt is obtained. 



III. We shall now give an inductive process by which we show that 

 if the formula holds for all values of n and p combined with the values 

 m— 1 and m of the third integer, it will hold for all values of n andp 

 combined with the value m + 1 of the third integer. 



For P JM+1 P„P J ,= ( r^ip^P,--^- P,.! } P M Pp 



I m + 1 m + 1 J 



= 2^+ l p f n±l + » j _^p iPn p 



m + 1 ' 12/1 + 1 2?i + l J m + 1 ^ 



Thus we can express j P^+iP^P^/t by means of cases of the formula 

 already established. If we now put 2s = m + n+p + l we thus obtain 

 for the definite integral j P OT+ iP«P;>^/* 



