1878.] Prof. J. C. Adams on Legendres Coefficients. 67 



1 .3.5..(2m-l) (n-m + l) (n-m + 2) ...n 



+ i.2.3... m ' (2w-2m + l) (2»-2m + 3) . . . (2»+l) 

 x[(2n-2m + l) P W _J 



And it remains to verify this observed law by proving that if it holds 

 good for two consecutive values of m, it likewise hold good for the 

 next higher value. 



13 5 (2 m I s ) 



If the function — — ' . ' ' ' v } be denoted by A(m), the general 



1 . 2 . 3 . . . m 



term of the above expression for P P n may be very conveniently repre- 

 sented by 



A(m-r) A(r)A(>-r) / 2^ + 2m-4r + l \ p 



' AO + m-r) \2n + 2m-2r + l) n+m ~' r 



r being an integer which varies from to m. 



The fundamental property of the function A is that 



i * . v 2m + 1 A / x 



A(m + 1) = —A(m) 



m+1 



or A(m) = A(m + 1) 



2m + 1 



We may interpret A(m) when m is zero or a negative integer, by 

 supposing this relation to hold good generally, so that putting m=0, 

 we have 



A(0)=A(1)=1 

 Similarly A(-1)=-5_A(0)=0 



and hence the value of A(m) when m is a negative integer will be 

 always zero. 



We will now proceed to the general proof of the theorem stated 

 above. 



Let Q Mj „ or Q m simply, denote the quantity of which the general 

 term is 



A(m— r) A(r) AQ— r) / 2n + 2m — + 1 \ p 



A.(n+m— r) \2n + 2m—2r + lJ n+m ~ 2r 



In this expression r is supposed to vary from to m, but it may be 

 remarked that if r be taken beyond those limits, for instance if 

 r=— l,or r='w+l, then in consequence of the property of the func- 

 tion A above stated, the coefficient of the corresponding term 

 will vanish. Hence practically we may consider r to be unrestricted in 

 value. 



Similarly, let Q m _! denote the quantity of which the general term 



is 



A(m — r) A(r — 1) A(n—r + l) / 2n + 2m — 4>r + 3 \ p 

 A(n + m-r) \2n+2m-2r + l) 



f 2 



