492 THE PRODUCT OF ANY TWO LEGENDRE'S COEFFICIENTS [60 



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

 good for two consecutive values of m, it likewise holds good for the next 

 higher value. 



If the function '-^ be denoted by A (m), the general 



1.2. 3...m v ' 



term of the above expression for P m P n may be very conveniently represented 



by 



A(m r)A(r)A (n r) /2n + 2m 



' 



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

 r being an integer which varies from to m. 



The fundamental property of the function A is that 





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 = Q, we 

 have 



Similarly A ( - l) = ~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 m denote the quantity of which the general term is 



A(m-r)A (r) A(n-r) /2?i + 2m-4r+ 1 

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



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= I, 

 or r = m+l, then in consequence of the property of the function A above 



