OF TWO LEGENDRE'S COEFFICIENTS. 355 



,,, 1.3.5...(m-r-l) 1 . 3 . 5 ... (ro + r) 

 Also let < = V 2) 2 . 4 .6...(n-r+l)' 



so that the operation or sign of functionality <f> has no meaning unless 

 the subject (r) of the operation be an even integer. 



We have evidently 



, 1.3.5 ...(m-r) 1.3.5 ... (n + r-l) 



~2.4.6...(m + r+l) 2 . 4 . 6 ... (n-r + 2) i' 



1.3.5...(m-r-2) 1 . 3. 5 ... (n + r+ 1) 

 = 2.4.6...(m + r+T) 2 . 4 . 6 ... (n-r) ' 



where r l, r+1 must be even and therefore r odd. 



We may observe that 



m-r 



. , , r, ^ (n + r+l} 

 and 4>(r+l)=f(r)-{- -=>. 



17 v ' [m + r + 3J 



Assume f (r) = \<f) (r 1) p.(f> (r + 1), 



// \ // \ f\ m-r n +r+ll 



and therefore / (r) = / (r) 4\ p. , f' 



j \ i j \ i ^ n _ r ^ 2 ^ m + r + 3 } 



_ m r n +r+l 



n-r+2 



and determine X and /a by the condition that p. is the same function of 

 r+ 1 that X is of r 1. 



This will evidently be the case if 



\ = (n-r + 2)(m + r+l)c and fj. = (m + r + 3) (n-r) c, 



where c is some quantity which remains the same when r 1 is changed 

 into r + 1 . 



Substituting, we have 



so that c is independent of r and =-, ^-, -r\. 



(m n) (m + n + 1 ) 



452 



