66 Prof. J. C. Adams on Legendres Coefficients. [Jan. 31. 



Hence, collecting the terms, we have 



1.3.5.7 (n + 1) (n + 2) (n + 3) (n + 4) p 

 r*r n - 1 2.3.4 (2^+1) (2»+3) (2» + 6) (2ra+7) n+4 



1.3.5 1 n(n + l) (n + 2) (n + 3) p 

 1.2 .3 '1 (2»-l) (2« + l) (2»>3)(2#i+-7) 



1.8 1-3 (n-1) »(»+!) (n+-2) 



n+2 



1.2 1.2 (2»— 3) 4 (2n-l) (2» + 3) (2» + 5) ■ 



1 1.3.5 (w-2) (n-l)n(n + l) p 

 1*1.2.3 (2n-5) (2m- 1) (2n'+. 1) (2n + 3) B " 3 



1.3.5.7 (»-3) (w-2) p 



1.2.3.4 (2n-5) (2»-3)-(2»-l) (2»+l) B " 4 



where the law of the terms is obvious, except perhaps as regards the 

 snccession of the factors in the several denominators. 



With respect to this it may be observed that the factors in the 

 denominator of any term P p are obtained by omitting the factor 2p + 1 

 from the regular succession of five factors (n+p — 3) (n+p — 1) 

 (n+p + 1) ( n +p + S) ( n +p + h). 



For instance, where p=n + 4t, 2p + l = 2n + 9, so that the factor 

 2n + 9 is to be omitted, arid we have 2n + l, 2w+3, 2w + 5 and 2n+7, 

 as the remaining factors, and so of the rest. 



Hence by induction we may write, supposing to fix the ideas that 

 m is not greater than n 



P P _ 1 • 3 . 5 . . (2m-l ) (7i + l) (n + 2) .... (w-j-m) 



1.2.3... m ' (2»+l) (2»+3) . .. (2» + 2m+l) 



+ 



x[(2^+2m+.l) P n+m ] 

 1.3.5.. (2m- 3) 1 n(n + l) . . . (n+m-1) 



1.2.3.. (m-1) 1 (2n-l) (2n+l) ... (2« + 2»i-l) 



x[(2^+2m-3) P n+m _ 2 ] 



+ &c. &c. 



1.3.5... (2w-2r-l ) 1.3.5... (2r-l) 

 1.2.3. ..(m-r) '1.2.3.... r 



x Qi— r + 1) (ot — r + 2) . . . (n—r + m) 



(2«-2r + l) (2n-2r + S) . . . (2n-2r + 2m + l) 



x[(2»+2w-4r+l) P n+ni _ 2r ] 



+ &c. &c. 



1 1.3. 5.. (27??— 3) (n-m + 2) (n-m + 3) . . . (n + 1) 

 1*1.2.3.. (m— 1) ' (2»— 2m+3) (2^-2m+5) . . . (2n + 3) 

 X [(27i-2m + 5) P„_ m+2 ] 



