1878.] Prof. J. C. Adams on Legendres Coefficients. 05 



5 p» + l)C*+2)p,+3) 

 r3ir " 2 (2ft+l) (2, 4 + 3; (2ft+5) w+3 



3 nCw+jj fw + 2) 

 + 2 (2ft-l) (2ft + l) (2»+5) "+ 



3 (»-l)n(n + l) 

 ~^2 (2»-8) (2» + l) (2»+3) - 1 



5 Q-2) Q-l> 



2 (2/1-3) (2»— 1) (2ft +1) 



7 3 



Again since P 4 =-/tP 3 — -P 2 



we have P^ W =^(P 3 PJ - |(P 2 P n ) 



'Whence by substituting the values found above for P a P„ and P 2 P„ 

 and again for yt*P n+3 , /* P n+1 , &c., we obtain 



■5.7 (n + l)(» + 2)^ + 3) / ft + 4 ft + 3 1 



4 n 2 . 4 (2^-j-l) (2ft+3) (2ft+5) 1 2»+7 "+* + 2»+7 "+* J 



3.7 7^ + 1) (n+g) J « + 2 n + 1 1 



+ 2 .4(2//,-!) (2» + l) (2» + 5) 1 2ft -f 3 1 * +2+ 2% + 8 - J 



3.7 (a-1) »(» + !) f ft ,«-l p 1 

 + 2.4(2ft-3) <2n+l) (2»+3)la»-r ,+ ^-l r *- a J 



5.7 (n-2) (»-!) » /^~ 2 p , ^-3 1 

 + 2.4(2ft-3) (2ft-l) (2w+l) 12^-5 "- 2 ~ h 2^-5 "" 4 J 



3.3 (ft+1) (n+2) P 3 n(n+ l) 



2.4(2ft+l) (2ft+3) 71+2 4 (2ft-l) (2ft-h3)^* 



3.3 (n-l)n 



2 .4(2« — l) (2ft+l) K " 2 



By reduction, the coefficient of P M+2 in this expression becomes 



5 ft (» + l) (ft+2) (ftj-3) 

 2(2ft-l) (2ft+l) (2ft-h3) (2ft+7) 



Similarly, the coefficient of P n _ 2 becomes 



5 (ft-2) + 



2 (2ft-5) (2/i-l) (2w + l) (2«H-3) 



and finally, the coefficient of P n becomes 



'3V (ft-1) »(n+l) (ft+2) 



v2/ (2»— 3) (2ft— 1) (2ft+3) (2ft + 5) 



YOL. XXVIT. 



