362 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



The factor 



[r (2n - 2r + 1 ) + (m - r + I ) (2w + 2m - 2r + 3)] = (m + I) (2n + 2m - 4r + 3). 



P / ,y_ 1.3.5...(2n-2r-l) 



B ~ 'rlm-r+lt 1 . 3 . 5 ... 



which is of the same form as before with m + 1 written in place of m, 

 so that the law of formation is generally true. 



17. We have seen above (p. 249) that 



( l m P 



- " - 



If m=0, 



If 1. 



Hence by successive integration we get 



/7 P 



(I - pY j^ =(-l) m (n-m + l)(n-m + 2)...(n + m) 

 up, 



d'"P 

 Hence the general term in (1 - /r)'" -y-^r o r ^ m ( see P- 25 ) 1S 



/ _ y+ r - 1-3.5. ..(2n -2r-l) _ 



r! (m-r)! (n-w)l 1 . 3 . 5 ... (2n + 2m-2r+ 1) 



x (2w + 2m - 4r + 1) P, 1+lft _ 5 , ....... (13). 



18. Adopting the notation of Section I., Art. 8 (p. 248), we have 



d m P fJ m P 



fynffnll ..2V " "1 



W^-U-ft, --- -- 



Integrating by parts between the limits 1 and 1 , the part outside the sign 

 of integration vanishes and we get by the above equation in Art. 17, 



