360 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



f 1 f 1 / 5 i s \ i 



and P 3 S.dfji = ( -- p." - - ^ + - /x 2 ) dp. = , as before. 



Jo Jov*4 iz 



Hence our results are confirmed. 



But we must be careful to note the paradoxical result that 

 1.3.5 ... (-!) = ! and 2.4...(0) = 1. 



If we call 1 . 3 . 5 ... m=f(m), the characteristic mark of/ is that 

 mf(m 2) =f(m) ; applying this when m=l, we have 1 x/( 1) =/(!) = 1, 

 .'. f( l) = l. Similarly in the other case n(f>(n 2)=<j)(n), make n = 2, 



15. We have seen above that the general term of the expression 



n dp. m will be 



/ _ , y m! 2n + 2m 4r + 1 



' 



rT(m-r)\ (2n-2r+ 

 r being taken from to m and ! being = 1 . 



Now generally (2x+ 



('" 

 Integrating each term of I P n dfj. m by means of this formula, we have 



fm+l 



P fJn m + l 

 *n ct P- - 



m ' P - P 



i / _ -i \r _ -*- n+m-zr+i -^ n+m-sr-1 



' 



/ _ i \r 



; 



_ _ 



rl(m-r)\ (2n-2r+l) (2i-2r + 3) ... (2n + 2m-2r+ 

 m \ p _ p 



'"" _ * n+m-w-i * n 



+m-zr-3 



_ __ 



\(m-r-l)l (2n-2r- 1) (2n- 2r+ 1) ... (2n + 2m-2r- 



The coefficient of P n + m -v-i i 

 ml r 



, . r 

 ' 



(r+l)!(m-r)I (2n-2r- 1) (2n-2r+ 1) ... (2n + 2m- 2r+ 1) 



