368 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



The last term of Q n , when m = n, is 



<FP n _d"P n ' 1 1 d* n (p*-l) n 

 Aiso ~dp n '' ~~ dp? '" 2" n ! dp" 



And the last term becomes 



2. 4. 6. ..(-2.) 



22 



["> ^-^"'^-(-ir 



J ^"^ " c^-' ' 



/; 



where P n dp m is so taken that at each successive integration the result 

 is divisible by 1 /u, 2 without remainder, we have 



m P n d'"P n ' 



- 

 ~ dp-'" ~(n + m)\) ( ^ dp m 



, ,,-. n 



(^m)l (] ~P> ~ M dp-" 



Hence Q n may be put under the symmetrical form 



We observe that 0! = 1 and ( l)! = =cc, so that there is no term before 



. . 2n\ 

 that involving -=- . 



0! d n+1 P 



Also there is no term after that involving - , since i-s+r = 0- 



' 2n\ dp n+ 



