352 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



10. We will now return to the consideration of the value of I S m P n dfi, 



Jo 



where S m = I P m dp., the integral vanishing when /u. = l. 

 First, consider the case when m = n, 



.'. f l S n P u dp=-(S n )\, ; 

 J o 



therefore as before, if n be even there will be no term in S n independent 

 of p. and therefore ($ ?l ) M=0 vanishes ; 



.-. P&P.d/isO, 



Jo 



but if n be odd, the constant term in S H will be, as already shewn, 



(- 1 )"' r o'?'f '" 



' ' 2 



Next, suppose m = w + 1 and consider 



Similarly 



"I 



P +'^^*~' as before proved. 



= 2n+ 1 f. P - P ^ - 21^ 



= 2^TT J (jPn+l)2 ^ ~ 2^+T J 



