346 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



I 



Integrating repeatedly by parts we have 



Jm Jn 



pO'-irjpV-iW' 



,7m -1 ,]n /7 m ~ 2 f1 n + t 



-- v ~ ir <*'- D'-- &- 1) " <*'- 



--F-, o*' - 1 r 



since the following terms vanish. 





Now the last term is ( - 1)" ~ _, (p 2 - l)"(2*i)!. 



d" 1 ' 1 " 1 

 We may observe that all the quantities like , m - r -i (^ 2 ~ l)'" vanish 



when /x = 1 ; we have therefore only to find the values of the terms on 

 the right-hand side of the equation when /x = 0, and this with the sign 

 changed will be the value of 



5. If both of the quantities m and n be even or both be odd, we have 



f 1 1 f 1 



P m P H dp. = - P m P n dit = 0, unless m = n. 

 Jo 2 J _! 



In the case when m and n are both even or both odd the integrated 

 terms being of the form 



d" l+r ^n-r-i 



( 1 I ' 7 -4- (/^ 2 1 ) m 7 ~ (/^" 1 )" 



dp, CILL 



will consist of two factors, one consisting entirely of even and the other 

 entirely of odd powers of p., and therefore the product will consist entirely 

 of odd powers of /x and will vanish when /A 0. 



Also the last factor vanishes when /i=l. 



