364 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



Suppose Sq to be an increment of q corresponding to increments 

 8ju,j and 8<, then 



; 



Now let 8/A = e/A,(l-/r), 8^ = ^(1- 



and Sc = - e ( 1 - /r )' J ( 1 - /v) 4 sin <. 



Then 



Hence the increment of $ corresponding to these increments will be 



but if $ be regarded as a function of p., //,, and (j>, the same increment 

 will be represented by 



8 , c s , n s , 

 ^ O/A + -j 6/tj H ,-- orf> 

 /* a/*, d(f> 



C -"if +"(! -0 ^'-(1 -,f (I -^ sm . 



Hence equating the coefficients of e we have 



<' -* f -* (> -"'> t^d-^f ; - (i -,->' (i -rt sm 



Substituting in the above equation for Q n+l we have 



This equation may be made use of to prove that the general term of the 

 series for Q n is 



(rc-m)i 5. ^d m P n d m P n ' 



-' 



This expression may more readily be proved as follows : 



