530 Prof. Sylvester's Notes to the 



Hence, by expanding the denominator in a series proceeding 

 according to powers of (cos <f>) 2 , it is readily seen that the first 

 integral becomes 



To find the second integral, we must obtain the general term 

 in the expansion in a series of powers of / of— g ' T— - ' 



(where t stands for b cos </>), i. e. of , -. J efr(j — -^ \ 



i say of ft(Q = ^i- a ^. Now 



«-i-+ 



1 1 1.3 1.3.5 



Hence, writing 



. (l+y , l-<*) = io g 2+ K/+ . . .+ K 2l _ 2 f"- 2 + K 2j f» +&c, 

 5 v/l-i 2 



and equating the coefficients of t 2i , we obtain 



2i(2i-i) (K 2j -k 2( _ 2 ) + (2i- i)K 2j _ 2 = - 1, |;f , g; (28 ~ 1) , 



_ 2»-l lr 1.3.5...2J-3 



"2i — iv? — *2t-2' 



2i T2, ~ 2 2.4.6...(i-2)2i 



