370 ON THE DEFINITE INTEGKAL OF THE PRODUCT 



Expanding by the Binomial Theorem, we get 



1 f 12A(l-coBj), . 1.8|-2A(l-coe0-|' * 



r^t 2 (i-'O 2 ^2.4|_ (i-^) 2 J l H 



1.3.5 r2fc(l-cos 



-271T6L (i-hY 



the (r + 1 )th term being 



.....-- 



2.4.6...2r L (l-^) 2 -H+&C.J. 



Now multiply by c//x and integrate from //.= 1 to /* = !, observing that 



7T 



f 1 (i -n*y<in=2 (\i- l syd fJ .=2 !\smer +i de, [if ^=cos^] 



J-i Jo jo 



f 2r2r-2...2 

 or in irv fin = -7 



-u(a = - 



-i \2r+12r-l ... 3j 



Hence \ve have 



the (r+i)th term being 



(1-A) 



^2 sin- 



, T . - cos <4) 2 ,, 2 



Now if - ,,v~- -tan 2 ^ or 



we have 

 and 



/:, 



sn - < 



A 



/ _, 7 \ UC^AJ. I/ \_/i UCVI1 I/ 



( 1 - A)- 1 - h 



_2_ tang 



/i sin - ^> 



A 



Lt 



= - - {&}, as before found. 

 h sin <> 



