CHAP. VI.] VERIFICATION OF THE SUM. 299 



verifies itself. We have in fact 



Icos (0 sin 11) du = Idu (l ^ 1 , ^ \- &c.J ; 



and integrating from u to u TT, denoting by $ 2 , S# ^ 6 , &c. 

 the definite integrals 



we have 



Isirfudu, lsm*udu, I sin 6 u du, &c., 

 f fl* fi* f) 6 



(COS (0 Sin tt) &amp;lt;?M = 7T - W $ 2 + rj S 4 - w S t 4- &C., 



j 



it remains to determine $ 2 , ^ 4 , S 6 , &c. The term sin n u, n being 

 an even number, may be developed thus 



sin n u A n + B n cos 2u + C n cos ku + &c. 



Multiplying by du and integrating between the limits u = and 

 U = TT, we have simply I sin n u du = A n 7r, the other terms vanish. 



From the known formula for the development of the integral 

 powers of sines, we have 



A -- - A -! LL* A -L 4 5 6 



2 ~~ 2 2 1 ~~ 2 4 1 2 6 ~ 2 6 l 2 3 * 



Substituting these values of S# S^, S& &c., we find 



1 f 6 Z 6* Q Q 



- J cos (0 sin u) du=I-^ + ^fp - ^ ^ ^ + &c. 



We can make this result more general by taking, instead of 

 cos (t sin it), any function whatever (/&amp;gt; of t sin u. 



Suppose then that we have a function &amp;lt;j&amp;gt; (z) which may be 

 developed thus 



we shall have 



* 00 = &amp;lt; + f + f + f &quot; + &c. ; 



f t 3 



(f&amp;gt; (t sin u) = $ + (/&amp;gt; sin w + - $ sin 2 w + -5 c^&quot; sin 3 w + &c. 



X 

 and - |dw &amp;lt;f&amp;gt; (* sin w) = &amp;lt;f&amp;gt; + A 6 + /S! 2 &amp;lt;f&amp;gt;&quot; + S # 3 &amp;lt;#&amp;gt; &quot; -f &c. ,| (e). 



7TJ 25 o 



