CHAP. VI.] SUM OF A CERTAIN SERIES. 297 



Whence we conclude 



_ &c 

 * 2 





It is easy to ex^es^^e^lim of this series. To obtain the 



result, develope as follows the function cos_(a^siii#) in cosines of 

 multiple arcs. We have by known transformations ^\ 



o i \ iw*^ 1 -ae-* V= l , -^ae^~ l fcie-*^ 



2 cos (a sin x) e * e +e e^ , 



and denoting e x ~ l by o&amp;gt;, 



aw cut)&quot; 1 aw aw&quot; 1 



2 cos (a sin #) = e * e~ * + e~ a e 2 . 



Developing the second member according to powers of &&amp;gt;, we 

 find the term which does not contain w in the development of 

 2 cos (a sin x) to be 



The coefficients of a) 1 , o 3 , a&amp;gt; 5 , &c. are nothing, the same is the case 

 with the coefficients of the terms which contain of 1 , o&amp;gt;~ 3 , o&amp;gt;~ 5 , &c. ; 

 the coefficient of aT 2 is the same as that of o&amp;gt; 2 ; the coefficient of o&amp;gt; 4 is 



4.6.8 2 2 . 4. 6. 8. 10 ^ 



the coefficient of of 4 is the same as that of &&amp;gt; 4 . It is easy to express 

 the law according to which the coefficients succeed ; but without 

 stating it, let us write 2 cos 2a? instead of (o&amp;gt; 2 + o&amp;gt;~ 2 ), or 2 cos 4# in 

 stead of (ft) 4 + &)~ 4 ), and so on : hence the quantity 2 cos (a sin x} is 

 easily developed in a series of the form 



A + B cos 2x + Ccos 4# + D cos 6x + &c., 

 and the first coefficient A is equal to 



s fr ; , f t .*;.!. 



if we now compare the general equation which we gave formerly 

 2 TT &amp;lt;^&amp;gt;(a;) = ^ l&amp;lt;f)(x)dx + cos # |&amp;lt;^(a;) cos a?(?ic + &c, 



j f X 

 - 1 4 ( 



