SECT. VII.] APPLICATION OF THE THEORY. 211 



or 



l r n 



-TTV = /(*) da. (e^ sin x sin a 4- e~ 2v sin 2x sin 2a 

 * Jg 



+ e~ 5v sin 3^ sin 3x + &c.) 

 a is a new variable, which disappears after integration. 



If the sum of the series be determined, and if it be substituted 

 in the last equation, we have the value of v in a finite form. The 

 double of the series is equal to 



e~ v [cos (x - a) - cos (x + a)] + e~ Zy [cos 2 (x - a) - cos 2 (x + a)] 



+ e~ zv [cos 3 (a? - a) - cos 3 (x 4 a)] + &c. ; 

 denoting by F (y,p) the sum of the infinite series 



e~ v cosp -f e~^ cos 2^ -f e* v cos 3/&amp;gt; -f &a, 



we find 



TTl 1 



- f/W ^ 



- 



We have also 



,-(v+p\/-i) g-to-pV-i) 



J g-(i/+PV-l) 



or 



F(y t p) = *P- 



e v -2cos/?-fe-&amp;lt;&quot; 



cos (# 4- a) - e&quot; 



cos^? 

 whence 



2 cos (a? ct) 4- e v e 1 2 cos ^ -L ^ -j- &quot;~ v 

 or 



-) + e^] [e v - 2 cos ( 

 or, decomposing the coefficient into two fractions, 



TTU = 



fit f -J ^ -i 



J o /() ^ ^_ 2 cos (a? -*) + &amp;gt;-&quot; ~ ^-2cos(^+ a ) + ^j 



142 



