SECT. VI.] DEVELOPMENTS IN SERIES OP SINES. 183 



By means of these reductions equation (A) takes the following 

 form : 



sn x 



- J f (TT) + J &amp;lt; iv (7r) - J ^(TT) + &cj 

 - i sin 2* {&amp;lt;/&amp;gt; (TT) - I &amp;lt;&quot; (TT) + 1 4&amp;gt; lv (TT) - 1 &amp;lt;/&amp;gt; + &c. J 



sin 3* (/&amp;gt; (TT) - f (TT) + ^ (TT) - &amp;lt;^(TT) + &* 



- sn * c W - ^ (T) + r W - ^ W + & 



(B); 



or this, 

 5 



a?) = ^ (TT) ! sin x sin 2,r + sin 3x &c. h 



&amp;lt;t&amp;gt;&quot; (TT) | sin ^ ^ sin 2:c + ^ sin 3x &c. [ 



[ ^ o ) 



+ (/&amp;gt; IV (TT) -jsin x -^ sin 2x + ^ sin 3o? &c. ^ 



c/) vl (TT) ! sin x -^ sin 2x + ^? sin 3uC &c. [ 



+ &c. (C). 



218. We can apply one or other of these formulas as often as 

 we have to develope a proposed function in a series of sines of 

 multiple arcs. If, for example, the proposed function is e x e~* t 

 whose development contains only odd powers of x, we shall have 



1 (F . Q~* / 1 1 \ 



x TT - = f sin x -^ sin 2# + sin 3^ &c. J 



^ *Vu (sin a; ^ sin 2ic + ^ sin 3a; &c. ) 



*% ! i 



*t*3 + ( sm ^ B sm 2ic + o5 sin 3x &c. J 

 i 



( sin x yj sin 2x + ^ sin 3, &c. J 



