180 THEORY OF HEAT. [CHAP. HI. 



values (0), &amp;lt;J&amp;gt;&quot; (0), ( v (0), &amp;lt; vii (0), &amp;lt; lx (0), &c., we have the general 

 equation 



jjf 



+ &C. 



We may make use of the preceding series to reduce into 

 a series of sines of multiple arcs any proposed function whose 

 development contains only odd powers of the variable. 



216. The first case which presents itself is that in which 

 4&amp;gt; (as) = ?; we find then &amp;lt;/&amp;gt; (0) = 1, &amp;lt;&quot; (0) = 0, &amp;lt; v () = 0, &c., and so 

 for the rest, We have therefore the series 



x on = sin x n sin 2x + ^ sin 3x -r sin 4# + &c., 

 4j . &quot; 2 o 4 



which has been given by Euler. 



If we suppose the proposed function to be x*, we shall have 



&amp;lt; (0) = 0, f &quot;(0) = [3, $ (0) = 0, &amp;lt;/&amp;gt; ((&amp;gt;) = 0, &o., 

 which gives the equation 



- a? = \7r z - -j= J sin x - (TT* - L= J s i n 2cc -}- ^7r 2 - -^J g sin 3ic -f &c. 



(A). 



