SECT. VI.] SERIES OF SINES OF MULTIPLE ARCS. 160 



variable. Denoting such a function by &amp;lt; (x), we arrange the 

 equation 



(j) (x) = a sin x + b sin 2x -f c sin 3x + d sin 4&amp;lt;x -f . . . &c., 

 in which it is required to determine the value of the coefficients 

 a, b, c, d, &c. First we write the equation ^ 



&amp;lt;^(^) = ^Xo) + |V Xo)+^f Xo) + ^^o) + ^xo)+..^W M 



If. !_ l_ 2. 



in which &amp;lt; (0), &amp;lt;&quot;(0), ^ &quot;(0), &amp;lt; lv (0), &c. denote the values taken 

 by the coefficients 



(x) 



* c 



dx dx* da? dx 



when we suppose x in them. Thus, representing the develop 

 ment according to powers of x by the equation 



we have &amp;lt;j&amp;gt; (0) = 0, and &amp;lt;f&amp;gt; (0) = A, 



&c. &c. 



If now we compare the preceding equation with the equation 

 &amp;lt;j)(x) = a sin x + b sin 2x + c sin 3# + J sin 4&amp;lt;x + e sin 5^ -|- &c., 



developing the second member with respect to powers of x, we 

 have the equations 



A = a + 2Z&amp;gt; + 3c + 4d + 5e + &c., 

 = a + 2 3 6 + 3 3 c + tfd + 5 3 e + &c., 

 (7= a + 2 5 ^ + 3 5 c + 4 5 cZ + 5 5 e + &c., 

 D = a + 2 7 6 + 3 7 c + 4 7 d + 5 7 e + &c., 



These equations serve to find the coefficients a, b, c, d, e, 

 &c., whose number is infinite. To determine them, we first re 

 gard the number of unknowns as finite and equal to m ; thus 

 we suppress all the equations which follow the first m equations, 



