232 THEORY OF HEAT. [CHAP. IV. 



can therefore express the general term a m by the equation 

 a m = A sin mu + B sin (m 1) u, 



determining suitably the quantities A, B, and u. First we find 

 A and B by supposing m equal to and then equal to 1, which 

 gives a = B sin w, and a l = A sin it, and consequently 



a i / i\ 

 a m = , sin ?WM r sin (m 1) u. 



sin M 

 Substituting then the values of 



a ,n&amp;gt; -! &amp;lt;W &C - 



in the general equation 



M = m -lfe + 2 )-&amp;lt;V 2 &amp;gt; 



we find 



sin mu = (&amp;lt; f 2) sin (m 1) M sin (m 2) w, 



comparing which equation with the next, 



sin mu 2 cos u sin (m 1) u sin (w 2) u, 



which expresses a known property of the sines of arcs increasing 

 in arithmetic, progression, we conclude that q -f 2 = cos u, or 

 q = 2 versin w ; it remains only to determine the value of the 

 arcw. 



The general value of a m being 



-r- 1 - [sin ?m sin (m - 1) w], 

 sin u L 



we must have, in order to satisfy the condition a n+l =^ a n9 the 

 equation 



sin (n -f 1) u sin u = sin ?m - sin (n 1) u t 



TT 



whence we deduce sin nu = 0, or u = i , TT being the semi- 

 circumference and i any integer, such as 0, 1, 2, 3, 4, ... ( 1) ; 

 thence we deduce the n values of q or -y- . Thus all the roots 



K 



of the equation in h, which give the values of h } ti, h&quot;, li \ &c. 

 are real and negative, and are furnished by the equations 



