242 THEORY OF HEAT. [CHAP. IV. 



temperatures. We see by this solution that, if the initial tem 

 peratures j, a 2 , a 2 , ... a n , were proportional to the sines 



27T -, 2-7T 27T . . - N 2?T 



sm , sin 1 , sin 2 , ... sin (n - 1) , 

 n n n n 



they would remain continually proportional to the same sines, and 

 we should have the equations 





, 2& . 2&amp;lt;7T 



where h = versin - 

 m n 



For this reason, if the masses which are situated at equal dis 

 tances on the circumference of a circle had initial temperatures 

 proportional to the perpendiculars let fall on the diameter 

 which passes through the first point, the temperatures would 

 vary with the time, but remain always proportional to those per 

 pendiculars, and the temperatures would diminish simultaneously 

 as the terms of a geometrical progression whose ratio is the 



-S versin 



fraction e n n . 



263. To form the general solution, we may remark in the 

 first place that we could take, instead of & 15 5 2 , b 3 , ... b n , the n 

 cosines corresponding to the points of division of the circumference 

 divided into n equal parts. The quantities cos Ou, cos \u, cos 2w,... 



cos (n 1) u, in which u denotes the arc , form also a recurring 



Yl 



series whose scale of relation consists of two terms, 2 cos u and 1, 

 for which reason we could satisfy the differential equations by 

 means of the following equations, 



- versin 



otj = cos Oue 7 , 



KM 



versin u 



2 = cos lue , 



Zkt 

 versin u 



a = cos 2ue m 



n =r cos (n l)ue 



