216 THEORY OF HEAT. [CHAP. IV. 



be the case if the initial temperatures were represented by the 

 function a sin 1- b cos , i being any integer, a and b any co 

 efficients whatever. 



240. Let us pass now to the general case in which the initial 

 temperatures have not the relations which we have just supposed, 

 but are represented by any function whatever F(x). Let us give 



(x\ I ic\ 



- ) , so that we have F (as) &amp;lt;j&amp;gt; ( - j , and 



imagine the function &amp;lt;/&amp;gt;(-) to be decomposed into a series of 



sines or cosines of multiple arcs affected by suitable coefficients. 

 We write down the equation 



* p sin (O - ) + a, sin (l X ] + a 2 sin (2 *} + &c. 

 \ r) \ rj \ rj 



I 



+ & c 



The numbers a , a lt a a ..., 6 , ^, 6 2 ... are regarded as known 

 and calculated beforehand. It is evident that the value of u will 

 then be represented by the equation 



fc - 



u =* 



. X 



a, sm - 



-L 



o, cos - 



r * sin 2 - 



&amp;gt; cos 2- 

 2 r 



. &c. 

 x 



In fact, 1st, this value of u satisfies the equation -7- = k -7- j, 



dt d/x 



since it is the sum of several particular values ; 2nd, it does not 

 change when we increase the distance x by any multiple whatever 

 of the circumference of the ring ; 3rd, it satisfies the initial state, 

 since on making t = 0, we find the equation (e). Hence all the 

 conditions of the problem are fulfilled, and it remains only to 

 multiply the value of u by e~ ht . 



241. As the time increases, each of the terms which compose 

 the value of u becomes smaller and smaller ; the system of tem 

 peratures tends therefore continually towards the regular and con- 



