260 THEORY OF HEAT. [CHAP. IV. 



As to the initial temperatures a lt a 2 , a 3 ...a n , they depend on 

 the value of the arc x, and regarding these temperatures as the 

 successive states of the same variable, the general value a t repre 

 sents an arbitrary function of x. The index i must then be 



x 

 replaced by -y- . With respect to the quantities a lt a g , a 3 , ..., 



these are variable temperatures depending on two quantities 

 x and t Denoting the variable by v, we have v = $ (x, t). The 

 index j t which marks the place occupied by one of the bodies, 



99 



should be replaced by -y-. Thus, to apply the previous analysis to 



the case of an infinite number of layers, forming a continuous 

 body in the form of a ring, we must substitute for the quanti 

 ties n, m, Ic, a it i, a j} /, their corresponding quantities, namely, 



-y- , dx, ff . f(x\ -j- , 4&amp;gt; (x. t\ -7- . Let these substitutions be 

 dx dx J ^ J) dx Y ^ &quot; dx 



made in equation (e) Art. 273, and let ^ dx* be written instead 

 of versin dx, and i and j instead of i 1 and j 1. The first 



term - 2o ( becomes the value of the integral ~ \f(x) dx taken from 

 n %Tr) J 



07 = to 7=27r; the quantity sm(j-l)^ becomes smjdx or 



n 



sin x ; the value of cos (/I) -y- is cos x ; that of - 2a 4 sin (i 1) - 



dx ft n 



is -\ f(x] sin JPC&P, the integral being taken from x = to x=2jr : 

 irj 



and the value of - 2a^ cos (i - 1) - r is -If () cos # cZx, the 



integral being taken between the same limits. Thus we obtain 

 the equation 



-f - ( sin x I / (x) sin xdx -f cos x If (x} cos xdx }e- ffnt 

 f n \j J / 



4- - f sin 2# lf(x)sinZ 



cos 

 (E) 



