SECT. I.] FURTHER APPLICATION. 219 



temperatures/ cools in air after the suppression of the source of 

 heat. 



243. In order to make a second application of the general 

 equation (E), we shall suppose the initial heat to be so distributed 

 that half the ring included between x = and x = TT has through 

 out the temperature 1, the other half having the temperature 0. 

 It is required to determine the state of the ring after the lapse of 

 a time t. 



The function /(#), which represents the initial state, is in this 

 case such that its value is 1 so long as the variable is included 

 between and TT. It follows from this that we must suppose 

 f(x) = 1, and take the integrals only from x = to x = TT, the 

 other parts of the integrals being nothing by hypothesis. We 

 &quot;obtain first the following equation, which gives the development 

 of the function proposed, whose value is 1 from x = Q to X = TT and 

 nothing from x = TT to x = 2w, 



f( x ) = o + ( sm x + o s i n % x + ^ sin oaj + = sin 7-z + &c. ) . 



A 7T \ O O / / 



If now we substitute in the general equation the values which 

 we have just found for the constant coefficients, we shall have the 

 equation 



x TTV = e~ ht t-77r + sin xe~ kt + ^$m 3xe~ kt +^ sin oxe~ 5ZJct + &c 

 2i \4 o o 



which expresses the law according to which the temperature at 

 each point of the ring varies, and indicates its state after any 

 given time : we shall limit ourselves to the two preceding applica 

 tions, and add only some observations on the general solution 

 expressed by the equation^ (E). 



244. 1st. If k is supposed infinite, the state of the ring is 

 expressed thus, 7rrv = e~ ht ^lf(x)dx ) or, denoting by M the 



mean initial temperature (Art. 241), v = e~ M M. The temperature 

 at every point becomes suddenly equal to the mean temperature, 

 and all the different points retain always equal temperatures, 

 which is a necessary consequence of the hypothesis in which we 

 admit infinite conducibility. 



