238 THEORY OF HEAT. [CHAP. IV. 



temperature ; those which are situated on one side of the middle, 

 all have an excessive temperature, which surpasses the mean 

 temperature the more, according as they are more distant from 

 the middle ; the bodies which are placed on the other side, all 

 have a temperature lower than the mean temperature, and they 

 differ from it as much as those on the opposite side, but in con 

 trary sense. Lastly, these differences, whether positive or negative, 

 all decrease at the same time, proportionally to the successive 

 powers of the same fraction ; so that they do not cease to be repre 

 sented at the same instant by the values of the cosines of the 

 same semi-circumference. Such in general, singular cases ex- 

 cepted, is the law to which the ultimate temperatures are subject. 

 The initial state of the system does not change these results. We 

 proceed now to deal with a third problem of the same kind as the 

 preceding, the solution of which will furnish us with many useful 

 remarks. 



\. 



259. Suppose n equal prismatic masses to be placed at equal 

 distances on the circumference of a circle. All these bodies, 

 enjoying perfect conducibility, have known actual temperatures, 

 different for each of them ; they do not permit any part of the 

 heat which they contain to escape at their surface ; an infinitely 

 thin layer is separated from the first mass to be united to the 

 second, which is situated towards the right ; at the same time a 

 parallel layer is separated from the second mass, carried from left 

 to right, and joined to the third; the same is the case with all the 

 other masses, from each of which an infinitely thin layer is sepa 

 rated at the same instant, and joined to the following mass. 

 Lastly, the same layers return immediately afterwards, and are 

 united to the bodies from which they had been detached. 



Heat is supposed to be propagated between the masses by 

 means of these alternate movements, which are accomplished 

 twice during each instant of equal duration; the problem is to 

 find according to what law the temperatures vary : that is to say, 

 the initial values of the temperatures being given, it is required to^ 

 ascertain after any given time the new temperature of each of the 

 masses. 



We shall denote by a iy a z , a Jz ,...a i ...o Jn the initial temperatures 

 whose values are arbitrary, and by a v a 2 , a s ...a i ...& n the values of 



