224 THEORY OF HEAT. [CHAP. IV. 



it differs from it in that the number of points which have always 

 the same temperature equal to the mean temperature of the ring 

 is not 2 only, but in general equal to 2i. Each of these points or 

 nodes separates two adjacent portions of the ring which are in 

 a similar state, but opposite in sign. The circumference is thus 

 found to be divided into several equal parts whose state is alter 

 nately positive and negative. The flow of heat is the greatest 

 possible in the nodes, and is directed towards that portion which 

 is in the negative state, and it is nothing at the points which are 

 equidistant from two consecutive nodes. The ratios which exist 

 then between the temperatures are preserved during the whole of 

 the cooling, and the temperatures vary together very rapidly in 

 proportion to the successive powers of the fraction 



If we give successively to i the values 0, 1, 2, 3, &c., we shall 

 ascertain all the regular and elementary states which heat can 

 assume whilst it is propagated in a solid ring. When one of these 

 simple modes is once established, it is maintained of itself, and the 

 ratios which exist between the temperatures do not change; but 

 whatever the primitive ratios may be, and in whatever manner 

 the ring may have been heated, the movement of heat can be de 

 composed into several simple movements, similar to those which 

 we have just described, and which are accomplished all together 

 without disturbing each other. In each of these states the tempe 

 rature is proportional to the sine of a certain multiple of the dis 

 tance from a fixed point. The sum of all these partial temperatures, 

 taken for a single point at the same instant, is the actual tempera 

 ture of that point. Now some of the parts which compose this 

 sum decrease very much more rapidly than the others. It follows 

 from this that the elementary states of the ring which correspond 

 to different values of i, and whose superposition determines the 

 total movement of heat, disappear in a manner one after the 

 other. They cease soon to have any sensible influence on the 

 value of the temperature, and leave only the first among them to 

 exist, in which i is the least of all. In this manner we form an 

 exact idea of the law according to which heat is distributed in 

 a ring, and is dissipated at its surface. The state of the ring be 

 comes more and more symmetrical; it soon becomes confounded 



