SECT. II.] CONCLUDING TEMPERATURES. 237 



257. If we divide the semi-circumference into n equal parts, 

 and, having drawn the sines, take the difference between two 

 consecutive sines, the n differences are proportional to the co- 



_ versin u 



efficients of e r , or to the second terms of the values of 

 a, @, 7,...&). For this reason the later values of , & y...w are 

 such that the differences between the final temperatures and the 



mean initial temperature - (a + b + c + &c.) are always propor 

 tional to the differences of consecutive sines. In whatever 

 manner the masses have first been heated, the distribution of 

 heat is effected finally according to a constant law. If we 

 measured the temperatures in the last stage, when they differ 

 little from the mean temperature, we should observe that the 

 difference between the temperature of any mass whatever and the 

 mean temperature decreases continually according to the succes 

 sive powers of the same fraction ; and comparing amongst them 

 selves the temperatures of the different masses taken at the same 

 instant, we should see that the differences between the actual 

 temperatures and the mean temperature are proportional to the 

 differences of consecutive sines, the semi-circumference having 

 been divided into n equal parts. 



258. If we suppose the masses which communicate heat to each 

 other to be infinite in number, we find for the arc u an infinitely 

 small value ; hence the differences of consecutive sines, taken on 

 the circle, are proportional to the cosines of the corresponding 



, sin mu sin (m l)u. , 



arcs; for : is equal to cos mil, when the 



sin \JL 



arc u is infinitely small. In this case, the quantities whose tem 

 peratures taken at the same instant differ from the mean tempera 

 ture to which they all must tend, are proportional to the cosines 

 which correspond to different points of the circumference divided 

 into an infinite number of equal parts. If the masses which 

 transmit heat are situated at equal distances from each other on 

 the perimeter of the semi-circumference TT, the cosine of the arc at 

 the end of which any one mass is placed is the measure of the 

 quantity by which the temperature of that mass differs yet from 

 the mean temperature. Thus the body placed in the middle of 

 all the others is that which arrives most quickly at that mean 



