24 THEORY OF HEAT. [CHAP. I. 



submit these grand phenomena to calculation, to discover the 

 mathematical laws of the propagation of heat in the interior of 

 masses. 



19. It will be perceived, on reading this work, that heat at 

 tains in bodies a regular disposition independent of the original 

 distribution, which may be regarded as arbitrary. 



In whatever manner the heat was at first distributed, the 

 system of temperatures altering more and more, tends to coincide 

 sensibly with a definite state which depends only on the form of 

 the solid. In the ultimate state the temperatures of all the points 

 are lowered in the same time, but preserve amongst each other the 

 same ratios : in order to express this property the analytical for 

 mulae contain terms composed of exponentials and of quantities 

 analogous to trigonometric functions. 



Several problems of mechanics present analogous results, such as 

 the isochronism of oscillations, the multiple resonance of sonorous 

 bodies. Common experiments had made these results remarked, 

 and analysis afterwards demonstrated their true cause. As to 

 those results which depend on changes of temperature, they could 

 not have been recognised except by very exact experiments ; but 

 mathematical analysis has outrun observation, it has supplemented 

 our senses, and has made us in a manner witnesses of regular and 

 harmonic vibrations in the interior of bodies. 



20. These considerations present a singular example of the 

 relations which exist between the abstract science of numbers 

 and natural causes. 



When a metal bar is exposed at one end to the constant action 

 of a source of heat, and every point of it has attained its highest 

 temperature, the system of fixed temperatures corresponds exactly 

 to a table of logarithms ; the numbers are the elevations of ther 

 mometers placed at the different points, and the logarithms are 

 the distances of these points from the source. In general heat 

 distributes itself in the interior of solids according to a simple law 

 expressed by a partial differential equation common to physical 

 problems of different order. The irradiation of heat has an evident 

 relation to the tables of sines, for the rays which depart from the 

 same point of a heated surface, differ very much from each other, 



