6 THEORY OF HEAT. 



equations, the laws of such a compound effect ; and what is the 

 resulting change in the general equations of hydrodynamics ? 



Such are the chief problems which I have solved, and which 

 have never yet been submitted to calculation. If we consider 

 further the manifold relations of this mathematical theory to 

 civil uses and the technical arts, we shall recognize completely 

 the extent of its applications. It is evident that it includes an 

 entire series of distinct phenomena, and that the study of it 

 cannot be omitted without losing a notable part of the science of 

 nature. 



The principles of the theory are derived, as are those of 

 rational mechanics, from a very small number of primary facts, 

 the causes of which are not considered by geometers, but which 

 they admit as the results of common observations confirmed by all 

 experiment. 



The differential equations of the propagation of heat express 

 the most general conditions, and reduce the physical questions to 

 problems of pure analysis, and this is the proper object of theory. 

 They are not less rigorously established than the general equations 

 of equilibrium and motion. In order to make this comparison 

 more perceptible, we have always preferred demonstrations ana 

 logous to those of the theorems which serve as the foundation 

 of statics and dynamics. These equations still exist, but receive 

 a different form, when they express the distribution of luminous 

 heat in transparent bodies, or the movements which the changes 

 of temperature and density occasion in the interior of fluids. 

 The coefficients which they contain are subject to variations whose 

 exact measure is not yet known ; but in all the natural problems 

 which it most concerns us to consider, the limits of temperature 

 differ so little that we may omit the variations of these co 

 efficients. 



The equations of the movement of heat, like those which 

 express the vibrations of sonorous bodies, or the ultimate oscilla 

 tions of liquids, belong to one of the most recently discovered 

 branches of analysis, which it is very important to perfect. After 

 having established these differential equations their integrals must 

 be obtained ; this process consists in passing from a common 

 expression to a particular solution subject to all the given con 

 ditions. This difficult investigation requires a special analysis 



