104 THEORY OF HEAT. [CHAP. II. 



3rd. If in the function v which contains x, y, z, t, we make 

 t 0, whatever be the values of x, y, and z, we ought to have, 

 according to hypothesis, v = A, which is the initial and common 

 value of the temperature. 



131. The equation arrived at in the preceding problem 

 represents the movement of heat in the interior of all solids. 

 Whatever, in fact, the form of the body may be, it is evident that, 

 by decomposing it into prismatic molecules, we shall obtain this 

 result. We may therefore limit ourselves to demonstrating in 

 this manner the equation of the propagation of heat. But in 

 order to make the exhibition of principles more complete, and 

 that we may collect into a small number of consecutive articles 

 the theorems which serve to establish the general equation of the 

 propagation of heat in the interior of solids, and the equations 

 which relate to the state of the surface, we shall proceed, in the 

 two following sections, to the investigation of these equations, 

 independently of any particular problem, and without reverting 

 to the elementary propositions which we have explained in the 

 introduction. 



SECTION VI. 



General equation of the propagation of heat in the interior of solids. 



132. THEOREM I. If the different points of a homogeneous 

 solid mass, enclosed between six planes at right angles, have actual 

 temperatures determined by the linear equation 



v = A ax by cz, (a), 



and if the molecules situated at the external surface on the six 

 planes which bound the prism are maintained, by any cause what 

 ever, at the temperature expressed by the equation (a) : all the 

 molecules situated in the interior of the mass will of themselves 

 retain their actual temperatures, so that there will be no change in 

 the state of the prism. 



v denotes the actual temperature of the point whose co 

 ordinates are x, y, z ; A, a, b, c, are constant coefficients. 



To prove this proposition, consider in the solid any three 

 points whatever wJ//z, situated on the same straight line m^, 



