ll4 THEORY OF HEAT. [CHAP. II. 



of heat required to raise from to 1 the molecule whose volume 

 is dx dydz. Hence dividing by this product the quantity of 

 heat which the molecule has just acquired, we shall have its 

 increase of temperature. Thus we obtain the general equation 



^ - J^ (^ JL ^ + &1 



which is the equation of the propagation of heat in the interior 

 of all solid bodies. 



143. Independently of this equation the system of tempera 

 tures is often subject to several definite conditions, of which no 

 general expression can be given, since they depend on the nature 

 of the problem. 



If the dimensions of the mass in which heat is propagated are 

 finite, and if the surface is maintained by some special cause in a 

 given state ; for example, if all its points retain, by virtue of that 

 cause, the constant temperature 0, we shall have, denoting the 

 unknown function v by (f&amp;gt; (x, y, z, t}, the equation of condition 

 (j&amp;gt; (x, y, 2, t) = ; which must be satisfied by all values of x, y, z 

 which belong to points of the external surface, whatever be the 

 value of t. Further, if we suppose the initial temperatures of the 

 body to be expressed by the known function F (x, y, z), we have 

 also the equation &amp;lt;f&amp;gt; (x, y, z, 0) = F (x, y, z) ; the condition ex 

 pressed by this equation must be fulfilled by all values of the 

 co-ordinates x, y } z which belong to any point whatever of the 

 solid. 



144. Instead of submitting the surface of the body to a con 

 stant temperature, we may suppose the temperature not to be 

 the same at different points of the surface, and that it varies with 

 the time according to a given law ; which is what takes place in 

 the problem of terrestrial temperature. In this case the equation 

 relative to the surface contains the variable t. 



145. In order to examine by itself, and from a very general 

 point of view, the problem of the propagation of heat, the solid 

 whose initial state is given must be supposed to have all its 

 dimensions infinite; no special condition disturbs then the dif- 



