334 THEORY OF HEAT. [CHAP. IX. 



been heated, all other points of the solid having the initial tem 

 perature 0. The laws of the distribution of heat in an infinite 

 solid mass ought to have a simple and remarkable character ; 

 since the movement is not disturbed by the obstacle of surfaces, 

 or by the action of a medium. 



343. The position of each point being referred to three rect 

 angular axes, on which we measure the co-ordinates x, y, z, the 

 temperature sought is a function of the variables x, y, z, and of 

 the time t. This function v or &amp;lt; (x, y, z, t) satisfies the general 

 equation 



dv _ K fd z v d*v d z v\ , . 



dt~ C7)(dx 2+ d^ + dz 2 ) 



Further, it must necessarily represent the initial state which is 

 arbitrary; thus, denoting by F(x, y, z) the given value of the 

 temperature at any point, taken when the time is nothing, that is 

 to say, at the moment when the diffusion begins, we must have 



&amp;lt;(*, y, z, 0) = F(x, y, z) (5). 



Hence we must find a function v of the four variables x, y, z, t, 

 which satisfies the differential equation (a) and the definite equa 

 tion (&). 



In the problems which we previously discussed, the integral is 

 subject to a third condition which depends on the state of the 

 surface : for which reason the analysis is more complex, and the 

 solution requires the employment of exponential terms. The 

 form of the integral is very much more simple, when it need only 

 satisfy the initial state; and it would be easy to determine at 

 once the movement of heat in three dimensions. But in order to 

 explain this part of the theory, and to ascertain according to what 

 law the diffusion is effected, it is preferable to consider first the 

 linear movement, resolving it into the two following problems : we 

 shall see in the sequel how they are applied to the case of three 

 dimensions. 



344. First problem : a part a b of an infinite line is raised at 

 all points to the temperature 1 ; the other points of the line are at 

 the actual temperature ; it is assumed that the heat cannot be 

 dispersed into the surrounding medium; we have to determine 



