31 G THEORY OF HEAT. [CHAP. VII. 



the temperature is necessarily nothing. The same would not be 

 the case, if after having given to each point within the solid whose 

 co-ordinates are x, y t z the initial temperature ty(x, y, z), we gave 

 to all points of the section at the origin the temperature 0. We 

 see clearly, and without calculation, that in the latter case the 

 state of the solid would change continually, and that the original 

 heat which it contains would be dissipated little by little into the 

 air, and into the cold mass which maintains the end at the tem 

 perature 0. This result depends on the form of the . function 

 ty(x, y, z), which becomes nothing when x has an infinite value as 

 the problem supposes. 



A similar effect would exist if the initial temperatures instead 

 of being + ty (x, y, z) were -^ (#, y, z] at all the internal points 

 of the prism ; provided the section at the origin be maintained 

 always at the temperature 0. In each case, the initial tempera 

 tures would continually approach the constant temperature of the 

 medium, which is ; and the final temperatures would all be nul. 



327. These preliminaries arranged, consider the movement of 

 heat in two prisms exactly equal to that which was the subject of 

 the problem. For the first solid suppose the initial temperatures 

 to be + ^(a?, y, s), and that the section at origin A is maintained 

 at the fixed temperature 1. For the second solid suppose the 

 initial temperatures to be ^ (x, y, z), and that at the origin A 

 all points of the section are maintained at the temperature 0. It 

 is evident that in the first prism the system of temperatures can 

 not change, and that in the second this system varies continually 

 up to that at which all the temperatures become nul. 



If now we make the two different states coincide in the same 

 solid, the movement of heat is effected freely, as if each system 

 alone existed. In the initial state formed of the two united 

 systems, each point of the solid has zero temperature, except the 

 points of the section A, in accordance with the hypothesis. Now 

 the temperatures of the second system change more and more, 

 and vanish entirely, whilst those of the first remain unchanged. 

 Hence after an infinite time, the permanent system of tempera 

 tures becomes that represented by equation E, or v = ^r(#, y, z]. 

 It must be remarked that this result depends on the condition 

 relative to the initial state ; it occurs whenever the initial heat 



