164 THEORY OF HEAT. [CHAP. III. 



It follows from this that the molecule m of the third solid 

 acquires, during the instant dt, an increase of temperature equal 

 to the sum of the two increments which the same point would 

 have gained in the two first solids. Hence at the end of the 

 first instant, the original hypothesis will again hold, since any 

 molecule whatever of the third solid has a temperature equal 

 to the sum of those which it has in the two others. Thus the 

 same relation exists at the beginning of each instant, that is to 

 say, the variable state of the third solid can always be represented 

 by the equation 



203. The preceding proposition is applicable to all problems 

 relative to the uniform or varied movement oinea^7 It shews 

 that the movement can always be decomposed into several others, 

 each of which is effected separately as if it alone existed. This 

 superposition of simple effects is one of the fundamental elements 

 in the theory of heat. It is expressed in the investigation, by 

 the very nature of the general equations, and derives its origin 

 from the principle of the communication of heat. 



Let now v &amp;lt; (x, y] be the equation (a) which expresses the 

 permanent state of the solid plate BAG, heated at its end A, and 

 whose edges B and C preserve the temperature i; the initial state 

 of the plate is such, according to hypothesis, that all its points 

 have a nul temperature, except those of the base A, whose tem 

 perature is 1. The initial state can then be considered as formed 

 of two others, namely : a first, in which the initial temperatures are 

 (j&amp;gt;(x, y), the three edges being maintained at the temperature 0, 

 and a second state, in which the initial temperatures are + &amp;lt;j&amp;gt;(x,y), 

 the two edges B and C preserving the temperature 0, and the 

 base A the temperature 1; the superposition of these two states 

 produces the initial state which results from the hypothesis. It 

 remains then only to examine the movement of heat in each one 

 of the two partial states. Now, in the second, the system of tem 

 peratures can undergo no change ; and in the first, it has been 

 remarked in Article 201 that the temperatures vary continually, 

 and end with being nul. Hence the final state, properly so called, 

 is that which is represented by v = $ (x, y] or equation (a). 



