396 THEOKY OF HEAT. [CHAP. IX. 



may be reduced to ~ , since the factors which we omit coincide with 



L 



unity. Consequently we find 

 V 



V = 



The integral Ida Idb ldcf(a, b, c) represents the quantity of 



the initial heat : the volume of the sphere whose radius is r is 



4 



K 7rr s , so that denoting by / the temperature which each molecule 

 o 



of this sphere would receive, if we distributed amongst its parts 

 all the initial heat, we shall have v = A/ $f. 



The results which we have developed in this chapter indicate 

 the law according to which the heat contained in a definite portion 

 of an infinite solid progressively penetrates all the other parts 

 whose initial temperature was nothing. This problem is solved 

 more simply than that of the preceding Chapters, since by 

 attributing to the solid infinite dimensions, we make the con 

 ditions relative to the surface disappear, and the chief difficulty 

 consists in the employment of those conditions. The general 

 results of the movement of heat in a boundless solid mass are 

 very remarkable, since the movement is not disturbed by the 

 obstacle of surfaces. It is accomplished freely by means of the 

 natural properties of heat. This investigation is, properly 

 speaking, that of the irradiation of heat within the material 

 solid. 



SECTION IV. 



Comparison of the integrals. 



396. The integral of the equation of the propagation of heat 

 presents itself under different forms, which it is necessary to com 

 pare. It is easy, as we have seen in the second section of this 

 Chapter, Articles 372 and 376, to refer the case of three dimen 

 sions to that of the linear movement ; it is sufficient therefore to 

 integrate the equation 



** JL &* 



dt~~ CDdx* 



