368 THEORY OF HEAT. [CHAP. IX. 



The value of the series represents, as we have seen previously, 

 any function whatever of x + 2q? *Jkt ; hence the general integral 

 can be expressed thus 



= / 



The integral of the equation -^- &^ 2 may besides be pre 

 sented under diverse other forms 1 . All these expressions are 

 necessarily identical. 



SECTION II. 

 Of the free movement of heat in an infinite solid. 



372. The integral of the equation ,, = -^ -j- 9 (a) furnishes 



immediately that of the equation with four variables 

 dv 



, , , 

 ......... 



as we have already remarked in treating the question of the pro 

 pagation of heat in a solid cube. For which reason it is sufficient 

 in general to consider the effect of the diffusion in the linear 

 case. When the dimensions of bodies are not infinite, the distri 

 bution of heat is continually disturbed by the passage from the 

 solid medium to the elastic medium; or, to employ the expres 

 sions proper to analysis, the function which determines the 

 temperature must not only satisfy the partial differential equa 

 tion and the initial state, but is further subjected to conditions 

 which depend on the form of the surface. In this case the integral 

 has a form more difficult to ascertain, and we must examine the 

 problem with very much more care in order to pass from the case 

 of one linear co-ordinate to that of three orthogonal co-ordinates : 

 but when the solid mass is not interrupted, no accidental condition 

 opposes itself to the free diffusion of heat. Its movement is the 

 same in all directions. 



1 See an article by Sir \V. Thomson, &quot; On the Linear Motion of Heat,&quot; Part I, 

 Camb. Math. Journal, Vol. in. pp. 170174. [A. F.] 



