62 Prof. Thomson on the Mathematical Theoi^ of 



have to consider the case iu which A is subject to no external 

 influence, we must suppose every part of the surface of B to be 

 very far from A. The most general problem we can contemplate 

 in electricity (exclusively of the case in which the insulating 

 medium is heterogeneous, and exercises a special action, which 

 will be alluded to below), is to determine the potential at any 

 point when A, instead of being a single conductor, is a group of 

 separate insulated conductors charged to different degrees, and 

 when there are non-conductors electrified in a given manner, 

 placed in the insulating medium, in the neighbourhood. The 

 conditions of equilibrium will still be that the potential at each 

 surface due to all the free electricity must be constant, and the 

 theorems stated above will still be true : thus the attraction will 

 be nothing in the interior of each portion of A, and without the 

 surface of B ; and the whole quantity of induced electricity on 

 the latter surface will be the algebraic sum of the charges of all 

 the interior bodies with its sign changed. When the potential 

 due to such a system is determined for every point, the compo- 

 nent of the resultant force at any point P, in any direction PL, 

 may be found by differentiation, being the limit of the differ- 

 ence between the values of the potential at P, and at a point Q, 

 in PL, divided by PQ, when P moves up towards and ultimately 

 coincides with P, and the direction of the force, on a negative 

 particle, being that in which the potential increases. By Cou- 

 lomb's theorem, the intensity at any point in one of the con- 

 ducting surfaces is equal to the attraction (on a negative unit), 

 at that point, divided by 47r. 



Now if we wish to consider the corresponding problem in the 

 theory of heat, we must suppose the space between A and B, 

 instead of being filled with a dielectric medium (that is, a non- 

 conductor for electricity), to be occupied by any homogeneous 

 solid body, and sources of heat or cold to be so distributed over 

 the terminating surfaces, or the interior surface of B and the 

 surface of A, that the permanent temperature at the first surface 

 may be zero, and at the second shall have a certain constant 

 value, the same as that of the potential in the case of electricity. 

 If A consist of different isolated portions, the temperature at the 

 surface of each will have a constant value, which is not neces- 

 sarily the same for the different portions. The problem of 

 distributing sources of heat, according to tliese conditions, is 

 mathematically identical with the problem of distributing elec- 

 tricity in equilibrium on the surfaces of A and B. In the case 

 of heat, the permanent temperature at any point replaces the 

 potential at the coiTcsponding point in the electrical system, and 

 consequently the resultant flux of heat replaces the resultant 

 attraction of the electrified bodies, in direction and magnitude. 



