58 ELECTRICAL EQUILIBRIUM. 



It is evident, moreover, that the action of the system is not 

 appreciable at great distances, and that the temperature /, which 

 it determines, is zero at an infinite distance. 



Hence, for every point of the medium and for the limits, the 

 function / satisfies the same conditions as the function V. It is 

 seen further that if the constant k is equal to unity, the numerical 

 values of the flow of electrical force, and of the flow of heat during 

 unit time, are identical in every point in the two problems. 



71. Let us now consider the corresponding hydrodynamical 

 problem. Let us imagine that the space originally occupied by the 

 dielectric is filled by a frictionless and incompressible liquid ; let us 

 imagine, moreover, that -the conductors are replaced by porous 

 surfaces, so that the liquid has at each point of such surfaces, a 

 normal velocity, equal to the original value of the electrical force at 

 this point. The whole of the trajectories of the molecules which 

 at the same moment have traversed the element dS of the surface 

 of a conductor, form a liquid thread which issues perpendicularly, 

 and yields the same supply in all sections. As there is nowhere any 

 accumulation of liquid, the flow passing through a volume element 

 dxdydz taken at any point P, is equal to that which emerges ; now 

 if #, v, w, denote the components of the velocity at the point P, this 

 condition is expressed by the equation 



~+ = 0. 



oy cz 



The motion is further inappreciable at an infinite distance ; we thus 

 see that the velocity at each point, depends on a function of the 

 co-ordinates which satisfies the same conditions as the potential or 

 the temperature. Lines of flow will coincide everywhere with the 

 lines of force of the corresponding electrical problem, and at each 

 point the electrical force and the velocity of the liquid will have the 

 same numerical value. 



The correlation which we have established is of great interest ; 

 for if it is clear that the analytical difficulties are exactly the same 

 in the three kinds of problems, it is no less true that certain 

 consequences present themselves more naturally in one order of 

 ideas than in another, and it is clear that any result obtained in one 

 case may be directly transferred, with its special interpretation, 

 into another. We shall meet with many instances of this in the 

 sequel 



