ANALOGIES OF THE PROBLEM OF ELECTRICAL EQUILIBRIUM. 55 



and the n - 2 other coefficients relative to the conductor A p , y^, 



7 2 P 7njp W0uld be nul1 ' 



70. ANALOGIES OF THE PROBLEM OF ELECTRICAL EQUILIBRIUM. 

 It is interesting to compare with the problem, which we have just 

 treated, two other problems relating to phenomena which are entirely 

 different, but which, analytically, present the most complete analogy 

 that of the uniform propagation of heat in a homogeneous medium, 

 and that of the steady motion of an incompressible and frictionless 

 liquid. 



In short, the electrical problem is characterised by the existence 

 of a function of the co-ordinates, which, vanishing at an infinite 

 distance, has a constant value on each of the conductors, and for 

 each point of the dielectric satisfies the ratio 



AV=0, 



the physical signification of which is very simple. X, Y and Z being 

 the components of the force at a point P, the quantity - AWz; 

 represents the total flow of force which proceeds from an element of 

 volume dv taken at this point, and equation AV = expresses that 

 this flow is nothing in the dielectric, or in the interior of a con- 

 ductor ; that is to say, where there is no electricity. 



Let us now suppose that in a problem of statical electricity the 

 insulating medium is replaced by a medium which conducts heat, 

 and which is homogeneous, and isotropic\ that is to say, which has 

 the same properties in all directions; and let us suppose each of 

 the electrified conductors replaced by sources which emit or which 

 absorb heat, so as to maintain constant temperatures on the surfaces 

 which are respectively equal in numerical value to the initial potentials, 

 so that for each of the conductors /=V. 



When once equilibrium is established, every point of the medium 

 will be at a definite temperature, and isothermal surfaces can be 

 traced ; that is to say, surfaces of equal temperature, or of equal 

 thermal level. It is clear that the temperature of a point P, com- 

 prised between two isothermal surfaces S and S', is independent of 

 the situation of the sources, and that it will remain the same when 

 these sources are suppressed, if the temperatures / and /' of these 

 two surfaces are kept constant in any other way. 



Fourier's hypothesis consists in assuming, what indeed may be 

 regarded as the simple expression of facts, that heat travels from 

 layer to layer ; that the thermal effect of a point has no appreciable 

 influence except on very near points; and that the hotter points 



