CONDUCTORS OF ANY GIVEN FORM. 199 



for determining the currents in the various branches of the circuit. 

 Any modification in the resistances, or in the electromotive forces 

 which produces no change in the equations, will obviously be without 

 influence on the currents. For instance : 



i st. The resistance of a branch in which the current is null may 

 be modified at will. 



2nd. In all conductors which terminate in the same point, we 

 may introduce equal electromotive forces tending to produce currents 

 which all approach or recede from the point in question, these elec- 

 tromotive forces neutralising each other in pairs in all the closed 

 contours which pass through that point 



212, CONDUCTORS OF ANY GIVEN FORM. ELECTRODES. The 

 analogy of electrical conductivity with thermal conductivity, and 

 of this latter with the phenomena of statical electricity, enables us 

 to establish directly some theorems relative to the propagation of 

 electricity. 



Let us consider, in the first place, a single isotropic unlimited 

 medium. Let us suppose that, for the three different orders of phe- 

 nomena, the temperature on the one hand and the potential on the 



other are kept constant on different closed surfaces Sj, S 2 , S n , 



and that on each of the surfaces the temperatures and the potentials 



/! and V p / 2 and V 2 are represented by the same numbers, or 



by proportional numbers. In the thermal problem, these surfaces 

 will represent sources of heat ; in the problem of statical electricity, 

 the conductors; in the problem of the propagation of electricity, 

 they are called the electrodes. 



The temperature and potential of any point of the medium are 

 functions of the co-ordinates defined by the condition that these 

 functions acquire determinate values on the bounding surfaces, and 

 in the interval of these surfaces satisfy the condition 



A/=0, or AV = 0. 



The temperature and the potential at every point will therefore have 

 values which are either equal or are in a constant ratio. The iso- 

 thermal and the equipotential surfaces will be identical throughout 

 the whole extent of the medium, and therefore the tubes of flow are 

 identical. 



Across an element dS of any given surface, the flow of heat (70) 



is - kd , the flow of electrostatic force is - d$ , and the elec- 



9* av ^ n 



trical current is - cd$ - , k and c being the coefficients of thermal 



