PROPAGATION OF ELECTRICITY IN A WIRE. 209 



The loss, which takes place at the surface is no more than a flow 

 of electricity in the external medium ; it is therefore proportional to 

 the flow of electrical force (201). For a length dx of the wire, the 

 charge is yVdx and the flow of electrostatic force is ^iryVdx. If c' 

 is the coefficient of conductivity of the medium, the strength of the 

 lateral current would therefore be c'^TryVdx. As this strength is 

 also equal to the quotient of the potential V by the resistance of the 

 medium, from the lateral surface in question, to the points where the 



potential is zero, the resistance relative to the- length dx is ; 



T & 4*fyi* 



and the resistance p' for unit length is equal to . 



fgpy 



The permanent state being established, the total flow of electricity 

 through the surface S of the volume-element dx should be equal to 

 the sum of the flows by the opposite surface S' and by the lateral 

 surface, which gives 



dx \ dx dx* 



that is to say 



dx* cSp' ' ' 



The product <:S represents also the reciprocal of the resistance p 



of the wire for unit length ; putting ft 2 = = , we get 



cSp p 



This is Fourier's equation for the propagation of heat in a 

 cylindrical bar. The integral of this equation may be put under 

 the form 



To determine the constants A and B, we must know the poten- 

 tials V and V x at two points P and P l at a distance of / from each 

 other. The value of the potential at a point P, situate at a distance 



p 



