DURATION OF THE RELATIVE PROPAGATION. 217 



Consider an insulated wire, whose loss by the surface may be 

 neglected, originally in the neutral state, and of an unlimited length, 

 or at any rate of a length such that the condition at a point is not 

 appreciably modified by that of the most distant end. At the end 

 of the wire a constant potential V is maintained. At the end of a 

 time /, the potential at a distance x is defined by the equation 



For a second wire placed in the same conditions as the first, and 

 the material of which is defined by another coefficient a', we shall 

 have in like manner 



Let us put x' = mx^ t' = nt, m and n being constants ; we may 

 then consider the potential V as a function of the variables x and t, 

 and equation (5) will become 



If the coefficients m and n are chosen so that 



that is to say 



/' 



the potentials V and V satisfy the same differential equation (2), 

 and the same limiting conditions; they represent then the same 

 function of x and of /. 



227. Hence, for unlimited wires, which is practically equivalent 

 to wires so long that the duration of the propagation has an appre- 



a?x 2 

 ciable value, the potential V does not change when the ratio 



has the same value ; it is, therefore, a function of this ratio. 



