224 VARIABLE STATE. 



the electrification begins. The element ^U of the potential at the dis- 

 tance x and at the time /, which corresponds to the potential V set up 

 during the time dO and to the time at the origin of the wire is equal 



(229) to M </>(/-#) dO. If the total duration of the electrification 



VTT 

 is r, we shall have for the resultant potential 



U = -= 



It may, however, be remarked, that this expression has no meaning 

 except for values of /, greater than T. 



If the potential at the origin of the wire varies periodically 

 according to a simple law if it is represented, for instance, by 

 V sin 2nt, the ultimate electrical state of the wire at each point 

 varies obviously according to the same period. The potential at 

 the distance x may be expressed by the formula 



in which b is a constant, and A a function of x. Substituting this, 

 in equation (2) we get finally 



(9) V = V Q e- ax ^ n sm(2nt-xa f Jn). 



Hence a definite phase of the potential at the origin is trans- 

 mitted along the wire with a constant velocity equal to - . In 



this case it may be said that there is a regular velocity of propa- 

 gation, but this velocity depends on the period of electrical 

 oscillation; the time necessary to traverse a definite length is 

 proportional to a, and not to a 2 , as has been found for the relative 

 length of propagation considered above (225). 



If the electrification at the origin, comes under a more or less 

 complex law, and if the expression for the initial potential is de- 

 composable into a series of simple periodic terms, the potential 

 in the wire will be represented by a series of corresponding ele- 

 mentary waves; but these waves will be propagated with different 

 velocities, and a kind of electrical dispersion will be produced 

 analogous to the phenomenon of the dispersion of light in a re- 

 fracting medium. 



