214 VARIABLE STATE. 



In this case it is still possible that the flow of electricity at each 

 point is proportional to the actual electrical force, but this force will 

 not depend solely on the position of the acting masses it will 

 depend also on the velocity of these masses, and the effects may 

 be very different, according as the velocity of displacement of the 

 acting masses is, or is not, of the same order of magnitude as the 

 velocity of propagation of the potential. 



We shall see lastly, in connection with the phenomena of electro- 

 dynamic induction, that the displacement of electrical currents and 

 their changes of strength, produces new electromotive forces, which 

 can be calculated in a certain number of cases, and which may greatly 

 modify the results relative to the variable state. 



The two effects which we have mentioned are perhaps produced 

 by the same mechanism; we shall not for the present take them 

 into account. 



With this reservation we can again apply Fourier's formulas. 

 In any case, the results to which they lead must be so much 

 nearer the truth the slower are the changes in the variable state ; 

 in fact, these results represent very approximately the propagation 

 of electricity in submarine cables, and apply rigorously to Gaugain's 

 experiments on the propagation of electricity in bodies of great 

 resistance, such as cotton threads, or columns of oil. 



224. VARIABLE STATE IN A CYLINDRICAL CONDUCTOR. Let 

 us consider, then, in a cylindrical conductor the volume-element dx 

 comprised between two infinitely near sections S and S' (Fig. 55). 

 The potential at a point P is no longer a simple function of x that 

 is to say, of the position of this point but it is also a function of the 

 time /. During the time dt, the amount of electricity which this 

 volume-element gains, is equal to the excess of the flow through the 

 section S, over the flow which issues by the section S', together with 

 the loss by the external surface that is to say : 



-V dxdt. 



I- -vl. 



I p*bx 2 p 



/ 1 <) 2 V I \ 



The increase of charge ( -^~2~~^ )& for unit length, will 

 \pDx* p ) 



produce a variation of potential dV or dt\ if then we assume 

 that the ratio of the charge to the potential, remains equal to the 



