308 Steady Currents in Linear Conductors [CH. ix 



We can continue in this way, until finally we find as the whole resistance 



from A to earth, 



1 I ! 1 1 



If the currents or potentials are required, it will be found best to attack 

 the problem in a different manner. 



Let V A , TI, TJ, ... be the potentials at the points A, F-^ F^..., then, by 

 Ohm's Law, 



F_ F 



the current from 



F 8 through the fault = ^ . 

 Hence, by Kirchhoff s first law, 



V-V 



S+1 



r s r s+l 8 



or V s+l r s+ r l - V s (Er 1 + ry 1 + r.+r 1 ) + T-i rr 1 = 0, 



and from this and the similar system of equations, the potentials may be 

 found. 



If all the It's are the same, and also all the rs are the same, the equation 

 reduces to a difference equation with constant coefficients. These conditions 

 might arise approximately if the line were supported by a series of similar 

 imperfect insulators at equal distances apart. The difference equation is in 

 this case seen to be 



and if we put 1 + ^ = cosh a, 



Zit 



the solution is known to be 



V 8 = Acoshsa+ sinhsa ..................... (280), 



in which A and B are constants which must be determined from the 

 conditions at the ends of the line. For instance to express that the end B 

 is to earth, we have V n +i = 0, and therefore 



III. Submarine cable imperfectly insulated. 



354. If we pass to the limiting case of an infinite number of faults, we 

 have the analysis appropriate to a line from which there is leakage at every 

 point. The conditions now contemplated may be supposed to be realised in 

 a submarine cable in which, owing to the imperfection of the insulating 

 sheath the current leaks through to the sea at every point. 



