68 Mr. 0. Heaviside on the 



features of a natural fault. If properly constructed it would 

 be of constant resistance, or only varying with the tempera- 

 ture, would contain no electromotive force of polarization, 

 would not deteriorate, and would considerably accelerate the 

 speed of working. The best position for a single fault would 

 be the centre of the line; and perhaps ^ of the line's resistance 

 would not be too low for the fault. 



9. In the cable, the potential v at any point schas to satisfy 

 the differential equation 



d?= ch ii> (1) 



and the current at x is 



_lrh 

 7 ~ k civ 



The particular solutions in paragraphs 5 and 6 regarding the 

 strength of the received current when reversals are made with 

 a battery at the sending-end are derived from the simple har- 

 monic solution 



v = ff (A cos + B sin) (— + y) 



+ e /(A'cos+B'sm)(— - y J 



of the above equation (1). TVhen there are faults, each of 

 the sections into which they divide the cable has a solution of 

 the above form. In the case of a single fault, there are four 

 conditions (namely, two for the fault and one for each end of 

 the line) which suffice to determine the eight constants. But 

 to determine the maximum strength of the received current, 

 it is only necessary to find the sum of the squares of two of 

 the constants. This shortens the labour, which is again greatly 

 shortened by neglecting e~ n in comparison with 1. 



10. The calculation of arrival-curves demands an entirely 

 different method of proceeding. The general problem may be 

 thus stated. Given a cable Avith faults in it, also the con- 

 nexions at the ends, resistances, condensers, &c., and given 

 also the electrical state of the whole system at a certain time : 

 to find its state at any time after, the system being left to 

 itself, and the action of the known laws regulating the poten- 

 tial, current, &c. In fig. 4 let P Q be a cable, of length /, resist- 

 ance kj and electrostatic capacity c per unit of length. Also, 

 let the terminal connexions be as shown, viz. at the begin- 

 ning P a resistance R : and a condenser of capacity Cx shunted 

 by a resistance S 1? with a similar arrangement at the end Q. 

 This includes the cases of signalling either with or without 



