On the Theory of Faults in Cables. 163 



(1 action than as a mere speculation. And even independently 

 of all other considerations, it would seem more reasonable to 

 look to a dynamical interpretation of the motions of the stellar 

 masses, than to regard them as drifting indiscriminately among 

 each other with the absence of all recognized purpose. 



XX. On the Theory of Faults in Cables. 

 By Oliver Heaviside. 

 [Concluded from p. 74.] 

 16. npHERE is no difficulty in finding formulae from the 

 J- preceding results which will correspond to any par- 

 ticular example considered. Such formulae, however, have, 

 save to the mathematically curious, little value or interest 

 unless they are interpreted numerically. Even then the labour 

 involved is, save in special cases, out of proportion to the de- 

 rived benefit. I shall confine myself to the simple cases of 

 direct working without condensers, and with condensers, with 

 a single fault in the centre of the line. 



Suppose the signalling is made by means of a battery at P 

 and' a receiving instrument at Q, both of negligible resistance, 

 and to earth direct. Then 



= mi = m 2 = n l = n 2 . 



Let there be a single fault of resistance zld at the centre of 



the line. Then 



tan 5=0, 



1 . 2 a 

 sin a -\ sin - = 0, 



za 2 7 



by (10) and (11). The latter splits up into 



sin | = and tan | = - 2za. . . . (21) 



Therefore, when i is even, a i =i ( n", and when i is odd, a { lies 

 between iir and (2 + 1)77-. The denominator of (19) is 

 i • . « 



A/ 7\ f^ " 2 (LV 7 , ii • C{X . % ' (& 1\ 1 2 7 



I A sin a\ 

 Therefore, by (20), 



A..=-^ 



fl'-sin di 



