Theory of Faults in Cables. 177 



rem for periodic functions ; that is, solving the linear equation 

 (63), its solution is found to be 



f( X ) = 2,Asm(^™ + b\ (64) 



Hence it is concluded that an arbitrary function f(os) may be 

 expanded in such a series as the right-hand side of (64), 

 though this proof of the possibility does not tell us how to do 

 it. Mr. O'Kinealy, however, completes the solution in the 

 usual way, leading to 



1 I 17T7j k VTTOC 



/(#) = 2l ) ./W dx+i^ cos — | /O) cos — dx 



+ j2sm-j-\ /(aj)sin-p. . (65) 



Similarly, if we start from equation (61), which is linear, 

 with constant coefficients, and includes the above case, we may 

 easily prove that its solution is (59), with the condition that 

 the a's therein are the + roots, real and imaginary, of (60), 

 the A's and 6's being undetermined. Or we may get the 

 same result from {62), the a's being now found from 



= k ia —k 5 a 3 + k 5 a 5 — (66) 



It will be observed that (60) or (66) have numerically equal 

 + and — roots, each pair of which go to a single term of (59). 



Here again the proof, if it may be now called a proof, gives 

 us no information as to how to find the coefficients settling 

 the amplitudes; and even the phases are undefined without 

 further knowledge. But in working out practical problems 

 requiring arbitrary functions to satisfy certain conditions 

 when expanded in a harmonic series, the physical nature of a 

 particular problem will usually suggest, step by step, the ne- 

 cessary procedure to render the solution complete, as in the 

 last paragraph 24 ; and the completion of a solution is of far 

 greater importance than any proof that the solution is possible. 



With respect to the periodic series (65), it is only applicable 

 to a cable when the ends are joined so as to make a closed 

 circuit, changing 21 into I ; and there must be no external 

 electrical connexions with the cable. If there are connexions 

 at a point, or at several points, even without interrupting the 

 continuity of the cable, although the potential of the cable will 

 now repeat itself every time x is increased by I or 21 &c, yet 

 the periodic form (65) will obviously not be suitable. - The 

 proper series are of course more general, and pass into the 

 form (65) in limiting cases. 



