Self-induction of Wires. 345 



the first two of these being the terminal conditions, and 

 ~R! m + U m p being merely a convenient way of writing the real 

 complex expressions; (equation (QS), with e m = 0). It is clear 

 that the only cases in which the m's become clear of the jo's 

 are the before-mentioned five cases, equivalent to Z and Z x 

 being zero or infinite, and the line closed upon itself, which 

 is a sort of combination of both. Considering only the four, 

 they are summed up in this, VC = at the terminals, or the 

 line cut off from receiving or losing energy at the ends. We 

 have then the series of m's, 0, it /I, 27r//, &c. ; or \tt\1, | 7t//, §7t//, 

 &c. ; and every m 2 has its own infinite series of p's through 

 the third equation (128). These, though very special, are 

 certainly important cases, as well as being the most simple. 

 We can definitely effect the expansions of the initial states in 

 the normal functions, and obtain the complete solutions in 

 every particular. 



Although rather laborious, it is well to verify the above 

 results by direct integration of the proper expressions for the 

 electric and magnetic energies of normal systems throughout 

 the whole line. Thus, let 



rH 1 + s^H 1 =0, where —s 2 l =4:7rfjL 1 k 1 p 1 + m 2 ) 



^H 2 + s^H 2 = 0, where — sl-=k r rr\x l k x 'p 2 - i rm\) 



in the inner conductor. We shall find 



(s\ -jg) ( ai K 1 K 2 rdr=S7r(C 1 T 2 -Q 2 T 1 ), 



as H! = = H 2 at r — a Q \ I\ and T 2 being the longitudinal 

 current-densities at r=ai. Similarly for the outer conductor, 



(s'l-s'l) ( a3 W 1 -E' 2 rdr= -8^(0^-0^) 



if Ci, C 2 still be the currents in the inner conductor; the 

 accents merely meaning changes produced by the altered p, 

 and k in the outer conductor. H^ = = H 2 ' at r = a z in this 

 case. Then, thirdly, for the intermediate space, 



f a2 H/'H 2 "^r= C^, X 4 log ^- 



Therefore the total mutual magnetic energy of the two distri- 

 Phil. Mag. S. 5. Vol. 22. No. 137. Oct. 1886. 2 A 



d 



1 d f 



dr 



r dr ' 



d 



1 d 



dr 



r dr 



