Self-induction of Wires, 129 



As there is to be no current beyond the sheath, 73=0, or 

 H 3 =0, at r=a s . This gives 



B 3 =-A 3 i(* 3 a s ) (20) 



This, and the conditions y 1 =y 2) and p$T z =p 2 T 2 , at r=a 2 , 

 give us 



( A 2 J x + B 2 K 1 )(s 2 a 2 ) = A 3 J Ji(s 3 « 2 ) - j^{>3«3)Ki(s 3 a 2 ) [ > 



: 



(21) 



(P2V/°3S3)(A 2 J + B 2 K )(s2a2)=A3| J (s 3 a 2 )— ^-(s 3 a 3 )K (5 3 a 2 ) | 



whence, eliminating A 3 by division, and putting for A 2 and 

 B 2 their values in terms of A x through (19), we obtain the 

 determinantal equation of the p J s for a particular value of m 2 . 

 It is 



p3$z Jo (g 3 flg) K x (s B a s ) — Ji (s s a B ) K (s 3 a 2 ) 



p 2 s 2 J 1 (s3a 2 )K 1 (s 3 a 3 ) — J x (s 3 a 3 ) Kj (s 3 a 2 ) 



J 1 (g 1 a 1 )K (.s 2 a 1 )-(^ 1 g 1 /p 2 g 2 )Jo(g 1 a 1 )K 1 (g 2 a 1 ) j ^ | K fc 3 ) ■ 

 _ (p 1 g 1 //3 2 g 2 )J (g 1 a 1 )J 1 (g 2 a 1 )— J 1 (g 1 a 1 )J (g 2 a 1 ) oV 2 2) o{ 2 2; ^^ 



where the dots indicate repetition of the fraction immediately 

 over them. 



Before proceeding to practical simplifications, we may in 

 outline continue the process of finding the complete solution 

 to correspond to any given initial state. The m's must be 

 found from the terminal conditions. Suppose, for example, 

 that the wire, of length I, forms a closed circuit, and that the 

 sheath and the dielectric are similarly closed on themselves. 

 Then clearly we shall have Fourier periodic series, with 



m = 0, 'lirjl, 477-/Z, Qir/l, &c. 



If, again, we desire to make the sheath the return to the 

 wire, without external resistance, join them at the end 2=0 

 by a conducting-plate of no resistance, placed perpendicular 

 to the axis ; and do the same at the other end, where z=l. 

 This will make 



y=0 at #=0 and at #=Z ; 

 will make the d's vanish, and 



m = 0, it /I, 2ir/l, Stt/I, &c. 

 Each of these ra's has its infinite series of p's, by the equa- 

 tion (22). 



Now, as regards the initial state, the electric field and the 

 magnetic field must be both given. For, although the quan- 

 tity H, fully expressed, alone settles the complete state of the 



Phil. Mag. S. 5. Vol. 22. No. 135. August 1886. K 



