Self-induction of Wires. 347 



a function of p ; or more explicitly, put V — Sp(W -f Up) for 

 m, and then differentiate to p. 



Given, then, the initial state to be V = Y , a function of z, 

 and H = H 01 in the inner conductor, H 02 in the dielectric, and 

 H 03 in the outer conductor, functions of r and z, and that 

 this system is left without impressed force, subject to VC = at 

 both ends, the state at time t later will be given by 



the summations to include every p, with similar expressions 

 for H, r, 7, &c, the magnetic force and two components of 

 current, by substituting for u or w the proper corresponding 

 normal functions ; the coefficient A being given by the frac- 

 tion whose denominator is the expression M in (130), and 

 whose numerator is the excess of the mutual electric energy 

 of the initial and the normal system over their mutual mag- 

 netic energy, expressed by 



P 



1 Y sin (mz + 6)dz 



— \ cos (mz + 6) dzi 1 fi i R iG 1 'dr+ l ^ 2 H 02 C'<ir 



+ p^HosCs'^ } , . (131) 

 where c , = | ^(^-(J^^K^a,)}; 



and 0/ is the same with r put for a 1} and C 3 ' is the same with 

 r put for a 1? a B for a , and s 3 for s y It should not be forgotten 

 that in the case m = 0, the denominator (130) requires to be 

 doubled, J I becoming I. Also that R", or B/ + Up, contains 

 ra 2 , and must not be the m=0 expressions for the same. 



To check, take the initial state to be e (l — z/l), with no 

 magnetic force, and that V = at both ends. We find imme- 

 diately, by (130) and (131), that at time t, 



tt 2^1. ^ (m 2 /Sp 2 )eP« , 1W 



+ W+TJp) 



m d /m' 2 



dp\8p 



where the wi's are to be ir/l, 2tt/1, 37r/Z, &c. ; the first summa- 

 tion being with respect to m, and the second for the p's of a 

 particular m. 

 But, initially, 



*-+(*-fh?*z* 



sin mz. 

 m 



2 A2 



