Self-induction of Wires. 437 



from which inertia has disappeared. Here V is given by 

 (188) below. The process amounts to taking one half the 

 terms of the summation in (183), and joining them on to the 

 preceding term to make up e/R, which is quite arbitrary. An 

 alternative form of (185) is 



V dz 2 r> 



C = J -~j- + g- 2 cos mz I e cos mz dz . e -^/ RS . . (186) 



On the other hand, there is no such peculiarity connected 

 with the Y solution in the act of abolishing inertia. The 

 m=0 term is 



— sin mz 1 edz\ = 1 



Rl\Sp 

 because m is zero and p finite. Therefore V rises thus, 



m sin mz I e cos mz dz 



v 4 s -fr-k+»ig) (1 ~ 6P °' • (187) 



before abolition of inertia. But as L is made zero, the deno- 

 minator becomes m 2 for the electrostatic p, and oo for the 

 other ; thus one half the terms vanish, leaving 



v _2^ sin me 

 I m 



( l ecosmzdz{l-e- m2t l™), . (188) 



where L = 0, without any of the curious manipulation to which 

 the current formula was subjected. 



Next let us consider the transition from the combined 

 elasticity and inertia solution to inertia alone (of course with 

 resistance in both cases, as in the preceding transition). It 

 is usual to wholly ignore electrostatic induction in investiga- 

 tions relating to linear circuits. This is equivalent to taking 

 S = 0, stopping elastic displacement, and compelling the cur- 

 rent to keep in the wires always, i. e. when the insulation is 

 perfect, as will be here assumed. We then have, by (145), 



-S=°> «-S= R "°- • • • < 189 ) 



By integrating the second of these with respect to z we get 

 rid of Y, and obtain the differential equation of C, 



\ l edz=( ( l ~R"dz + Z 1 -Z \c = cl> 1 C, say, . (190) 



whence follows this manner of rise of the current, when e is 

 Phil. Mag. S. 5. Vol. 22. No. 138. Nov. 1886. 2 H 



