134 Mr. 0. Heaviside on the 



than i/u-j/Ro, when the return is so distant, or the retardation 

 (fj^i^af) of the wire is so small that a steady current subsides 

 with very nearly uniform current- density, being very slightly 

 less at the boundary than at the axis. It is not, however, to 

 be inferred that the subsidence of the "current in the wire " 

 is delayed. It is accelerated, at least at first. 

 (29) may be written 



(fi 1 / : L )J (s 1 a 1 ) = ±s 1 a 1 J 1 (s 1 a 1 ), .... (31) 



the appropriate form when a full investigation is desired. 

 Draw the curves y x — right member, and y 2 — left member, 

 the abscissa being s^i. Their intersections will give the 

 values of ajOj satisfying (31). The first root has been already 

 considered, when /*i/L is very small. The rest, under the 

 same circumstances, will be nearly those of Ji(si%) = 0. But 

 if the wire is of iron ^i/L may be very large, and there will 

 be no approach to the linear theory. Many normal systems 

 must be taken into account to get numerical solutions. Simi- 

 larly if the sheath be close to the wire, whether it be magnetic 

 or not. 



Electrostatic charge being ignored, join the wire and sheath 

 to make a closed circuit, in which insert a steady impressed 

 force e at time t = 0. Let T be the current at distance r from 

 the axis at time t. (There is no y now). The rise of Y to 

 the final steady value, say T , is given by 



r=r fi-sA JoM^l, . . (32) 



where q = ~L a- l /2fi 1 . The values of sxOi are to be got by (31). 

 The total current C, or the current in the wire, in ordinary 

 language, rises thus to its final value C : — 



The boundary condition of V is that, at r = a l , 



r+g g£=0, .-. J(»i«0=M- • • • (34) 



Considering the first term only in the summation in (33), 

 as may be done except in the first stage of the rise, when the 

 linear theory is nearly followed, put — p~ l = (L + L 1 ) /U 0) 

 where Lj must be very small compared with L ; then 



When the current is started, by a steady impressed force in 



