Also ^ m —mr^)s + mr'j.s' — 0, 



Currents in Concentric Cables. 65 



27rm the total capacity of the inner dielectric measured 



between the two conductors ; 

 2im the total capacity of the outer dielectric measured 



between the outer conductor and the water. 



tnn 



Then 2ir is the capacity of the central conductor, the 



m-\-n i j ■> 



outer one being insulated ; and '2irn the capacity of the outer 



conductor, the inner one being insulated. 



u u, C C are the potentials of and the currents in the two 



conductors at point P and time t, and s s' the density of the 



charges on the conductors. All these quantities being referred 



to a unit of length subtending unit angle at the centre, then 



dC (du du'\ dC (du' du\ , du 



-de^dt-w)' ~To= m \-dt-dl) +n -dt ; 



s + s'=nu' : and s = m(ii — u'). 



Id 1 d\ 



\aW- mr dt) 



and (§p- (m + n)tJ d^ S ' + mr Jt S = ' 

 Hence s and s' satisfy the equation 



/ cP d\/ d* d\ 



\»w*-dt)vw-it) s=0 > 



where ,u, v are roots of mnrr'x* — (mr + (m + n)r')x+ 1 = 0. 



§3. A uniform conductor Q, along which the propagation 

 of charges due to any electrical disturbance is determined by 

 the differential equation 



/ d 2 d\( cP d\ 

 (*W*-dt)VaW-It) s=0 > 



is arranged in a circle as in fig. 2. Its total resistance is 

 2irr, and its total capacity 2irc, Any disturbance will be 

 propagated along the conductor in two distinct systems, one 

 determined by the differential equation 



{*m - !)* =0 ' the other b ? { v w - Ji} =0 > 



and the disturbance produced at any given point will be the 

 sum of two disturbances conveyed by the two systems. 

 Phil. Mag. S. 5. Vol. U. No. 266. July 1«97. F 



