CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
309 
are related to each other as in a plane plate ; during its decay also, P follows the 
same law as in that case. 
When the shell is infinitely thin, the effect, on an external point, of the currents 
excited in it may be represented by the following system of images, which constitute 
a generalisation of those of Maxwell. Divide the time into an infinite number of 
equal intervals, and at the commencement of any of these let a positive image of the 
system be formed in the place occupied by its electric image at the surface. Let the 
parts of this image move towards the centre in straight lines so that the logarithmic 
It 
decrement of their distances from the centre is constant and equal to -— (It being; 
1 2?ra v & 
the resistance of the shell and a its radius), and let the intensity of the image increase 
Ii 
at each point with a constant logarithmic rate —At the end of the interval let an 
exactly equal but negative image be formed in the place of the former and move 
towards the centre in the same manner, and let these operations be repeated at the 
commencement and end of every interval during which the external system is varying ; 
the action of the sheet on external points will be that due to the above train of 
images. The action on a point within the sheet may be represented in a somewhat 
similar manner. 
When the shell possesses a finite thickness, or is a solid sphere, it is not possible to 
express its effect so simply. The variations in the external system produce continually 
new systems of currents, the law of whose decay may be exhibited by expressing P in 
a series of terms containing each the product of a tesseral harmonic, a “ spherical ” 
function of the radius, and an exponential the coefficients of which are to be 
found by known methods. 
When the shell degrades into an infinite plate, the “ spherical” function becomes 
an exponential or circular function, and the tesseral harmonic becomes the product of 
a factor cos m<f> or sin mcf) by a Bessel’s function J m (i<p). The coefficients might then 
be found by means of Neumann’s theorem for expanding ./(sc, y) in Bessel’s functions ; 
but their deduction from the corresponding problem of spherical harmonic analysis 
throws an interesting light upon Neumann’s expansion, and especially on the meaning 
of the symbol co in the limits of integration. 
When a symmetrical conductor revolves uniformly about its axis of symmetry for 
a sufficient length of time, the currents and electric distribution become steady, and 
the total currents are then identical with the currents of conduction. In the case of 
a plate or spherical shell, the vector potential and currents are expressible in the same 
manner as before in terms of two functions P and <f>, which are still related to each 
other as formerly. The equation which now determines P is ~ V 2 P = & / 
The general results of calculation verify Maxwell’s theorem of the spiral trail of 
images due to an infinitely thin plate. The theorem is also extended to a spherical 
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