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V. On Ellipsoidal Current-Sheets. 
By Horace Lamb, M.A., F.R.S., Professor of Pure Mathematics in Owens College, 
Victoria University. 
Received Marcia 2,—Read March 24, 1887. 
It is a problem of some interest in Electromagnetism to determine the natural modes 
of decay, and the corresponding persistencies, of free currents in a given conductor. 
When this has been solved it is an easy matter to find the currents induced by given 
varying electromotive forces. 
The general theory for a system of linear circuits is of course well known. If the 
variables y x , y z , . . . y n , which specify the currents, be so chosen that the electro- 
kinetic energy T and the dissipation-function F are both expressed by sums of 
squares, say 
2T = Lp/p + L yyp + . . + L n y„ 2 , 
2F = R x yp + Rpp 2 + . . + R»y« 3 , 
then y x , y 2 , . . y n are for the present purpose the “ normal coordinates ” of the 
system; and the equations of motion of electricity are of the form 
L y 4- % = E, 
where E is the external electromotive force of the type in question. In the case of 
tree currents, E = 0, and consequently 
y = Ae~ Kf , 
where 
X = R/L. 
If we put 
r= X- 1 = L/R, 
then r may be called the “ modulus of decay,” or the “ persistency,” of free currents 
of this type. 
In considering the effect of varying electromotive forces, it is convenient to suppose 
these expressed, as regards the time, in a series of simple harmonic terms, each of 
which may be taken separately. Assuming, then, that E oc e ipt , we have, for the 
induced current, 
