322 Prof. A. Schuster on Magnetic Precession, 



This equation connects the force-function <!>' with the cor- 

 responding current-function <J>. 



Now let there be two current-functions, a^ and «<J> 2 , 

 both of degree n and differing only in so far as <3>! contains 

 the factors cos crcf), and a function of the time, and <E> 2 contains 

 the factors sin acf>, and some other function of the time. 



The current-function {£& x will call forth a force-function 

 2ua) , 



— ITT^ 1 ot ' tne same typ e as ^2? and the current- 



in . n — r* j. q 



function a<J> 2 a force -function f- «(J> 2 of the same 



type as ^ In. n + l 



So that substituting in (11), 



Sna d&i _ 2/ulco 1 



2n + l"rfT _ = T~ ' n. n + l 2 ' 



87T<X d<& 2 2yLtft) 1 _ 



3*1 . 



2n + l dr t n . n + l 



It follows from this, that ^ and <I> 2 are proportional to 

 cos X2t and — sin Ot where 



fl^it. J^+ll.iL (12) 



at n . n + l Sir v 



This gives the periodicity of the forces. The complete 

 current-function will now be proportional to 



cos (T(j> cos Or— sin <r<fr sin Ot = cos(<t0 — fir), 



= cos c-(<p ); 



so that the angular velocity with which the currents revolve 

 is X2/<r. 



8. I have so far only dealt with electric currents confined 

 to a spherical shell, as this problem adapts itself most easily 

 to mathematical treatment. But in the simplest and most 

 important case we may without difficulty obtain the solution 

 for currents which circulate in the body of a conducting 

 sphere. We take the case that the currents are such as 

 would produce outside magnetic effects which can be repre- 

 sented by a magnetic potential of degree — (n + 1). Dividing 

 the sphere into concentric shells we may neglect radial 

 currents, and the currents within each shell may be repre- 

 sented by a current-function which is proportional to a 

 surface harmonic of degree n. If we w T rite t<P instead of O 

 in the previous investigation, the differential coefficients of 

 this function will give current-densities instead of currents 



