Prof. A. Schuster on Magnetic Precession. 321 



7. To obtain the angular velocity of the system o£ currents, 

 we may proceed as in the simple case, which has already been 

 discussed. 



A current-function a& n of degree n produces a magnetic 

 potential which inside the sphere is equal to 



— ^ lira l - ) <1>, 



and in the outer space 



2n + 1 \rj 



The energy of magnetic stress is easily calculated from 

 this and found to be 



n .n+l f 



47ra ^"T x |^>^S 



If there is a force-function <!>' from which the electric 

 forces are derived in the same way as the currents from the 

 current-function, the rate of doing work in a rectangular 

 element bounded by the linear elements add and a sin 6d<p 

 will be 



/ d<5> d& d<P d&\ 



\sin 0d<j> sin 6d(j> d6 dd ) 

 But 



J sm 6dcf> sin 0d<p J sin- 6 



as <I> and <&' only contain (j> in the form of a factor cos <7<£ 

 or sin cr<£. 



By partial integration, if again \ = cos#, 



f +l . d<S> d& n f^ 1 d . 2 „d<S>'. 



1 sin 2 6'^— . — =- dX= — 1 <P ,-siir^-— d\; 

 J_! aX. dX J_ 1 dA, </\ 



so that the total rate of doing work will be 



J'H sin 2 



°" 2 ^r d • 9 ad®'') to 



sin 2 d\ dX 



which by the characteristic equation of tesseral harmonics 

 becomes 



n . n + 1 f<I><E>'dS. 



The rate of doing work is equal to the rate of increase of 

 energy; hence 



in + ldr (1L> 



