Prof. A. Schuster's Electrical Notes. 9 



Turning back to the system of currents considered in § 7, 

 it is seen that, leaving out of account the straight-line currents, 

 the above theorem becomes applicable, for the electric currents 

 are now deducible from a potential (ft. Hence that part of 

 the energy which depends on displacement currents vanishes, 

 and it is therefore sufficient to consider the convection currents 

 alone. These vanish everywhere except at the surface of the 



sphere, where the vector-potential reduces to Aj = -r~— . Hence 



_ fiq 2 w 



'6a 



This is a result identical with that deduced by Heaviside, 

 as also with that obtained by J. J. Thomson in his '' Recent 

 Researches/ The different value obtained by Thomson in his 

 first paper is partly due to the wrong value of the vector- 

 potential he adopted, and partly to his omission of the con- 

 vection curents in calculating the energy. If the correct 

 vector-potential had been taken, the neglect of the convection 

 currents would have led to the result that the total energy is 

 zero. 



9. We may extend the investigation to the case where the 

 charge of the sphere varies according to some spherical 

 surface harmonic Y n . Let the surface-density be given by 



47raV=(2;i + l)Y n . 



The electric potential due to such a surface distribution will 



r 4 . a 4 



be n+1 Y n inside the sphere, and ~p[Y n outside the sphere. 



Let B be the vector-potential of electric currents which pro- 

 ceed along the lines of force with current intensities nume- 

 rically equal and opposite everywhere to the displacements, 

 that is to say, such currents as in unit time would destroy 

 the charge of the sphere ; then by the same reasoning as in 

 § 7 we find that the vector-potential produced by the dis- 

 placement and convection currents of a sphere charged as 

 above, and moving with velocity w in the direction of the 

 axis of Z, would be given by 



. dB l dB 2 dB 3 t (WS 



— — represents the effect of the convection 



