4 Prof, A. Schuster's Electrical Notes. 



the pole of H. The force which acts on the moving sphere 

 will be equal and opposite to that which acts on S, and there- 

 fore its components are 



X=-$aad8, Y=-j/3ar/S, Z = 0. 



The values of a and /3 at the surface S are surface harmonics 

 of the first degree, and the same holds for o\ Hence if C 

 and A are the values of a and a at the pole of H, the 

 surface integrals may be written down at once, and we thus 

 obtain 



X = ^ ! CA. 



If the axis of Y is chosen so that the axis of H lies in the 

 plane of YZ, and 6 is the angle between it and the direc- 

 tion of motion of the sphere, 



X=^g?tfH sin 0, 



Y=0, Z = 0. 



This is Heaviside's result. 



4. To bring out clearly the cause of the different result ob- 

 tained by J. J. Thomson it is necessary to form expressions 

 for the mutual energy of the sphere and an outside magnetic 

 system. 



The potential energy of a magnet placed in a magnetic 

 field is 



W=-$(Aa + Bl3 + C 7 )dxch,dz, ... (2) 



where A, B, C represent the components of intensity of mag- 

 netization and a ? /3, 7, those of the magnetic forces acting on 

 the magnet. If for a and /3 we substitute their values we 

 find 



where F 3 is the .--component of the vector-potential at the 

 centre of the moving sphere. 



We may also consider the energy as kinetic and use the 

 equation 



T=^\\Uaa+!3b + vc)da:dydz, ... (4) 



a, b, c being the components of magnetic induction. The 

 expression is deduced under the supposition that the magnetic 

 forces are all due to electric currents. In isotropic media 



