180 HENRY A. KOWLAND 



magnetic force at any point is dependent on the electrical force at that 

 same point or, in other words, that all the equipotential surfaces have 

 the same magnetic action. Hence, when we shield a needle from elec- 

 trostatic action, we also shield it from magnetic action. 



This theorem only applies to irrotational motion, and assumes that 

 the elementary law for the magnetic action of electric convection is the 

 same as the most simple elementary law for closed circuits. Hence we 

 see that, provided the earth were uniformly electrified on the exterior 

 of the atmosphere, there would be no magnetic action on the earth's 

 surface due to mere motion of translation through space. 



In calculating the magnetic action due to the rotation, I have taken 

 the most favorable case, and so have assumed the earth to be a sphere 

 of magnetic material of great permeability, ft. It does not seem prob- 

 able that it would make much difference whether the inside sphere 

 rotated or was stationary; or at least the magnetic action would be 

 greatest in the latter case; and hence by considering it stationary we 

 should get the superior limit to the amount of magnetism. 



Let a be the radius of the sphere moving with angular velocity w, 

 and let a be its surface-density in electrostatic measure, and n the ratio 

 of the electromagnetic to the electrostatic unit of electricity. Then the 

 current-function will be 



<p we? I sin Odd = wa? cos . 



n J n 



Hence (Maxwell's ' Treatise/ 672) the magnetic potential inside the 

 sphere is 



8:: ff 

 u = 



and outside the sphere 



= -TT - war cos , 

 o n 



^ n r 2 



The magnetic force in the interior of the sphere is thus 



F=i* wa. 



n 



or the field is uniform. If the electric potential of the sphere on the 

 electrostatic system is V, we may write 



^T 

 which is independent of the dimensions of the sphere. 





