of the Earth 9 s Magnetism. 103 



cause a deflection when the electrification of the condenser 

 was reversed. On trying the experiment in the most careful 

 manner, there was not the slightest trace of action after all 

 sources of error had been eliminated. 



But the experiment did not satisfy me, as I saw there was 

 some electricity on the metal case surrounding the needle. 

 And so I attacked the problem analytically, and arrived at the 

 curious result that if an electrified system moves forward with- 

 out rotation through space, the 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, w T e 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 favourable case, and so have assumed the earth 

 to be a sphere of magnetic material of great permeability, /jl. 

 It does not seem probable 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 velo- 

 city ic, and let cr 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 



<f> = - tea 2 1 sin 6 old— wa 2 cos 6. 



T n J n 



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



inside the sphere is 



r\ 8-7T cr n 



12 = s — tear cos a, 



3 n 7 



and outside the sphere 



cr , cos 



xy= 



7r - tea 



3 n ?' 2 



The magnetic force in the interior of the sphere is thus 



„ 8 <r 

 6 n ; 



I 2 



