402 Professors Perry and Ayrton on a new 



independently of its interior constitution. And precisely similar 

 reasoning, of course, proves that outside the earth's surface 

 there would also be a magnetic field. ( Vide addition at the 

 end of the paper.) To determine the strength of this field we 

 have the following relationship to start with. 



In § 526, Clerk Maxwell's 'Electricity,' it is shown that 

 an element of current C, of length ST, acts upon a unit mag- 

 netic pole at a point P with a force 



c|| 2 sinPST, 



in a direction at right angles to PS and ST. Combining this 

 with the experiments referred to above, we may assume that 

 if a charge of static electricity (measured in electromagnetic 

 units) Q, at the point S, moves in a direction ST with a velo- 

 city v relative to a point P, it produces on a unit magnetic 

 pole at P a force 



jgsinPST, 



in a direction at right angles to PS and ST ; and this is the 

 only assumption employed in the following investigation. 



Sow suppose the earth to have the uniform density of elec- 

 tricity g over its surface, and let its radius be unity. Consider 

 the force produced, by the rotation of the electricity at a point 

 S on the surface having coordinates r, 0, <p, at a point inside 

 the sphere having the coordinates r 1? 6 1} fa. Then if the 

 sphere be rotating with an angular velocity w round the axis 

 of z, and if 6 be the angle between this axis and a radius, 

 while </> is the angle between the axis of x and the projection 

 of a radius, the velocity of S relative to P will have for its 

 coordinates 



u or —w (sin 6 sin $ — r sin X sin fa) parallel to as, 

 t or w (sin 6 cos (f> — r sin 1 cos fa) parallel to y ; 



PS 2 = Z 2 + m 2 + n 2 , 



also 

 where 



Z= sin 6 cos (j>—r sin 6 1 cos fa, 



m = sin 6 sin <f> — r sin 6 1 sin fa, 



7i= cos6—rcos6 v 



Now the direction-cosines of PS are proportional to I, m, and 

 n ; and the direction-cosines of ST, the direction of motion of 

 S relatively to P, are proportional to u, v, or to —icm, wl, 

 and 0. Consequently PS is perpendicular to ST. Also, if X, 



