Theory of Electric Inertia. 245 



sign of integration for t. And since u may in this case be 

 written 



du . , dru c 

 u = u -t dt +±P -^ — &c., 



di( 

 the result must contain not only — e but also u , , &c. It 



may, indeed, still be a very near approximation, but whether it 

 is so or not will depend on the rate at which u varies with the 

 time. The electric inertia calculated by this method cannot 

 in general be completely expressed as a function o£ e only 

 as proposed. 



5. Lodge subsequently employs another method of calcu- 

 lating the inertia, the method, namely, of the magnetic 

 reactions. This rests on the principle that any change of 

 velocity, as ~du. impressed on the moving sphere while at S, 

 and also the change of its position while moving at S with 

 the velocity u. gives rise at any point P in space after 

 the interval of time SP/t* to a change in the magnetic force 

 H at P. And this change of magnetic force gives rise after 



PS / 



the further interval of time to an electric force at another 



v 



point S'. And if S' be suitably chosen on the axis, this electric 

 force will take effect at S' at the instant when the moving- 

 sphere arrives at S', and so affect the sphere. 



6. S 1 being chosen to satisfy this condition, let us describe 



an ellipse with S and 8' for foci and r£ = i(SP + PS') = -SS' 



for <emi-major axi-. and let it revolve about the axis SS' de- 

 scribing a prolate ellipsoid of revolution. Then all the electric 

 forces arising from the change in the magnetic force taking- 

 place on the surface of the ellipsoid affect the charged sphere 

 as it passes S'. And if S" be a further point in the axis 

 distant udt from S', we may describe in like manner another 

 ellipsoid of revolution having S and S" for foci, and v(t-\-dt) 

 for semi-major axis. Then, since v>u, the two ellipsoids 

 cannot intersect each other. And all the electric forces arising 

 from changes in the magnetic force at points in the ellipsoidal 

 shell between the two ellipsoids affect the charged sphere in 

 its course between S' and 8", that is in the limit at S'. And 

 do other of the electric forces considered affects it at that 

 point or between S' and S". 



7. If P bo any point on the ellipsoid and PX the per- 

 pendicular from P on the axis, PX =rsin 6. And if wo describe 

 a circh' with X as centre and radius NP, the magnetic force 



