of an Electrified Ellipsoid. 335 



moving charges. When they move parallel to each other 

 with the speed of light the force between them vanishes. 



If the ellipsoid is more oblate than Heaviside's the limiting 

 internal surface of ellipsoidal form, whose action is the same 

 as that of the ellipsoid, is a disk of radius v5 2 -a 2 /a, the 

 axis of the disk coinciding with the axis of x. 



The form of the lines of the electric force E due to an 

 ellipsoid of revolution is easily found. Putting p 2 for y 2 + z 2 , 

 the equilibrium surfaces are given by 



P _ 



-^+^ = 1 (13; 



a- -f aX b" + X v 



Now the mechanical force P is normal to this surface, and 

 therefore 





F p p(a? + x\) 

 F,~ x(b 2 -i-\) ? 





where 



F P 2 = F 2 2 + F 3 2 . 





But by (5), 

 so that 



E, = Fj and E„ = 



E p _ 1 p(a 2 + a\) 

 E x a. x{b 2 +\) 



*> 







Now consider 



the conic 





(14) 



-^-+7^=1 (15) 



a- + a.v b- + v ' 



The tangent of the angle which the geometrical tangent 

 makes with the axis of x is 



*(* + *) nn 



P {a 2 + xv) [Lb) 



But if the point x, p lies on both (13) and (15), it follows 

 that 



x(b 2 + v) __ l p(a 2 + x\) 



p{a? + uv)~ a x(b' z + X) ' 



Hence by (14) and (16) the electric force is always tangential 

 to the conic (15). But this conic has exactly the same 

 equation as the equilibrium surfaces. Thus the single 

 equation (13) represents both the equilibrium surfaces and 

 the lines of electric force. 



If any point x, p be taken, there are two values of X which 

 will satisfy (13) considered as a quadratic in X. One value 



