MR. G. F. C. SEARLE ON PROBLEMS IN ELECTRIC CONVECTION. 
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in both cases. This result is not correct, for it can be shown* * * § that it is not a point- 
charge, but a uniformly charged line of length 2 au/v, which produces the same effect 
as the uniformly charged sphere.] 
Professor J. J. Thomson has also obtained the exact solution for a point-charge in 
two different ways. In his first treatment! he adopts Maxwell’s equations 
involving the Vector Potential, and an electrostatic potential ML In his last paperJ 
he finds the solution by the aid of his novel method of considering the phenomena of 
the electromagnetic field as being brought about by the motion of tubes of electric 
force. This paper may be considered as an attempt to take a step beyond Maxwell’s 
analytical theory, and to give a sort of material representation of the mechanism of 
the electromagnetic field. 
The result of all these investigations is that while the electric force due to a moving 
point-charge is still radial , the intensity of the force, for a given distance from the 
charge, gradually increases as the radius vector turns from the direction of motion to 
a perpendicular direction. There is also a distribution of magnetic force, in which 
the lines of force are circles centred on the axis of motion, the planes of the circles 
being perpendicular thereto. 
The fact that the electric force is radial led Mr. Heaviside to form the conclusion 
that the expression for the electric force due to a point-charge is the same as that due 
to a charged sphere in motion carrying an equal charge, the distribution on the 
sphere being such that cr = KE„/47t, where E„ is the electric force normal to the sur¬ 
face which would be due to the point-charge placed at the centre of the sphere. But 
the surface which gives rise to a field the same as that due to a point-charge is tin 
ellipsoid of revolution, whose minor axis, which is also the axis of figure, lies along 
the direction of motion, and whose axes are in the ratios 1 : 1 : (1 — u 2 /v 2 )% where u is 
the velocity of the point and v is the velocity of light through the dielectric.* The 
charge is distributed in the same way as if the ellipsoid were statically charged, i.e., 
the surface density is proportioned to the perpendicular from the centre on the 
tangent plane. This surface I call the “ Heaviside ” ellipsoid. 
Mr. Heaviside appears to have thought that if there is no disturbance within a 
closed surface, then the surface condition is that the electric force just outside the 
surface should be normal to the surface. As this led to the supposed equivalence of 
the sphere and the point, and as I convinced myself that this equivalence does not 
exist, I asked Mr. Heaviside about the matter. This led him to reconsider the 
conditions which obtain at a surface bounding a region of zero disturbance, and he 
showed§ that it is not the electric force which is perpendicular to the surface, but a 
certain vector F. This vector F, I have shown, is simply the mechanical force 
* This can he readily shown by the use of the auxiliary coordinates f, >/, f of § 16 below. 
f ‘ Phil. Mag.,’ July, 1889. 
X 1 Phil. Mag.,’ March, 1891, and ‘ Recent Researches in El. and Mag.,’ p. 16. 
§ ‘ Electrical Papers,” yol. 2, p. 514, and ‘ Electromagnetic Theory,’ vol. 1, p. 273. 
