MR. G. F. 0. SEARLE ON PROBLEMS IN ELECTRIC CONVECTION. 
687 
H = 
qu sin 6 (1 — w 2 /« 2 ) 
r" {1 — sin 2 
(47). 
From (46) we see that the electric force varies inversely as the square of the 
distance for any given direction, but that for any given distance it gradually increases 
as 6 increases from 0 to ^rr. As the speed increases the electric force tends to become 
more and more concentrated about the plane through the origin at right angles to the 
axis of x. When u — v, there is no electric force except in that plane ; we have, in 
fact, a plane electric wave moving forward at the speed of light. 
The expression for H shows that at low speeds, where u 2 fv z may be neglected in 
comparison with unity, the magnetic force is the same as that attributed by Ampere’s 
formula to a current element of “ moment ” uq. By the moment of an element is 
meant the product of its length by the strength of the current in it. When the 
speed of light is attained, the magnetic force is confined to the plane yz, and the lines 
of force are circles in that plane with their common centre at the origin. 
Mr. Heaviside has stated* the result when u is greater than v, but has not up to 
the present (March 14, 1896) divulged the manner in which he has obtained the 
solution in this case. I confine this paper to the case in which u is not greater than v. 
As the charge moves along, the electric displacement at each point varies, giving 
rise to a current, and I shall now investigate the form of the current lines in the case 
under consideration. The currents evidently flow in planes drawn through the axis 
of x, so that it will be sufficient to find the form of the current lines in the plane xy. 
The x and y components of the current at any point are and ^ > or > since 
, . . , K u dE, , K u dE 2 
the motion is steady, — — — and — 
current lines, we have 
47r dx 
47 t dt 47r dt 
Hence, if dyjdx refer to one of the 
dy dE 2 j dEj 
dx dx I dx 
Performing the differentiations, we find that for points in the plane xy 
the solution of which is 
or in polar coordinates 
dy 3 xy 
dx 2x 2 — ay 1 
cy 2 = ( X 2 + ay 2 )f . . . 
c sin 2 6 
u* 
1 — sin 2 # 
(48) , 
(49) , 
(50) . 
The form of the lines of flow is given by equation (49) or (50). 
* ‘ Electrical Papers,’ vol. 2, p. 516. 
