676 
MR. G. F. C. SEARLE ON PROBLEMS IN ELECTRIC CONVECTION, 
nature. Professor J. J. Thomson was, I believe, the first to consider a problem of 
this sort, his paper* giving the solution for the motion of a single electrical point- 
charge at a speed small compared with that of light. Mr. Oliver Heaviside next 
considered the question, and in his papert obtained an exact solution for the motion of 
a point-charge. He got the solution by means of the “ vector potential ” of the 
convection current formed by the moving charge. The mathematical analysis is 
of the symbolical kind, but it is shown that the result obtained by means of it 
satisfies all the necessary conditions. By integrations Mr. Heaviside obtains 
solutions for the motion of some simple cases of electrical distribution. Mr. Heavi¬ 
side’s expression for the vector potential is 
A = 
47T/3U 
which he re-writes in the form 
f/rf ~ V 2 
— 47rpu/V' 
A- 5 
1 _ PL 
r 2 V 2 
where A is the vector symbol for the vector potential, u the vector symbol for the 
velocity of the electricity, whose volume density is p, and v is the velocity of light, 
while i~r — u 2 d 2 Jdz 2 and v 3 = d 2 /dx 2 -j- cl 2 jdy 3 + cl 2 /dz 3 ; the motion is supposed to 
take place parallel to the axis of z. 
Mr. Heaviside performs the operation 1/v 3 first, and obtains 
so that 
The operation here indicated is then performed for the special case in which 
A 0 = ^u/r, corresponding to the motion of a single point-charge, and a correct value 
of A is obtained. 
[August 20, 1896.— But except in this simple case, there appears to be some 
difficulty in the interpretation of the two operations — and -I 1 — 
0_0 
-i 
when thev 
are performed separately. For if the operations are performed separately and in 
Mr. Heaviside’s order for a uniformly charged sphere of radius a, the result is the 
same as for a point-charge at its centre, since A 0 varies simply as the reciprocal of r 
* ‘ Phil. Mag.,’ April, 1881. 
t ‘ Phil. Mag.,’ April, 1889, or ‘ Electrical Papers,’ vol. 2, p, 504, 
