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PHYSICS: H. BATEMAN 
THE STRUCTURE OF AN ELECTROMAGNETIC FIELD 
By H. Bateman 
Throop College op Technology, Pasadena 
Communicated by A. A. Noyes, March 18, 1918 
1. Starting with the fundamental equations 
dE 
c rot H = — + pv, div E = p, 
at 
>vTT 
c rot E = - — , div H = 0, 
Of 
we adopt as an elementary solution 
_ 1 bA Ail b<I> 12>f 
H = rot A, E = — - — — V3>, ^ = div A + - — , p = — - — , pv = cV^, 
' c bt c bt c bt 
where 
Jot 
f(r) V log [s.r - c(t- r)] rfr, 
$ = ~ JjTW I log [S.r - C (t - r)] dr. 
In these expressions s is a unit vector depending on r, r denotes the radius 
vector from the point with co-ordinates £(t), r/(r) £* (t), to the point with co- 
ordinates x, y, z, while a is defined by the equation 
[x - £ (a)] 2 +[y- V (a)] 2 + [x - { (a)] 2 = c 2 (/ - a) 2 ; a ^ /. 
The function Sl> is given by the formula 
• *--/<«), 
where 
» = f(a) [x - f (a)] + ,'(a) [y - ,"(«)] + f (a) [s - - < 2 (* - «)• 
This elementary field corresponds to a state of affairs in which electric 
charges of a concentrated form are created and travel along straight lines with 
the velocity of light, the directions of these lines being specified by the different 
values of the unit vector s. Whenever a concentrated electric charge is cre- 
ated an amount of electricity which will just compensate it is fired out in all 
directions and provides an elementary 'aether' which is the seat of the electro- 
magnetic field of the concentrated charge. A concentrated electric charge 
and its elementary aether lie at any instant on a sphere whose centre is at the 
point where these charges originated 1 ; if now this point moves with a velocity 
