528 
PHYSICS: H. BATEMAN 
Proc. N. a. S. 
07V A DIFFERENTIAL EQUATION OCCURRING IN PAGE'S 
THEORY OF ELECTROMAGNETISM 
By H. Bateman 
California Institute of Technology, Pasadena 
Communicated by R. A. Millikan, June 7, 1920 
In a recent article Mr. Leigh Page^ has generaUzed the electromagnetic 
equations by introducing the idea of a rotation of the field round a moving 
electric pole. A line of electric force is supposed to be the locus of points 
traveling with constant velocity, c, in directions which vary according to 
a law giving rise to three differential equations of type 
dl 
k J- = c{f -\- vr ~ wq) - m{ch — gu + fv) + n{c'^q — fw hu) 
(JT 
- cl[l(f + vr — wq) + m{g -\- wp - ur) + n{h -\- uq - vp)], (1) 
where (c/, cm, cn) are the component velocities of the point which leaves 
the electric pole at time r ; {u, v, w), (/, g, h) and {p, q, r) are the components 
of the velocit}^ acceleration and angular velocity of the pole at this in- 
stant. The quantity, k, is defined by the equation k = c"^ — — w'^. 
These equations are of a type considered in my book on Differential 
Equations (Ex. 7, p. 71) and can be reduced to two Riccatian equations 
by means of the substitutions 
I -\- im I — im , , , 
1 -\- n 1 + 7z 
The lines of force can consequently be found by solving a Riccatian equa- 
tion just as in the usual electromagnetic theory. ^ 
It should be noticed that Page's expressions for the electric and magnetic 
vectors may be written in the forms 
1 d(T, a) b(r, d) 
cb{x,t) d{y,z)' 
(3) 
where 
b{y,z) c b{x,ty 
e = ^ - ^)p + {y - 'n)q + (g - 
4:7rk c'^(t — t) - m{x - ^) — v{y — 7}) — w{z — ^) 
^ = A - ^)/ + - 7/)g {z-^)h-\-k 
4:Tr c^{t - r) — u{x — ^) - v{y — t?) - w{z — ^) 
e is the electric charge associated with the moving electric pole, rj, 
are the coordinates of the pole at time r and r is defined in terms of {x, y, 
z, t) by the equation 
Ix - ^{r)Y +[y- vir)? + 1^ - tir)V = c^H - r^. r < t. 
