ponent velocities, -4re its electric charge, and M a quantity defined by the equations

M = {x— §(7)]§'(T)+[y−N(T)]N'(r)+[2−5(1)]5′(r) — c2(t−7),
=r(v2-c),

r2 = [x— §(r)]2+[y−n(r)]2+[z− Š(7)]2 = c2(t − r)2.

(22)

(23)

The last equation, combined with the inequality r<t, associates a time T with each space-time point (x, y, z, t). If the velocity v is always less than c the time associated with (x, y, z, t) is unique and increases as t increases if (x, y, z) remains stationary or moves with a velocity less than c. The point [(7), n(7), 5(7)] is called the effective position 16 of the pole for (x, y, z, t). The scalar potential y is now

and is connected with the potentials A and by the relation1

The potentials y, A and , moreover, satisfy the equations

Defining the field vectors E and H by means of the equations

(24)

(25)

(26)

(27)

(28)

where

81 and & are the two real unit vectors which satisfy the relations

E+(8×H)=8(8. E), H-(8XE) = 8(8. H)
(8XE) (8xH):

(8XE). (8×H)=0,

The vector 8 is in the direction of the radius from the effective position of the electric pole. The vector & does not generally admit of a simple geometrical interpretation, but when the pole moves with uniform velocity along a straight line, & is in the direction of the radius to that position of the pole which could be hit by a bullet travelling from (x, y, z, t) along a straight line with velocity c.

The field vectors may also be expressed in Hargreaves' form19

These expressions are useful for many purposes. It may be remarked in passing that

The field vectors may also be expressed in the form

where a and ẞ are certain functions of x, y, z and t. The equations a=const, ẞ=const may be regarded as those of a moving line of electric force.20 In the case of an electric pole moving in an arbitrary manner along the axis of x, we may write 21

απε

x-E-v(t-7)
́c2(t−7) — v(x — §)'

B=tan ̄y

(39)

The lines of electric force have been found in a number of other casesfor instance, in uniform circular motion, in uniform helical motion, in Born's hyperbolic motion and in motions that can be derived from these by means of transformations of certain types.

A line of electric force may be regarded as the locus of a series of points moving along straight lines with velocity c and projected from the moving pole at successive instants, the direction of projection varying according to a law which has been formulated by Leigh Page 22 and the author. If the unit vector s specifies the direction of projection at time, the components of s satisfy differential equations which are embodied in the vector equation

(c2 —v2)ds = (v—cs) (8 · v′)+v'[c— (s•v)]

dr

(40)

This equation may be replaced by a Riccation equation23 with the single complex dependent variable

Many interesting geometrical properties of the lines of force and the associated directions of projection have been given by Leigh Page,22 F. D. Murnaghan24 and the author.25

The rate of radiation of energy from a moving electric pole may be found by an extension of the method given by Liénard 13, use being now made of the more general tensor components defined in Section II. It is found that the rate of radiation is

the contribution of the electromagnetic radiation, as found by Liénard and Larmor15 being

the remaining part arising from the radiation which depends on the variation of the function Y.

In a periodic motion the total radiation of energy per period is zeroa result which is in accordance with the idea of non-radiating orbits that has been used so successfully in atomic theory. The total radiation may also be zero in a non-periodic orbit, if we integrate from apse to apse, an apse being defined as a point on the path where the velocity is a maximum or minimum.

V. THE REFLECTION OF LIGHT AT A MOVING PLANE MIRROR

Let the equation of the moving mirror be x=ut and let us consider in the first place the effect of the mirror on the field of a moving electric pole whose co-ordinates at time 7 are (§, 1, 5).

η,

We shall assume that the effect is the same as if an additional field were produced by an electric pole at the image of the moving pole, the image of (§, n, Š, 7) being supposed to be determined by the equations 26

and so furnish the ordinary laws of reflexion for points that move with the same velocity as the mirror. If (x*, y*, z*, t*) are derived from (x, y, z, t) by the same set of equations, we have

(x*— ¿*)2+(y*—n*)2+(≈*—¿*)2— c2(t*—7*) 2

= (x− )2+(y-n)2+(2-5)2 — c2(t-T)2

}

(46)

Placing a charge e at (§, n, Š, 7) and a charge −e at (§*, n*, 3*, 7*), it is easily verified that, with the notation of Section IV

d(r*, 0*) = −d(1, 0)

where the symbol d(7, σ) is used for an expression of type drdo-dodt,
d's and 's referring to independent increments.
The above equation may be written in the form

1*)

(47)

the

H*d(y*, z*) + H„d(x*, x*) + H2(x*, y*)+cE‡d(x*, t*)+cE„d(y*, +cE¿d(z*, t*) + HÅd(y, z) + H„d(z, x) + H¿d(x, y) + cЕ¿d(x, t) {(48) +cE,d(y, t) + cE,d(z, t) = 0.

At points of the moving mirror, however, we have x*=x, y*=y, 2*=2, t*=t, and the above equation becomes

H2d(y, z)+Ã ̧d(z, x)+H2d(x, y)+cĒ¿d(x, t)+cĒ,d(y,t)
+cE,d(z, t)=0

(49)

where Ē and H are the electric and magnetic forces in the total field. This is the condition to be satisfied at the surface of a perfect conductor or perfect reflector. We also have =0. These conditions hold for arbitrary fields, as may be seen by superposition of the fields of a number of moving electric poles.

We may deduce from equation (48) that

These equations give us the usual boundary conditions at the surface of a perfect conductor, viz.,

H‚=0, E‚=“H., E.+"H‚=0.

These equations and the equation y=0 indicate that in the total
field there is no flow of energy across the surface of the mirror.
We also have

Ezd(y*, 2*)+E«d(2*,x*)+Ed(x*,y*)—cH„d(x*, t*)—cH¶(y*, t*)
-cHd(2*, t*) = E„d(y, z)+E,d(z, x)+E ̧d(x, y)—cH,d(x, t)
-cH,d(y, t)-cH,d(z, t) (51)

or

d(a*, ß*)=d(a, ß).

(52)

This equation may be interpreted to mean that the lines of electric force of the pole (*, n*, *, *) are the images in the moving mirror of the lines of electric force of the moving pole (, n, 5, 7). It is important to notice that if the point (§, 7, 8, 7) moves with a velocity less than c, the point (§*, n*, ¿*, r*) does also. The image in the mirror of a stationary observer is an observer moving with velocity

in the direction of the axis of x. Referred to the stationary axes this observer will suffer from the Lorentz-Fitzgerald contraction

v2
1-
1.

as is easily seen from the equations.

All observations made by the moving observer may be regarded as images in the mirror of corresponding observation's made by the stationary observer and we may deduce the whole theory of the relativity transformation from reflections in moving mirrors.