EMISSION BY A THIN PLATE. 83

We shall begin by taking into account only the component of the
electric vibrations in a certain direction h perpendicular to OP.

Let us choose the point 0 as origin of coordinates, drawing the
axis of z along OP, that of x in the direction A, and denoting the
distance OP by r. According to what has been found in § 39,
a single electron, moving with the velocity V in the part of the plate
considered, will produce at P a dielectric displacement whose first

component is given by

e dvx
~ 4«c*r dt '

if we take the value of the differential coefficient for the proper
instant.

On account of our assumption as to the thickness of the plate,
this instant may be represented for all the electrons in the portion &<d

**

by t------, if t is the time for which we wish to determine the state

j c )

of things at the point P. We may therefore write for the first
component of the dielectric displacement at P

dv.

The flow of energy through a> per unit of time will be

Since the motion of the electrons between the metallic atoms is
highly irregular, we shall have, at rapidly succeeding instants, a large
number of impacts in which the changes of the velocity are widely
different. The state at P, which is due to all these impacts, will
show the same irregularity. Nevertheless, we must try to deduce
from the formulae relating to it, results concerning those quantities
that can make themselves felt in actual experiments.

Results of this kind are obtained by considering the mean values
of the variable quantities calculated for a sufficiently long lapse of
time. We shall suppose this time to extend from t ~ 0 io t •= 4K
If the mean value of d* is denoted by d|, we shall have for the flow
of energy through «' that is accessible to our means of observation

' (135)

64. The introduction of this long time # is also very useful
for the application of Fourier's theorem. Whatever be the way in
which dg changes from one instant to the next, we can always expand
it in a series by the formulaconsidered the radiation from the body J£