212
DOC.
7
PROBABILITY CALCULUS
independently
of
each
other;
for since
there
is
no
limit to
the
distance
of
this surface
from
the
test
point,
there
is
no
limit to its
total extension either. Into these radiation
elements
arising
from the
individual surface
elements
we again
introduce
a
higher
principle
of order
in
that
we
conceive
of
all
of these radiation elements
as having
the
same
form and
differing
only
in
their
temporal
phases; or,
in
mathematical
terms:
The
coefficients
of the Fourier series that
represent
the radiation of the
individual surface
elements
shall
be the
same
for
all
the
surface
elements,
and
only
the
initial times shall
differ
from
element
to
element. If
equation (1) can
be
proven
on
the
basis
of these
principles
of
order,
then
it will
hold
a
fortiori
in
the
case
where these
principles
have
been
dropped.
If
the index
s
denotes the
individual surface
element,
then the radiation
emitted there
will
be of the
form
E(n)
a
sin 2itn
n T
Hence,
the total radiation
we are
considering
will
be
represented
by
the double
sums
(2)
£ £
ün
\
ft
t
t
sin
2itn
cos
2itn


cos
2itn

sin
2itn

T T T
Comparison
of
(2)
and
(1)
leads thus to
the
expressions
(3)
An
=
a
p
cos
2icn,
n JLJ
j1
Ei
n
sin
t
rj*7
where
n
is
a
very large
number,
and
ts
can assume any
value
between
0
and
T,
so
that the
individual summands
t
cos
2%n

T
t
and
sin
2tcw
 T
are
randomly
distributed between 1 and
+1,
and
are as
likely
to be
positive
as
negative.
If
we can prove
the
general
validity
of
our equation (1)
for
a
combination of
sums
of
such
quantities,
we
will also have
proved thereby
the
impossibility
of
introducing
any
order
principle
into the radiation
propagated
in
empty
space.
§2.
Formulation of
the
General Mathematical Problem
We thus
set ourselves
the
following
mathematical
problem:
We
are
given
a
very large
number of elements
whose
numerical
values
a
(corresponding
to
ts)
follow
a
known
statistical law.
From each of
these
numerical values
we
build certain
functions
[5]
[6]