362
THE
RADIATION
PROBLEM
Proceeding
from
this
definition,
one can
show
that the
entropy S must
[21]
satisfy the
equation
S
=
jjj
lg ¥
+
const.,
where
the
constant
is
the
same
for all
states
of the
same
energy.
b)
Neither
Mr.
Boltzmann
nor
Mr.
Planck
gave
a
definition
of
W.
They
put purely formally
W =
number
of
complexions
of the
state
under
consideration.
If
one now
demands
that
these
complexions
be
equally probable,
where
the
probability of
the
complexion
is defined
in
the
same
way
that
we
have
defined
the
probability
of the
state under
(a),
one
will obtain
precisely
the defini-
tion for the
probability of
a
state
given
under
(a);
however,
the
logically
unnecessary
element
"complexion"
has been used in
the definition.
Even
though
the indicated relation
between
S
and
W
is valid
only
if
the
probability of
a
complexion
is defined in the
manner
indicated
or
in
a
manner
equivalent
to
it, neither
Mr. Boltzmann
nor
Mr.
Planck
has
defined the
probability
of
a
complexion. But Mr.
Boltzmann
did
clearly
realize that the
molecular-theoretical
picture he had chosen
dictated his choice
of
complexions
in
a
quite
definite
manner;
he
discussed this
on
pages
404
and
405
of
his
paper
"Uber
die
Beziehung..."
["On
the
relation..."]
that
appeared
in the
Wiener
Sitzungsberichte
in 1877.1 Similarly,
Mr.
Planck
would
have had
no
freedom
in the choice
of
complexions
in the
resonator theory
of
radiation.
He
could
have been
permitted to
postulate
the
pair of
equations
S
=
J
lg
V
and
W
=
number
of
complexions
only
if
he had
appended
the
condition that the
complexions must
be chosen such
that
in
the theoretical
model
chosen
by
him
they had been found
to be equally
probable
on
the basis
of
statistical considerations.
In
this
way
he
would
[25] have
arrived
at
the formula
defended
by
Jeans.
Though
every
physicist
must
[22]
[23]
[24]
1Cf. L. Boltzman,
Vorlesungen
über
Gastheorie
(Lectures
on
the
theory
of
gases), Vol.
I,
p.
40,
lines
9-23.