DOC.
8
ANALYSIS
OF
A
RESONATOR'S MOTION
223
If
we
solve
(3)4
and take
into account
(4)
and
(5), we
obtain
V3
^ _
siny
/
=
-
TT3
V
Sill(PE"
An~-«»(*.
~
Y"
),
loTT
n
f
=
|^sinPEn
A^sin(T"
-
rj,
87r
AT
where
T
=
2ich
4
-
ö
AI
nn
AI7,
has
been assumed for the
sake
of
brevity,
and
yn
is
given by
the
equation
2
*v0
cotg y
=
2
n
r
v0
~
n3
o
-
r
Since,
further,
^1tcos2pcos(o5}' nylnsin2Tn,5
[11]
3z
ct
kx
appears as
the double
sum
kx
=
-
^-T2cos2psinpcoswV''
T/I
--yl
mcos(x
-
y
)sinc
x
o-.
T
iL/ Z^/
ai i An
v
ai
1 ai7
An
Ö7C
- ^-^sincpcoscoj}
Ari^lAjin(xn
-
Y")costm.
[12]
Because
the
phase
angles
ü
are
independent
of
each
other,
only
the
terms
n
=
m
need
be
considered
in
forming
the
average value,6
and
we
get7
4 M.
Planck,
loc.
cit.,
p.
114.
[10]
5
In
fact,
this
expression
for
dtjdz
as
well
as
the
one
for
y,
would have to be
supplemented
by
the
components
of the
wave
that
is
polarized
perpendicularly
to
the
wave
that
excites
the
oscillator; however,
it
is
obvious
that these
expressions
do not
contribute
anything
to
the
mean
value
of the
force because
their
phases are
independent
of those of the
oscillator.
6 This
independence
follows
from the result of the
preceding paper.
[13]
7
M.
Planck, l.c.,
p.
122.
[15]