DOC.
8
ANALYSIS OF
A
RESONATOR'S
MOTION
227
p
(2U7IO
2
JUI
/
\
~dx
~
^\~w
-
a
cos
r
(...)
+
onsin-^-(...)\/2'ii/i•i
r
/
2rcrc
(...)
-&:sinf^(...)
2
7172
+
fl"COS
r
But
the
quantities
an
+ an,
an
-
an...
are mutually
independent
and
of the
same type
as
the
quantities
denoted
by
s
in
the
preceding paper;
for such
quantities
it has
been
shown
that the
probability
for
a
combination
is
represented
as
the
product
of the
Gaussian
error
functions
of the
individual
quantities.
It
is
easy
to
see
from
what has
been
said
that
no
probability
relation of
any
kind
can
exist
between the
coefficients
of
the
expansions
of
tx
and
d&/dx.
Now
we
write down
tz
and
dt/cbc
in
the
form
of
Fourier
series:
[17]
ex
=
J
bncos
|2itnl
-
Ü
j,
5-
E" C.oos
(
t
r£
We then
get
f
3c3
rrq
D
t
-
/
=
t1
^
"
n
bcos
n3
\2iui-
[Tn
-
j
ft
-
v
Id«3
[18]
and
=
3c3
16*3
f
«fr
y^m
yr
c
b j
0
\2%{n
+
m)L
-
- yn
r
jl-j
/
^
m
n
sinYn
cos
n
-
cos{2n(n
-
m)t
÷
-
-
Integration over
t yields two summands with
the
factors 1/n+m and
1/n-m;
since n
and m
are
very large numbers,
the
first
summand
is
very small and can be neglected.
Thus one
arrives at
the
expression
(11)
/
=
-
3c3
t*
yr d
c b
smy"
Z-/
z-j
m rt
1
32*
n
n
-
m
cos5
sin7t(/2
-
m)~,j?7
mn
v
where,
for
brevity,
5
=
n(n
-
tri)-
+
$
-ft
-
y
mn v
'
rj-i
n
1 n
[19]