178
DOC.
14
PROOF OF
AMPERE'S
CURRENTS
701
cos
co
t
=
1
and for the
second
cos
co
t
=
-
1
so
that
we find,
using
(8)
4JUt
(11)
S€
Instead of
(6)
we
now get
the
equation
Bx
cos
tvt
=Qa+
Sa
-f-
(12)
the
periodic
solution
of which is
a =
-cos
(tot-
v),
(13)
u
if
the constants
u
and
v are
determined
by
u
COS
v
=
(co/-CO*)
Q
u
sin
v =
2
xco Q
(14)
Here
the
quantity u,
to
which
we
shall
give
the
positive
sign,
determines the
amplitude
whereas the
phase
of the oscillations is
given by
the
angle
v.
For the
amplitude,
which
we
shall denote
by
|a|,
we
find
|o(
= -=
U1'
....
(15)
CO
For
co
=
w0
it becomes
a
maximum
|a|m,
viz.
|a|m
=
2A
(16)
As to
the
phase,
we
first
remark
that
according to
(14)
v
=
x/2
for
w
=
w0.
If the
frequency
of the
alternating
current
is
higher
than that
of
the
cylinder,
we
have
v
x/2
and
in
the
opposite
case
v
x/2.
When
w
is made
to
differ
more
and
more
from
w0,
the
phase v approaches
the value
n
in the
first
case
and
0
in the
second.
If
the constant
of
damping x
is
small
we
may say
that
these
limiting
values
will
be
reached
at
rather
small
distances from
w0
already.
In
our
experiments
this
was really
the
case
and
we
may
therefore
say,
excepting
only
values
of
w
in
the
immediate
neigh-
bourhood of
w0
that
v
= n
for
w
w0
and
v =
0 for
w
w0.
Taking
into account what has
been
said about the
positive
direction
one
will
easily
see that,
if
the
current
i
and the deviation
a
had
the
same
phase,
the
cylinder
would at
every
moment
be deviated
in
the
direction
the current
in
the
coil
has
just
then.
In
reality
the