48
II
INDUCTION IN ROTATING SPHERES
Construc-
tion of the
lines of
flow.
u =
v =
W =
1
.
W
n + 1 x dwx
1 W
n + 1 x dw₂
1 W ΟΧ
· -·
n+ 1 x
and the current-function is
ป
=
P W
-
n+ 1 x
მთ.
· X'n⋅
2. We analyse Xn further and consider the term
=
Xni Anil
()
n
cos iwPni
·
R
To this belongs the current-function
Transfor-
mation of
the solu-
tion.
Yni
=
P wi
Ani
R
n + 1 K
n
sin iwPni:
Hence we get the following simple construction for the lines
of flow due to such a simple potential :—
Construct on any spherical sheet the equipotential lines
and turn the sheet through an angle π/2i; the lines now
represent the lines of flow produced by that potential.
For instance, when the sphere is rotating under the action
of a constant force perpendicular to the axis of rotation, the
external potential satisfies the required conditions and we
have n =
1, ¿ = 1. The equipotential lines on the sphere are
circles, and so also are the lines of flow. The planes of the
former are parallel to the axis of rotation and perpendicular to
the direction of the force, so that the planes of the latter are
parallel both to the axis of rotation and the direction of the
force.
3. We may give to a form which permits of summation
for all the spherical harmonics and makes the development of
the external potential in a series of them unnecessary.
Let n be positive, then
P
Xndp
P
=
n+1
1Xn⋅