II
55
INDUCTION IN ROTATING SPHERES
Thus
R\n+1
cos iw
P
become
ni
€¹², cos rn.
Pri (0) must become such a function of that its product
by e-"5 cos rn may satisfy the equation V2 = 0. Such a
function is cos s or sin se, provided
form
n² = p² + s².
Hence the spherical harmonics formerly used now take the
and related forms.
-
Arse cos rn cos st,
The external potential x must be represented as a series
of such forms. This is to be done by means of Fourier's
integrals.
For every term (element) of the development the solution
is at once obtained from those found before. We now put
tan &
=
·
2πη α
10'
n
where a denotes the velocity of the plate, and find
Ω
+A,, "sin & sin (rn - 8) cos s,
=
=
↓ =
- Ae" sin & sin (rn - 8) cos sĘ,
1
2π
A,, sin & sin (rn - 8) cos s§.
By summation of all the terms we obtain the general
solution of the problem. The summation may be performed
in the case when a/k becomes infinite. Then we have
π
=L
8
-2'
sin 8 = 1,
therefore
ประ
=
=
-
X,
1
2π
--
Solution.