II
57
=
INDUCTION IN ROTATING SPHERES
Ωμ
2πα ° ' +
Σπα
-
=
k
მუ
k
an
Ω.
Q_(-5)= −2+(5)
1
= Ω
2π
+ ·
Second
form of the
solution.
This series leads to as accurate a result as we please, if
only it be carried far enough; in fact it is only the develop-
ment of the result in ascending powers of 2πa/k, as we may
show in the following way.
In the spherical shell the part of N, corresponding to
X(-n-1 may be represented in the form (p. 52)
d
1 (½ 2 x + x²x).
Ωρ
=
1+h² i dw
h= =
4Rwi
(2n+1)k
If we again introduce the substitutions to be made in the
case of a plane plate, develop
1
=
1-h²+h-h® + ...,
1+h2
and put for h its value
2πα τ
k n
we get
Ω
2πα 1 δχτε
2πα
+
k
n an
k
1 3xx - (2x) = x + (2x) =
Xrs
k ns an
θη
2πα 4 με
+
k
from which development the preceding one follows when we
use the relations
Xrs d5 = Xrs,
22 Xrs
- p² Xrs ›
an²
n
and sum for all values of r and s.
L