84
II
INDUCTION IN ROTATING SPHERES
On differentiation these become
or
30
2n-1
=
2n
-
σφα
= σφι
d
do
o2n+2 d
do
o)-
d
o2n+2
d
do
.0).
do
2n
4"-274
-
Φι
2n
= -
Φα
P2" -P2' = $1.
If we put =2n+1, we get for the equations
+
2n+2-
Φι
σ
2n+2.
$
Þ₂" +
σ
= -
=
$2
= $1,
which equations again lead to the p's and q's.
The constants may be determined in the same way as
before, and then we find
-
=
-
2n+1
2(n+1)(2n+1)¸2n+3
-
-
In-1(XS)In( − λo) – In-1( − λS)In(λo)
In-1(AS)In+1(As) - In-1 - AS)In+1(As)
-
The formula becomes especially simple for the case where
S∞, that is, when we have to deal with a spherical hollow
in an infinitely extended mass.
Then In-1(AS) = 0, and we get
-
=
-
- 2(n + 1)(2n+1)02n+
=
s2n+3
In (λo)
In+1(As)
If we may neglect upσ in comparison with unity, on
account of the small value of w, we may put for the q's their
values for small arguments (p. 77). We then obtain
-
$1 √ − 1 − $₂ = X²e ¯ Mo − s),