74
II
INDUCTION IN ROTATING SPHERES
2n
=
22
S
a²n+²p₂(λa)da + o²n+1 Jap (xa)da
2n
= 27-o²+1pm-1 (AS) - 24 s²++¹p₂-1(28) + 2n+1st+¹p (A6)
Pn-1(AS)
2n
=
。²n+¹Pn−1(XS) -
–
λ
-
λε
82n+3
2(n + 1) P₂+1(As).
2
_2n+1 2+1 p₂ (X)
እያ
2n+1
እ።
_o2n+1p₁(λo).
-
When we substitute these and the similar expressions for q
in the equations, and remember that = x², and λ= -1,
the p's and q's cancel as they must do, and we are left with
equations of the form
0 = (const), +
(const),
2n+1
ρ
which are the solutions of the equation
d
_d [- 2 (~²+1)] = 0
do
-2n
σ
do
-(2n+1)
»>]=0.
The constants occurring here must vanish separately;
remembering that
1
x²
=
-
equations for A, B, C, D
2n+1
2n
=
=
1
λ
we thus obtain the following four
APn-1(S)+BP-1(λ₂S) + C£n-1(S)+D¶n-1(λS),
-
· AP₂-1(S) — BPn-1(^₂S) + C£n-1(λ,S) — D£n-1(λ₂S),
0 = APn+1(λ₁8) + BP₂+1(8) + C£n+1(8) + D¶n+1(λ₂8),
-
-
0 = APn+1(8) — BPn+1(28) + C£n+1(8) — D£n+1(№8).
These equations are easily solved and give
A =
2n+1
4n
C =
-
In+1(M8)
Pn-1(S)In+1(8) − Pn+1(M³)In-1(MS)
2n+1
Pn+1(8)
4n 'Pn−1(~S)£n+1(~18) − Pn+1(µ³)In−1(S)
>