II
71
INDUCTION IN ROTATING SPHERES
Φισ = 1+
1
S
(20+ 1)puri ( [ampa, da+ forvraga, da },
(2n+1)2n+1
1
φοσ =
(2n+1)2n+1
These give by differentiation
-2n
α
+(-2nd
do
d
do
2n
S
24₁a. da +2n+1
φα.
-Jornada. da}
(02n+14;)) = -
-
σφα
2 (0-2 d² (+14₂)) = $1.
do
The form of the functions 1, P, depends only on n; in
the constant of integration μ, s, S are involved.
The above equations may be written
2n+2
"1+ Φ.
σ
= -
- Φ2,
2n+2
"2+
φ', = Φι
σ
As differential equations these are exactly equivalent to
the following:-
$₂ = ±$1√-1,
2n+2
-'₁±1√√1 = 0.
For all solutions of the latter system satisfy the former,
and the general solution of the latter involves 2 x 2 arbitrary
constants, and is therefore also the general solution of the
former system.
λ
Let us put = -1, where λ is that root whose real part
is positive, so that
~= √5. (1 + √ − 1),
λ = √†. (1 - √ − 1).
-