II
77
INDUCTION IN ROTATING SPHERES
Hence follow the equations
Po
σ
2
P1
=
€ +
2
etc.
,
σ
For large values of o p and q approach the values
€
-
In = ( − 2)". \n .
•
== +1
P„(0) = −¶n( − σ).
-
For very small values of σ we get
In(0)
=
-
( − 2)". n. 1. 3 . . . (2n − 1)
2n+1
σ
-
-
The equation -p(o) = q(o)+q( − σ) here has no longer any
meaning; for when σ = 0, q = ±∞. In order to find p for
very small values of σ also, we expand it in a series of ascend-
ing powers of σ. This is easily done by substituting for ev
its expansion in the integral representing p; integrating each
term we obtain
2(243)
P₁(0) =
2n+1. In
1.3...(2n+1)
1+
2(2n+3)
+
+....
2.4. (2n+3)(2n+5)
The following formula is of importance for our further
investigations. We have
In(O)In-₂( − o) – In( −σ)In-¸(0)
-
(n) = − 4x(n = 1) —
-
{In-1(0)2n-2( — 0) — 2n-1( — 0)2n-2(0)}
n.
In
N-
4
-
σ
which equality is easily demonstrated by means of the recur-
ring formula found for q.
In all the properties of p and q considered we notice
The p's and
q's for large
and small
values of
the argu-
ment.