76
II.
INDUCTION IN ROTATING SPHERES
in a finite form.
Չ the
We may and will denote by p and
functions thus calculated. Then also q's with negative argu-
ments are admissible, and this equation holds
− Pn(p) = In(P)+9n( − p),
whence follows
Pn(p) = Pn( − p).
For let the indefinite integral
then we have
[(1 − v²)" e¯°dv=V(∞, ∞),
In(0) = V(∞, ∞) – V(0,1),
P.(0) = [(1 − v²)"e˜˜dv + [(1 − v²)"e"dv
=
-
-
- V( − σ,1) – V( − 0,0)
+V(0,1) – V(0,0).
But for integral values of n
The first
q's.
V(∞, ∞) = 0, V( − 0, 0) =
-
-
·V(0, 0),
and thus the statement made follows.
Hence the simplest integrals of the original equations
are
In(σ) and In( - ).
-
The following are the expressions for the first few q's;
the first one has been determined directly, the others by the
recurring formula.
=
Lo
2€
σ
2 - -2 = (1+1)
12
22 - 22 - 2
·
23.3
3
σ
1+
3
(1 + 2 + 3 ),
1+
6 15 15
+ +
G2 0-3
35
etc.
,