II
73
INDUCTION IN ROTATING SPHERES
d
do
[-
l
-2n
do
+1
= λ(1 − v²)" (oλv²+2(n+1)v)e^ººdv
-1
= λέση, (λσ).
The last equation shows that P is a solution of the
equation under discussion.
and
The preceding equations when integrated give
λσ)
2n
Sop (λoxlo = 2 P-1(10),
[ otp₁-1(10)do = 10²+1p (A0),
0²"Pn-1(10)do
2n
whence by differentiating we get these recurring formulæ
Pn(λo) = 2n p'n-1(λo)
2n+1
λ
Pa-1(Ao) = 2n + ¹p¸(\o) +2=p' (xo),
λσ
2n
Pn(λo)+
2n
whence
2n+1
Pn-1
=
2
Pn
=
-P₁₂+
λ202
2n 4n(n + 1)Pn+1 ›
-
-
x² z{2n(n − 1)Pn-2 − n(2n − 1)Pn-1}.
Exactly similar calculations may be performed for the q's.
The results are got by replacing p by q; hence
o¶n(λo)do =
2n
for (10)do - 21 12-1(10),
(λo)do
1
=
¹In(λo), etc.
2n
With the use of these formulæ it is easy to perform the
necessary integrations; for instance, we get by help of an
integration by parts
1 See footnote on p. 72.
Integral
and recurr-
ing for-
mulæ for
p and q.