II
89
INDUCTION IN ROTATING SPHERES
V ===
W =
-
A
2R
A.
R
{sin (i+1)w. Pri+1
cos ia. Pri
-
-
− (n + i)(n − i + 1). sin (i-1)w. Pai-1},
Thus u, v, w are developed in terms of spherical harmonics.
We suppose the u, v, w expanded in a similar form for all the
terms, and then form the expression
f(x²+
u² + v²+w²)ds.
On integration, terms involving products of spherical harmonics
of different orders vanish, so that we may determine W, sepa-
rately for each, and add the results. A closer consideration
then shows that we may also determine the heat separately
for each and again add the results. It is true that all
the integrals do not vanish which correspond to combinations
of different ni's; but those integrals in 2ds for which this
Jwds
occurs will be destroyed by corresponding ones in
We now get for the
above quoted
Wnik (u²+v² + w²)ds
=
f
kA2
v2ds.
=
4R2
(P,
Pn‚i+1)²ds
+ (n + i)² (n − i + 1)² | (Pa,i-1)²ds +41² (cos iwP¡)²ds
-
- /
eds},
which gives, by well-known formula and simple reductions
W k¸² 2πn(n+1)(n − i + 1)(n − i + 2) . . . (n + i)
=
ni
2n+1
= kA²(n,i),
-
-
where (n,i) is an easily intelligible abbreviation.
Since we
have further
W = ΣΣW
ni,
n i