II
39
INDUCTION IN ROTATING SPHERES
Xn
ical harmonics, then X, is to denote the term involving p" as a
factor, and this notation is, unless specially excepted, to apply
to n negative.
Xn
In the further analysis of X let this notation be used:-
for positive n
for negative n
n
Xn = p²Y₂
Xn=p"Y-n-1
Y₂ = ¿¡(Ani
Σi(Ani cos iw + Bni sin iw)Pri(0);
for every n these equations hold:-
v²X₂ = 0,
Xxn
X-
дх
dx~+Y dy
dxn+xəz
=
nxn
Xn
The mth differential coefficients with regard to x, y, z of
are spherical harmonics of order n -m, unless a preceding
one should be of order zero. The expressions
Xn
>
მა
Xn Xn
მა
Tw₂
are spherical harmonics of order n.
Further
▼²(pmYn) = (m − n)(m + n + 1)pm – ²Yn
-
d(pmYn) a(pmYn)
+2
dy
m
+y-
дх
a(pmYn)
др
小
​¥ds
=
m
Əz
·(pmYn).
P
=
mpmyn
5. Let be the current-function of a spherical shell of Theorems
radius R, and let Y =
on the flow
ds be the potential of a mass of in spherical
matter distributed over the shell with density ; then the
potential of the current sheet is
shells.
1 a
Ω = -
(Yp),
Rap