66
II
INDUCTION IN ROTATING SPHERES
and at the boundary
аф
др
= -
(m+n+1) p" (p²
m Xn
Əz
-
nzXn
Now satisfies these conditions. For we have, firstly,
ə
v³¢ = − œ[ v³(p»+s³X« ) − 2 n = (p″xn) – nov"(p™xn)]
- -
-
az
--∞[pm
Jxn
Əz
{(m+2)(m+2n + 1) − 2n}
-
Deduc-
tions.
− nzp™-2X₂{m(m+2n+1)+2m}
-
m-
-
− − ∞{m(m + 2n + 3)p*-*(p?x« – n=x«n) + 2(n+1)p?x=},
=
– n²x»)
Əz
- -
so that the first condition is satisfied; secondly, is the pro-
duct of p+n+1 by a function of the angles, thus
Эф
a$ _m+n+¹p,
მ
ap
P
From this
so that the second condition also is satisfied.
correct value of the values of u', v', w' follow by the
original differential equations, at first in a more complicated
form. But the same form has already occurred on p. 43,
and has already been shown to be identical with the one given
above.
This theorem leads to the following propositions:-
1. In the theorem we may replace pm by a series of
powers of p, each power multiplied by an arbitrary constant,
that is, for pm we may substitute any arbitrary function of p.
And again, we may replace Xn by a series of spherical har-
monics of different degrees with arbitrary coefficients; for n is
without effect on the final result. Hence we get the following
generalisation of the theorem :-
If x is an arbitrary function, and if
U = ox
მა
V = x W
dwy
then the currents u', v', w' induced by U,
u'
=
-
wax v =
K dwz
ω
- @ax'
K dwy
=
го
ax
dw₂
V, W are
= -
ω όχι
ལེ་
K Jwz