52
II
INDUCTION IN ROTATING SPHERES
If h is a proper fraction, the series involved in ; con-
verges
and we get
Ωρ
=
n
1+h(sin iw - h cos iw)Pri›
R/1+h2
P h
N₁ = Ani
Ani
=
-
-
n
n+1
2n+1
R\n+1 1.
-
52(sin iw — h cos iw)Pnt ›
1+h2
h (sin iw-h cos iw)Pri
4π (n + 1) An² 1 + h²
"1+h²
If h>1, the series occurring in , diverges and it is no
longer allowable to regard the phenomenon as a series of suc-
cessive inductions, since each one would be larger than the
preceding one.
Nevertheless the formulæ given hold for every value of h,
as we may easily verify a posteriori, and deduce by the same
considerations that we shall have to employ in the case of
spherical shells of finite thickness. Since I propose again to
deduce the above formulæ from the general ones. I shall not
now consider them further.
We write
and now
h=tan 8,
Solution.
Ωρ
Construc-
lines of
flow.
=
n
Ω; = Από
R
P
sin & sin (iw-8)Pni:
Ani
-
-
n
n+1
2n+1
R\n+1
P
4π(n+1)
sin & sin (iw-8)Pni›
Ani sin 8 sin (iw - 8)Pni ·
Hence the result is as follows:-
1. A simple spherical harmonic in the inducing potential
tion of the induces a current-function which is a surface harmonic of its
own type. Hence we may here also retain the construction
previously given (§ 2, 2) for the lines of flow, but we must
suppose the spherical layer considered to be turned through a
certain angle /i in the direction of the rotation relative to the
1 A copper spherical shell of 50 mm. radius and 2 mm. thickness would
have to make about 87 revolutions per second in order that for i=1, n=1, h
might be equal to 1. [? 62 revolutions per second.—TR.]