60
IL
INDUCTION IN ROTATING SPHERES
Solution.
We use p, w, z as co-ordinates, where p now denotes the
perpendicular distance from the axis of rotation. In the
general formulæ we must replace
p by R+2,
• by R
w remains w,
and after making this substitution we must allow R to become
infinite. Then a simple spherical harmonic takes the form
- nz
Ante cos iwJ(np),
i
(and analogous ones), where J, denotes the Bessel's function
of order i. The given x is to be analysed into terms of this
form by means of integrals analogous to those of Fourier.
We treat each term separately.
If we put
2πω ί
tan &=
k n
then for the term in question the solution of the problem is
-nz
+= Anten sin 8 sin (iw - 8) J; (np),
Ω+
Ω.
зв
=
=
ni
nz
- Anie sin & sin (iw - 8) J, (np),
1
2π
Ani sin & sin (iw - 8) J, (np).
By summation we get the complete integrals.
We again attempt to obtain a development in powers of
2π/ by considering the successive inductions.
2πω/κ
same method as above we get
2πω (αχ dz +
- ) ....
n+=
k
მა
2πω
k
2
χ
dz2
aw2
2
Ω. (-2) = - Ω + (2),
By the
Second
form of the
solution.
10+
Ω
=
2π