11
79
INDUCTION IN ROTATING SPHERES
Using formula (a) (p. 77) we may divide out the q's and
thus obtain
1
$1+$2
-
1 =
δλ
1+
2n+1
But now we have
$8
4πRiw
=
= h,
2n+1 (2n+1)h
according to our previous notation, and hence
1
-
=
h
1
1+h-1
=
1+ h²
1+h 2
√ — 1,
which result agrees with the one previously obtained.
Thus we have on the one hand tested our formula by
means of a result already known; on the other hand we have
proved that the previously given formulæ hold for all values
of h, which proof still remained to be given.
velocities
2. Secondly, we apply our formula to the case where we Small
need only retain the first power of the angular velocity in f₁ of rotation.
and f. For simplicity we restrict the investigation to a solid
sphere.
In this case we found
2n+1 Pr(λo)
-
=
2n Pn-1(AS)
Expanding the p's and retaining only first powers we get
2+
-
ƒ₂+ƒ₂√− 1 =
2n+3
-
S2
2+
2n+1
A closer consideration of this equation shows that the
values of f,, f, thence found when substituted in merely
give the inductions of the first and second order, which we
have already calculated on p. 68. Here we only consider the