58
II
INDUCTION IN ROTATING SPHERES
In this connection it is natural to seek a development
in descending powers of 2πa/x for very large values of this
quantity.
When h>1, we have
hence
1
1+h2
=
Χτε
-
=
k n
1
h2
1
-
XXrs+
2πα γ δη
+
1
k \ 2 n²
:)
2πа 72 Xтs
3
k
n³ dv...
+
Σπα
pt on
It is true that the terms of this equation cannot, as is
shown by trial, be arranged in such a way as to at once permit
of summation for all Xrs; but if we suppose x to be sym-
metrical with respect to the 7-axis, so that in its development
only terms in cos rn appear, we have
n
-
- nxrs
=
οχτε
ô×
-
1 ax TS
Το θη
=
Xrsd
and then the summation can be performed, at any rate for the
terms of the first order in x/2а. If we confine ourselves to
these we get
Approxi-
mate solu-
tion for
large values
k
roxan.
2+
= -
X
2πα δζ
of the velo- and the very small resultant potential on the positive side is
city.
η
k
jax an⋅
2++x=
2πα ζ
0
But in addition to the condition mentioned, this equation
is subject to another one.
However large 2πа/ may be, yet for certain elements for
which r vanishes, h<1, and the development employed will
be invalid. This circumstance restricts the validity of the
expression obtained to a limited region, which however is
larger the greater 2πa/x. On this point I refer to an investi-
gation to follow immediately (p. 61).