62
II
INDUCTION IN ROTATING SPHERES
Approxi-
mation for
A development can also be given for large values of 2π/k.
large values We denote by Xo that part of x which is symmetrical about
of the velo- the axis of rotation, by x, x-x, the remainder. To Xo
corresponds for every velocity of rotation the value = 0.
Hence, assuming x to be symmetrical about the x-axis, we get
for large values of 2πw/k
city.
Ω
= -
ki fax,
2πω θα
do.
- X₁-
The formula is deduced in the same way as above. The
series may also be completely developed; and this too for
forms of x which are not symmetrical with respect to the
x-axis. I shall not here enter into further detail on the
point.
In conclusion let us determine p, the potential of the free
electricity. By the proper substitutions we get from the
general formulæ :-
1. Neglecting self-induction,
Potential of
the free
electricity.
∞
φ = ωρ
əx dz.
др
We must add to this value of a constant, whose value
is such as to make & vanish at infinity. The formula which
we have found has already been given by Jochmann for the
case in which x is symmetrical with respect to the axis of x.
It is seen to be generally true.
2. Taking self-induction into account we have
&= wp
0
Co
(a(x + n)
др
dz.
When is very large, we find, if x is symmetrical with
respect to the x-axis,
∞
&=wp
др
Əxo dz +
ki (axi dw.
2п др
The first term increases indefinitely with w.