11
61
INDUCTION IN ROTATING SPHERES
on this
But there is a limitation to the validity of these formulæ, Remarks
which we had not to impose on the previous analogous ones. form.
For their deduction presupposes that for each separate term
of the development of x it is allowable to regard the total
induction. as the sum of a series of successive inductions.
According to the results which we obtained for spheres this
condition is only satisfied for those terms for which 2πwi/kn
is a proper fraction. Now n may have any value from zero
to infinity; thus for a number of terms the necessary con-
dition is not satisfied, and the result can therefore only be
approximate. With reference to this point I remark :—
The
1. At a finite distance the terms for which n is very
small vanish relatively to those for which n is finite.
error committed in the above formula must have an appreciable
value for large values of ρ.
2. The quantity 27w/k may always be chosen so small
that the approximation may be any desired one within a
given region. For by diminishing 27w/k we diminish the
number of those terms which do not satisfy the required con-
dition a suitable diminution diminishes their number in any
desired degree.
There may possibly be difficulties in determining exactly
the region of validity for a given value of 2πw/k and a given
degree of approximation. For practical applications this de-
termination is of no importance: because, in the first place,
we are only concerned here with very small values of 2πw/k;
and, in the second place, we are only considering plates of
limited dimensions, and not infinite plates.
The equation
Ως
=
2πw (dx dz
k
მთ
is exact, apart from self-induction.
We see that, in order Possibility
of neglect-
that we may be allowed to neglect self-induction, it is neces- ing self.
sary not only that 2πw/k be small, but also that the investi- induction.
gation be limited to a certain finite region. The extent of
this region depends on 27w/k; beyond it not even an approxi-
mate determination of the current is possible without taking
self-induction into account. We shall meet with an exactly
analogous result at the end of
4.
SUIENUE ENVINCENSE IN