38
888
II
INDUCTION IN ROTATING SPHERES
be the magnetic permeability, and
mm+ mgr+
X
sec
Dimen-
sions of the
the magnetic potential, so far as it is directly due to magnets;
the magnetic potential of the currents is to be
Ω
mm* mgr+
sec
has no meaning inside the matter of the hollow sphere, and
therefore cannot be analytically continued through it, and is
one-valued in external and internal space.
We shall denote by
mm* mgr+
sec
the current function in an infinitely thin spherical shell. To
avoid all doubt as to sign we give this as the definition of :
when on traversing the space ds increases by dy, then d
is the quantity of electricity which in unit time crosses the
element traversed from left to right. In traversing the path
the feet are supposed directed to the centre of the shell, the
face forwards in the direction of motion.
Since in what follows we shall only deal with currents in
concentric spherical shells about the origin, we define
mgr¹
mm sec
more generally as a function of p, 0, w such that
da. Y(p=a)
denotes the current-function of the layer between pa and
p = a + da.
For convenience the units have been given along with the
magnitudes quoted.
3. Let the external radius of the spherical shell considered
sphere con- be R, its internal radius r, the angular velocity of rotation w.
4. When a function x, which throughout any given space
ment in satisfies the equation ²x = 0,¹ is developed in a series of spher-
sidered.
Develop-
spherical
harmonics.
¹ [In the original A is used where we use .—TR.]