40
II
INDUCTION IN ROTATING SPHERES
ཊྛ
and the quantities U, V, W are
U
-
V
W
yay
R dz
2 ay
=
R Əx
=
xay
Ray
-
-
1 ay
z ду
Roy Row'
x xx
R Əz
y ay
=
=
=
1 ду
R dwy
1 ay
Rdx R dwi
If y is a homogeneous function of x, y, z of the nth degree,
then
Əz მე
aw av
=
ду
Əz
av au
n + 1 ay
дх ду
au aw
R Əz'
n+1 ay
R dy'
n+1 ay
Ꭱ Əx
And we have always
av au
ΘΩ
дх
dy
dz'
au aw
an
=
Əz
Əx
ay'
aw
av an
Formulæ
for the
electro-
motive-
forces.
dy Əz dx
These formulæ are developed in Maxwell's Treatise on
Electricity, vol. ii. p. 276.¹ The signs in part are different,
because the system of coordinates there employed is not ours,
but a symmetrical one. The system used here is that to
which Helmholtz's formulæ apply.
6. The following are the expressions for the electromotive
forces which are produced by the components U, V, W of the
vector potential, supposed invariable, due to the element whose
component velocities are a, B, y:—
=
av
эх
-
au
ду
1 2nd ed. p. 280.
aw
Эх