64
11
INDUCTION IN ROTATING SPHERES
Theorem
which
forms the
basis of
what
follows.
If we remember that
Əx
au av aw
+ +
მყ Əz
=3
0,
we get for these conditions:-
In the material of the shell
av au
V² = 2w
+w(xV²V − yV²U) ;
-
дх
მიყ
and at the boundaries
24 P
av au
W
др
=
We shall first demonstrate the following theorem :—
If U, V, W have the forms
эх ду
m
U
=
py-
Əz
ду
V = p2
m
Xn
дх
#xt).
W
=
y
дх
m
X
ду
which forms satisfy the equation
au av aw
+ +
მყ Əz
მე
=
0,
then the solutions of the preceding differential equations are
φ
u'
=
-
opm(p??
= wpm+1 sin
=
=
V
=
-
W m
K
Y
dxn - nzxn
Əz
Əz
-
2
dy
მა