88
II
INDUCTION IN ROTATING SPHERES
Heat gener-
ated in a
Let the resistance of the shell be k: required the heat W
generated in it.
We determine the values of u, v, w belonging to , and in
particular to the term
Yni A sin iwPni
=
spherical When u, v, w have been found, the heat generated is
shell.
W=k(u²+v²+w²)ds,
the integral being extended over the whole surface of the sphere.
and to denote currents parallel to circles
the meridians in the direction of increasing
Introducing
of latitude and to
e and w, we have
น=
Ω
=
-
1 дв
R 00'
1
дов
-
R sin
dw'
u = cos e cos
+
sin w,
v = cos sin w
-
- Ω cos w,
w =
-
sin 0.
Substituting for ni the value given we get
u =
u =
w=
R
R
-
-
cos iw cos w. ¿Pni. cot - sin iw sin wP'ni}
cos iw sin w. ¿Pni cot + sin iw cos wP'ni},
i cos iw. P
A.
R
ni.
•
But now
¿Pni cot 0 = {Pn,i+1+(n + i)(n − i + 1)Pni-1},
-
-
P'ni = } { − Pni+1+(n + i)(n − i + 1)Pn,i-1}.
Substituted in u, v, w these give
-
A
2R
cos (i+1)w. Pn,i+1
-
+(n + i)(n − i + 1) cos (i-1)w. Pri-1},