90
II
INDUCTION IN ROTATING SPHERES
Analytical
formulæ.
our problem is solved.
It is easily seen that the result may
be expressed in the forms
30
or
k
nids,
- A+ Σn (n + 1) = f y² ²,
W
=
R2
W
=
k
R₂n(n+1)/4², ds.
R2
The preceding theorem might perhaps have been more
simply demonstrated by means of reasoning based on Green's
theorem. In the proof here given the following formulæ are
implicitly involved
2
[¥³
2nds,
[{(av: ) + ( a )" + (@w") } ds = n(n + 1) f 4%,de,
მთ. Jwy
a
(sin iwPni)
მა.
a
dwy
=
მა,
-cos (i+1)wP,i+1
− (n + i)(n − i + 1) cos (i-1)wPn,i-1
-
-
(sin iwP) sin (i+1)wP,,i+1
a
= -
-
+ } (n + i)(n − i + 1) sin (i − 1)wPn‚i - 1›
-
=
(sin iwPni) i cos iwP,
Jwz
ni,
and to these similar equations may easily be added. If w, 0
and ', ' refer to systems of polar co-ordinates with different
axes, the equations last quoted enable us to deduce integrals
of the form
cos iwPní(0) cos jw'P„¡(0')ds',
(in which the integrations are to be performed with respect
to w', ') from the well-known integral
4
[Pro(0) cos ja'P')ds' = 24 +1 cos jwP„(☺),
jw’P„¡(0′)ds' 2n+1
Generation but, it is true, only by laborious calculation.
of heat in I now proceed to determine the heat generated in the
the rotating
sphere.