72
II
INDUCTION IN ROTATING SPHERES
Definition
of p and q.
Then our equations become
Φα
φ, = - λάφι,
2n+2
"1+
-
-'1 − x²₁ = 0.
σ
The two particular integrals are
+1
fax- Jex-viper
(1 − v²)” cœìvdv, (1 − v²)*e¯¤¹ºdv,
which hold for real positive values of σ.
We shall prove farther on that these integrals satisfy the
equations. Since in our case n is a whole number the integra-
tions can be performed, and the solution expressed in a finite
form; but for simplicity we may retain the integral form.
Let us write
Pn(o)= (1-
-fa-
-1
- v²)¹€™dv,
In(0) = |(1 − v²)ˆ€˜º dv,
-
then clearly the solutions of the differential equations are—
$₁ = Ap„(λ₁σ)+£n(λ20)+C¶n(^10)+D¶n(^20),
− $2 = M²APn(Mo) +λ₂²Âƒ„(λ₂0)+λ²Сqn(~σ)+λ₂¸²Dqn(~20).
These solutions must be substituted in the integral equa-
tions so that the constants may be determined. The following
formulæ will serve to evaluate the integrals which occur:-
Pn(λo) =
+1
(1 − v²)³¿¹ovdv,
+1
σ-2n
d
do
+¹pn)=
Pn
=
-1
(1 − v²)^(oλv+2n+1)ť¹ººdv
= 2npn-1(λo),¹
1 The last members of the equations are got by transforming the preceding
integrals, especially by integration by parts.