II
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75
INDUCTION IN ROTATING SPHERES
B
=
2n+1
In+1(28)
4n Pn-1 (λS) In+1(8) − Pn+1(~28)In−1(S)'
2n+1
D
=
4n
Pn+1(8)
Pn-1(S)In+1(8) − Pn+1(λ28)In−1(S)
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We get the complete solution by substituting these values
in 4, and 2. It may be more simply exhibited thus: Since
and λ are conjugate, p (λσ) and p (λσ) also are conjugate;
in the same way A and B are conjugate, as is easily seen, and
hence
Ap₂ (λ¸σ)+Вp₂(λ20)
is equal to twice the real part of either expression.
In the same way
Ap₁(λσ) - Bp(20),
which expression occurs in 4,, is twice the imaginary part of
the first term. Remembering this, and also the values of A
and C, we easily recognise the truth of the equation
2n+1
2n
Pn(~¤)In+1(8) − In(10)Pn+1(~8)
Pn−1(^1S)In+1(s) − In-1(MS)Pn+1(8)
-
= $₁+$₂√ − 1 = ƒ₁+f₂√ −1.
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-
This equation is especially simple when s = 0, that is in the
case of a solid sphere. Then q+1 (s) is infinite, and thus
our equation becomes
2n+1 Pn(10)
2n Pn-1(S)
=
$1 + $₂√ - 1.
The quantities, a knowledge of which is of special interest
to us, are the angle σ = tan-f2f1, and the ratio of increase of
current, fi+f. These have a very simple analytical mean-
ing they are the amplitude and modulus of the complex
quantity on the left-hand side.
2.
The calculation may be carried still further by means of
the following remarks:-
Solution of
the equa
tions for the
f's by
means of
the p's and
q's.
Further
of the p's
The indefinite integrals defining p and q may be evaluated properties
for integral values of n, and thus p and q may be expressed and q's.