78
II
INDUCTION IN ROTATING SPHERES
their close relation to Bessel's functions; in fact we may
express J in terms of Pm++ and m+
m
We may now remove p from the formula expressing our
result, and thus we obtain¹
Final form
of the solu-
tion.
2n+1
2n
Its applica-
tion.
Thin spher-
ical shell.
-
-
In+1(λs)In( − λo) — In+1( − λs)In(λo)
In+1(As)In-1( — XS) — In+1( − λs)In−1(AS)
=
-
We shall apply this formula, which gives the exact solution,
to some special cases which admit of simplifications.
1. In the first place, let the spherical shell be very thin,
and let d be its thickness. Then S is only slightly different
from s. Let
S =s+δ, where now δ = μα.
For σ we may put any convenient value between s and S;
suppose σ = = S.
We substitute these values in the above formula. In the
denominator we employ the substitution
In-1 =
2n+1
2n
-In+
λέσε
4n(n+1
-In+1
and divide by the numerator.
Thus we obtain
$1+02
√-1
1
λ2s2
In+1(λs)In+1( − λS) − In+1( − λs)In+1(XS)•
-
1+
2(n+1)(2n+1) In+1(s)In(-AS) − In+1( − λs)In(XS)
We develop and put
In+1(XS) = In+1{λ(s+8)}
= In+1(λs) +λdg′n+1(As)
=2n+1(λs)+
8x2s
2(n + 2) In+2(As),
In+1(-AS) = In+1(−λs) +
8x2s
-
2(n + 2)²n+2( − λs).
1 From this point onwards we write A instead of λ₁; thus—
λ=√(1+√=1).