370
THE THEORY OF THE HELMHOLTZ RESONATOR
The first two series of P's on the right of (24) and (29) become div when /A = 1, or 0 = 0. To evaluate them we have
,(1 0 l a^ ,l- = l + gP1 + a'P,4...,.............
V i 1 — 2a cos v + a2}
so that l+Pi + P2+...=—r—5-----m = o • 1/5..........
V(2 — 2 cos u) Zsm^v
Again, by integration of (30),
^ l 3 2 "' J 0 A/J 1 — 2acos 6 + a2}
= log [a — cos 0 + V{1 - 2a cos # + a2}] - log [1 — cos 0] *,
so that 1 H- £PX + £ P2 + ... = log (1 + sin $0) - log sin \Q..........<
In much the same way we may sum the third series ^,n~1Pn. We
p + ap0 + a2p3 + ... = L------- 1
a. V [1 ~ 2«/u. + «-} a
/"a c?o ra dct
Jo a V{1 — ^a/a + a2} Jo a
We denote the right-hand member of this equation by/ and differe it with respect to p.
Thus
d/_ftt da a — fj, ^
d/j, Jo {(a —//,)2+1 —/i2]^ (1 — yti2) v'fl ~ 2aya + a2} 1— ^-' or when a = 1
dl 1 yU,
d/A ~ 4sin\0. cos2\6 1-~/72...................^
On integration
The constant is to be found by'putting p = 0, 0 = £TT. In this case
7 =
Thus
lfv2-10gtan8
* If we integrate this equation again with respect to a. between the limits 0 arid 1, we
1 Pi P
" sin2 J0 [log (1 + sin J0) - log sin £0].
When 6 is small, the more important part is form given by (4), the result applies to the exterior surface f the sphere.