and integrating between the limits [theta] = ±½[pi], the resultant, as
before, is 2[pi][rho]mU.
31. _Example 2.--Confocal Elliptic Cylinders._--Employ the elliptic
coordinates [eta], [xi], and [zeta] = [eta] + [xi]i, such that
z = c ch[zeta], x = c ch [eta] cos [xi], y = c sh [eta] sin [zeta]; (1)
then the curves for which [eta] and [xi] are constant are confocal
ellipses and hyperbolas, and
d(x, y)
J = -------, [xi]) = c²(ch²[eta] - cos² [xi])
d([eta]
= ½c²(ch 2[eta] - cos 2[xi]) = r1r2 = OD², (2)
if OD is the semi-diameter conjugate to OP, and r1, r2 the focal
distances,
r1, r2 = c(ch[eta] ± cos [xi]); (3)
r² = x² + y² = c²(ch²[eta] - sin² [xi])
= ½c²(ch 2[eta] + cos 2[xi]). (4)