and with y = b tan θ, r = b sec θ, this is
2ρmU dθ (1 − a2b−2 cos 3θ cos θ),
(9)
and integrating between the limits θ = ±½π, the resultant, as before, is 2πρmU.
31. Example 2.—Confocal Elliptic Cylinders.—Employ the elliptic coordinates η, ξ, and ζ = η + ξi, such that
z = c ch ζ, x = c ch η cos ξ, y = c sh η sin ζ;
(1)
then the curves for which η and ξ are constant are confocal ellipses and hyperbolas, and
| J = | d(x, y) | = c2 (ch2 η − cos2 ξ) |
| d(η, ξ) |
= (1/2)c2 (ch 2η − cos 2ξ) = r1r2 = OD2,