φ = m ch (η − α) cos (ξ − β), ψ = m sh (η − α) sin (ξ − β).
(6)
Then ψ = 0 over the ellipse η = α, and the hyperbola ξ = β, so that these may be taken as fixed boundaries; and ψ is a constant on a C4.
Over any ellipse η, moving with components U and V of velocity,
ψ′ = ψ + Uy − Vx = [ m sh (η − α) cos β + Uc sh η ] sin ξ
- [ m sh (η − α) sin β + Vc ch η ] cos ξ;
(7)
so that ψ′ = 0, if
| U = − | m | sh (η − α) | cos β, V = − | m | sh (η − α) | sin β, | ||
| c | sh η | c | ch η |
(8)
having a resultant in the direction PO, where P is the intersection of an ellipse η with the hyperbola β; and with this velocity the ellipse η can be swimming in the liquid, without distortion for an instant.