w = m ch([zeta] - [gamma]), [gamma] = [alpha] + [beta]i, (5)
[phi] = m ch([eta] - [alpha]) cos ([xi] - [beta]),
[psi] = m sh([eta] - [alpha]) sin ([xi] - [beta]). (6)
Then [psi] = 0 over the ellipse [eta] = [alpha], and the hyperbola
[xi] = [beta], so that these may be taken as fixed boundaries; and
[psi] is a constant on a C4.
Over any ellipse [eta], moving with components U and V of velocity,
[psi]´ = [psi] + Uy - Vx = [m sh([eta] - [alpha]) cos [beta] + Uc sh[eta]] sin [xi]
-[m sh ([eta] - [alpha]) sin [beta] + Vc ch [eta] cos [xi]; (7)
so that [psi]´ = 0, if
m sh([eta] - [alpha]) m sh([eta] - [alpha])
U = - --- ------------------- cos [beta], V = - --- - ------------------- sin [beta], (8)
c sh[eta] c ch[eta]
having a resultant in the direction PO, where P is the intersection of
an ellipse [eta] with the hyperbola [beta]; and with this velocity the
ellipse [eta] can be swimming in the liquid, without distortion for an
instant.