m m
U = - --- e^(-a) cos [beta] = - ----- cos [beta],
c a - b
m m
V = - --- e^(-a) sin [beta] = - ----- sin [beta], (9)
c a + b
a and b denoting the semi-axes of the ellipse [alpha]; so that the
liquid is streaming at infinity with velocity Q = m/(a + b) in the
direction of the asymptote of the hyperbola [beta].
An ellipse interior to [eta] = [alpha] will move in a direction
opposite to the exterior current; and when [eta] = 0, U = [oo], but V
= (m/c) sh [alpha] sin [beta].
Negative values of [eta] must be interpreted by a streaming motion on
a parallel plane at a level slightly different, as on a double Riemann
sheet, the stream passing from one sheet to the other across a cut SS´
joining the foci S, S´. A diagram has been drawn by Col. R. L.
Hippisley.
The components of the liquid velocity q, in the direction of the
normal of the ellipse [eta] and hyperbola [xi], are
-mJ^(-1)sh([eta] - [alpha]) cos([xi] - [beta]),
mJ^(-1)ch([eta] - [alpha]) sin ([xi] - [beta]). (10)
The velocity q is zero in a corner where the hyperbola [beta] cuts the
ellipse [alpha]; and round the ellipse [alpha] the velocity q reaches
a maximum when the tangent has turned through a right angle, and then
[root](ch 2[alpha] - cos 2[beta])
q = Qe^a ---------------------------------; (11)
sh 2[alpha]
and the condition can be inferred when cavitation begins.