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. At infinity 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. With [beta] = 0, the stream is parallel to x0, and [phi] = m ch([eta] - [alpha])cos [xi] = -Uc ch([eta] - [alpha])sh [eta] cos [xi]/sh([eta] - [alpha]) (12) over the cylinder [eta], and as in (12) § 29, [phi]1 = -Ux = -Uc ch [eta] cos [xi], (13) for liquid filling the cylinder; and [phi] th [eta] ------ = --------------------, (14) [phi]1 th ([eta] - [alpha]) over the surface of [eta]; so that parallel to Ox, the effective inertia of the cylinder [eta], displacing M´ liquid, is increased by M´th [eta]/th([eta]- [alpha]), reducing when [alpha] = [oo] to M´th [eta] = M´(b/a). Similarly, parallel to Oy, the increase of effective inertia is M´/th [eta] th([eta] - [alpha]), reducing to M´/th [eta] = M´(a/b), when [alpha] = [oo], and the liquid extends to infinity. 32. Next consider the motion given by