d²[theta] / F² F² \ F[xi] d[mu] C3--------- = ( -- - -- ) sin [theta] cos [theta] - ----- sin [theta] = F-----, (20) dt² \ c1 c2 / c1 dt _ _ d[theta] / | F² cos² [theta] F² sin² [theta] F[xi] | C3-------- = Fy = / | - --------------- - --------------- + 2----- cos [theta] + H |; (21) dt \/ |_ c1 c2 c1 _| so that cos [theta] and y is an elliptic function of the time. When [xi] is absent, dx/dt is always positive, and the centre of the body cannot describe loops; but with [xi], the influence may be great enough to make dx/dt change sign, and so loops occur, as shown in A. B. Basset's _Hydrodynamics_, i. 192, resembling the trochoidal curves, which can be looped, investigated in § 29 for the motion of a cylinder under gravity, when surrounded by a vortex. The branch of hydrodynamics which discusses wave motion in a liquid or gas is given now in the articles SOUND and WAVE; while the influence of viscosity is considered under HYDRAULICS. REFERENCES.--For the history and references to the original memoirs see _Report to the British Association_, by G. G. Stokes (1846), and W. M. Hicks (1882). See also the _Fortschritte der Mathematik_, and A. E. H. Love, "Hydrodynamik" in the _Encyklöpadie der mathematischen Wissenschaften_ (1901). (A. G. G.) HYDROMEDUSAE, a group of marine animals, recognized as belonging to the Hydrozoa (q.v.) by the following characters. (1) The polyp (hydropolyp) is of simple structure, typically much longer than broad, without ectodermal oesophagus or mesenteries, such as are seen in the anthopolyp (see article ANTHOZOA); the mouth is usually raised above the peristome on a short conical elevation or hypostome; the ectoderm is without cilia. (2) With very few exceptions, the polyp is not the only type of individual that occurs, but alternates in the life-cycle of a given species, with a distinct type, the medusa (q.v.), while in other cases the polyp-stage may be absent altogether, so that only medusa-individuals occur in the life-cycle.