| C3 | d2θ | = ( | F2 | − | F2 | ) sin θ cos θ − | Fξ | sin θ = F | dμ | , |
| dt2 | c1 | c2 | c1 | dt |
(20)
| C3 | dθ | = Fy = √ [ − | F2 cos2 θ | − | F2 sin2 θ | + 2 | Fξ | cos θ + H ]; |
| dt | c1 | c2 | c1 |
(21)
so that cos θ and y is an elliptic function of the time.
When ξ is absent, dx/dt is always positive, and the centre of the body cannot describe loops; but with ξ, 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.)