On the Motion of Two Spheres in a Liquid, and Allied Problems

Abstract

On the Motion of two Spheres, $c.369 face (ii.), so as to touch alternately the two ovals of this curve of intersection.Taking any portion of it, the tangent lines along it touch the surface (i.), and, when produced geodesically along it, form the corresponding arcs of the geodesic system of (i.), i.e. t those arcs which run in the same direction, in the sense that we can proceed by continuous change from the one to the other through the intermediate arcs.If, therefore, we take any two points A and B on (i.), the corresponding geodesic arcs drawn from them, and carried geodesically on to (ii.), finally run into one another and are continuous.Again, if we take any point on (ii.) and follow the two geodesic arcs of the system which cross each, other there, they will run on to (i.) so as to be non-corresponding geodesic arcs on it.If, therefore, we take any two pomts A and B on (i.), the noncorresponding geodesic arcs drawn from them and carried geodesically on to (ii.) will meet in a point P on (ii.), such that the sum or difference of AP and BP is constant, according to the relative position of A and B considered fixed, in away that is most easily illustrated by the analogy of Roberts' theorem for a singlo quadric.10.If the quadric (i.) is taken to be a focal conic, and the points A and B are taken on its boundary, all lines thixragh A and B are tangential to (i.), so that the condition of geodesic contact with (i.) imposes no restriction.Thus taking, for definiteness, (ii.) to be an ellipsoid, we see that a thread drawn tight over it between two points A and B on opposite quadrants of the focal hyperbola is free to slip all over it; and also that the locus of the space-foci, in the geodesic sense, of the lines of curvature is the focal hyperbola, which, of course, contains the umbilics.On the Motion of Two Spheres in a Liquid, and Allied Problems.