X. On the mathematical theory of stream-lines, especially those with four foci and upwards

Abstract

Object and Occasion of this Investigation .— A Stream-line is the line that is traced by a particle in a steady current of fluid. Each individual stream-line preserves its figure and position unchanged, and marks the track of a filament or continuous series of particles that follow each other. The motions in different parts of a steady current may be represented to the eye and to the mind by means of a group of stream-lines; for the direction of motion of a particle at a given point is that of a tangent to the stream-line which traverses that point; and when the fluid is of constant density, as is sensibly the case with liquids, the comparative velocities at different points are indicated by the comparative closeness of the stream-lines to each other. Even when the fluid is gaseous, the comparative mass-velocities are indicated by the closeness of the stream-lines—the term mass-velocity meaning the mass which traverses a unit of area in a unit of time. Gaseous fluids, however, will not be considered in the present paper. Stream-lines are important in connexion with naval architecture; for the curves which the particles of water describe relatively to a ship, in moving past her, are streamlines; and if the figure of a ship is such that the particles of water glide smoothly over her skin, that figure is a stream-line surface , being a surface which contains an indefinite number of stream-lines. The stream-lines of a current gliding past a circular cylinder in a direction transverse to its axis, and also those of a current gliding past a sphere, have long been known.