Page:Elementary Text-book of Physics (Anthony, 1897).djvu/156

142 of the front of the crest becomes still steeper because of the restraint which then is imposed upon the movement of the particles in the lower half of their paths, and at last the forward motion in the crest so much predominates that the wave curls over and breaks.

122. Vortices.—A series of most interesting results has been obtained by Helmholtz, Thomson, and others, from the discussion of the rotational motions of fluids. Though the proofs are of such a nature that they cannot be presented here, the results are so important that they will be briefly stated.

A vortex line is defined as the line which coincides at every point with the instantaneous axis of rotation of the fluid element at that point. A vortex filament is any portion of the fluid bounded by vortex lines.

A vortex is a vortex filament which has " contiguous to it over its whole boundary irrotationally moving fluid."

The theorems relating to this form of motion, as first proved by Helmholtz, in 1868, show that,— (1) A vortex in a perfect fluid always contains the same fluid elements, no matter what its motion through the surrounding fluid may be.

(2) The strength of a vortex, which is the product of its angular velocity by its cross-section, is constant; therefore the vortex in an infinite fluid must always be a closed curve, which, however, may be knotted and twisted in any way whatever.

(3) In a finite fluid the vortex may be open, its two ends terminating in the surface of the fluid.

(4) The irrotationally moving fluid around a vortex has a motion due to its presence, and transmits the influence of the motion of one vortex to another.

(5) If the vortices considered be infinitely long and rectilinear, any one of them, if alone in the fluid, will remain fixed in position.

(6) If two such vortices be present parallel to one another, they revolve about their common centre of mass.

(7) If the vortices be circular, any one of them, if alone, moves with a constant velocity along its axis, at right angles to the plane