It will be seen that this vortex-tube must be endless, that is, it must either return into itself, or be infinitely long in one or both directions. For if it were terminated anywhere within the fluid, it would be possible to form a surface, starting from a closed circuit round the tube, continued along the surface of the tube to the termination, and then closed by a cap situated beyond the termination. At no part of this surface would there be any rotation, and ΣωdS, which is equal to the circulation, would be zero for it; and of course this cannot be the case. Thus the tube cannot terminate within the fluid. It can, however, have both of its ends on the surface, or one on the bounding surface and the other at infinity, if the fluid is infinitely extended in one direction, but in that case the termination is only apparent. The section is widened out at the surface; some of the bounding lines pass across to the other apparent termination, when it also lies on the surface, while the other lines pass off to infinity along the surface, and correspond to other lines coming in from infinity to the other termination. Whether the surface is infinite or not, the vortex is spread out into what is called a vortex-sheet, that is, in a surface on the two sides of which the fluid moves with different tangential velocities. Through a vortex-ring or tube, the fluid circulates in closed lines of flow, each one of which is laced through the tube. The circulation along every line of flow which encloses the same system of vortex-tubes has the same value. If any surface be drawn cutting a vortex-tube, it is clear from the definition of the tube that the value of ΣωdS for every such surface must be the same. This Thomson calls the "rotation of the tube." As was pointed out first by von Helmholtz, vortex-filaments correspond to circuits carrying currents and the velocity in the surrounding fluid to magnetic field-intensity. The "rotation of the tube" corresponds to the strength of the current, and sources and sinks to positive and negative magnetic poles. Thomson made great use of this analogy in his papers on electromagnetism. Examples of vortex-tubes are indicated on p. 154; and the reader may experiment with vortices in liquids with water in a tea-cup, or in a river or pond, at pleasure. Air vortices may be experimentally studied by means of a simple apparatus devised by Professor Tait, which may be constructed by anyone.