On Gravitational Oscillations of Rotating Water. 109 



Thus, according to these definitions, the resultant impulse 

 of a vortex-filament of infinitely small cross section and of 

 unit circulation is equal to the resultant area of its curve. The 

 resultant axis of a vortex is the same as the resultant axis of 

 the curve ; and the rotational moment is equal to the resultant 

 areal moment of the curve. 



28. Consider for a moment a vortex-filament in an infinite 

 liquid with no disturbing influence of other vortices, or of 

 solids immersed in the liquid. We now see, from the con- 

 stancy of the impulse (proved generally in V. M. § 19), that 

 the resultant area, and the resultant areal moment of the curve 

 formed by the filament, remain constant however its curve 

 may become contorted ; and its resultant axis remains the same 

 line in space. Hence, whatever motions and contortions the 

 vortex-filament may experience, if it has any motion of trans- 

 lation through space this motion must be on the average along 

 the resultant axis. 



29. Consider now the actual vortex made up of an infinite 

 number of infinitely small vortex-filaments. If these be of 

 volumes inversely proportional to their vortex-densities (§ 25), 

 so that their circulations are equal, we now see from the con- 

 stancy of the impulse that the sum of the resultant areas of 

 all the vortex-filaments remains constant; and so does the sum 

 of their rotational moments : and the resultant areal axis of 

 them all regarded as one system is a fixed line in space. 

 Hence, as in the case of a vortex-filament, the translation, if 

 any, through space is on the average along its resultant axis. 

 All this, of course, is on the supposition that there is no other 

 vortex, and no solid immersed in the liquid, and no bounding 

 surface of the liquid near enough to produce any sensible in- 

 fluence on the given vortex. 



XVI. On Gravitational Oscillations of Rotating Water. 

 By Sir William Thomson.* 



THIS is really Laplace's subject in his Dynamical Theory 

 of the Tides; where it is dealt within its utmost generality 

 except one important restriction — the motion of each particle 

 to be infinitely nearly horizontal, and the velocity to be always 

 equal for all particles in the same vertical. This implies that 

 the greatest depth must be small in comparison with the dis- 

 tance that has to be travelled to find the deviation from level- 

 ness of the water-surface altered by a sensible fraction of its 

 maximum amount. In the present short communication I 



* From the Proceedings of the Royal Society of Edinburgh, March 17, 

 1879. Communicated by the Author. 



