ON THE VORTEX THEORY OF THE LUMINIFEEOUS jETHER. 



493 



fig. 1, wliere fhe small white and black circles represent cross sections of 

 the rings : the white where the rotation is opposite to, and the black 

 where it is in the same direction as, the rotation of the hands of a watch 

 placed on the diagram facing towards the spectator. Imagine first each 

 vortex-ring to be in a portion of the fluid contained within a rigid 

 rectangnlar box, of which four sides are indicated by the fine lines cross- 

 ing one another at right angles throughout the diagram ; and the other 

 pair are parallel to the paper, at any distance asunder we like to imagine. 

 Supposing the volume of rotationally moving portion of the fluid consti- 

 tuting the ring to be given, there is clearly one determinate shape, and 

 diametral magnitude, in which it must be given in oi'der that the motion 

 may be steady. Let it be so given, and fill space with such rectangular 

 boxes of vortices arranged facing one a.nother oppositely in the manner 

 shown in the diagram. Annul now the rigidity of the sides of the boxes. 

 The motion continues unchangedly steady. But is it stable, now that the 

 rigid partitions are done away with ? No proof has yet been given that 



Fig. 1. 



R 



d 



Q 



P 



W 



o 



Q 







it is. If it is, laminar waves, such as waves of light, could be propagated 

 through it ; and the velocity of propagation would be R V 2/3 if the 

 sides of the ideal boxes parallel to the undisturbed planes of the rings are 

 square (which makes ave ii^ = ave u'^), and if the distance between the 

 square sides of each box bears the proper ratio to the side of the square 

 to make ave v^ = ave u^ = ave lo^. 



23. Consider now, for example, plane waves, or laminar vibrations, in 

 planes perpendicular to the undisturbed planes of the rings. The change 

 of configuration of the vortices in the course of a quarter period of a 

 harmonic standing vibration, f{y,t) = sin wt cos i^g (which is more easily 

 illustrated diagrammatically than a wave or succession of waves), is illus- 

 trated in fig. 2, for a portion of the fluid on each side of 7/ = 0. The 



