1883.] On the Steady Motion of a Hollow Vortex, 307 



•of our knowledge respecting the properties of these atoms, or attempt 

 to find analogies even with the ordinary kinetic theory of gases. For 

 instance, the vortex-atoms are polar, and therefore do not behave 

 towards one another indifferently for all modes of approach. Clearly, 

 also, the temperature of a gas composed of vortex-atoms could not 

 depend on the translatory velocity of mean square, but would depend 

 in some way on the mean energy. In this connexion it is interesting 

 to notice that the time of vibration of a ring in class (2), when at 

 least the ring is moving steadily, is independent of its energy, 

 depends in fact only on the constants of the ring, the fluid, and the 

 inverse square root of the number of crimps in a cross-section. If 

 relations are to be sought between spectral lines they would arise 

 from classes 1, 2, 5. But from Mr. Thomson's investigations it would 

 appear that in the case of a solid tore, the time of vibration in 

 class (1) would depend on the temperature. 



Section I of the paper is devoted to a consideration of the functions 

 employed in the investigation. In Section II is considered the motion 

 of a rigid tore in fluid moving parallel to its straight axis. In 

 Section III, the problem of the steady motion of a hollow vortex is 

 taken up, together with the small vibrations of classes (2) and (4) 

 above. It will be sufficient to give here a short abstract of the 

 results arrived at in Section III. The cross-section of a ring is 

 throughout considered as small compared with the aperture, and the 

 expressions giving the form of the hollow and the velocity of trans- 

 lation are carried to a second approximation, the quantity by which 

 the approximation proceeds being the ratio r/{R + ^/(R 3 — r 3 ) } = Jc, 

 where r, R — r, are the radii of the mean cross-section and aperture 

 respectively ; when the ring is small this is very approximately 

 r/2R. The condition that the hollow must be a free surface, 

 gives a relation which the volume of the hollow must satisfy, which 

 for very small rings reduces to the constancy of the radius of the 

 hollow. For a solid ring the corresponding condition is, of course, 

 the constancy of volume. This makes an essential difference between 

 the two theories. 



To a second approximation the velocity of translation is unaltered 

 and is given by 



V=_£_f31ogi-2\ 

 167ra\ k ) 



whilst to the second approximation the surface velocity, relative to 

 the hollow itself, is 



^irak\ 2 /' 



where a is the radius of the " critical " circle — or the leno-th of a 



x 2 



