1895.] System of the Periods of a Holloio Vortex Ring. 1 55 



III. "The Complete System of the Periods of a Hollow Vortex 

 Ring." By H. C. POCKLINGTON, B.A., Fellow of St. Johii's 

 College, Cambridge. Communicated by Professor LARMOR. 

 F.R.S. Received April 23. 1895. 



(Abstract.) 



The author discusses the stability of a hollow annular vortex in an 

 infinite perfect liquid, and also the effect of an electric charge on the 

 steady motion and the stability of such a vortex. It is known that a 

 hollow vortex ring (without electric charge) is stable for such small 

 deformations as are symmetrical about the axis of symmetry of the 

 ring, and for such as consist in displacement of the axis of the hollow 

 without alteration of the size or shape of its cross section. This 

 investigation shows that, in addition to the fluted and sinuous vibra- 

 tions above referred to, the vortex is capable of beaded vibi-ations, in 

 which the hollow is enlarged and contracted at regular intervals 

 along its length, and also of vibration of a more general type, in 

 which the displacement at any moment consists of waves on the 

 surface of the hollow, of which the crests are circles parallel to the 

 axis of the hollow, and the amplitude a sine or cosine of a multiple 

 of the azimuth angle. The periods of these vibrations are found and 

 proved to be real. Since, as is easily seen, any displacement of the 

 surface can be compounded by displacement of the various types here 

 mentioned, the vortex is stable for all displacements of its surface. 



When an attempt is made to explain matter as composed of atoms 

 which consist of such vortex rings, a difficulty is found at the outset 

 if the theory is applied to gases. On the kinetic theory of gases, a 

 theory which, over a wide range, gives results in accordance with 

 those of experiment, the energy of an atom varies as the square of its 

 velocity. The energy of a vortex ring, however, decreases as its 

 velocity increases. Thus the single vortex atom theory is likely to 

 yield results that disagree with experimental results when applied to 

 gases. If, however, the electric charge is taken into account, this 

 defect can be, to a certain extent, remedied. 



The case where the electricity resides on the surface, supposed 

 conducting, of the hollow of the vortex ring is worked out on the 

 hypothesis that the period of an electrical oscillation is so small that 

 the electricity has at any time its equilibrium distribution. It is 

 found that, in the case of such a ring, however small the charge may 

 be, the velocity of the ring can be decreased, made to vanish, and 

 finally to change sign by decreasing the radius of the ring. At the 

 same time, the energy diminishes, attains a minimum when the 

 velocity of the vortex is zero, and then increases. If therefore the 



