18 Sir William Thomson on Vortex Atoms. 



possible description of infinitesimal vibration, and intended to 



include them in a mathematical paper which he hoped soon to 



be able to communicate to the Royal Society. One very simple 



result which he could now state is the following. Let such a 



vortex be given with its section differing from exact circular 



figure by an infinitesimal harmonic deviation of order i. This 



form will travel as waves round the axis of the cylinder in the same 



direction as the vortex rotation, with an angular velocity equal to 



i— 1 



— r— of the angular velocity of this rotation. Hence, as the 



number of crests in a whole circumference is equal to i } for an 

 harmonic deviation of order i there are 2 — 1 periods of vibra- 

 tion in the period of revolution of the vortex. For the case 

 2 = 1 there is no vibration, and the solution expresses merely an 

 infinitesimally displaced vortex with its circular form unchanged. 

 The case i=2 corresponds to elliptic deformation of the circular 

 section ; and for it the period of vibration is, therefore, simply 

 the period of revolution. These results are, of course, applicable 

 to the Helmholtz ring when the diameter of the approximately 

 circular section is small in comparison with the diameter of the 

 ring, as it is in the smoke-rings exhibited to the Society. The 

 lowest fundamental modes of the two kinds of transverse vibra- 

 tions of a ring, such as the vibrations that were seen in the ex- 

 periments, must be much graver than the elliptic vibration of 

 section. It is probable that the vibrations which constitute the 

 incandescence of sodium-vapour are analogous to those which the 

 smoke-rings had exhibited ; and it is therefore probable that the 

 period of each vortex rotation of the atoms of sodium- vapour is 

 much less than j\-^ of the millionth of the millionth of a second, 

 this being approximately the period of vibration of the yellow 

 sodium light. Further, inasmuch as this light consists of two 

 sets of vibrations coexistent in slightly different periods, equal 

 approximately to the time just stated, and of as nearly as can 

 be perceived equal intensities, the sodium atom must have two 

 fundamental modes of vibration, having those for their respec- 

 tive periods, and being about equally excitable by such forces as 

 the atom experiences in the incandescent vapour. This last 

 condition renders it probable that the two fundamental modes 

 concerned are approximately similar (and not merely different 

 orders of different series chancing to concur very nearly in their 

 periods of vibration). In an approximately circular and uniform 

 disk of elastic solid the fundamental modes of transverse vibra- 

 tion, with nodal division into quadrants, fulfils both the condi- 

 tions. In an approximately circular and uniform ring of elastic 

 solid these conditions are fulfilled for the flexural vibrations in 

 its plane, and also in its transverse vibrations perpendicular to 



