97 
of Edinburgh, Session 1866 - 67 . 
infinitesimal vibration, and intended to include them in a mathe- 
matical 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 infinitesmal 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-l 
i 
of the angular velocity of this 
rotation. Hence, as the number of crests in a whole circumference 
is equal to i, for a harmonic deviation of order i, there are i — 1 
periods of vibration, in the period of revolution of the vortex. 
For the case i = 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 
simply, therefore, 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 trans- 
verse vibrations of a ring, such as the vibrations that were seen in 
the experiments, 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 the vortex rotations of the atoms of sodium vapour are 
much less than 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 co-existent in slightly different periods, equal ap- 
proximately to the time just stated, and of as nearly as can be 
perceived equal intensities ; the sodium atom must have two fun- 
damental modes of vibration, having those for their respective 
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 
VOL. VI. 
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