156 ' System of the Periods of a Hollow Vortex Ring. [May 16, 



vortex atom always has nearly the size that corresponds to minimum 

 energy, its energy, neglecting a constant term, varies as the square 

 of its velocity. This relation, however, only holds through a small 

 range. On investigation of the periods of the electrified vortex it is 

 found that it is always unstable for Rome types of disturbance. 



The method employed throughout the investigation is to refer the 

 circumstances of the motion to toroidal co-ordinates. The steady 

 motion of the fluid is then expressed by means of a Stokes' current 

 function, and the disturbances of the steady motion, in the case of 

 vibration, by means of a potential function. The periods of the 

 different oscillations, when real, are given as 27r/p, where p is 

 .given by 



p- + 2pn -A, +n(n l) ( H -- , \ ( ) = 

 27T6- I ir/jcffi? J \27re~J 



in the case of the general oscillation, where there are n waves in the 

 circumference of the hollow ; 



in the case of beaded vibrations, where m is the number of enlarge- 

 ments in the circumference of the ring, L' = log 8 o/e and 



7/;j 4(1 + 1/3 + 1/5 + .... l/(2m 1)}; 



* ; 1+ ^u 



J 2 7> L TTpd'U," J 



in the case of the pulsations, where p, is the circulation of the ring, 

 a its vadius, e the radius of the cross-section of the hollow, p the 

 density of the liquid, E the electric charge on the ring. In the case 

 of the general oscillation, in which the disturbance consists of n 

 Avaves parallel to the axis of the ring, the amplitudes of which vary 

 as cosine or sine of m times the azimuth angle, the value of p must 

 be modified if m is large. The formula for p then is 



. 2mlK' n (2mb} 



E 2 _ 



2mbK' n (2mb) . K(2m6) 



