PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
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begin to form when the radius of the ring is 4mll/ p\d-\-d'). So long as the core is 
simply continuous the volume is constant, and therefore the sectional radius varies 
inversely as the square root of the radius of the ring. When there is no core it was 
shown in the former paper that the sectional radius of the hollow remained constant. 
In the general case, after a hollow is formed the sectional radius of the core changes 
more slowly, and the additional circulations add to this tendency. The outer section 
always decreases as the aperture increases, but when the hollow becomes large this 
decrease is very small, and the sectional radius of the core remains almost constant. 
The sectional radius of the hollow also increases with the aperture. In cases where a 
hollow begins to form the sectional radius at that time is equal to v^l+cr times its 
ultimate value ; or being the density of the core with respect to the surrounding fluid. 
The expansibility of the ring due to the presence of a hollow has a marked effect 
on the variation of the velocity of translation with increasing aperture, the tendency 
being to make the variation smaller. 
With an internal additional circulation the ring will possess internal energy com¬ 
parable with that of the external fluid. It will, however, decrease as the whole energy 
is increased. This is of importance for the general theory of gases. 
The fluted vibrations in general consist of two sets of two, travelling in opposite 
directions round the core, the modes being defined by the number of flutings. For a 
single continuous core there are two sets ; for a hollow core without an internal 
additional circulation, there are three sets together with a standing wave (where the 
time of vibration is infinite); for two additional circulations we get four sets, the 
times being determined by a biquadratic, which I have not succeeded in solving in 
general terms. When there is no rotational core the motion is always stable. When 
there is a simple continuous core, whose density referred to the outer fluid is cr, and 
no additional inner circulation, the ratio of the outer circulation to that due to the 
core must be > cr )/(lfl-cr)|. When there is no additional circulation, or 
no slip over the core, the ring cannot be always stable unless cr< v /2. These con¬ 
ditions hold until a hollow has formed. When there is a hollow and no internal 
additional circulation, the simple ring usually considered is still stable. But if the 
core is denser than the surrounding fluid, it is always stable only when the outer 
additional circulation is larger than a certain critical value depending on the densities 
and the circulations. If it is less than this critical value the ring becomes unstable at 
some point as the aperture increases. When the density of the core is very large, of 
the order 1(F (p large) this critical value is •y/f- ] 0 p/3 times the circulation of the core 
itself. 
The condition of stability when there is an inner additional circulation depends on 
the reality of the roots of a biquadratic equation, and the general conditions are not 
discussed, but the same property of the outer additional circulation preserving the 
stability clearly holds good. The motion is always stable for pulsations. 
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