SMALL VIBRATIONS OF A HOLLOW VORTEX. 
163 
the first few terms in the expansions of the first four orders of P, Q, R, T, are given. 
Section II. is devoted to the consideration of the motion of a rigid ring in fluid, when 
it moves parallel to its straight axis. The functions for the motion apply directly to 
the case considered afterwards of the vortex. The points of division of the stream, 
the quantity of fluid carried forward, and the energy of the motion are considered. 
In Section III. the problem of the steady motion of a hollow vortex is treated, 
together with the small vibrations when the hollow is fluted, and when it pulsates. 
The section of a ring is throughout considered as small compared with the aperture, 
and the expressions giving the form of the hollow, the surface velocity, velocity of 
translation and energy, are carried to a second approximation, the quantity by which 
the approximation proceeds being the ratio r/{R+ v /(R 2 — r 2 )}=k where r, Pt — r 
denote the radii of the mean section and aperture respectively ; when the ring is very 
small, this is very approximately r/2R. The condition that the hollow must be a free 
surface over which the pressure is constant gives a relation which R, r must always 
satisfy, which for very small rings reduces to the constancy of the radius of the hollow. 
For a solid ring the corresponding condition is, of course, the constancy of volume. 
This makes an essential difference between the two theories. To a second approxi¬ 
mation the velocity of translation is unaltered, and is given by”' 
whilst to the second approximation the surface velocity, relative to the hollow itself, is 
U= 4 ^{l-i(log|+5)F} 
where a is the radius of the “critical” circle—or the length of a tangent from the 
centre to the ring, and is therefore equal to R for small rings—and /x is the cyclic 
constant. 
In the steady motion considered, the fluid carried forward with the ring forms a 
single mass, without aperture even for extremely small tores, though not for infinitely 
small ones. For values of R/r>10 2 there will be no aperture, whilst for less values 
the fluid carried forward will be ring-shaped. To a first approximation the energy due 
to the cyclic motion is the most important, and is the same as for a rigid ring at rest 
of the same size. It does not depend on the velocity of translation, except in so far 
as this determines the size of the aperture ; as entering in this way the principal term 
varies inversely as the velocity of translation, and thus increases with diminished 
* [April, 1884.—Owing to an error in § 3, tlie values given in the Proceedings require correction.] 
