726 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
is not merely one of mathematical interest, for the vortex atom theory of matter 
has—so far as it has* yet been developed—shown such claims on our consideration 
that anything throwing light on it will be of value. The supposition of a dense core 
may possibly be necessary to account for the different masses of the various elements. 
As soon as the existence of a core is postulated the ring at once becomes more 
complex, depending on the density (or even the arrangement of density) of its core, 
on its vorticity, and on the presence or absence of additional circulations. In what 
follows the vorticity has been taken uniform ; this not only greatly simplifies the 
mathematical methods, but is also the case we should naturally choose first to investi¬ 
gate. In the general investigation the density is taken to be different from that of 
the surrounding fluid. The ring is supposed hollow, with an additional circulation 
round it, and another additional circulation round the outer boundary of the core. It 
is evident that the presence of the former circulation necessitates the perpetual 
existence of the hollow. It is shown that the presence of the latter circulation is 
necessary to render the ring stable when its density is greater than that of the rest 
of the fluid. 
As in the former paper, the investigation is divided into three sections. The first 
is preliminary and deals with the necessary functions and their approximate values. 
The second is devoted to the consideration of the state of steady motion. Here the 
approximations are carried in the beginning so far as to include the second order of 
infinitesimals, but this necessitates in certain parts approximation to the fourth order. 
It has been carried to this order for future work ; the reader however may, so far as 
the results of the present communication are concerned, without any loss of intelligi¬ 
bility, pass over the parts giving the calculation of the highest orders. The third 
section discusses the question of fluted vibrations, of pulsations, and of stability. 
When the motion is steady the sectional centre * of the hollow lies outside that of 
the core. In general (§9) if C x is its position (with given outside boundary) when the 
inner additional circulation is very large, and Co when the same quantity is zero, Cj is 
outside of C 3 and the position of C when the additional circulation is general, is the 
centre of gravity of masses proportional to the added circulation at C : and the circula¬ 
tion due to the core itself at C 3 . When the hollow is just zero, the distance of C 2 from 
the sectional centre of the core bears to the sectional radius the ratio 5r/8(X where r 
is the sectional radius, and a the radius of the ring. This, therefore, is the point where 
the hollow begins to form when the energy is sufficiently increased. If with the same 
outer boundary the mass of the core be lessened (or size of hollow increased) C 3 moves 
in and ultimately coincides with the centre of the outer section. The position of C x 
alters in the same manner, only in this case the hollow can never vanish. 
If m be the volume of the core, II the pressure at an infinite distance, p the circula¬ 
tion when there are no additional ones, and d, d' the densities, then (§ 10) a hollow will 
* By sectional centre is meant the centre of the cross-section ; by apertural centre is meant the centre 
of the aperture. 
