448 Researches on the Theory of Vortex Rings. [June 18, 



perpetual existence of the hollow. It is shown that the presence of 

 the latter is necessary to render the ring stable when its density is 

 greater than that of the surrounding fluid. 



The investigation is divided into three sections. The first is pre- 

 liminary, and deals with the necessary functions and their approxi- 

 mate values. The second is devoted to the state of steady motion ; and 

 the third to the small symmetrical vibrations and the question of 

 'stability. 



When the motion is steady the sectional centre* of the hollow lies 

 outside that of the core. In general if Ci is its position when the 

 inner additional circulation is very large, and C 2 when the same quan- 

 tity is zero, Cx is outside of C 2 , and the position of C when the addi- 

 tional circulation, is anything else is the centre of gravity of masses 

 proportional to the added circulation at and the circulation due to 

 the core itself at C 2 . 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 hr/8a where r is the sectional radius and a the aperture radius. 

 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), both Ci and 

 C 2 move in, and ultimately coincide with the centre of the outer 

 section. 



So long as the core is only doubly continuous the volume is constant, 

 and therefore the sectional radius varies inversely as the square root 

 of the aperture radius. 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 small, 

 and the sectional radius of the core remains almost constant. The 

 sectional radius of the hollow, however, increases with the aperture 

 when the hollow is small, but decreases beyond a certain critical value. 

 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 comparable with that of the external fluid. It will, however, 

 decrease as the whole energy is increased. The fluted vibrations in 

 general consist of two pairs, those of one pair travelling with the 

 cyclic motion, and the other against, the mode being determined by 

 the number of flutings. For a simply continuous, core there are two 

 times of vibration for each order ; for a hollow core without internal 



* By "sectional" centre, &c, is meant the centre of the cross section j by 

 " aperture " centre is meant the centre of the aperture, &c. 



