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LIX. The Mass carried forward by a Vortex. 

 By W. M. Hicks, F.R.S* 



TTTHEN a vortex aggregate is moving steadily through 

 ▼ T an ir rotational liquid we can in general distinguish 

 three definite regions of fluid motion, (1) that of the ring or 

 aggregate itself which is in rotational motion and which 

 keeps its identity and constituents throughout however the 

 energy may alter, (2) the portion in irrotational cyclic motion 

 surrounding the first, which also keeps its identity and 

 volume so long as the energy is constant, and which travels 

 uniformly through the liquid like a solid, (3) the irrotational 

 acyclic motion, outside the second region which remains at 

 rest at infinity and no portion of which is ever displaced by 

 more than a small amount. The distinction between the 

 rotational region (1) and the irrotational (2), (3) is funda- 

 mental and well known. Less attention than it deserves, 

 however, appears to have been devoted to the discussion of 

 the relationships between the 2nd and 3rd regions. In this 

 regard the following note may be interesting. 



When any such aggregate is travelling uniformly through 

 an unlimited fluid with translatory velocity U relative to the 

 fluid at a great distance, we may in order to study the 

 relative motion suppose the ring brought to rest by imposing 

 everywhere a velocity equal and opposite to U. In this case 

 the boundaries between the three regions appear as fixed. 

 The first is always a ring surface, even when the aggregate 

 closes up, or the aperture diminishes to zero. The boundary 

 between (2) and (3) may be a ring surface, or it may appear 

 as a singly-connected surface, past which the outer liquid 

 streams. This is what an observer travelling with the ring 

 sees, and we can in this way determine the boundary mass 

 and energy of the portion which goes bodily through the 

 liquid, and the energy of the external part or region (3). 

 Take as an instance a circular unicyclic t vortex ring. 

 When the ratio of the cross section of ring to aperture is very 

 small, it is well known that the velocity at the centre is less 

 than that of translation. Consequently when the system is 

 brought to rest the flow at the centre is in the opposite 

 direction to that of the original translation. In this cast 1 the 

 boundary (2), (3) must cut the equatorial plane somewhere 



* Communicated by the Author. 



t I.e. without bicyclic or gyrostatic quality. 



Phil. Mag. S. if. Vol. 38. No. 227. Nov. 1919. 2 T 



