66 JREPORT— 1881. 



the surface over any surface whose boundary is the curve. This gives at 

 once Helmholtz's theorem that the surface-integral of the same quantity 

 over any closed surface is zero. Looking at the general question of the 

 production of cyclic motion, and noticing that the diaphragm closing the 

 aperture of a ring may be of any shape, it is easy to see that the impulse 

 of the motion of a closed filament of infinitely small section is the 

 resultant of a uniform distribution of pressure ( = cyclic constant) over 

 three plane areas which are the projections of the core of the ring on 

 any three planes at right angles. 



The general question of the steady motion and stability of vortex- 

 rings has also been considered by Sir W. Thomson in a paper before the 

 same Society, of which, unfortunately, only an abstract ' has been published. 

 The theory is based on the proposition (only to be stated for its truth to 

 be evident,) that if when the vorticity ■^ and impulse are given the kinetic 

 energy is a maximum or minimum, the motion is steady and stable ; if it 

 is a maximum-minimum (minimax), the motion is steady, but may be 

 stable or unstable. Unfortunately, the simple circular Helmholtz ring has 

 its energy a minimax, so that from this it is not possible to rigorously 

 decide the question of its stability for all possible displacements, although 

 when the aperture is not too small compared with the section of the core, 

 experiment would lead us to believe that it is stable. For displacements, 

 symmetrical about its axis, it is clear that for some determinate distribu- 

 tion of the vorticity the energy must be a maximum, and the motion 

 stable. Very interesting too are the illustrations given of the steady 

 motion of non-plane vortex-filaments. When the number of twists in 

 the axis of a vortex is large, the core is approximately a helix wound on a 

 circular tore, and approximate expressions are obtained in this case for 

 the radii of the axis and section of the tore in terms of the number of 

 twists, the circulation and the components of the given impulse. 



The method inti-oduced of considering the question of stability as a 

 problem of maximum and minimum energy enables us to arrive at many 

 general results which at first sight appear very remarkable, as, for 

 example, the complete annulment of the energy in certain cases by opera- 

 tions on the boundary alone. An extremely interesting illustration of the 

 transformation of a vortex motion with stability of maximum energy to 

 one with stability of minimum energy by operations on the boundary 

 alone, which withdraw energy, was given by the same author before 

 the British Association at its meeting at Swansea^ in 1880. Approach- 

 ing the subject from another point of view Lamb** had proved that in 

 steady motion it is possible to draw a system of surfaces, each of which 

 is covered by a network of vortex lines and stream lines, and that 

 between any two near surfaces of the system ^ w sin 6. v is constant 

 where q denotes the velocity along a stream line, w is the rotation, d the 



' 'Vortex Statics.' Proc. JRny. Soc. Edin.., ix. p. 69 ; and Phil. Mar/., (5) x. p. 97. 



2 The ' vorticity ' of a vortex is given, when supposing it analysed into an intinite 

 number of infinitely small filaments, the volume of each filament and its circulation 

 are given. This does not suppose tliat the arrangement of the filaments is given. 



' ' On maxiraum and minimum energy in vortex motion,' Pritisk Association, 

 BejutHs for 1880, p. 473 ; also Nature, xxii. p. 618. See also a practical illustration, 

 given at the same meeting of the British Association, ' On an experimental illustration 

 of minimum energy in vortex motion,' Po'it. Ass. Pejj., 1880, p. 491 : and Nature, 

 xxiii. p. 69. . 



* ' On the conditions of steady motion of a fluid,' Proc. Loud. Math, Soc. ix. p. 

 91 (1878). 



