The Vortex Ring Theory of Gases. 



25 



Thus for the same relative changes of a and d the changes in the 

 velocities are in the ratio of log 8a/ d to 1, and as log 8a/ d is very 

 great, we may neglect the change in the velocity of the ring produced 

 by the alteration in the distance between the loops in comparison 

 with that produced by the alteration in the size of the ring. 



The kinetic energy of a quantity of fluid containing vortex rings of 

 this kind may conveniently be divided into several parts. The first 

 part consists of the kinetic energy of the irrotationally moving fluid 

 surrounding the ring, the second part of the kinetic energy of the 

 rotationally moving fluid ; this again may conveniently be divided 

 into two parts, one part being the kinetic energy due to the rotation 

 in the core, and the other that due to the trans] ational velocity of the 

 vortex core. 



The kinetic energy of the irrotationally moving liquid surrounding 

 the ring may be expressed in several ways ; it is equal to the strength 

 of the ring multiplied by the rate of flow of the fluid through it ; the 

 most convenient expression for our purpose, however, is 



Am 2 , 



where v is the velocity of translation of the vortex ring resolved 

 along the normal to its plane, a is the radius of the ring and A a 

 constant. 



(See p. 12 of my " Treatise on the Motion of Vortex Rings.") 

 The energy due to the rotation of the vortex core is 



\mfipm^a^ 



where n is the number of loops in the ring, p the density of the fluid, 

 and m the strength of the ring. 



The kinetic energy due to the translational velocity of the ring is 



+ + 



where M is the mass of fluid in the ring and u 2 + v 2 -\-w 2 the square of 

 the velocity of the ring. 



Thus if T be the whole kinetic energy due to the ring — 



T = Ava* + \n^ P m?a + JM O 3 + v% + . 



Let us consider a vortex ring placed in a fluid where there is a 

 velocity potential Q independent of that due to the vortex ring itself, 

 the value of O is supposed to be known at every point of the fluid. 



We have to fix the position, size, and motion of the ring. We can 

 do this if we know the coordinates (x, y, z) of its centre, its radius 

 (a), the direction cosines (I, m, n) of its plane, and Y that part of the 

 velocity at the ring which is due to the ring itself. V is not neces- 

 sarily the actual velocity of the ring, for this latter quantity is the 



