The Vortex Ring Theory of Gases. 



27 



, d d , d , d 



where = j + w +n—. 



all ax ay dz 



so tliat if Sa, SI, 8m, 8n be the changes in I, m, n respectively in the 

 small time t then — 



. ! dm 



ca— — -ka T 



2 dW ' 



,d*Q d 2 Q\ 



\ dh 2 dhdxj 



d*Q d*Q > 

 dh 2 dhdijj 



. / <&Q d*Q\ 

 cm— I m - — It, 



( W - — It, 



and if &b, By, 8z are the changes in x, y, z in the time t, 



cx=ut, 

 8y=vr, 

 Bz=wt, 



where u, v, iv are the component velocities of the centre of the 

 vortex ring. 



Let x\ y r , z' , to', rj', £' be the values of x, y, z, w, rj, £ respec- 

 tively after the time t, then — 



x'=x-\-ut, 



y'=y + vr, 



z' = z + wr, 



to' =ZUJ -\- Bit), 



but since w=Yi, 



S ( v = qV<i- 1 6V. 



Now the change in V will be due to the change in the shape and 

 size of the ring, and as we saw before that this is due almost entirely 

 to the change in the radius, thus the change in V will be —Y8a/a, or 

 substituting for ha its value — 



