Prof. J. Trowbridge on Liquid Vortex-Rings. 293 

 Helmkoltz's notation, we have, if -ty is a function of x, y, z, t, 



%-%+•%+*$ +-5f w> 



Calling f , 77, £ the components of the angular velocity, we can 

 obtain their variations with the time by substituting them in 

 succession in equation (10). If we eliminate X, Y, Z from 

 equations (2) by the help of equations (5), and introduce the 

 values of f, ??, J from equations (9), supposing that A is a 

 function of x, y, z, t } we obtain 



?l /<iy d«A ^ 1 y ^. . JL /^ f?£ _ dh dp\ n 1 * 



St ' \dy + dz)~ i " n dx + ^dx + W \dz dy dy d~z)> (il) 



and similar expressions for the variations of 77 and f. It will 

 be seen that in this case terms of the form 



1 (dh dp dh dp\ 

 2tf\dzfy~fydz)> 



independent of f , 77, 5, and depending upon the variations of h 

 and p, enter into the expressions for the variations of the an- 

 gular velocities ; and therefore a vortex movement is to be 

 expected in a process of diffusion by a variation of density and 

 pressure without initial angular velocities. This condition can 

 be shown experimentally by dropping a solution of one of the 

 aniline colours into a mixture of glycerine and water. The 

 original ring, after ceasing to move downward in the mixture, 

 breaks up gradually into segments, wilich in their turn slowly 

 assume the ring form. A mixture of water and glycerine is 

 not necessary ; peculiar cup-like figures, indicating the first 

 stage of vortical movement, can be seen whenever a thin stra- 

 tum of one liquid slowly diffuses itself through another liquid 

 of different density. 



By a consideration of the equations 



e^+(u 1 -u)dt=eh+ |f dt), 

 erj + (y 1 —v)dt=€ h}+ ^ dt\ 



ei; + (w 1 -w)dt=€(s+ J dt\ 



given by Helmholtz, from which he draws the conclusion that 



" each vortex-line remains continually composed of the same 



elements of fluid and swims forward with them in the fluid." 



isS We see, on introducing the new expressions which we have 



