J. Trowbridge— Vortex Rings in Liquids. 



dy dx 

 and therefore " the existence of i 

 tent with the existence of rotati< 

 have seen from the equation of strain that the existence of a 

 strain potential is inconsistent with the rotation of a material 

 particle. Let us now see if vortex movement can arise in a 

 liquid from variation of density and pressure. Following Helm- 

 holtz's notation, we have, if ip is a function of x, y, z, t, 



6,p_dtp drb dib d+ ( . 



W-W + U dx- + V dy- +W d7 Eq - (l0) ' 



Calling B, x y, 8,, the components of the angular velocity, we can 

 obtain their variations by substituting them in succession in 

 Eq. (10). If we eliminate X, Y, Z, from Eqs. (3) by the help 

 of Eqs. (6), supposing that h and p are functions of x, y, a, t, we 

 obtain, introducing the values of £, y, Z 7 from Eqs. (9) : 

 w #„%** */«& , dw\ , dv , Jho , 1 Idh dp dhdp\ 



and similar expressions for the variations of y and Z. If the 

 variation of h is infinitely small, we obtain by the aid of Eq. (5) : 



85 JLu , dv , Jlw 



itely small, we have the term 



J^/dJ^dp _< 

 2h i \dz~ dy < 



which is independent of $, y, 8,, and de^ 

 tion of h and p. This term enters into the expressions f 

 variations in the angn] -hows, therefore, that 



a vortex movement can arise in a process of diffusion by a 

 variation in density and pressure, without the aid of initial 

 angular velocities. This condition can be shown experiment- 

 ally by dropping a somewhat dense solution of one of the aniline 

 colors into a mixture of glycerine and water. The original 

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

 up gradually into segments, which slowly in their turn assume 

 the ring form. A mixture of water and glycerine is not neces- 

 sary : peculiar cusp-like figures indicating the first stage of vor- 

 tical movement can be seen whenever a thin stratum of one 



