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



dh dh dh dh __hh 

 dt dx dy dz ht ' 



du dv dw __~ 

 dx dy dz ' 



(3) 



(4) 



, X, Y, Z are 



in which p is the pressure at the point x, y, z 

 the components of the external forces acting on a unit of mass 

 and /* is the density at the point x, y, z. The forces X, Y, Z 

 are considered to have a potential V, so that 



dY „ dY /rx 



.... (5) 



v dY 



dy 9 dz' 



and the velocities u, v } w a velocity-potential cj>, so that 



d<b d(f> deb 



ii ^^ — i— it —— — i/i —= — 1-» 



dy' 



dx' 



dz 



(6) 



or 



udx + vdy + wdz = d<j) ; 

 and <j> satisfies the equation 



cP± #£ <P± 



d# 2 dy 5 



^ 



which is equation (4) under the conditions expressed in equa- 

 tion (6). We must also have 



dv 



dv 



dz 



dw 



dw 

 dx 



du 

 dz' 



00 



du _ 



dy dx dz dy 



equations similar to equations (1). Helmholtz has shown 

 that in the case of rotation of a fluid-element, equations (8) 

 become 



dv dw _%,. 



dl~~dy~rs> 



dw du _p^ 

 ~dx~~dz r v > 



du dv _hy. 

 d~y~~dx~~^ ; 



(9) 



" and therefore the existence of a velocity-potential is inconsis- 

 tent with the existence of rotation of the fluid -element." We 

 have also seen from the equations of strains that the existence 

 of a strain-potential is inconsistent with the rotation of a ma- 

 terial particle ; and therefore, from the conditions of impact, 

 the particles of a drop of diffusing material are in a condition 

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

 liquid from variation of density and pressure. Following 



