Clebsch has shown that the components of the velocity of a fluid u , v , w , parallel to rectangular axes x , y , z , may always be expressed thus u = dχ / dx + λ dΨ / dx , v = dχ / dy + λ dΨ / dy , w = dχ / dz + λ dΨ / dz ; where λ, Ψ are systems of surfaces whose intersections determine the vortex lines; and the pressure satisfies an equation which is equivalent to the following p / ρ + V = – dχ / dt –½{( dχ / dx ) 2 + ( dχ / dy ) 2 +( dχ / dz ) 2 } + ½ λ2 {( dΨ / dx ) 2 +( dΨ / dy ) 2 +( dΨ / dz ) 2 } where p is the pressure, ρ the density, and V the potential of the forces acting on the liquid. It is shown in this paper that an equation of a complicated nature in λ only can be obtained in the following cases (that is to say, as in cases of irrotational motion, the determination of the motion depends on the solution of a single equation only):— (1.) Plane motion, referred to rectangular coordinates x , y . The equation is somewhat simpler when the vortex surfaces are of invariable form, and move parallel to one of the axes of coordinates with arbitrary velocity.