Putting
ω2 = ξ2 + η2 + ζ2,
(12)
a vortex line is defined to be such that the tangent is in the direction of ω, the resultant of ξ, η, ζ, called the components of molecular rotation. A small sphere of the fluid, if frozen suddenly, would retain this angular velocity.
If ω vanishes throughout the fluid at any instant, equation (11) shows that it will always be zero, and the fluid motion is then called irrotational; and a function φ exists, called the velocity function, such that
u dx + v dy + w dz = −dφ,
(13)
and then the velocity in any direction is the space-decrease or downward gradient of φ.
25. But in the most general case it is possible to have three functions φ, ψ, m of x, y, z, such that
u dx + v dy + w dz = −dφ − m dψ,