[omega]² = [xi]² + [eta]² + [zeta]², (12)
a _vortex line_ is defined to be such that the tangent is in the
direction of [omega], the resultant of [xi], [eta], [zeta], called the
components of molecular rotation. A small sphere of the fluid, if
frozen suddenly, would retain this angular velocity.
If [omega] 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 [phi] exists, called the _velocity
function_, such that
udx + vdy + wdz = -d[phi], (13)
and then the velocity in any direction is the space-decrease or
downward gradient of [phi].
25. But in the most general case it is possible to have three
functions [phi], [psi], m of x, y, z, such that
udx + vdy + wdz = -d[phi] - md[psi], (1)
as A. Clebsch has shown, from purely analytical considerations
(_Crelle_, lvi.); and then
d([psi], m) d([psi], m) d([psi], m)
[xi] = ½ -----------, [eta] = ½ -----------, [zeta] = ½ -----------, (2)
d(y, z) d(z, x) d(x, y)
and
d[psi] d[psi] d[psi] dm dm dm
[xi]------ + [eta]------ + [zeta]------ = 0, [xi]-- + [eta]-- + [zeta]-- = 0, (3)
dx dy dz dx dy dz
so that, at any instant, the surfaces over which [psi] and m are
constant intersect in the vortex lines.