du dH -- - 2v[zeta] + 2w[eta] + -- = 0, (4) dt dx dv dH -- - 2w[xi] + 2u[zeta] + -- = 0, (5) dt dy dw dH -- - 2u[eta] + 2w[xi] + -- = 0, (6) dt dz where _ / H = | dp/[rho] + V + ½q², (7) _/ q² = u² + v² + w², (8) and the three terms in H may be called the pressure head, potential head, and head of velocity, when the gravitation unit is employed and ½q² is replaced by ½q²/g. Eliminating H between (5) and (6) D[xi] du dv dw / du dv dw \ ----- - [xi]-- - [eta]-- - [zeta]-- + [xi]( -- + -- + -- ) = 0, (9) dt dx dx dx \ dx dy dz / and combining this with the equation of continuity 1 D[rho] du dv dw ----- ------ + -- + -- + -- = 0, (10) [rho] dt dx dy dz we have D /[xi] \ [xi] du [eta] dv [zeta] dw -- ( ----- ) - ----- -- - ----- -- - ------ -- = 0, (11) dt \[rho]/ [rho] dx [rho] dx [rho] dx with two similar equations. Putting [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. Putting d[phi] d[psi] H - ------ - m ------ = K, (4) dt dt the equations of motion (4), (5), (6) § 24 can be written