dK d([psi],m) -- - 2u[zeta] + 2w[eta] - ---------- = 0, ..., ...; (5) dx d(x,t) and therefore dK dK dK [xi]-- + [eta]-- + [zeta]-- = 0. (6) dx dy dz Equation (5) becomes, by a rearrangement, dK d[psi] /dm dm dm dm \ -- - ------ ( -- + u-- + v-- + w-- ) dx dx \dt dx dy dz / dm / d[psi] d[psi] d[psi] d[psi] \ + -- ( ------ + u------ + v------ + w------ ) = 0, ..., ..., (7) dx \ dt dx dy dz / dK d[psi] Dm dm D[psi] -- - ------ -- + -- ------ = 0, ..., ..., (8) dx dx dt dx dt and as we prove subsequently (§ 37) that the vortex lines are composed of the same fluid particles throughout the motion, the surface m and [psi] satisfies the condition of (6) § 23; so that K is uniform throughout the fluid at any instant, and changes with the time only, and so may be replaced by F(t). 26. When the motion is _steady_, that is, when the velocity at any point of space does not change with the time, dK -- - 2v[zeta] + 2w[eta] = 0, ..., ... (1) dx dK dK dK dK dK dK [xi]-- + [eta]-- + [zeta]-- = 0, u-- + v-- + w-- = 0, (2) dx dy dz dx dy dz and _ / K = | dp/[rho] + V + ½q² = H (3) _/ is constant along a vortex line, and a _stream line_, the path of a fluid particle, so that the fluid is traversed by a series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is irrotational H is a constant throughout the fluid. Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = 0, [xi] = 0; and in steady motion the equations reduce to dH/d[nu] = 2v[zeta] - 2w[eta] = 2q[omega] sin [theta], (4) where [theta] is the angle between the stream line and vortex line; and this holds for their projection on any plane to which d[nu] is drawn perpendicular. In plane motion (4) reduces to dH / dQ q \ ----- = 2q[zeta] = q ( -- + --- ), (5) d[nu] \ dv r / if r denotes the radius of curvature of the stream line, so that 1 dp dV dH d½q² q² ----- ----- + ----- = ----- - ----- = ---, (6) [rho] d[nu] d[nu] d[nu] d[nu] r the normal acceleration. The osculating plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are du du du d½q² d½q² dH u-- + v-- + w-- = ---- - 2v[zeta] + 2w[eta] = ---- - --, ..., (7) dx dy dz dx dx dx