d[phi] d[psi]
H - ------ - m ------ = K, (4)
dt dt
the equations of motion (4), (5), (6) § 24 can be written
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.