d[phi] d[psi] d[phi] d[psi] d[phi] d[psi]
u = - ------ - m------, v = - ------ - m------, w = - ------ - m------, (6)
dx dx dy dy dz dz
as in §25 (1), a first integral of the equations in (5) may be written
_
/ dp d[phi] d[psi] / d[phi] d[psi] \
| ----- + V + ½q² - ------ - m------ + (u - u´) ( ------ + m------ )
_/ [rho] dt dt \ dx dx /
/ d[phi] d[psi] \ / d[phi] d[psi] \
+ (v - v´)( ------ + m------ ) + (w - w´)( ------ + m------ ) = F(t), (7)
\ dy dy / \ dz dz /
in which
d[phi] d[phi] d[phi] d[phi]
------ - (u - u´)------ - (v -v´)------ - (w - w´)------
dt dx dy dz
d[phi] d[phi] d[phi] d[phi]
= ------ - (U - yR + zQ)------ - (V - zP + xR)------ - (W - xQ + yP)------ (8)
dt dx dy dz
is the time-rate of change of [phi] at a point fixed in space, which
is left behind with velocity components u - u´, v - v´, w - w´.
In the case of a steady motion of homogeneous liquid symmetrical about
Ox, where O is advancing with velocity U, the equation (5) of § 34
p/[rho] + V + ½q´² - [f]([psi]´) = constant (9)
becomes transformed into
p U d[psi]
----- + V + ½q² - --- ------ + ½U² - [f]([psi] + ½Uy²) = constant, (10)
[rho] y dy
[psi]´ = [psi] + ¼U², (11)
subject to the condition, from (4) §34,
y^(-2)[nabla]²[psi]´ = -[f]´([psi]´),
y^(-2)[nabla]²[psi] = -[f]´([psi] + ½Uy²). (12)