(7)
| dK | − | dψ | Dm | + | dm | Dψ | = 0, ..., ..., | ||
| dx | dx | dt | dx | dt |
(8)
and as we prove subsequently (§ 37) that the vortex lines are composed of the same fluid particles throughout the motion, the surface m and ψ 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 | − 2vζ + 2wη = 0, ..., ... |
| dx |
(1)
| ξ | dK | + η | dK | + ζ | dK | = 0, u | dK | + v | dK | + w | dK | = 0, |
| dx | dy | dz | dx | dy | dz |
(2)
and