(1)
by Green’s transformation, dν denoting an elementary step along the normal to the exterior of the surface; so that dφ/dν = 0 over the surface makes T = 0, and then
| ( | dφ | ) | 2 | + ( | dφ | ) | 2 | + ( | dφ | ) | 2 | = 0, | dφ | = 0, | dφ | = 0, | dφ | = 0. |
| dx | dy | dz | dx | dy | dz |
(2)
If the actual motion at any instant is supposed to be generated instantaneously from rest by the application of pressure impulse over the surface, or suddenly reduced to rest again, then, since no natural forces can act impulsively throughout the liquid, the pressure impulse ῶ satisfies the equations
| 1 | dῶ | = −u, | 1 | dῶ | = −v, | 1 | dῶ | = ῶ, | |||
| ρ | dx | ρ | dy | ρ | dz |
(3)
ῶ = ρφ + a constant,
(4)
and the constant may be ignored; and Green’s transformation of the energy T amounts to the theorem that the work done by an impulse is the product of the impulse and average velocity, or half the velocity from rest.