if r denotes the radius of curvature of the stream line, so that
| 1 | dp | + | dV | = | dH | − | d ½q2 | = | q2 | , | |
| ρ | dν | dν | dν | dν | r |
(6)
the normal acceleration.
The osculating plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are
| u | du | + v | du | + w | du | = | d ½q2 | − 2vζ + 2wη = | d ½q2 | − | dH | , ..., |
| dx | dy | dz | dx | dx | dx |
(7)
and when q is stationary, the acceleration is normal to the surface H = constant, and the stream line is a geodesic.
Calling the sum of the pressure and potential head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces touch where the velocity is stationary.
Equation (3) is called Bernoulli’s equation, and may be interpreted as the balance-sheet of the energy which enters and leaves a given tube of flow.