so that PT = c/½π, and the curve AP is the tractrix; and the coefficient of contraction, or
| breadth of the jet | = | π | . |
| breadth of the orifice | π + 2 |
(4)
A change of Ω and θ into nΩ and nθ will give the solution for two walls converging symmetrically to the orifice AA1 at an angle π/n. With n = ½, the reentrant walls are given of Borda’s mouthpiece, and the coefficient of contraction becomes ½. Generally, by making a′ = −∞, the line x′A′ may be taken as a straight stream line of infinite length, forming an axis of symmetry; and then by duplication the result can be obtained, with assigned n, a, and b, of the efflux from a symmetrical converging mouthpiece, or of the flow of water through the arches of a bridge, with wedge-shaped piers to divide the stream.
| Fig. 5. | Fig. 6. |
42. Other arrangements of the constants n, a, b, a′ will give the results of special problems considered by J. M. Michell, Phil. Trans. 1890.
Thus with a′ = 0, a stream is split symmetrically by a wedge of angle π/n as in Bobyleff’s problem; and, by making a = ∞, the wedge extends to infinity; then
| ch nΩ = √ | b | , sh nΩ = √ | n | . |
| b − u | b − u |
(1)
Over the jet surface ψ = m, q = Q,