/b - a´ /a - u ch n[Omega] = cos n[theta] = / ------ / ------, (36) \/ a - a´ \/ b - u´ /a - b /a´ - u sh n[Omega] = i sin n[theta] = i / ------ / ------, (37) \/ a - a´ \/ b - u´ a´ > u = a´e^([pi]/sc) > -[oo], (38) giving the intrinsic equation. 41. The first problem of this kind, worked out by H. v. Helmholtz, of the efflux of a jet between two edges A and A1 in an infinite wall, is obtained by the symmetrical duplication of the above, with n = 1, b = 0, a´ = -[oo], as in fig. 5, /u - a / -a ch [Omega] = / -----, sh [Omega] = / ---; (1) \/ u \/ u and along the jet APJ, [oo] > u = ae^([pi]s/c) > a, sh [Omega] = i sin [theta] - i[root](a/u) = ie^(-½[pi]s/c), (2) _ _ / [oo] / c c PM = | sin [theta] ds = | e^(-½[pi]s/c) ds = ----- e^(-½[pi]s/c) = ----- sin [theta], (3) _/ s _/ ½[pi] ½[pi] so that PT = c/½[pi], and the curve AP is the tractrix; and the coefficient of contraction, or breadth of the jet [pi] ---------------------- = --------. (4) breadth of the orifice [pi] + 2 A change of [Omega] and [theta] into n[Omega] and n[theta] will give the solution for two walls converging symmetrically to the orifice AA1 at an angle [pi]/n. With n = ½, the reentrant walls are given of Borda's mouthpiece, and the coefficient of contraction becomes ½. Generally, by making a´ = - [oo], 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. [Illustration: FIG. 5.] [Illustration: 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 [pi]/n as in Bobyleff's problem; and, by making a = [oo], the wedge extends to infinity; then / b / n ch n[Omega] = / -----, sh n[Omega] = / -----. (1) \/ b - u \/ b - u Over the jet surface [psi] = m, q = Q, u = - e^([pi][phi]/m) = - be^([pi]²/c),