/ 1 / e^([pi]²/c) ch [Omega] = cos n[theta] = / ---------------, sh [Omega] = i sin n[theta] = i / ---------------, (2) \/ e^([pi]²/c) + 1 \/ e^([pi]²/c) + 1 ½[pi] ds 2n e^(½[pi]²/c) = tan n[theta], ----- -------- = -------------. (3) c d[theta] sin 2n[theta] For a jet impinging normally on an infinite plane, as in fig. 6, n = 1, e^(½[pi]²/c) = tan [theta], ch (½[pi]s/c) sin 2[theta] = 1, (4) sh ½[pi]x/c = cot [theta], sh ½[pi]y/c = tan [theta], sh ½[pi]x/c sh ½[pi]y/c = 1, e^(½[pi](x + y)/c) = e^(½[pi]x/c) + e^(½[pi]y/c) + 1. (5) With n = ½, the jet is reversed in direction, and the profile is the catenary of equal strength. In Bobyleff's problem of the wedge of finite breadth, / b /u - a /b - a / u ch n[Omega] = / --- / -----, sh n[Omega] = / ----- / -----, (6) \/ a \/ u - b \/ a \/ u - b / b /a - b cos n[alpha] = / ---, sin n[alpha] = / -----, (7) \/ a \/ a and along the free surface APJ, q = Q, [psi] = 0, u = e^(-[pi][phi]/m) = ae^([pi]s/c), / e^([pi]²/c) - 1 cos n[theta] = cos n[alpha] / ---------------------------, \/ e^([pi]²/c) - cos² n[alpha] cos² n[alpha] sin² n[theta] e^([pi]²/c) = -----------------------------, (8) sin² n[theta] - sin² n[alpha] the intrinsic equation, the other free surface A´P´J´ being given by cos² n[alpha] sin² n[theta] e^([pi]²/c) = -----------------------------. (9) sin² n[alpha] - sin² n[theta] Putting n = 1 gives the case of a stream of finite breadth disturbed by a transverse plane, a particular case of Fig. 7. When a = b, [alpha] = 0, and the stream is very broad compared with the wedge or lamina; so, putting w = w´(a - b)/a in the penultimate case, and u = ae^(-w) [asympt] a - (a - b)w´, (10) /w´ + 1 / 1 ch n[Omega] = / ------, sh n[Omega] = / --------, (11) \/ w´ \/ [root]w´ in which we may write w´ = [phi] + [psi]i. (12) Along the stream line xABPJ, [psi] = 0; and along the jet surface APJ, -1 > [phi] > -[oo]; and putting [phi] = -[pi]s/c - 1, the intrinsic equation is [pi]s/c = cot² n[theta], (13) which for n = 1 is the evolute of a catenary. [Illustration: FIG. 7.]