d[Omega] A [root](b - a·b - a´) -------- = ---------------------------, where A = --------------------. (11) du (u - b)[root](u - a·u - a´) 2n And the w polygon has a zero angle at u = 0, [oo], where [psi] changes from 0 to m and back again, so that w changes by im, and dw B m -- = ---, where B = - ----. (12) du u [pi] Along the stream line xBAPJ, [psi] = 0, u = ae^(-[pi][phi]/m); (13) and over the jet surface JPA, where the skin velocity is Q, d[phi] ------ = -q = -Q, u = ae^([pi]sQ/m) = ae^([pi]s/c), (14) ds denoting the arc AP by s, starting at u = a; /b - a´ /u - a´ ch n[Omega] = cos n[theta] = / ----- / ------ (15) \/ a - a´ \/ u - b´ /a - b /u - a´ sh n[Omega] = i sin n[theta] = i / ------ / ------ (16) \/ a - a´ \/ u - b´ [oo] > u = ae^([pi]s/c) > a, (17) and this gives the intrinsic equation of the jet, and then the radius of curvature ds 1 d[phi] i dw i dw / d[Omega] [rho] = - -------- = --- -------- = --- ------- = --- -- / ------- d[theta] Q d[theta] Q d[Omega] Q du / du c u - b [root](u - a·u - a´) = ----·2n----- --------------------, (18) [pi] u [root](a - b·b - a´) not requiring the integration of (11) and (12) If [theta] = [alpha] across the end JJ´ of the jet, where u = [oo], q = Q, /b - a´ /a - b ch n[Omega] = cos n[alpha] = /-------, sh n[Omega] = i sin n[alpha]= / ------, (19) \/ a - a´ \/ a - a´ Then a - b·b - a´ a - a´ cos 2n[alpha] - cos 2n[theta] = 2------------ = ½ sin² 2n[alpha]------ a - a´·u - b u - b [root](a - b.b - a´)[root](u - a·u - b´) sin 2n[theta] = 2---------------------------------------- (20) a - a´·u - b [root](a - a·b - a´) = sin 2n[alpha]--------------------; u - b 2n c / b \ [root](a - b·b - a´) ----- ----- = ( 1 + ----- ) -------------------- (21) [phi] [rho] \ u - b/ [root](u - a·u - a´) a - a´ + (a + a´) cos 2n[alpha] - [a + a´ + (a - a´) cos 2n[alpha] cos 2n[theta] = -------------------------------------------------------------------------------- (a - a´) sin² 2n[alpha]