The fluid is bounded by one or more stream lines, over which is constant, and represented by parallel lines in the w diagram, and the angle a in the w polygon is either + or, so that which can be resolved into a quotient and partial fractions of the form the 2 polygon is composed either of parallel lines of constant and variable log- corresponding to a boundary or barrier of z, Q or else a line at right angles of constant velocity q=Q, making Q log = 0, while 0 is variable, as over the free surface of a jet. Here is the advantage of = log over $= = diagram, stant qQ would give an arc of a circle on the and would be constant along a radius; and so the procedure of Helmholtz and Kirchhoff in employing is not the simplest, and was much improved by Planck's idea of using log = . Begin with the application to Kirchhoff's problem, where a plane barrier AA' like an aeroplane is placed at an angle a across an infinite current of fluid, moving when undisturbed with velocity Q. In the disturbed motion a wake is formed in rear of AA', which may be supposed still or turbulent, but at constant atmospheric pressure; and the wake is bounded by the two free surfaces AJ, A'J', extending to infinity at J and J', over which the pressure is constant and atmospheric. M.F. D The fluid is bounded by the single stream line JABA'J', over which we take 0 (Fig. 10). = At the branch point B, where the stream divides, the velocity is The w diagram consists of the single straight line = 0, but doubled back on itself at ub, so that coming from infinity at j along the under side of the line with the area to the right, a turn to port must be made on arriving at b by starboarding the helm and the turn must be made through two right angles to return along the upper side of the line, making a = — π. As u = b at the only corner of the w diagram, The dotted line bi in the prolongation corresponds to the part of the stream line = 0 along the curved dividing line BI in the stream, but this does not form part of the boundary, and along it, ww is negative, Q q = Q, log =0, and 0 diminishes from a to 0, so that the line ja is described. 0 = Passing from a to b, 0, and ab is described at right angles, and extending to infinity, since q = 0 at b, log Q q As u passes through b, changes suddenly from 0 to 7, SO that the 2 diagram continues in a straight line b a' at a height above ab, and arrives at a' on the line aj a' where log Q = 0. q Beyond a', from a' to j, log Q q = 0, and 0 diminishes from to a, so that j' rejoins j, and the circuit is complete. Then in accordance with the fundamental theorem (1) of conformal representation, ΦΩ is composed of the factors (u — b) ̄', (u — a) ̃ ̄ 1, (u — a') ̄ 1 ; du In the neighbourhood of ub, when the chief variation is due to u-b, we may replace u-a, u-a' by b-a, b-a', and put and since 2 increases by as u diminishes through b, equation (14) shows that log = a (ab. u a') + √(a' - b.ua) √ (a — a'. u — b) √ (a — b. u — a') + √ (a' — b. u — a) - v (ab.ua') - √(a' - b.ua) by theorems, of the Integral Calculus, which ought to be familiar to the student of this subject. At uj, (28) ai, Over the plane AA', a >u> a', 0 = 0, dz = dx, a - a' u - b" and integrating with respect to u, from a' to a, = cos2a (aa')2 + sin2 a (a a') + sin aπ (a — a')2, To express any length such as A'Px, as a function of u, it is convenient to introduce a variable angle (not to be confused with the velocity function ), as shown in the diagram (Fig. 11); = = Qdx = (a — a')2 sin2 ø. dx sin2 (a + AA' = 1 + a' = (a — a') sin2 4, (a + 4) sin & d þ ) sin & do sin a § sin 6 – § cos a cos o sin ☀ + ‡ sin a sin2 Þd & is obtained as in (41), Lecture I., by putting & = a. As the velocity diminishes from Q on the skin of the jet at atmospheric pressure to q something less in the interior, the |