An experiment was devised by Lord Kelvin for demonstrating this, in which the difference of steadiness was shown of a copper shell filled with liquid and spun gyroscopically, according as the shell was slightly oblate or prolate. According to the theory above the stability is regained when the length is more than three diameters, so that a modern projectile with a cavity more than three diameters long should fly steadily when filled with water; while the old-fashioned type, not so elongated, would be highly unsteady; and for the same reason the gas bags of a dirigible balloon should be over rather than under three diameters long. 40. _A Liquid Jet._--By the use of the complex variable and its conjugate functions, an attempt can be made to give a mathematical interpretation of problems such as the efflux of water in a jet or of smoke from a chimney, the discharge through a weir, the flow of water through the piers of a bridge, or past the side of a ship, the wind blowing on a sail or aeroplane, or against a wall, or impinging jets of gas or water; cases where a surface of discontinuity is observable, more or less distinct, which separates the running stream from the dead water or air. Uniplanar motion alone is so far amenable to analysis; the velocity function [phi] and stream function [psi] are given as conjugate functions of the coordinates x, y by w = [f](z) where z = x + yi, w = [phi] + [psi]i, (1) and then dw d[phi] d[psi] -- = ------ + i------ = -u + vi; (2) dz dx dx so that, with u = q cos [theta], v = q sin [theta], the function dz Q Q Q [zeta] = -Q -- = -------- = ---(u + vi) = --- (cos [theta] + i sin [theta]), (3) dw (u - vi) q² q gives [zeta] as a vector representing the reciprocal of the velocity q in direction and magnitude, in terms of some standard velocity Q. To determine the motion of a jet which issues from a vessel with plane walls, the vector [zeta] must be Constructed so as to have a constant direction [theta] along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant. It is convenient to introduce the function [Omega] = log [zeta] = log(Q/q) + [theta]i (4) so that the polygon representing [Omega] conformally has a boundary given by straight lines parallel to the coordinate axes; and then to determine [Omega] and w as functions of a variable u (not to be confused with the velocity component of q), such that in the conformal representation the boundary of the [Omega] and w polygon is made to coincide with the real axis of u. [Illustration: FIG. 4.] It will be sufficient to give a few illustrations.