Considered by itself, with the cylinders held fixed, the vortex sets up a circumferential velocity m/r on a radius r, so that the angular momentum of a circular filament of annular cross section dA is [rho]mdA, and of the whole vortex is [rho]m[pi](b² - a²). Any circular filament can be started from rest by the application of a circumferential impulse [pi][rho]mdr at each end of a diameter; so that a mechanism attached to the cylinders, which can set up a uniform distributed impulse [pi][rho]m across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple -[rho]m[pi]a² and [rho]m[pi]b², having resultant [rho]m[pi](b² - a²), and this couple is infinite when b = [oo], as the angular momentum of the vortex is infinite. Round the cylinder r = a held fixed in the U current the liquid streams past with velocity q´ = 2U sin [theta] + m/a; (2) and the loss of head due to this increase of velocity from U to q´ is q´² - U² (2U sin [theta] + m/a)² - U² -------- = ----------------------------, (3) 2g 2g so that cavitation will take place, unless the head at a great distance exceeds this loss. The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin[theta]/ga, and its thrust is 2[pi][rho]mU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r = a with velocity U reversed, and the cylinder is surrounded by a vortex. Similarly, the streaming velocity V reversed will give rise to a thrust 2[pi][rho]mV in the direction xC. Now if the cylinder is released, and the components U and V are reversed so as to become the velocity of the cylinder with respect to space filled with liquid, and at rest at infinity, the cylinder will experience components of force per unit length (i.) - 2[pi][rho]mV, 2[pi][rho]mU, due to the vortex motion; (ii.) - [pi][rho]a² dU/dt, -[pi][rho]a² dV/dt, due to the kinetic reaction of the liquid; (iii.) 0, -[pi]([sigma] - [rho])a²g, due to gravity, taking Oy vertically upward, and denoting the density of the cylinder by [sigma]; so that the equations of motion are dU dU [pi][rho]a²-- = - [pi][rho]a²-- - 2[pi][rho]mV, (4) dt dt dV dV [pi][rho]a²-- = - [pi][rho]a²-- + 2[pi][rho]mV - [pi]([sigma] - [rho])a²g, (5) dt dt or, putting m = a²[omega], so that the vortex velocity is due to an angular velocity [omega] at a radius a, ([sigma] + [rho])dU/dt + 2[rho][omega]V = 0, (6) ([sigma] + [rho])dV/dt - 2[rho][omega]U + ([sigma] - [rho])g = 0. (7)