N1 = [ -([alpha]M´U´ + [xi]) sin ([theta] - Rt) + [beta]M´V´ cos ([theta] - Rt)OO´ = [ -([alpha] - [beta])´Q cos ([theta] - Rt) sin ([theta] - Rt) - [xi] sin ([theta] - Rt)]Qt. (5) The components of force, X, Y, and N, acting on the liquid at O, and reacting on the body, are then X = lt. X1/t = ([alpha] - [beta])M´QR sin [theta] = ([alpha] - [beta])M´VR, (6) Y = lt. Y1/t = ([alpha] - [beta])M´QR cos [theta] + [xi]R = ([alpha] - [beta])M´UR + [xi]R, (7) Z = lt. Z1/t = -([alpha] - [beta])M´Q² sin [theta]cos[theta] - [xi]Q sin [theta] = [-([alpha] - [beta])M´U + [xi]]V. (8) Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid / dU \ / dV \ dR -[alpha]M´ ( -- - VR ), -[beta]M´ ( -- + UR ), [epsilon]C´--, (9) \ dt / \ dt / dt so that its equations of motion are / dU \ / dU \ M ( -- -VR ) = -[alpha]M´ ( -- - VR ) - ([alpha] - [beta])M´VR, (10) \ dt / \ dt / / dV \ / dV \ M ( -- + UR ) = -[beta]M´ ( -- + UR ) - ([alpha] - [beta])M´UR - [xi]R, (11) \ dt / \ dt / dR dR C-- = -[epsilon]C´-- + ([alpha] - [beta])M´UV + [xi]V; (12) dt dt and putting as before M + [alpha]M´ = c1, M + [beta]M´ = c2, C + [epsilon]C´ = C3, (13) dU c1-- - c2 VR = 0, (14) dt dV c2-- + (c1 U + [xi])R = 0, (15) dt dR c3-- - (c1U + [xi] - c2 U)V = 0; (16) dt showing the modification of the equations of plane motion, due to the component [xi] of the circulation. The integral of (14) and (15) may be written c1U + [xi] = F cos [theta], c2V = - F sin [theta], (17) dx F cos² [theta] F sin² [theta] [xi] -- = U cos [theta] - V sin [theta] = -------------- + -------------- - ---- cos [theta], (18) dt c1 c2 c1 d[mu] / F F \ [xi] ----- = U sin [theta] + V cos [theta] = ( -- - -- ) sin [theta] cos [theta] - ---- sin [theta], (19) dt \ c1 c2 / c1