a² / b² \ b² / a²\ [psi] = -U ------- ( --- - r ) sin [theta] - U1------- ( r - --- ) sin [theta]; (16) b² - a² \ r / b² - a² \ r / and similarly, with velocity components V and V1 along Oy a² / b² \ b² / a²\ [phi] = V ------- ( --- + r ) sin [theta] - V1------- ( r + --- ) sin [theta], (17) b² - a² \ r / b² - a² \ r / a² / b² \ b² / a²\ [psi] = V ------- ( --- - r ) cos [theta] + V1------- ( r - --- ) cos [theta], (18) b² - a² \ r / b² - a² \ r / and then for the resultant motion a² z a²b² U + Vi w = (U² + V²) ------- ------ + ------- ------ b² - a² U + Vi b² - a² z a² z a²b² U1 + V1i -(U1² + V1²) ------- -------- + ------- --------. (19) b² - a² U1 + V1i b² - a² z The resultant impulse of the liquid on the cylinder is given by the component, over r = a (§ 36), _ / / b² + a² 2b² \ X = | [rho][phi] cos [theta]·ad[theta] = [pi][rho]a² ( U ------- - U1 ------- ); (20) _/ \ b² - a² b² - a²/ and over r = b _ / / 2a² b² + a² \ X1 = | [rho][phi] cos [theta]·bd[theta] = [pi][rho]b² ( U ------ - U1------- ), (21) _/ \ b² - a² b² - a² / and the difference X - X1 is the component momentum of the liquid in the interspace; with similar expressions for Y and Y1. Then, if the outside cylinder is free to move V1 2a² b² - a² X1 = 0, -- = -------, X = [pi][rho]a²U -------. (22) U b² + a² b² + a² But if the outside cylinder is moved with velocity U1, and the inside cylinder is solid or filled with liquid of density [sigma], U1 2[rho]b² X = -[pi][rho]a²U, -- = --------------------------------, U [rho](b² + a²) + [sigma](b² - a²) U - U1 ([rho] - [sigma])(b² - a²) ------ = ---------------------------------, (23) U1 [rho](b² + a²) + [sigma](b² - a²) and the inside cylinder starts forward or backward with respect to the outside cylinder, according as [rho] > or < [sigma]. 30. The expression for [omega] in (1) § 29 may be increased by the addition of the term im log z = -m[theta] + im log r, (1) representing vortex motion circulating round the annulus of liquid.