When b = 0, U1 = [oo]; and when b = [oo], U1 = -U, so that at infinity the liquid is streaming in the direction xO with velocity U. If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by [omega] = -Uz, we are left with [omega] = Ua²/z, (6) [phi] = U(a²/r) cos [theta] = Ua²x/(x² + y²), (7) [psi] = -U(a²/r) sin [theta] = -Ua²y/(x² + y²), (8) giving the motion due to the passage of the Cylinder r = a with velocity U through the origin O in the direction Ox. If the direction of motion makes an angle [theta]´ with Ox, d[phi] / d[phi] 2xy tan[theta]´ = ----- / ----- = ------ = tan 2[theta], [theta] = ½[theta]´, (9) dy / dx x² - y² and the velocity is Ua²/r². Along the path of a particle, defined by the C3 of (3), y² y(y - c) sin² ½[theta]´ = ------- = -------, (10) x² + y² a² d[theta]´ 2y - c dy ½ sin [theta]´ --------- = ------ --, (11) ds a² ds on the radius of curvature is ¼a²/(y - ½c), which shows that the curve is an Elastica or Lintearia. (J. C. Maxwell, _Collected Works_, ii. 208.) If [phi]1 denotes the velocity function of the liquid filling the cylinder r = b, and moving bodily with it with velocity U1, [phi]1 = -U1x, (12) and over the separating surface r = b [phi] U / a²\ a² + b² --------- = - -- ( 1 + -- ) = -------, (13) [phi]1 U1 \ b²/ a² - b² and this, by § 36, is also the ratio of the kinetic energy in the annular interspace between the two cylinders to the kinetic energy of the liquid moving bodily inside r = b. Consequently the inertia to overcome in moving the cylinder r = b, solid or liquid, is its own inertia, increased by the inertia of liquid (a² + b²)/(a² - b²) times the volume of the cylinder r = b; this total inertia is called the effective inertia of the cylinder r = b, at the instant the two cylinders are concentric. With liquid of density [rho], this gives rise to a kinetic reaction to acceleration dU/dt, given by a² + b² dU a² + b² dU [pi][rho]b² ------- -- = ------- M´--, (14) a² - b² dt a² - b² dt if M´ denotes the mass of liquid displaced by unit length of the cylinder r = b. In particular, when a = [oo], the extra inertia is M´. When the cylinder r = a is moved with velocity U and r = b with velocity U1 along Ox, a² / b² \ b² / a²\ [phi] = U ------- ( --- + r ) cos [theta] - U1------- ( r + --- ) cos [theta], (15) b² - a² \ r / b² - a² \ r /