[phi] + [psi]i = F(tan ½[theta]·e^([lambda]i)). (14) 28. _Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it._--A stream-function [psi] must be determined to satisfy the conditions [Nabla]²[psi] = 0, throughout the liquid; (1) [psi] = constant, over any fixed boundary; (2) d[psi]/ds = normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity U in the direction Ox, d[psi]/ds = -Udy/ds, or [psi] + Uy = constant over the moving cylinder; and [psi] + Uy = [psi]´ is the stream function of the relative motion of the liquid past the cylinder, and similarly [psi] - Vx for the component velocity V along Oy; and generally [psi]´ = [psi] + Uy - Vx (4) is the relative stream-function, constant over a solid boundary moving with components U and V of velocity. If the liquid is stirred up by the rotation R of a cylindrical body, d[psi]/ds = normal velocity reversed dx dy = -Rx-- - Ry--, (5) ds ds [psi] + ½R(x² + y²) = [psi]´, (6) a constant over the boundary; and [psi]´ is the current-function of the relative motion past the cylinder, but now V²[psi]´ + 2R = 0, (7) throughout the liquid. Inside an equilateral triangle, for instance, of height h, [psi]´ = -2R[alpha][beta][gamma]/h, (8) where [alpha], [beta], [gamma] are the perpendiculars on the sides of the triangle. In the general case [psi]´ = [psi] + Uy - Vx + ½R(x² + y²) is the relative stream function for velocity components, U, V, R. 29. _Example 1._--Liquid motion past a circular cylinder. Consider the motion given by [omega] = U(z + a²/z), (1) so that / a²\ / a² \ [phi] = U ( r + -- ) cos [theta] = U ( 1 + -- )x, (2) \ r / \ r² / / a²\ / a² \ [psi] = U ( r - -- ) sin [theta] = U ( 1 - -- )y. \ r / \ r² / Then [psi] = 0 over the cylinder r = a, which may be considered a fixed post; and a stream line past it along which [psi] = Uc, a constant, is the curve / a²\ ( r - -- ) sin [theta] = c, (x² + y²)(y - c) - a²y = 0 (3) \ r / a cubic curve (C3). Over a concentric cylinder, external or internal, of radius r = b, / a²\ [psi]´ = [psi] + U1y = [U ( 1 - --- ) + U1] y, (4) \ b²/ and [psi]´ is zero if U1/U = (a² - b²)/b²; (5) so that the cylinder may swim for an instant in the liquid without distortion, with this velocity U1, and [omega] in (1) will give the liquid motion in the interspace between the fixed cylinder r = a and the concentric cylinder r = b, moving with velocity U1.