where Pn denotes the zonal harmonic of the nth order; also, in the exceptional case of [psi] = A0 cos[theta], [phi] = A0/r; [psi] = B0r, [phi] = -B0 log tan ½[theta] = -½B0 sh(-1) x/y. (11) Thus cos[theta] is the Stokes' function of a point source at O, and PA - PB of a line source AB. The stream function [psi] of the liquid motion set up by the passage of a solid of revolution, moving with axial velocity U, is such that 1 d[psi] dy --- ------ = -U --, [psi] + ½Uy² = constant, (12) y ds ds over the surface of the solid; and [psi] must be replaced by [psi]´ = [psi] + ½Uy² in the general equations of steady motion above to obtain the steady relative motion of the liquid past the solid. For instance, with n = 1 in equation (9), the relative stream function is obtained for a sphere of radius a, by making it [psi]´ = [psi] + ½Uy² = ½U(r² - a³/r) sin² [theta], [psi] = -½Ua³ sin² [theta]/r; (13) and then [phi]´ = Ux(1 + ½a³/r²), [phi] = ½Ua³ cos [theta]/r², (14) d[phi] a³ d[phi] a³ - ------ = U -- cos [theta], - --------- = ½U -- sin [theta], (15) dr r³ rd[theta] r³ so that, if the direction of motion makes an angle [psi] with Ox, tan ([psi] - [theta]) = ½ tan [theta], tan [psi] = 3 tan [theta]/(2 - tan² [theta]), (16) Along the path of a liquid particle [psi]´ is constant, and putting it equal to ½Uc², (r² - a³/r) sin² [theta] = c², sin² [theta] = c²r/(r³ - a³), (17) the polar equation; or y² = c²r³/(r³ - a³), r³ = a³y²/(y² - c²), (18) a curve of the 10th degree (C10). In the absolute path in space cos [psi] = (2 - 3 sin² [theta])/[root](4 - sin² [theta]), and sin³ [theta] = (y³ - c²y)/a³, (19) which leads to no simple relation. The velocity past the surface of the sphere is 1 d[psi]´ / a³ \ sin² [theta] ------------ ------- = ½U ( 2r + -- ) ------------- = 3/2 U sin [theta], when r = a; (20) r sin[theta] dr \ r² / r sin [theta] so that the loss of head is (9/4 sin² [theta] - 1) U²/2g, having a maximum 5/4 U²/2g, (21) which must be less than the head at infinite distance to avoid cavitation at the surface of the sphere. With n = 2, a state of motion is given by [psi] = -½Uy²a^4[mu]/r^4, [psi]´ = ½Uy²(1 - a^4[mu]/r^4), (22) [phi]´ = Ux + [phi], [phi] = -1/3 U(a^4/r³)P2, P2 = 3/2 [mu]² - ½, (23) representing a stream past the surface r^4 = a^4[mu].