34. _Motion symmetrical about an Axis._--When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function [psi] can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2[pi]([psi] - [psi]0); and, as before, if d[psi] is the increase in [psi] due to a displacement of P to P´, then k the component of velocity normal to the surface swept out by PP´ is such that 2[pi]d[psi] = 2[pi]yk.PP´; and taking PP´ parallel to Oy and Ox, u = -d[psi]/ydy, v = d[psi]/ydx, (1) and [psi] is called after the inventor, "Stokes's stream or current function," as it is constant along a stream line (_Trans. Camb. Phil. Soc._, 1842; "Stokes's Current Function," R. A. Sampson, _Phil. Trans._, 1892); and d[psi]/yds is the component velocity across ds in a direction turned through a right angle forward. In this symmetrical motion d / 1 d[psi] \ d / 1 d[psi]\ [xi] = 0, [eta] = 0, 2[zeta] = -- ( --- ------ ) + -- ( --- ------ ) dx \ y dx / dy \ y dy / 1 /d²[psi] d²[psi] 1 d[psi]\ 1 = --- ( ------- + ------- - --- ------ ) = - ---[nabla]²[psi], (2) y \ dx² dy² y dy / y suppose; and in steady motion, dH 1 d[psi] dH 1 d[psi] -- + --- ----- [nabla]²[psi] = 0, -- + --- ------ [nabla]²[psi] = 0, (3) dx y² dx dy y² dy so that 2[zeta]/y = -y^(-2)[nabla]²[psi] = dH/d[psi] (4) is a function of [psi], say [f]´([psi]), and constant along a stream line; dH/dv = 2q[zeta], H - [f]([psi]) = constant, (5) throughout the liquid. When the motion is irrotational, d[phi] 1 d[psi] d[phi] 1 d[psi] [zeta] = 0, u = - ------ = - --- ------, v = - ------ = --- ------, (6) dx y dy dy y dx d²[psi] d²[psi] 1 d[psi] [nabla]²[psi] = 0, or ------- + ------- - --- ------ = 0. (7) dx² dy² y dy Changing to polar coordinates, x = r cos[theta], y = r sin[theta], the equation (2) becomes, with cos[theta] = [mu], d²[psi] d²[psi] r²------- + (1 - [mu]²) ------- = 2[zeta]r³ sin [theta], (8) dr² d[mu]² of which a solution, when [zeta] = 0, is / B \ dPn / B \ dPn [psi] = ( Ar^(n+1) + --- ) (1 - [mu]²) ----- = ( Ar^(n-1) + ------- ) y²-----, (9) \ r^n / d[mu] \ r^(n+2) / d[mu] [phi] = {(n + 1)Ar^n - nBr^(-n-1)} Pn, (10)