(4a²y² - (y - c)^4 4a²y² + (y-c)^4
y - c = [root][2a(r - x)], x = -------------------, r = ---------------, (7)
(4a(y - c)² 4a(y - c)²
and in the absolute space curve given by [psi],
dy (y - c)² 2ac
-- = - --------, x = - ----- 2a log (y - c). (8)
dx 2ay y - c
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.