(2)
an elliptic-function Fourier series; with a similar expression for ψ2 with x and y, a and b interchanged; and thence ψ = ψ1 + ψ2.
Example 4.—Parabolic cylinder, axial advance, and liquid streaming past.
The polar equation of the cross-section being
r1/2 cos ½θ = a1/2, or r + x = 2a,
(3)
the conditions are satisfied by
ψ′ = Ur sin θ − 2Ua1/2r1/2 sin ½θ = 2Ur1/2 sin ½θ (r1/2 cos ½θ − a1/2),
(4)
ψ = 2Ua1/2r1/2 sin ½θ = −U √ [ 2a (r − x) ],