(6)
φ = U (a2/r) cos θ = Ua2x/(x2 + y2),
(7)
ψ = −U (a2/r) sin θ = −Ua2y/(x2 + y2),
(8)
giving the motion due to the passage of the Cylinder r = a with velocity U through the origin O in the direction Ox.
If the direction of motion makes an angle θ′ with Ox,
| tan θ′ = | dφ | / | dφ | = | 2xy | = tan 2θ, θ = ½θ′, |
| dy | dx | x2 − y2 |
(9)
and the velocity is Ua2/r2.