For instance, with n = 1 in equation (9), the relative stream function is obtained for a sphere of radius a, by making it
ψ′ = ψ + ½Uy2 = ½U (r2 − a3/r) sin2 θ, ψ = −½Ua3 sin2 θ/r;
(13)
and then
φ′ = Ux (1 + ½a3/r2), φ = ½Ua3 cos θ/r2,
(14)
| − | dφ | = U | a3 | cos θ, − | dφ | = ½U | a3 | sin θ, |
| dr | r3 | r dθ | r3 |
(15)
so that, if the direction of motion makes an angle ψ with Ox,
tan (ψ − θ) = ½ tan θ, tan ψ = 3 tan θ/(2 − tan2 θ),