(22)
If A, B have the same sign, this is equivalent to
au = cosh mθ,
(23)
if the origin of θ be suitably adjusted; hence r has a maximum value α, and the particle ultimately approaches the pole asymptotically by an infinite number of convolutions. If A, B have opposite signs the form is
au = sinh mθ,
(24)
this has an asymptote parallel to θ = 0, but the path near the origin has the same general form as in the case of (23). If A or B vanish we have an equiangular spiral, and the velocity at infinity is zero. In the critical case of h2 = μ, we have d2u/dθ2 = 0, and
u = Aθ + B;
(25)