this is a minimum for θ = 0 only so long as ω² < g/a. For greater values of ω the only position of “permanent” stability is that in which the particle rotates with the bowl at an angular distance cos−1 (g/ω²a) from the lowest point. To examine the motion in the neighbourhood of the lowest point, when frictional forces are taken into account, we may take fixed ones, in a horizontal plane, through the lowest point. Assuming that the friction varies as the relative velocity, we have
ẍ = −p²x − k (ẋ + ωy),
ÿ = −p²y − k (ẏ − ωx),
(3)
where p² = g/a. These combine into
z¨ + kz˙ + (p² − ikω) z = 0,
(4)
where z = x + iy, i = √−1. Assuming z = Ceλt, we find
λ = −½k(1 ∓ ω/p) ± ip,
(5)
if the square of k be neglected. The complete solution is then