The onset of hydrodynamical instability induced by impulsive spin-down to rest in a cylinder containing a Newtonian fluid is analyzed by using propagation theory. It is well-known that the primary transient swirl flow is laminar, but with initial high velocities secondary motion sets in at a certain time. The dimensionless critical time τ_c to mark the onset of instability is presented here as a function of the Reynolds number Re. Available experimental data indicate that for large Re deviation of the velocity profiles from their momentum diffusion occurs starting from a certain time τ ≒ 4 τ_c. This means that secondary motion is detected at this characteristic time. It seems evident that during τ_c ≤ τ ≤ 4 τ_c, secondary motion is relatively very weak and the primary diffusive momentum transfer is dominant.