  The feasibility of registering seconds using the frictionless motion of a
point-like particle that slides under gravity on an inverted conical surface is
studied. Depending on the integer part of the relation between the angular and
radial frequencies of the particle trajectory, only an angular interval for the
cone is available for this purpose. For each one of these possible angles,
there exists a unique trajectory that has the capability of registering
seconds. The method to obtain the geometrical properties of these trajectories
and the necessary initial conditions to reach them are then established.
