EK : EH :: OC : OD.
Let vE be the linear velocity of the point E fixed in the plane of axes AOB. Then
vK = α · EK.
Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone. That is to say,
β · EH = vE = α · EK;
therefore
α : β :: EH : EK :: OD : OC,
which is the remainder of the solution.
The path of a point P in or attached to the rolling cone is a spherical epitrochoid traced on the surface of a sphere of the radius OP. From P draw PQ perpendicular to the instantaneous axis. Then the motion of P is perpendicular to the plane OPQ, and its velocity is
vP = γ · PQ.