64 C. M. liKOOMAI.I, : 



tion of motion and cannot influence the velocity of the axis, 

 to every level of the axis there corresponds a certain velocity 

 over and above any initial velocity the axis may have had. 

 This additional velocity results from the interchano^e of poten- 

 tial and kinetic energy which is constantly occurring as the 

 axis rises and falls. Let us assume in what follows that no 

 initial velocity existed. Since the rotation of the wheel is 

 becoming slower, and consequently the deviating force a 

 smaller and smaller function of the velocity of the axis, obvi- 

 ously a time must come when this force will be so weakened 

 at the higher levels, where the velocity of the axis is small, 

 as to be unable to bring the path to a cusp. A reversal of 

 curv'ature will take place and the axis pass through a rounded 

 summit, beginning the second descent with a certain initial 

 velocity. The elevation of the summit will be less than the 

 original cusp since onlj- by a fall could the energy be obtained 

 for this initial velocity. The axis henceforth will be expected 

 to describe a series of prolate cycloids. The prolate cycloids 

 thus resulting, however, tecome flatter and flatter with suc- 

 cessive arcs because the decreasing gyroscopic force is less 

 and less able to raise the axis from its lowest level in each 

 loop, (Figure 4). Obviously the effect of the decreasing 

 gyroscopic force is to ultimately produce a horizontal path, 

 since sooner or later the deflecting force will be only just suf- 

 ficient to support the axis when moving horizontally with its 

 maximum speed without being able to deflect it upward. The 

 attainment of this path is delayed by the fact that the succes- 

 sive summits being lower and lower there is a continual increase 

 of horizontal velocity. The effect of this is to prevent the 

 deflective force from decreasing as rapidly as it otherwise 

 would. The ultimate result, however, is the production of a 

 horizontal path, the velocity being now uniform and equal to 

 the critical value. If henceforth no further diminution of 

 rotational velocity ensued the subsequent path of the axis 

 would be horizontal, supposing the absence of pivotal and 

 atmospheric friction, the o^-roscopic force just balancing the 



