Till': c.vKoscoi'i^. 59 



The deviating effect of the gyroscopic force continues until 

 the end of the axis just comes to rest with the tangent to its 

 path vertical. The axis is now at the same level as it was 

 when first released, all the velocity'' gained in its fall having 

 been lost in bringing the mass up to the same level again. 

 The end of the axis now begins to fall again, the gyroscopic 

 couple once more exerts its deviating force and another arc is 

 described. The free end of the axis henceforth describes a 

 series of small cycloidal arcs, the axis coming up to the same 

 level after each arc, ( P'igure 2). Under ordinary circum- 

 stances with high wheel velocity the nutations of the axis are 

 too small to be perceived. The motion of the end of the axis 

 is analogous to that of a vibrating pendulum with its constant 

 interchange of potential and kinetic energy, the gyroscopic 

 force taking the place of the supporting rod or cord in the 

 pendulum. In the operation of the apparatus no exhaustion 

 of energy of rotation of the wheel occurs, since no external 

 work is done and the average orbital velocity is constant. 

 The axis necessarily moves with constant average orbital velo- 

 city because the loops of the path are equal and similar, a 

 necessary conse(iuence of being described under similar cir- 

 cumstances. 



The two halves of each cycloidal arc are similar and equal, 

 as may be shown by the following reasoning : The velocity of 

 the end of the axis at any point, resulting from the conver- 

 sion of potential into kinetic energy, is a function only of the 

 distance the axis has fallen. The deviating gyroscopic force 

 being always at right angles to the path of the axis cannot 

 alter the velocity in any way. On the upward arc the loss of 

 velocity at any point is a function of the distance the axis 

 has risen, or in other words, the velocity remaining at any 

 point on the upward arc is a function of the distance it has 

 yet to rise to regain the original level. The total energy of 

 the system being constant and equal to the sum of the poten- 

 tial and kinetic energies, it is evident that for every elevation 

 of the axis there must correspond a given velocity of the 



