146 V. OSCILLATIONS AND CYCLIC MOTIONS. 



If the particle be allowed to ascend after passing the lowest 

 point of the cycloid it will rise to the same height from which it 

 fell and the motion being repeated, the time of a complete to- and 

 fro -oscillation is 



Thus the time of an oscillation of whatever amplitude on a cycloid 

 is the same as that of the infinitesimal oscillation of a circular 

 pendulum of length 4 a or twice the diameter of the rolling circle 

 which generates the cycloid. Since the time of oscillation is 

 independent of its amplitude we are led to the question of whether 

 the motion is harmonic. 



We may more generally inquire whether an isochronous or 

 tautochronous vibration is necessarily harmonic, that is: Is the 

 elongation of a particle, performing a vibration whose period is 

 independent of the amplitude, necessarily represented by a sine or 

 cosine function of the time? 



Let the distance along the path from the point to which the 

 motion is tautochronous be s. Then if the system is conservative 

 the force will be a function of s. Suppose 



dt* ds 



Multiplying by -rr and integrating we obtain the equation of energy 



*0 



where S is the initial value from which the particle started from 

 rest. The time of the motion from s = 5 to 5 = will be 



16) <=/y^ 



or putting 5 = S O M, 



o 



ds 



t = JL C s du 



y^J VF(S,U}- 



Differentiating by 5 0? 

 o 



- F(s}-F(s) - 

 W 



