OBSERVATION OF OSCILLATIONS. 33 



If the movable body is left to itself at the time /= 0, at a distance 

 a from its position of equilibrium, equation (25) gives then 



(28) 



^= 

 dt 



The movable body approaches its position of equilibrium 

 asymptotically; the velocity, which starts from zero, reaches its 



maximum at the time /=-, and then decreases to zero. 



e 



For the case of an initial velocity - o> directed towards the 

 position of equilibrium, equation (26) gives 



***-<.[ -(- oc)/], 



(29) 



it will be seen, that the movable body will only pass beyond its 

 position of equilibrium if we have w>ae. 



Lastly, when the movable body is thrown from its position of 

 equilibrium with the velocity w, we get by equation (27) 



The elongation is reached at the time / = - , and its value is 



689. OBSERVATION OF OSCILLATIONS. It follows from the 

 preceding discussion that all the circumstances of oscillating motion 

 will be defined by the time of the successive elongations, and by the 

 magnitude of the angle of deflection. 



VOL. II. D 



