NEWTON S PRINCIPIA. 245 



.*. u = L cos n (t A). 



Supposing the time to be counted from the epoch when the 

 pendulum began its first complete oscillation from its 

 highest point, then X = 0. 



_ X / 



. * . s = L s 5Tn cos n t 



Hence we learn, 



1. That the oscillations will be isochronal, and the time 

 of oscillation 





This differs from the time of oscillation in a vacuum ; if 

 x be small, and T the time in vacuo, the difference will be 

 very nearly 



x 2 / 



= 8mV 



If we neglect terms so small as this, the time will be 

 unaltered. 



2. The arcs of each successive oscillation will decrease 

 in geometrical progression, and if L be any one such arc, 



X X 



the next will be L s ~ / . 



A particle is constrained to oscillate in a cycloid wlwse axis is 

 vertical under the action of gravity, and is resisted by the 

 medium in which it moves in the ratio of the square of the 

 velocity. Determine the motion. Newt. -xxix. 



Following the same notation as before, it is manifest 

 that the equation of motion will be 



d v x 



v -j- = ir s v 



as m 



R 3 



