154 OF DEFLECTIVE FORCES. 



^t=:f--^dKv=-^ Iv^d--^ ^x = 0. Hence 



^ asjijaz as/ydz a^/yaz 



dx dx 



•^azzcl — -— -r- Zxf which vanishing for the whole 



a^Jydz ais/ydz 



d?/ 

 curve and for all its parts, as d -^ was shown to vanish 



dx 

 before, it follows that^ — -— j— must be a constant quantity ; 



which is the property of the cycloid. 



290. Theorem. " 262/' The time of 

 vibration of a simple circular pendulum, in 

 a small arc, is ultimately the same as that of 

 a cycloidal pendulum of the same length ; 

 " but in larger arcs the times are greater/' 

 (280). 



In small cycloidal arcs the radius of curvature is very 

 nearly constant ; but at greater distances from the lowest 

 point, the circular arc falls without the cycloidal, andis less 

 inclined to the horizon, so that the force is smaller, and 

 consequently the velocity is smaller. 



291. Theorem. '^263.'' If a body sus- 

 pended by a thread revolve freely round 

 the vertical line, the times of revolution will 

 be the same, when the height of the point of 

 suspension above the plane of revolution is 

 the same, whatever be the length of the 

 thread. 



For, by the resolution of forces, the force urging the body 

 towards the vertical line is to that of gravity as the dis- 



