.V THE TAUTOCHRONISM OF THE CYCLOID. 327 



latter point, it will be variable. In either case, the path of the 

 point selected obviously becomes a cycloid, and it is easily seen 

 that the velocity of the point towards the lowest point of the 

 circle is destroyed by the motion of the circle itself, while the 

 velocity at right angles to this direction is doubled : consequently 

 the whole velocity of the point will have, for its square, 2gz or 

 2gz, and we have thus a perfect representation of cycloidal mo- 

 tion under the action of gravity. But it is obvious from the 

 fundamental hypothesis that the two points will reach the lowest 

 point of the circle at the same time: that is to say, the descent 

 to the lowest point of the cycloid is tautochronous. 



The analogy to the descent to the lowest point of a circle 

 along its chords is in the essential point complete, but here the 

 motion is not along the chord but along the arc, and is compli- 

 cated with the motion of the circle itself. In both cases it is 

 easily seen that a medium, the resistance of which varies as the 

 velocity, does not affect the tautochronism. 



