ON THE TAUTOCHRONISM OF THE 

 CYCLOID*. 



CONCEIVE two points, not acted on by gravity, to move on 

 the circumference of a stationary circle towards its lowest point : 

 the plane of the circle we will suppose to be vertical. Let their 

 motion be such that the ratio between their distances from the 

 lowest point may be invariable. Then their velocities towards 

 that point must be in that invariable ratio. Their heights above 

 it are in the duplicate ratio: their initial heights above it were 

 also in the duplicate ratio: so likewise therefore are their ver- 

 tical descents towards it. In other words, the squares of their 

 velocities towards the lowest point are as their vertical descents. 



Conceive one of the points to be, when the other starts, at 

 the highest point of the circle, and to move with a c6nstant 

 velocity (#a) J , a being the radius of the circle. Then, when it 

 has descended through a vertical space z from its initial position, 

 the square of its velocity towards the lowest point of the circle 

 is equal to \gz. On this supposition with respect to one point, 

 it appears, from what has been said before, that the square of 

 the velocity of the other point towards the lowest point of the 

 circle is similarly equal to \gz ', z being the quantity correspond- 

 ing to z, viz. its vertical descent below its initial position. 



Now suppose the circle to move horizontally in its own plane 

 with a velocity equal at every instant to the velocity, along the 

 arc, of one of the points, the direction of the motion of the circle 

 being towards the right or the left accordingly as the point is to 

 the right or the left of the vertical diameter. If the former point 

 be chosen, then the velocity of the circle will be constant, if the 



* Walton's Probkms in Ekmentary Mechanics, p. 245. 



