ON CONFIXED MOTION. 45 



The absolute time of the descent or ascent of a pendulum, in a cycloid, is to 

 the time in which any heavy body would fall through one half of the length 

 of the thread, as half the circumference of a circle to its diameter. It ia 

 therefore nearly equal to the time required for the descent of a body through 

 ^ of the length of the thread; and if we suffer the pendulum to descend, at 

 the same moment that a body falls, from a point elevated one fourth of the 

 length of the thread above the point of suspension, this body will meet the 

 pendulum at the lowest point of its vibration. (Plate II. Fig. Sil4.) 



Hence it may readily be inferred, that since the times of falling through any 

 spaces, are as the square roots of those spaces, the times of vibration of differ- 

 ent pendulums are as the square roots of their lengths. Thus, the times of 

 vibration of pendulums of 1 foot and 4 foot in length, will be as 1 to 2 : the 

 time of vibration of a pendulum 39 '4v inches in length is one second; the 

 length of a pendulum vibrating in two seconds must be four times as great. 



The velocity, with which a pendulous body moves, at each point of a cy- 

 cloidal curve, may be represented, by supposing another pendulum to revolve 

 imiformly in a circle, setting out from the lowest point, at the same time 

 that the first pendulum begins to move, and completing its revolution in the 

 time of two vibrations; then the height, acquired by the pendulum revolving 

 equably, will always be equal to the space described by the pendulum vibrat- 

 ing in the cycloid. (Plate II. Fig. 24.) 



It may also be shown, that if the pendulum vibrate through the whole curve, 

 it will everywhere move with the same velocity as the point of the circle 

 which is supposed to have originally described the cycloid, provided that the 

 circle roll onwards with an equable motion. 



All these properties depend on this circumstance, that the relative force, 

 urging the body to descend along the curve, is always proportional to the dis- 

 tance from the lowest point; and it happens in many other instances of the 

 action of various forces, that a similar law prevails: in all such cases, the vi- 

 brations are isochronous, and the space described corresponds to the versed 

 sine of a circular are increasing uniformly, that is to tlie height of any point 



