150 OF DEFLECTIVE FORCES. 



formly, its half CBD must also increase uniformly. And 

 if the motion begin at any other point of the curve, it 

 follows, from the former part of the demonstration, that 

 the velocity will be in a constant ratio to the velocity in 

 similar points of the whole cycloid. It is also obvious 

 that the arc of ascent will be equal to the arc of descent, 

 and described in an equal time, supposing the motion 

 without friction. 



288. Theorem. "260.'' The times of 

 vibration of different cycloidal pendulums 

 are as the square roots of their lengths. 



For the times of falling through half their lengths are in 

 the ratio of the square roots of these halves, or of the 

 wholes. 



Scholium. Major Kater has ascertained, by a great 

 number of very accurate experiments, performed with an 

 apparatus of his own invention, that the length of the 

 pendulum vibrating in a second in London, on the level of 

 the Thames, and in a vacuum, is 39'14 inches, very nearly. 

 Hence the time of falling through 19*57 inches will be 



V 



— , and the space described in a second 19*57 x cr^. Now 



TT 



Log 31415922== -9943 and Log 19*57 =1*291 6; their sum 

 2-2859 is the logarithm of 193*15 inches, or 16*096 feet, 

 the space described by a heavy body in the first second of 

 its descent. More accurately the numbers are 39.1387 

 and 16.095. 



289. Theorem. " 261." The cycloid is 

 the curve of swiftest descent between any two 

 points not in the same vertical line. 



