THE PENDULUM. 127 



tions, each having an amplitude of 5, then if the arc 

 of vibration were doubled, the same pendulum would 

 make only 998^ vibrations in the same time, and 

 if the amplitude were four times as great (20), it 

 would only vibrate 993 times in that time, making in 

 the first case 1^, in the second 7 vibrations less. In 

 our experiments, however, greater inaccuracies arise 

 from the difficulty of making both pendulums precisely 

 of equal length, than from the differences in their 

 amplitudes. 



Next let two pendulums be suspended as in fig. 93, 

 one of which is precisely four times as long as the other, 

 and let them be started at the same instant in the posi- 

 tions shown respectively at a and a lt The short 

 pendulum vibrates more quickly than the long one. It 

 arrives at bi at the instant when the long pendulum is 

 at c, and when the latter arrives at b the former has 

 returned to a^. While the long pendulum returns 

 from b to a, the short one completes again a whole 

 oscillation to and fro, so that both arrive now simultane- 

 ously at a and cti respectively. Two pendulums which 

 vibrate in unequal times cannot easily be observed 

 together at any intermediate points of their paths, 

 but the simultaneous return to the point of starting 

 after each whole oscillation of the long pendulum 

 can be accurately observed, and it may thus be ascer- 

 tained that the short pendulum makes twice as many 

 vibrations as the long one. If the experiment be re- 

 peated with longer pendulums, which may be swung 

 between door-posts, one pendulum may be made 9 times, 

 md again 16 times as long as the other, and it will be 

 Tound that in the former case 3, in the latter 4 vibra- 



