Hates of a Rocking Watch. 601 



rates being respectively increasingly slower and diminishingly 

 faster than the chronometer. In fig. 3, the curve of long- 

 periods marches rapidly to -co, the other is asymptotic to 

 a horizontal line at 5 sec. /hour, the normal rate. The tendency 

 of the positive rates to reach a limit on the right is borne out 

 by the observations. The remarkably linear character of the 

 negative rates, however, is somewhat perplexing ; for if both 

 curves be prolonged to the left, the negative rates should tend 

 toward the asymptote + 5 sec/hour (watch at rest), while the 

 positive rates with continually decreasing ballast should tend 

 towards positive infinity. When the component period of 

 the pendulum falls below that of the balance-wheel, both 

 compound rates would soon be positive and the larger positive 

 rates become the more variable. The appearance of the 

 observation curves (fig. 3) is such as would lead one to 

 suspect an intersection when Sk/k— — '15; i.e. when the 

 inertia of the watch has been reduced 15 per cent. [ regret 

 that the character of the mounting did not enable me to 

 enter the negative field with the object of exhibiting the 

 curves in full, in both directions. 



(4) In the case of the long period no beating was observed 

 or producible. The component periods are increasingly 

 different as 8k increases. The overwhelming inertia of the 

 watch in. this case completely controls the motion of the 

 balance-wheel. To obtain beats the balance-wheel must be 

 given an advantage by starting the pendulum as nearly as 

 possible from rest. In the case of the short period started 

 with a moderate arc of vibration, beating is almost an 

 invariable accompaniment of the incipient motion. As a 

 first approximation the system may be compared with the 

 case of a single degree of freedom, undergoing forced 

 vibrations imposed by the escapement, eventually to respond 

 permanently to the modified period of the balance-wheel. 

 Meanwhile, however, the independent simple harmonic 

 motion of the pendulum evoked on starting is vanishing 

 exponentially. The observed beating is an interference of the 

 two sets of vibrations of nearly the same period. When the 

 pendulum vibrations proper have run their coarse beating 

 ceases. 



This explanation can only be partial, since it implies the 

 existence of but one compound period instead of two, and 

 since beats are particularly marked when the pendulum is 

 started from rest. They occur during the period of mutual 

 reaction, while the balance-wheel is stimulating the clock to 

 vibrate permanently in the smaller period, and they appear 

 as a hovering around the period of equilibrium which is very 



Phil. Mag. S. 5. Vol. 50. No. 307. Dec. 1900. 2 U 



