GOO Prof. C. Barus on the 



§§ 112-118. The inferences to be drawn when the system is 

 assimilated to the case of a single degree of freedom are 

 adduced in the third chapter. Inasmuch as I purpose merely 

 to present the above observations as an interesting illustration 

 of the general theory, having a possible practical application, 

 a few remarks will suffice for guidance. 



The present system of two degrees of freedom may vibrate 

 permanently in one of two ways : either the balance-wheel 

 and the pendulum (watch) vibrate in the same phase, in 

 which case the torque of the hair-spring is slowly paid out, 

 and there will be a long- period and a long swing of the 

 pendulum ; or the two elements vibrate in opposed phases, 

 in which case torque is rapidly expended, the common period 

 is short, and the arc of vibration small. The Ions; period may 

 usually be established from a sweeping swing of the pendulum, 

 while the short period is reached spontaneously from rest. 

 The adjustment is often difficult ; as a consequence the two 

 periods occur in the curves in irregular succession, the 

 attempted predisposition having failed {cf. fig. 1). The two 

 compound periods found were invariably different. If the two 

 component periods are not equal, there can be no proportion- 

 ality between corresponding terms in the Lagrangian function. 

 If they are equal this can take place only for one arc of 

 vibration of the pendulum. Beyond this the case is actually 

 one of forced vibrations, and the energy of the system receives 

 a regular accession from the escapement. 



The treatment of the problem involves the consideration of 

 four frequencies ; the two belonging to the compound free 

 system as observed above, and the two individual frequencies 

 of the component elements, the balance-wheel and the pendu- 

 lum, when vibrating alone. I have already called these com- 

 pound and component frequencies respectively. They must 

 be so grouped in relative magnitude that the two latter lie 

 within the limits of the two former, while an increase of 

 the inertia of the system (ballast) increases both periods. 



In the observations above the period of the pendulum 

 alone is successively increased, whereas the period of the 

 balance alone is left unchanged. Hence the long period of 

 the system, which must exceed that of the watch, is suc- 

 cessively pushed forward to infinity. The short period of 

 the system, remaining ever smaller than that of the balance 

 alone, is also increased gradually to reach the limit given by 

 the fixed period of the balance-wheel stated : i. e., the limit 

 corresponding to the normal rate of the watch at rest. In 

 the experiments, the occurrence of the two compound periods 

 and their variations is remarkably well shown, the compound 



