Rates of a Rocking Watch. 



599 



been taken at equal intervals of time apart. The figure 

 brings oat the long period (negative rate) and the short 

 period (positive rate) very well. The relative increment, 

 8k/k, of moment of inertia is inscribed on each carve. 

 Considerable fluctuation is observed in all the individual 

 rates, due to the imperfect watch, for they remain when the 

 watch is fixed. Sometimes definitely intermediate values 

 (as under 8k/k = 0) are in evidence, due to the spontaneous 

 change of the mode of vibration from the long to the short 

 period, the nature of which I will indicate below. 



Table II. — 'Relation of 8k/ k to rate corresponding to the 

 long period and to the short period. 



Sk/k. 



Rate of long 



period : 



sec. /hour. 



Ratio of 

 increments*. 



Rate of short 



period : 



sec/hour. 



Ratio of 

 increments *. 









•ooo 



- 39 8 



-42 



+ 17-5 



+ •6 



•013 



- 397 





18-5 





•023 







17-1 





•030 



- 460 





161 





•046 



- 58-3 





150 





•091 



- 75-3 





132 





•240 



-1350 





50 





•380 . 



- 1993 





5-0 





Finally, if the successive rates be plotted in their dependence 

 on the change of moment of inertia (8k/k) due to the ballast f 

 added, the curves fig. 3 will be obtained. Within the limits 

 imposed by the errors of the watch and the errors of observa- 

 tion both are nearly straight lines, the values for the long 

 period being much more clear-cut than those for the short 

 period. These values are also given in Table II., from which 

 it appears that numerically the rate for the long period 

 increases 4V2 seconds per hour for each per cent, of relative 

 increment (8k/ fc) of the moment of inertia of the watch. 

 The rate for the short period decreases on the average about 

 "6 sec. /hour for each percent, of 8k/k, within the greater part 

 of the range observed. Here, however, the approach to a 

 limit is definitely manifested. 



(3) The theory of a system of two degrees of freedom is 

 given by Lord Eayleigh in the fifth chapter of ' Sound,' 



* 8k/ Jc in per cents. 



t I have expressed the rates in terms of the increments of moment of 

 inertia for convenience. It should not be overlooked that ballast increases 

 the gravitational torque. 



