REDUCTION OF PENDULUM OBSERVATIONS. 
127 
may be made in order, without actually counting the oscil¬ 
lations, not to miss one. Repeating this operation, the 
period may be found to any desired degree of accuracy, at 
least up to the point where the uncertainties from other 
causes overshadow those from the clock and the pendulum 
itself. 
From two minutes’ observations the period of pendulum 
No. 4 was determined at Pakaoao to be l s .007, with an uncer¬ 
tainty of less than s .001. The pendulum was now allowed 
to swing 500 oscillations, the uncertainty only accumulating 
in this time to half a second. From this interval the period 
was found to be l s .0072, the uncertainty being less than 
s, 0001. With this data an interval of half an hour gives a 
value of F.00720 correct to the nearest one hundred thou¬ 
sandth of a second, and this value may be used in counting 
the oscillations for intervals of several hours. Each new 
approximation increases the accuracy tenfold and decreases 
the probable error of the result in an equal ratio. Hence 
the rule is to take sixty transits, allow ten minutes to elapse, 
and take forty more ; allow thirty minutes more to pass, and 
take forty additional transits. This gives sufficient data for 
the determination of the period. 
Whether the reductions are made on the principle of the 
reversible or the invariable pendulum, it becomes necessary 
to measure the distance of the center of mass from the two 
knife edges. The accuracy necessary here depends entirely 
on the relation between the times of oscillation in the two 
positions of the pendulum. Letting a be the difference in 
these times and h d — h a the difference of the distances, and 
h n the distance of the center of mass from the knife edge at 
the heavy end, the effect on the time of one oscillation will 
be given by the differential of the fraction 
<*K 
K—K 
For pendulum No. 4 
a= - s * 000031; 
hence for this pendulum an error of one millimeter in the 
location of the position of the center of mass will introduce 
