16 THE ANTARCTIC MANUAL. 
The correction to the period which arises from this is given by 
the formula 
_ PM sin(¢ + ¢') sin (¢ — ¢) 
32 log sin @ — log sin ¢’ 
Correction = 
when ¢ and ¢’ are the observed amplitudes, P the period and M 
the modulus of common logarithms, 0°4343. 
Rate Correction.—This is necessary in order to correct for the rate 
of the clock. The correction is 
+ 0°00001157 RP 
where R is the observed rate in sidereal time in seconds per day, 
P the period. R is positive if the clock is losing. 
Flexure Correction.—The stand is not perfeetly rigid, and in con- 
sequence of its yielding a correction is necessary to the observed 
period. In the instrument two of the pendulums B, C, are mounted 
so that they swing in the same plane on opposite sides of the support ; 
the plane of vibration of the third pendulum is at right angles to that 
of B and C. 
If the stand were perfectly rigid and B were set in motion, A and 
C being at rest, the motion of B would not be transmitted to C and it 
would remain at rest. In consequence of the want of rigidity C is 
set in motion, and since the period of C is nearly the same as that of 
B a slight yielding of the stand will produce a considerable motion 
in C. From observations on the motion of C the correction to the 
period of B can be determined. 
To determine this correction pendulum B is set swinging, pen- 
dulum C being at rest; after a short time it will be found that C is 
in motion. Observations are made of its amplitude at equal intervals 
of time—the length of the interval will depend on the amount of 
yielding—and it will be found that the amplitude of C gradually 
increases while that of B decreases. After a time the amplitude of 
C reaches a maximum, it then decreases again, passing through a 
minimum and so on in a regular cycle. 
A series of observations are made of the time and the corresponding 
arcs of vibration a, a’ of the two pendulums B and C respectively, 
then if s, s' be the period of the two, it can be shown that the flexural 
correction to s is given by the expression 
, 
oa b>) cosee -S 2" 
2 88 
8 
Correction for Damping.— Unless the air can be reduced to a low 
