the Times of Vibration of Watch Balances. 
*57 
Then, \/ - 
If X n + 1 
/cxn+ i — 
zb + znc 
x into a circular arc, of which the sine is 
parts of a second. 
O.O9955O7O 
vX X into a circular arc, of which the sine 
*// 
is / =L=^= , - - - 0.00047174 
y/ + n c 1 
Time of a semivibration in the arc BO = 
The time shewn by the watch in 24 11 = l o ^ 2 ~ 
4o".6o, giving a daily rate of 19".40 slow. 
This variation of the daily rate is not to be considered as 
10002244 
= 23 h 59' 
affecting the regularity of the watch, as it is either compen- 
sated by adjustments when the watch is regulated to mean 
time, or taken as the established rate. A more material point 
is next to be determined ; admitting the deviation of O and 
Q (fig. 4. and 8.) to be O Q — i°, the same as in the former 
case ; suppose the semiarc of vibration to be diminished from 
135 0 to 125 0 . If the points of quiescence O and Q were coinci- 
dent, this diminution of the arc of vibration would cause no 
alteration (page 128) in the time of a semivibration, because 
the forces of acceleration of both springs would be as the an- 
gular distances from the same quiescent point O ; but since, 
these points are separated by an arc of i°,.a diminution of io° 
in the semiarc of vibration, will cause a change in the daily 
rate, which will be obtained from the general theorem (page. 
140) by making the following substitutions : 
a = 1.5707963 = an arc of 90° 
b = 2.1816616 = an arc of 125 0 
c = 2.1642083 = an arc of 124° 
b — c =.d = 0.0174533 = an arc of i°. . 
