128 Mr. Atwood's Investigations for determining 
It is observable that the semiarc of vibration B O = b, does 
not enter into these expressions for the time of a semh ' oration ; 
if therefore instead of the semiarc B O, an arc of any other 
length LO, terminating in the point of quiescence O, (fig. 1.) 
should be substituted in the preceding investigation, the time 
of describing L O would be still = v / > or y/ZJtlJL- 
equal to the time of describing the other semiarc B O ; con- 
sequently, whether the balance vibrates in the largest or 
smallest arcs, the times of vibration will be the same. 
From the preceding investigations it appears, that when 
the force by which the circumference of the balance is acce- 
lerated at the given angular distance c° from the quiescent 
position is = F, the time of a semivibration t = ^/^JLLL 1 _ ; 
and conversely, when the time of a semivibration is = t, the 
force which accelerates the circumference at the given angu- 
, . . . , •1-' p* r c° 
lar distance c° from the quiescent position, that is r = § u £ x ~ i teP m 
Since watches and time-keepers are usually adjusted to 
Namely, W = 42 gr. =, the weight of the balance, including the parts which vibrate 
with it. 
p — 24 gr. — the force at the circumference of the balance, which counter- 
poises the force of the spring when wound to the distance 90°. 1 
r — 1.125 inch, and parts the radius of the balance* 
/ — 192 inches — the space described in one second of time by bodies which 
descend freely from rest by the acceleration of gravity. 
p. — 3.14x59, &c. = the circumference of a circle of which the diameter = 1 ; 
s~ pts. of a second. 
the time of a semivibration t zz y/ - X = 0*0994 
32 /\ 2^ 193 
The balance, when adjusted to mean time, makes 5 vibrations in a 
second ; the actual time of a semivibration is therefore 
Difference between the actual time and the time by the calculation - 
0.0006 
