the Times of Vibration of Watch Balances. 
3 39 
vX * * * t0 radius ~ 1 ; which is the time of describing 
the arc B H. When x = c, the time of describing the arc 
B 0 , is y/ - ■ a — . x into a circular arc of which the sine = 
if x » + i 
c X n+ I 
z b + 2 n c 
The time of describing the remaining part of the semiarc 
Q (%. 4- ) is next to be determined. In the cases to which 
this investigation is applied, the auxiliary spring ceases en- 
tirely to act on the balance after it has described the arc B Q. 
This being stated, while the balance describes the remaining 
arc of the semivibration Q O, it will be impelled by the ba- 
lance spring only. To ascertain the time of describing the 
arc Q O, it is first to be observed, that when the circumfe- 
rence has described the arc B Q, it will have acquired a velo- 
city equal to that of a body which has fallen from rest by the 
acceleration of gravity through a space * = ^ x 2 be — c 2 -j-nc\ 
Suppose the balance to have proceeded through the arc Q R, 
(fig. 4.) and let O R = x; the force by which the circumfe- 
rence is accelerated at R = 4 ^- ; and if u is the space through 
which a body falls freely from rest by the acceleration of gra- 
vity to acquire the velocity of the circumference in the point 
R, w = - xx ; taking the fluents so that u may become =s 
- V f nf when x — fu — L x fbc — c z + n c z 
+ d* — x x ; or because dh = V — 2 b c 4 - c a ; u= — xb z 4 - nc x 
za 1 
* When the circumference has described the arc BH -r, it will have acquired a 
velocity equal to that of a body which has fallen freely from rest by the acceleration of 
gravity through a space ~ x z b x — x z z c n x — nx z } as appears from the in- 
vestigation in page 138 ; and when x — c, this space becomes — x 2 b c — c‘ -f- 
— f - 
2 c z n _ n c z — — X 2 b c — c z + nc z . 
za 1 
T * 
