the Times of Vibration of Watch Balances. 143 
In the preceding pages, the time has been investigated in 
which a balance vibrates in the semiarc B O, (fig. 4.) when 
the point of quiescence Q, of the auxiliary spring u, is si- 
tuated between B and O, that is in the first semiarc of vibra- 
tion. In this case, in order that the time of describing the 
latter semiarc O E, by retardation, may be equal to that of 
describing the first semiarc B O by accelerated motion, the 
quiescent point of the auxiliary spring v must be placed at N, 
between O and E, so that E N may be equal to B Q ; accord- 
ing to this disposition of the auxiliary springs, the point of 
quiescence Q of the auxiliary spring u will be situated in the 
first semiarc of vibration, considering the balance as vibrating 
from B to E ; and the point of the quiescence N of the auxiliary 
spring v, will be in the first semiarc of vibration, considering 
the balance as vibrating from E to B. 
The third case which remains to be more fully investigated 
is, when the points of quiescence of the auxiliary springs are 
situated in the latter semiarcs of the respective vibrations ; 
that is, when the quiescent point Q of the auxiliary spring u 
is situated in the latter arc OE, (fig. 5.) while the balance 
vibrates from B to E, and the point of quiescence N of the 
auxiliary spring v is situated in the latter arc O B, while the 
balance vibrates from E to B. In this case it is evident, that' 
while the balance vibrates from B to O, it describes the arc 
BN by the acceleration of two springs ; namely, the balance 
spring, of which the point of quiescence is O, and the auxi- 
liary spring u, of which the point of quiescence is O. And 
the balance describes the arc N O by the combined action of 
three springs ; namely, by that of the balance spring, of which 
the point of quiescence is O, that of the auxiliary spring u , of 
