the Times of Vibration of Watch Balances. 1 37 
thought foreign to the present subject, especially as it may 
contribute to elucidate some circumstances respecting the ef- 
fect of springs on the vibrations of balances, which at the first 
view are not at all obvious. 
Let two spiral springs be applied to act on a watch balance 
in the same direction ; if the two springs in unwinding them- 
selves, by turning the balance come to the same point of qui- 
escence, or, in other words, if the accelerative forces of both 
springs cease at the middle point of the vibration, whatever be 
the relative strength of the two springs, they will act on the 
balance precisely in the same manner as if one spring only had 
been applied, of equal strength with both (the springs being 
here supposed similar, in respect of the law of the elastic for- 
ces and tensions). But when the points of quiescence of the 
two springs do not coincide, that is, when one spring continues 
to accelerate the balance in its vibration, after the acceleration 
of the other spring has ceased, the time of a semivibration 
must be obtained from a separate investigation. 
Let the circumference of a balance (fig. 4.) be impelled 
by the action of a spiral spring through the semiarc of vibra- 
tion BO, the forces of this spring being always in propor- 
tion to the angular distances from the point of quiescence O ; 
let a secondary or auxiliary spring also act on the balance 
from the extremity of the semiarc B as far as the point O, at 
which point all acceleration of the auxiliary spring ceases, 
the forces of the auxiliary spring varying as the angular dis- 
tances from the point of quiescence Q. Suppose that the ac- 
celerative force of the principal or balance spring on the cir- 
cumference of the balance at the distance from quiescence 
O D is =f take Q d = O D, and let the accelerative force of 
the balance spring on the circumference of the balance be to 
that of the auxiliary spring, when both springs are wound to 
MDCCXCIV. T 
