144 Mr. Atwood's Investigations for determining 
which the point of quiescence is O, both of which accelerate 
the balance, and that of the auxiliary spring v, of which the 
point of quiescence is N, acting by retarding the balance. 
The time in which the balance describes the semiarc B O, 
by the combined action of these three springs, will be obtained 
by the following investigation. 
Resuming the former notation, let DO (fig. 5.) = a — Qd 
= N*, BO = b, BO = c, OQ = ON — d—c—br The accele- 
rative force of the balance spring on the circumference of the 
balance at the tension O D =/; the accelerative force of the 
auxiliary springs at the same tension Q d or N e = nf. From 
page 138 and 139, it appears that the time in which the balance 
describes the arc BN by the joint action of the balance spring 
and auxiliary spring u = x a circular arc of which 
the sine = ” + 1 ; and the velocity acquired by the 
circumference at N is equal to that of a body which has fallen 
freely from rest by the acceleration of gravity through a space* 
x (g _j_ nb* — 3 « -f 211b d. To find the time of 
describing the arc N O, suppose the balance to have proceeded 
in its vibration from N to R, (fig. 5-) and make 
* In the investigation, page 1 38, it is shewn, that when the balance has described the 
arc B H — x, the space through which a body must fall freely from rest to acquire the 
f r 
velocity of the circumference at the point H, or = — y.zb x — x 1 + 2 c n x — nx , 
the expression being the same, whether the point CL is on one side of O or on the 
other, provided BQj= c. In the present case, c — b -f d, and when the circumference 
has described the arc B N, x - b - d ; wherefore if b + d is substituted for c, and 
b — d for x in the equation, u — ^ x 2 b x — x z + 2 c n x — n x z , the result will 
= - X 6*- d z + nb z - 3 nd z + znbd- 
2 a 
