d — 
the Times of Vibration of Watch Balances. 145 
The accelerative force of the balance spring on the cir- 
cumference of the balance at the point R is -J- x 
The accelerative force of the auxiliary spring 
u, at the tension or distance from quiescence 
Qd = a being nf the force of the same 
spring at the tension or distance QR = 2 d— x, 
is - - 
«/ 
PX2 d — X 
The force of the auxiliary spring v, at the 
tension N e — a being nf, at the tension N R, 
the force of this spring acting by retardation 
will be - - - - — — x x. 
a 
Sum of the forces acting on the circumfe- 
rence of the balance, when it has described the 
arc N R = - - — x d — x -f- 2 dn — 2 nx. 
a 1 
Let u be the space through which a body falls freely from 
rest by the acceleration of gravity, to acquire the velocity of 
the circumference at R ; the principles of acceleration give this 
equation ; u == x d x — x x -{- 2 d n x — 2 n x x ; and tak- 
ing the fluents so that when x = 0, u, may be = ~ x 
— d* -f n V — 3 n d* -f 2 nb d ; u = ^ x V ~ d z + n b z — 
3nd z -\-2nbd-{-2 d-{-^ndxx — i-f s»xi‘; if if is put 
to represent the time of describing the arc NR,/ = \/ ~^— x 
2 if 
and 
+ 2 n X. X 2 
v/V - d z + 71b 7 - — 3nd x + znbd+ 2 d-\- 4 .nd. 
taking the fluents so that when x = o,t = o, and making x — d, 
the time of describing the arc NO ==>/■ 
2 If X 1 + 2 71 
x a circular 
arc of which the sine 
d 7 - 1 + 27 1 
b 7 - + 71 X b + d 2 — 2nd 7 - 
u 
, or because 
MDCCXCIV. 
