the Times of Vibration of Watch Balances. 123 
The determination of the time in which the balance vi- 
brates, from the theory of motion, requires the following par- 
ticulars to be known. 
1st. The spring's elastic force, which impels the circumfe- 
rence of the balance when it is at a given angular distance 
O D (fig. 1.) from the quiescent point O. 
2dly. The law or ratio observed in the variation of the 
spring's force, while the balance is impelled from the extremity 
of the semiarc B to the point of quiescence O, where all acce- 
leration ceases. 
3dly. The weight of the balance, including the parts which 
vibrate with it. 
4thly. The radius of the balance C O, and the distance of 
the centre of gyration from the axis of motion C G. 
5thly. The length of the semiarc B O. 
Suppose the plane of the balance to be placed vertically, 
and let a weight P (fig. 2.) be applied by means of a line 
suspended freely from the circumference at T, to counterpoise 
the elastic force of the spring when the balance is wound 
through an angle from quiescence O C D. This weight P (the 
weight of the line being allowed for) will be the force of the 
spiral spring which impels the circumference of the balance, 
when at the angular distance O D, from the quiescent po- 
sition. 
It appears from many experiments, that the weights neces- 
sary to counterpoise a spiral spring's elastic force, when the ba- 
lance is wound to the several distances from the quiescent point, 
represented* by the arcs OG, OH, OI, (fig. 2.) &c. are nearly 
in the ratio of those several arcs. It also appears, that the shape, 
the length, and number of turns of the spiral may be so adjusted 
* Berthoud Traite des Horloges marines, p. 49. 
R 2 
