282 PHILOSOPHICAL TRANSACTIONS. [aNNO 1794. 



impelled by the spring's clastic force either in one direction or the other. If the 

 balance should be turned through any angle ocB, the spiral spring, being wound 

 through the same angle, endeavours by its elastic force to restore itself; and 

 when at liberty, impels the balance through the arc bo with an accelerated velo- 

 city till it arrives at the position o, where the force of acceleration ceases ; with 

 the velocity acquired at o, the balance proceeds in its vibration, describing the 

 arc OE with a retarded motion. 



The elastic forces of the spring at equal distances on the opposite sides of the 

 point o, are assumed to be equal ; it is also assumed that the effects of friction, 

 and other irregular resistances which retard the motion of the balance, are com- 

 pensated by the maintaining power, so that the time of describing the first arc of 

 vibration bo by an accelerated motion, shall be equal to the time of describing 

 the latter arc oe by a retarded motion, and that the entire arc of vibration boe 

 is bisected by the point o. 



To render the construction of fig. 13, more distinct, the fixed circle odbe is 

 represented to be at a small distance from the circumference of the balance, but 

 is to be considered as co-incident with it, so that the arc bo subtending the angle 

 Bco, may be of the same length with an arc of the circumference of the balance 

 which subtends the same angle bco : on this principle co or ca may be taken in- 

 differently as the radius of the balance. The determination of the time in which 

 the balance vibrates, from the theory of motion, requires the following parti- 

 culars to be known, ist. The spring's elastic force, which impels the circum- 

 ference of the balance when it is at a given angular distance on (fig. 13.) 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 semi-arc b to the 

 point of quiescence o, where all acceleration ceases. 3dly. The weight of the 

 balance, including the parts which vibrate with it. 4thly. The radius of the 

 balance co, and the distance of the centre of gyration from the axis of motion 

 CG. 5thly. The length of the semi-arc BO. 



Suppose the plane of the balance to be placed vertically, and let a weight p 

 (fig. 14.) 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 ocd. This weight p, the weight of the line 

 being allowed for, will be the force of the spiral spring which impels the circum- 

 ference of the balance, when at the angular distance on, from the quiescent po- 

 sition. It appears from many experiments, that the weights necessary to counter- 

 poise a spiral spring's elastic force, when the balance is wound to the several dis- 

 tances from thcquiesccnt point, represented* by the arcs og, oh, oi, fig. 14, &c, 

 are nearly in the ratio of those several arcs. It also appears, that the sh;ipe, the 

 * Berthoud Traite des Horloges marines, p. U). 



