the Times of Vibration of Watch Balances. 1 25 
F = Zll ; suppose the radius C A commencing a vibration from 
the point B to have described the arc B H, and let OH = j; 
since the force which accelerates the circumference at the 
angular distance from quiescence O D is = F, and the forces 
of acceleration are supposed to vary in the proportion of the 
angular distances from the quiescent point O, the force 
which accelerates the circumference of the balance at the 
point H will be = ^ » let u be the space through which a 
body fails freely from rest by the acceleration of gravity to 
acquire the velocity of the circumference, at the point H ; the 
principles of acceleration give this equation, * u 
* Newton 1 1 Princ. Vol. I. prop, xxxix. Let a body 
describe the line AC by the acceleration of a force varying 
in any ratio of the distances from a centre C. Let ano- 
ther body describe the line EH by the acceleration of a 
constant or uniform force. Suppose the velocity at O to 
be equal to the velocity at D, and let O G and D F be 
the evanescent spaces, or increments of space in which 
equal velocities are generated ; so that E D may represent 
a line through which a body must fall from rest by the ac- 
celeration of the constant or uniform force, to acquire the 
velocity of the other body at O. It is to be proved that 
the increment of space O G is to the increment of space 
D F, as the force of acceleration at D to the force of acce- 
leration at O. Let the former of these forces, i. e. at D be denoted by G, and the latter 
force at O by H. Let ED —u, and let A O — x. Also let D F — u, and O G — x. 
Because the increments of velocity are always as the forces of acceleration and the 
elementary times in which they act jointly, it follows, that when the increments of velo- 
city are equal, the forces are in the inverse ratio of the elementary times in which they 
act; that is (the velocities of describing the evanescent spaces O G, D F being equal by 
the supposition), the forces are in the inverse ratio of those spaces ; and consequently 
the force at D (G) is to the force at O (H) as O G to D F; that is, according to the 
preceding notation, G : FI : : x : u or u — The constant force G being as- 
sumed equal to that of gravity, may be denoted by any constant quantity, such as 
unity. By substituting therefore i for G, the equation will become u — Hi.. In this 
Fxx 
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