124 Mr. Atwood's Investigations for determining 
to each other, that the forces of elasticity shall be counterpoised 
by weights which are in the precise ratio of the angular dis- 
tances from the quiescent position, or, as it is sometimes ex- 
pressed, in the ratio of the spring's tensions ; at least as nearly 
as can be ascertained by experiment : this law of elastic force is 
assumed in the subsequent investigation. 
The position of the centre of gyration may be always deter- 
mined when the figure of the vibrating body is regular, by 
calculating the sum of the products which arise from multi- 
plying each particle into the square of its distance from the 
axis of motion, and dividing the sum by the weight of the 
vibrating body ; the square root of the result will be the dis- 
tance of the centre of gyration from the axis of motion. When 
the figure of the vibrating body is irregular, recourse may be 
had to experimental * methods, in order to determine the po- 
sition of the centre of gyration. 
Let the radius of the balance CA or CO = r, (fig. 1.) the 
semiarc BO — b ; let the spring's elastic force, acting on the 
circumference'of the balance, when wound to any given angle 
O C D from the quiescent position be = P, and let the arc 
OD = a; the weight of the balance, and the parts which vi- 
brate with it = W ; the distance of the centre of gyration 
from the axis of motion CG=^. These notations being 
premised, the resistance of inertia by which the mass con- 
tained in the balance opposes the communication of motion to 
the circumference is ^JL : and consequently the force which 
accelerates the circumference at the angular distance O C D 
from the quiescent position is This quantity remaining 
invariably the same, while the balance describes the arc of 
vibration B O E, may be denoted by the letter F, so that 
* Treatise on the Rectilinear Motion and Rotation of Bodies, p. 226 and 301. 
