VOL. LXXXir.] PHILOSOPHICAL TRANSACTIONS. 383 



length, and number of turns of the spiral may be so adjusted to each other, that 

 the forces of elasticity shall be counterpoised by weights which are in the precise 

 ratio of the angular distances from the quiescent position, or, as it is sometimes 

 expressed, in the ratio of the spring's tensions; at least as nearly as can be ascer- 

 tained by experiment. This law of elastic force is assumed in the subsequent in- 

 vestigation. 



The position of the centre of gyration may be always determined when the 

 figure of the vibrating body is regular, by calculating the sum of the products 

 which arise from multiplying 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 distance 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 position of the 

 centre of gyration. 



Let the radius of the balance ca or co = r, fig. 13, the semi-arc bo = Z; ; let 

 the spring's elastic force, acting on the circumference of the balance, when wound 

 to any given angle ocd from the quiescent position be := p ; and let the arc od 

 = a ; the weight of the balance, and the parts which vibrate with it = w ; the 

 distance of the centre of gyration from the axis of motion cg = g. These nota- 

 tions being premised, the resistance of inertia by which the mass contained in the 



balance opposes the communication of motion to the circumference is ~^~ : and 



consequently the force which accelerates the circumference at the angular distance 



ocD from the quiescent position is — ^. This quantity remaining invariably the 



same, while the balance describes the arc of vibration bob, may be denoted by 



the letter f, so that f =: — 5-; suppose the radius ca commencing a vibration 



from the point b to have described the arc bh, and let oh = j- ; since the force 

 which accelerates the circumference at the angular distance from quiescence od 

 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 v be the space 

 through which a body falls freely from rest by the acceleration of gravity to ac- 

 quire the velocity of the circumference at the point h ; the principles of accelera- 

 tion give this equation, « = ; and taking the fluents, while x decreases 



from b to X, u = — ^ : if therefore / be made = IQ3 inches, being the 



space which bodies falling freely from rest by the force of gravity near the earth's 

 * Treatise on the Rectilinear Motion and Rotation of Bodies, p. 226 and 301. — Orig. 



