b.sG2 + A GA2 + B. • GB2 + c Gc2 &c; which being substituted in the numerator of the foregoing value of so, gives SO = b. SG2 + A GA2+ B GB2 + &c b. SG 245. Corol. 3. Hence the distance of the centre of percussion always exceeds the distance of the centre of gravity, A. GAB. GB2 &c and the excess is always Go = 246. And hence also, SG 60= · that is SG. Go is always the same constant quantity, wherever the point of suspension s is placed; since the point G and the bodies A, B, &c, are constant. Or Go is always reciprocally as SG, that is Go is less, as SG is greater; and consequently the point o rises upwards and approaches towards the point G, as the point s is removed to the greater distance; and they coincide when SG is infinite. But when s coincides with G, then Go is infinite, or o is at an infinite distance. PROPOSITION XLVIII. 247. If a Body A, at the Distance SA from an axis passing throughs, be made to revolve about that axis by any Force acting at P in the Line SP, Perpendicular to the Axis of Motion: It is required to determine the Quantity or Matter of another Body a, which being placed at P, the Point where the Force acts, it shall be accelerated in the Same Manner, as when A revolved at the Distance SA; and consequently, that the Angular Velocity of ▲ and Q about s, may be the Same in Both Cases. By the nature of the lever, SA: SP :: f SP f, the effect of the force f, acting at P, on the body at ▲; that is, the force ƒ acting at P, will have the same effect on the body A, as the forcef, acting directly at the point A. But SA P A as as ASP revolves altogether about the axis at s, the absolute velocities of the points a and s, or of the bodies A and Q, will be as the radii sa, sp, of the circles described by them. Here then we have two bodies A and a, which being urged SP directly by the forces ƒ and f, acquire velocities which SA are as SP and sa. And since the motive forces of bodies are as their mass and velocity: therefore SP =f:f:: A. SA: Q. SP, and sp2 : s▲2 :: A : Q = SA any SA2 which therefore expresses the mass of matter which, being placed at P, would receive the same angular motion from the action of force at P, as the body a receives. So that the resistance of any body A, to a force acting at any point P, is directly as the square of its distance sa from the axis of motion, and reciprocally as the square of the distance sp of the point where the force acts. ƒ. sp2 248. Corol. 1. Hence the force which accelerates the point P, is to the force of gravity, as to A. SA3. A SA2 249. Corol. 2. If any number of bodies A, B, C, be put in motion, about a fixed axis passing through s, by a force acting at P; the point P will be accelerated in the same manner, and consequently the whole system will have the to 1, or as ƒ. sp3 A same angular velocity, if instead of the bodies A, B, C, placed at the distances SA, SB, SC, there be substituted the bodies SA2 SB2 Sc2 A, Sp2B, Sp2 -c; these being collected into the point P. And hence, the moving force being f, and the matter moved being A. SA2+ B. SB2 + c SC2 SP2 is the accelerating force; which therefore is to the accelerating force of gravity, as f. Sp2 to A. SA2 + B . SB2 + c.sc2. 250. Corol. 3. The angular velocity of the whole system lute velocity of the point P, is as the accelerating force, or directly as the motive force f, and inversely as the mass A SA2 &c : but the angular velocity is as the absolute velo Sp2 city directly, and the radius SP inversely; therefore the angular velocity of P, or of the whole system, which is the same f. SB thing, is as A. SA2 + B SB2+C. SC2 251. To determine the Centre of Oscillation of any Compound Mass or Body MN, or of any System of Bodies A, B, C, &c. LET MN be the plane of vibration, to which let all the matter be reduced, by letting fall perpendiculars from every particle, to this plane. Let I 9n/Pm M G be the centre of gravity, sp, Also, the angular veloc. any particle rates in the system, by its weight, is B b 3, prop. 48, the angular A. sp + B. sq 2 C ST SA2+ B. SB2+c.sc2 placed in o, gene p. sq2 or , S02 SG SO because of the similar triangles SGm, son. But, by the problem, the vibrations are performed alike in both cases, and therefore these two expressions must be equal to But, by cor. 2, pr. 41, the sum A Sp+B sq (A + B + c). sm; therefore the distance so = A SA2 + B SB2 + c . Sc2 A SA2 + B (A+B+C) = Sr by prop. 42, which is the distance of the centre of oscillation o, below the axis of suspension; where any of the products A. sa, B sb, must be negative, when a, b, &c, lie on the other side of s. So that this is the same expression as that for the distance of the centre of percussion, found in prop. 47. Hence it appears, that the centres of percussion and of oscilliation, are in the very same point. And therefore the properties in all the corollaries there found for the former, are to be here understood of the latter. 252. Corol. 1. If p be any particle of a body b, and d its distance from the axis of motion s; also G, O the centres of gravity and oscillation. Then the distance of the centre of oscillation of the body, from the axis of motion, is 253. Corol. 2. If b denote the matter in any compound body, whose centres of gravity and oscillation are G and o; the body P, which being placed at P, where the force acts as in the last proposition, and which receives the same motion from that force as the compound body b, is P = A SG For, by corol. 2, prop. 47, this body p is = P= 254. By the method of Fluxions, the centre of oscillation, for a regular body, will be found from cor. 1. But for an irregular one; suspend it at the given point; and hang up also a simple pendulum of such a length, that making them both vibrate, they may keep time together. Then the length of of the simple pendulum, is equal to the distance of the centre of oscillation of the body, below the point of suspension. 255. Or it will be still better found thus: Suspend the body very freely by the given point, and make it vibrate in small arcs, counting the number of vibrations it makes in any time, as a minute, by a good stop watch; and let that number of vibrations made in a minute be called n: Then 140850 shall the distance of the centre of oscillation, be so = nn inches. For, the length of the pendulum vibrating seconds, or 60 times in a minute, being 39 inches; and the lengths of pendulums being reciprocally as the square of the number of vibrations made in the same time; therefore 140850; the length of the n n pendulum which vibrates n times in a minute, or the distance of the centre of oscillation below the axis of motion. 256. The foregoing determination of the point, into which all the matter of a body being collected, it shall oscillate in the same manner as before, only respects the case in which the body is put in motion by the gravity of its own particles, and the point is the centre of oscillation: but when the body is put in motion by some other extraneous force, instead of its gravity, then the point is different from the former, and is called the Centre of Gyration; which is determined in the following manner : PROPOSITION L. 257. To determine the Centre of Gyration of a Compound Body or of a System of Bodies. LET R be the centre of gyration, or the point into which all the particles A, B, C, &c, being collected, it shall receive the same angular motion from a force f acting at P, as the whole system receives. Now, by cor. 3, pr. 47, the angular. velocity generated in the system by the f. sp ; and force f, is as. A SA2 + B. S2 &c ́ ́ B by |