What has been here observed of the effects of particles of matter placed upon rigid wire will be equally applicable to the particles of a solid body. Those which are nearer to the axis are urged forward by those which are more remote, and are in their turn retarded by them; and as with the particles placed upon the wire, there is a certain particle of the body at which the effects are mutually neutralised, and which vibrates in the same time as it would if it were unconnected with the other parts of the body, and simply connected by a fine thread to the axis. By this centre of oscillation the calculations respecting the vibration of a solid body are rendered as simple as those of a molecule of inconsiderable magnitude. All the properties which have been explained as belonging to a simple pendulum may thus be transferred to a vibrating body of any magnitude and figure, by considering it as equivalent to a single particle of matter vibrating at its centre of oscillation. (213.) It follows from this reasoning, that the virtual length of a pendulum is to be estimated by the distance of its centre of oscillation from the axis of suspension, and therefore that the times of vibration of different pendulums are in the same proportion as the square roots of the distances of their centres of oscillation from their axes. The investigation of the position of the centre of oscillation is, in most cases, a subject of intricate mathematical calculation. It depends on the magnitude and figure of the pendulous body, the manner in which the mass is distributed through its volume, or the density of its several parts, and the position of the axis on which it swings. The place of the centre of oscillation may be determined when the position of the centre of gravity and the centre of gyration are known; for the distance of the centre of oscillation from the axis will always be obtained by dividing the square of the radius of gyration (186.) by the distance of the centre of gravity from the axis. Thus if 6 be the radius of gyration, and 9 the distance of gravity from the axis, 36 divided by 9, which is 4, will be the distance of the centre of oscillation from the axis. Hence it may be inferred generally, that the greater the proportion which the radius of gyration bears to the distance of the centre of gravity from the axis, the greater will be the distance of the centre of oscillation.