of perfectly elastic bodies, other formulæ have place which express the motion of each body separately; as in the following proposition. 4. If the impact of two perfectly elastic bodies be direct, their relative velocities will be the same both before and after impact, or they will recede from each other with the same velocity with which they met; that is, they will be equally distant, in equal times, both before and after their collision, although the absolute velocity of each may be changed. The circumstances attending this change of motion in the two bodies, using the above notation, are expressed in the two following formulæ : which needs no modification, when the motion of b is in the same direction with that of B. 5. In the other case of b's motion, the general formulæ be when b moves in a contrary direction to that of B, which arises from taking v negative. And (B — b) v B+ b 2 B V B+b the velocity of B the velocity of b when b was at rest before impact, that is, when v = 0. 6. If a perfectly hard body в, impinge obliquely upon a perfectly hard and immoveable plane A D, it will after collision move along the plane in the direction c a. And its velocity before impact Is to its velocity after impact E Is to the cosine of the angle B C D. But if the body be elastic, it will rebound from the plane in the direction CE, with the same velocity, and at the same angle with which it met it, that is, the angle A C E will be equal to the angle B C D A C 7. In the case of direct impact, if в be the striking body, b the body struck, v and v their respective velocities before impact, u and u their velocities afterwards; then the two following are general formulæ : viz. In these, if n == 1, they serve for non-elastic bodies; if n 2, for bodies perfectly elastic. If the bodies be imperfectly elastic, n has some intermediate value. When the body struck is at rest, the preceding equations become, from which the value of n may be determined experimentally. H S 8. In the usual apparatus for experiments on collision, balls of different sizes and of various substances are hung from different points of suspension on a horizontal bar. H R M N is an arc of a circle whose centre is s; and its graduations, 1, 2, 3, 4, 5, &c. indicate the lengths of cords, as measured from the lowest point. Any ball, therefore, as P, may be drawn from the vertical, and made to strike another ball hanging at the lowest point, with any assigned velocities, the height to which the ball struck ascends on the side A м furnishes a measure of its velocity; and from that the value of n may be found from the M B last equation. Balls not required in an individual experiment may be put behind the frame, as shown at A and B. The cup c may be attached to a cord, and carry a ball of clay, &c. when required. Example.-Suppose that a ball weighing 4 ounces strikes another ball of the same substance weighing 3 ounces, with a velocity of 10, and communicates to it a velocity of 84: what in that case, will be the value of n? index of the degree of elasticity; perfect elasticity being indi cated by 2. Principles of Chronometers. 1. Clockwork, regulated by a simple balance, is inadequate to the accurate mensuration of time. 2. Clockwork, regulated by a pendulum vibrating in the arch of a circle, is of itself inadequate to the accurate mensuration of time. 1st. Because the vibrations in greater or smaller arches are not performed in equal times. 2dly. Because the length of the pendulum is varied by heat and cold. 3. Clockwork, regulated by a pendulum vibrating in the arch of a cycloid, is inadequate to the accurate mensuration of time. The isochronism of the vibrations of a cycloidal pendulum in greater and smaller arches is true only on the hypothesis, that the pendulum moves in a non-resisting medium, and that the whole mass of the pendulum is concentrated in a point, both of which positions are false. For these reasons the application of the cycloid in practice has been entirely relinquished. 4. Modern time-keepers owe almost the whole of their superiority over those formerly made to two things; 1st, the application of a thermometer; 2dly, the particular construction of the escapement. 5. Metals expand by heat and contract by cold. This is proved experimentally by the pyrometer. Metallic bars of the same kind are found to expand in proportion to their length. Metals of different kinds expand in different proportions: thus the expansion of iron and steel are as 3, copper 41, brass 5, tin 6, lead 7. Hence pendulum rods, expanding and contracting by the successive changes of temperature, affect the going of the clocks to which they are applied. Various have been the contrivances to correct the errors of pendulums from their contraction and expansion by heat and cold; the principal of these are described under the subject of pendulums (p. 253). 6. The balance of a watch is analogous to the pendulum in its properties and use. The simple balance is a circular annulus, equally heavy in all its parts, and concentrical with the pivots of the axis on which it is mounted. This balance is moved by a spiral spring called the balance-spring, the invention of the ingenious Mr. Hook. 7. The pendulum requires a less maintaining power than the balance. Hence the natural isochronism of the pendulum is less disturbed by the relatively small inequalities of the maintaining power. 8. The spring's elastic force which impels the circumference of the balance, is directly as the tension of the spring; that is, the weights necessary to counterpoise a spiral spring's elastic force, when the balance is wound to different distances from the quiescent point, are in the direct ratio of the arcs through which it is wound. 9. The vibrations of a balance, whether through great or small arches, are performed in the same time. For the accelerating force is directly as the distance from the point of quiescence; hence, therefore, the motion of the balance is analogous to that of a pendulum, vibrating in cycloidal arches. 10. The time of the vibration of a balance is the same as if a quantity of matter, whose inertia is equal to that by which the mass contained in the balance opposes the communication of motion to the circumference, described a cycloid whose length is equal to the arc of vibration described by the circumference, the accelerating force being equal to that of the balance. Because in both cases the spaces described would be equal, as also the accelerating forces in corresponding points, and therefore the times of description. 11. If 1 denote the accelerating force of gravity, L the length of a pendulum vibrating seconds in a cycloid, a the semi-arc of vibration of the balance, T the time of vibration, and the accelerating force of the balance, then will T = a LX F 12. Let be the space which a body falling freely from a state of rest describes in 1", and p = 3.141593 the circumference of a circle whose diameter is unity, then will In this expression for the time of vibration, the letter a denotes the length of the semi-arc of vibration; if this arc should be expressed by a number of degrees, co, and r be the radius of the balance, then a will be Prco 1800 ; and this quan tity being substituted for a, the time of a vibration will be T = let the given arc be 90°, in this case T = pc 2 g F 13. If the spring's elastic force, when wound through the given angle or arc a= 90° from the quiescent position, be P; the weight of the balance, and the parts which vibrate with it =w, the distance of the centre of gyration from the These are expressions for the time of a vibration, whatever may be the figure of the balance, the other conditions remaining the same as above stated. If the balance be an annulus or a cylindrical plate, ≤ = and the time of vibration T = |w p3r 4 Pg 14. The times of vibration of different balances are in a ratio compounded of the direct subduplicate ratios of their weights and semidiameters, and the inverse subduplicate ratio of the tensions of the springs or of the weights which counterpoise them, when wound through a given angle. 15. The times of vibration of different balances are in a ratio compounded of the direct simple ratio of the radii, and direct subduplicate ratio of their weights, and the inverse subduplicate ratio of the absolute forces of the springs at a given tension. 16. Hence the absolute force of the balance spring, the diameter and weight of the balance being the same, is inversely as the square of the time of one vibration. 17. The absolute force or strength of the balance spring, the time of one vibration, and the weight of the balance being the same, is as the square of the diameter and the balance. 18. The weight of the balance, the strength of the spring and time of vibration being the same, is inversely as the square of the diameter. Hence a large balance vibrating in the same time, with the same spring, will be much lighter than a small one. 19. If the rim of the balance be always of the same breadth and thickness, so that the weight shall be as the radius, the |