preffion on an obftacle that may be opposed to it whilft vibrating; for if the obftacle be opposed to it at different distances from the point of suspension, the ftroke, or percuffion, will not be equally powerful; and it will foon appear that this centre of percuffion does not coincide with the centre of gravity.

Let the body AB, fig. 1, Plate V. N. I, confifting of two equal balls faftened to a stiff rod, move in a direction parallel to itself, and it is evident that the two balls muft have equal momentums, fince their quantities of matter are equal, and they move with equal velocities. Now if in its way, as at N. II, an obstacle C be opposed exactly against its middle E, the body will thereby be effectually ftopped, nor can either end of it move forwards, for they exactly balance each other, the middle of this body being its centre of gravity. Now fhould an obftacle be oppofed to this body, not against its middle, but nearer to one end, as at N. III, then the ftroke being not in the direction of the centre of gravity, is in fact an oblique ftroke, in which case, agreeably to the laws of congrefs which have been delivered in chap. VIII. a part only of the momentum will be spent upon the obstacle, and the body advancing the end A, which is fartheft from the obftacle, as fhewn by the dotted reprefentation, will proceed with that part of the momentum which has not been spent upon the obftacle; confequently in this cafe the percuffion is not fo powerful as in the foregoing. Therefore there is a certain point in a

moving body which makes a ftronger impreffion on an obftacle than any other part of it.In the prefent case, indeed, this point coincides with the centre of gravity; because the two ends of the body before the ftroke moved with equal velocities. But in a pendulum the cafe is different; for let the fame body of fig. 1, Plate V. be fufpended by the addition of a line AS, fig. 2, Plate V. which line we shall suppose to be void of weight and flexibility, and let it vibrate round the point of fufpenfion S. It is evident that now the two balls will not move with equal velocities; for the ball B, by defcribing a longer arch than the ball A in the fame time, will have a greater momentum; and of courfe the point where the forces of the two balls balance each other, which is the centre of percuffion, lies nearer to the lower ball B; confequently this point does not coincide with the centre of gravity of the body AB; but it is that point wherein the forces of all the parts of the body may be conceived to be concentrated. Hence the centre of ofcillation and the centre of percuffion coincide; or rather they are exactly the fame point, whose two names only allude, the former to the time of vibration, and the latter to its ftriking force.

If in fig. 1, Plate V. the balls A and B be not equal, their commón centre of gravity will not be in the middle at E, but it will lie nearer to the heavier body, as at D, fuppofing B to be the heavier body; fo that the distances BD, AD, may bẹ in

versely

verfely as the weights of those bodies. Now when the above-mentioned body is formed into a pendu lum, as in fig. 2, though the weights A and B be equal, yet by their moving in different arches, viz. with different velocities, their forces or momentums become actually unequal; therefore in order to find the point where the forces balance each other, fo that when an obftacle is opposed to that point, the moving pendulum may be effectually stopped, and no part of it may preponderate, in which cafe the obftacle will receive the greatest impreffion; we must find firft the momentums of the two bodies A and B, then the diftances of those bodies from the centre of percuffion, or of equal forces, must be inversely as thofe momentums. Thus the velocities of A and B are reprefented by the fimilar arches which they describe, and those arches are as the radii SA, SB. Therefore the momentum of A is the product of its quantity of matter multiplied by SA, and the momentum of B is the product of its quantity of matter multiplied by SB; confequently AD must be to BD, as the weight of B multiplied by SB is to the weight of A multiplied by AS. Then D is the centre of percuffion. And fince, when four quantities are proportional, the product of the two extremes is equal to the product of the two means; therefore if the weight of A multiplied by AS, be again multiplied by AD, the product must be equal to the product of the weight of B multiplied by BS, and again multiplied

by

by BD; that is, the product of the body on one fide of the centre of ofcillation multiplied by both its diftance from the point of fufpenfion and its diftance from the centre of ofcillation, is equal to the product of the body on the other fide of the centre of ofcillation, multiplied both by its diftance from the point of fufpenfion, and its diftance from the centre of ofcillation.

The fame reasoning may evidently be applied to a pendulum confifting of more than two bodies connected together, or to the different parts of the fame pendulous body; hence we form the following general law.

If the weight of each part of a fimple or compound pendulum be multiplied both by its diftance from the centre of fufpenfion, and its diftance from the centre of ofcillation or percuffion, the fums of the products, on each fide of the centre of ofcillation, will be equal to each other.

From this law the rule for determining the diftance of the centre of ofcillation from the point of fufpenfion is easily deduced; but the application of it is attended with confiderable difficulty, on which account we shall fubjoin it in the note (1), and fhall now proceed to fhew an experimental or mechanical

(1) Let a pendulum confift of any number of parts or fmall bodies A, B, C, D, E, joined together; let a, b, c, d, e, ftand for their respective distances from the point of fufpenfion; and x for the distance of the centre of ofcillation from the point of suspension.

The

chanical method of finding the centre of ofcillation, which method is general and easy, at the fame time that it admits of fufficient accuracy.

>

The

The distances of thofe parts, or bodies, from the centre of ofcillation will be x •A, X · b, x—c, d — x, e— x; D and E being fuppofed to be farther from the point of fufpenfion, than the centre of ofcillation is. By multiplying every one of those bodies, both by its distance from the centre of suspenfion and its diftance from the centre of ofcillation, we have, agreeably to the above-mentioned law, the equation Aax-Aaa+Bbx- Bbb + Ccx- Ccc Ddd Ddx+ Eee-Eex; which, by tranfpofition and divifion, is refolved into the following; viz.

Aaa + Bbb + Ccc + Ddd + Eee

x= Aa + Bb + Cc + Dd + Ee

:

Should any of the bodies, as for inftance A and B, in the preceding inftance, be fituated above the centre of fufpenfion, then their diftances will be negative, viz. -a, - by though their squares aa, bb, are always pofitive. In this cafe the value of x is = Aaa + Bbb + Ccc + Ddd + Eee

· Bb + Cc + Dd + Ee

Since the centre of gravity of a body or fyftem of bodies, is that point wherein all their matter may be conceived to be condensed, therefore the product of all the matter or fum of the different weights A, B, C, D, E, multiplied by the distance of the common centre of gravity from the point of fufpenfion, is equal to the fum of the products, of each body multiplied by its diftance from the point of fufpenfion. Hence the above ftated value of x becomes Aaa + Bbb + Ccc + Dad + Eee divided by the product of the whole body or fum of the weights, multiplied by the distance of

the