inftance, if the moving body A weigh one pound, and move at the rate of one foot per minute, whilst the body B at reft weigh one pound alfo, the velocity after the concurfe will be half a foot per minute; half a foot being the quotient of one foot, divided by the fum of their quantities of matter; viz. 2 pounds. If cæteris paribus B weigh 10 pounds, then the velocity after the concurfe will be the 11th part of a foot per minute. If B weigh 100000 pounds, then the velocity after the concurfe will be the 100001th part of a foot per minute; and in fhort, when B is infinitely bigger than A, the velocity after the concurfe will be infinitely finall, which is the fame thing as to fay, that in that cafe, after the 'ftroke, the bodies will remain at rest. And fuch is the cafe when a non-elastic body strikes against an immoveable obftacle.

III. If a body in motion ftrikes directly against another body, the magnitude of the ftroke is proportional to the momentum loft, at the concurfe, by the more powerful body.

According to the third law of motion, action and re-action are equal and contrary to each other; therefore whatever momentum is loft by one of the bodies, is acquired by the other. Or the magnitude of this acquired momentum (which is the effect of the ftroke) is as the momentum loft by the more powerful body; it being by the quantity

of

of the effect that we measure the quantity of the action.

IV. When a given body frikes directly against another given body, if the latter be at reft, the quantity of the ftroke is proportional to the velocity of the former body.-If the fecond body be moving in the fame direction with the first, but at a flower rate, the magnitude. of the ftroke will be the fame as if the fecond body flood Still, and the first impinged upon it with a velocity equal to the difference of their velocities.And lastly, if the bodies move directly towards each other, the magnitude of the ftroke is the fame as if one of the bodies stood at reft, and the other ftruck it with the fum of their velocities.

The momentum of a given body is proportionate to its velocity; for with a double velocity the momentum is double, with a treble velocity the momentum is treble, and fo on; therefore, as long as the body remains the fame, the magnitude of the stroke, being proportional to the momentum, must likewise be proportional to the velocity. And when one of the bodies is at reft, the magnitude of the ftroke is evidently proportional to the velocity of the moving body.

V. It follows from the foregoing theory, that the mutual actions of bodies, which are inclofed in a certain Space, are exactly the fame, whether that space be at reft or move on uniformly and directly.

For if the motion of the space adds to the velocity of those bodies within it, which move the fame

4. It is called perfectly elastic when it recovers its original figure entirely, and with the fame force with which it loft it; otherwise it is called imperfectly elaftic.

5. One body is faid to ftrike directly on another body, when the right line, in which it moves, paffes through the centre of gravity of the other body, and is perpendicular to the furface of that other body.

Though there are innumerable gradations from a body perfectly hard, to one perfectly foft; or between the latter and a body perfectly elastic; yet we cannot fay with certainty that a body perfectly poffeffed of any of the above mentioned qualities does actually exift. It is however certain that our endeavours have not been able to deprive certain bodies of the leaft degree of their elafticity, by mechanical means.

The object of the theory of percutient bodies is to determine the momentums, the velocities, and the directions of bodies after their meeting; which we fhall lay down, and explain, in the following propofitions. But it must be obferved, that throughout this chapter we only speak of bodies which move with equable motion, that is, of such as defcribe equal spaces in equal portions of time; and we do likewise fuppofe that the bodies move in a non-refifting medium, and that they are not influenced by any other action, excepting the fingle impulfe, which puts them in motion: for though fuch

fuch fimple and regular movements never take place in nature; yet when their theory is once established, the complicated cafes, wherein the refiftances of mediums and other interfering caufes, are comprehended, may be more commodioufly examined, and proper allowances may be made agreeably to the nature of those causes.

I. If bodies moving in the fame Araight line, firike against each other, the state of their common centre of gravity will not thereby be altered; viz. it will either remain at reft, or it will continue to move in the fame fraight line, exactly as it did before the meeting of the bodies.

This propofition is fo evidently deduced from the properties of the centre of gravity, as mentioned at N II and III. in the preceding chapter, that nothing more needs be faid about it in this place.

II. Let there be two non-elaftic bodies; and if one of them move in a straight line, whilst the other is at reft in that line, or is moving in the fame direction, but at a flower rate, or is moving in the contrary direction; viz. towards the body first mentioned; then those bodies must neceffarily meet or ftrike directly against each other, and after the ftroke they will either remain at reft, or they will move on together, conjointly with their common centre of gravity.-Their momentum after the ftroke will be equal to the fum of their momentums before the Atroke, if they both moved in the fame direction, but it will be equal to the difference of their momentums if they

moved in contrary directions.-Their velocity after the ftroke will be equal to the quotient that arifes from dividing the fum of their momentums, if they both moved the fame way, or the difference of their momentums, if they moved in contrary directions, by the fum of their quantities of matter.

That in any of the above mentioned cafes the two bodies must meet, and ftrike against each other, is fo very evident as not to require any farther illuftration.

That after the stroke those two bodies muft either remain at reft, or they must move together, conjointly with their common centre of gravity, is likewise evident; for as the bodies are not elaftic, there exifts no power that can occafion their feparation.

With respect to the mcmentum, it may be obferved, that when the two bodies meet, whatever portion of momentum is loft by one of them must be acquired by the other; fince, according to the third law of motion, action and re-action are always equal and contrary to each other; therefore, if be-. fore the ftroke the bodies moved the fame way, their joint momentum after the ftroke will be equal to the fum of their momentums before the ftroke. If one of the bodies was at reft, then, as its momentum is equal to nothing, the joint momentum will be equal to the momentum of the other body before the meeting. If the bodies moved towards each other, then their momentum after the meet

9

ing