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 causes, 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 ftraight line, ftrike 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 ftraight 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-elastic bodies; and if one of them move in a straight line, whilst the other is at rest 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 thofe 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 ftroke 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 elastic, there exifts no power that can occafion their feparation. With respect to the momentum, 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 before the ftroke the bodies moved the fame way, their joint momentum after the stroke 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 ing will be equal to the difference of their former momentums; and if in this cafe their momentums are equal, then their difference vanishes; hence the bodies will remain motionless after their meeting. The laft part of the propofition is likewise evident; fince it has already been fhewn, that in equable motion, the velocity is equal to the quotient of the momentum divided by the quantity of matter. When the weights and velocities of the two bodies before their meeting are known, their velocity after the meeting may be determined by the following general method. Let A and B, in fig. 4, 5, 6, and 7, of Plate II. which represent the above mentioned cafes, be the two bodies; let C be their common centre of gravity, and D the place of their meeting. Make DE equal to DC; fo that the point D may be between C and E; then DE will reprefent the velocity of the two bodies after their meeting; for, fince the bodies after the concurse move together conjointly with their common centre of gravity; and fince it has been proved in the preceding propofition, and at N° II. of the preceding chapter, that the ftate of the common centre of gravity of the two bodies is not altered by their mutual action upon each other; therefore the velocity of their common centre of gravity after their meeting, muft be equal to its velocity before the meeting; viz. DE must be equal to C D, and is the fame as the velocity velocity of the two bodies after their meeting, because then they move together with their common centre of gravity. Of the above mentioned figures, it may be easily perceived, that the 4th fhews when both the bodies move the fame way; the 5th represents the cafe in which B is at reft before the ftroke, and of course the two points B and D coincide; the 6th fhews when the two bodies move towards each other; and the 7th fhews when the two bodies. move towards each other with equal momentums, in which cafe, after their meeting, they will remain at reft. The refpective velocities of those two bodies are reprefented in all the four figures, by AD and BD; for they run over those distances in the fame time; and AB is the difference of thofe velocities. Alfo their refpective momentums are reprefented by the product of the weight of A multiplied by AD, and the product of the weight of B multiplied by B D. The momentum of both the bodies together after their meeting, is reprefented by the product to their joint weight multiplied by DE (1). Since (1) The following is an example of the numerical computation of the firft cafe, fig. 4, which will be fufficient to indicate the manner of calculating the other cafes. Let A weigh 10 pounds, and move'at the rate of 4 feet per minute. Let Since when one of the bodies is at reft, the velocity after the meeting is equal to the quotient of the velocity of the moving body, divided by the fum of the quantities of matter of both the bodies; it follows that the larger the body at reft is, the smaller will the velocity be after the meeting. For Let B weigh 6 pounds, and move at the rate of 2 feet per minute. And let the distance A B be 32 feet. ́ The centre of gravity is found by saying 16: 32:: 10: 20 feet; hence AC 12 feet. (See BC = page 75.) 32 X 10 Put BDx, and AD will be equal to 32+x. Then the time employed by A in moving from A to D, is equal to the quotient of the space 32+x, divided by its velocity; 32+ x viz. it is And the time employed by B in mov 4 ing from B to D, is equal to the quotient of the space x, divided by the velocity of B; viz. it is. But fince the 2 bodies meet at D, thofe times must be equal; that is, 32 + * = *; hence 64 + 2x=4x; and x=32=BD. 4 Then DE DB+BC 32+20= 52 feet; that is, af ter the meeting, the two bodies will move from D to E; (viz. over 52 feet) in as much time as each of them employed in going to D; that is, 16 minutes. Therefore, to find how many feet per minute the bodies will run over after the meeting, divide 52 by 16, and the quotient 3 thews that they will move at the rate of 3 feet per |