1821.] Phenomena of Heat, Gases, Gravitation, S^c. 289 



motion acquired by B and the motion lost by A. Whence the 

 motion of A after the stroke is A a — (a — 6) A = 6 A, and 

 that of B is B 6 H- A « — A 6. 



Cor. 1 . — By this proposition, the direction in which a body 

 overtakes and strikes ancjther being given, as well as the motions 

 and directions of the bodies before the stroke, the motions and 

 directions of the bodies after the stroke may 

 be found. Let A B be the motion and direc- 

 tion of the body A before the impact, and let 

 B C be the same of B, and let E B be the di- 

 rection in which the impulse is made by A, 

 and D B the intensity of it, or the quantity of 

 motion with which A in the line E B overtakes 

 and strikes the body B. Then join A D and 

 D C, and they shall be respectively as the 

 quantities of motions and directions of the bodies A and B after 

 the stroke. ^ 



Cor. 2. — Draw AC; then, since A C is the motion com- 

 pounded of the motions A D and D C, and likewise of A B 

 and B C, it follows that the aggregate motions of the bodies 

 before and after the stroke, reduced to the same direction, are 

 the same : and, consequently, the motion of the common centre 

 of gravity of the bodies remains unaffected by the impulse. 



Prop.V. 



If two perfectly hard bodies, moving in the same right line 

 but towards opposite parts, come in contact, an exchange 

 of motion will lake place ; or each body will retrace its path 

 with the motion which the other had before the contact. 



Let A and B be the two bodies, moving in opposite directions 

 with the velocities a and b. Then, because A « is the motion 

 with which A advances towards the parts B is leaving, and B b 

 is the motion with which B advances towards the parts A is 

 leaving, the sum A a + B b o£ these momenta is the motion 

 with which the two bodies approach ; and, therefore, the motion, 

 or force, with which their surfaces come in contact. But the 

 force with which the surfaces come in contact is the force with 

 which each surface, or body, is acted on at the time of contact 

 in a direction opposite to that in which the body was moving. 

 Therefore, at the time of contact, each body is acted on by two 

 opposite forces ; one its momentum ; and the other, the force of 

 contact, or the sum of the momenta of the two. Consequently, 

 the difference between these forces, or the momentum of the 

 other body, is the motion with which either of them is impelled 

 backwards, and retraces its path after the stroke. 



Cor. 1. — Hence if one of the bodies be. at rest before the 

 stroke, the other will be at rest afterwards ; and that which was 

 at rest will go on after the stroke with a motion equal to what 

 the other had before. These things coincide with what we have 

 deduced in our third proposition ; but the proof here given is 

 New Series, vol. i, t 



