288 Mr. Herapath on the Causes, Laws, and principal [April, 



if the ratio of A' to B be assignable, the motion of A' after the 

 stroke will be unassignably small ; that is, the body A will remain 

 at rest. And because b^ A' is indefinitely small compared to 

 a* A% the intensity a' A' — b' A' of the impulse will likewise be 

 equal to the momentum a' A' of the moving body before the 

 stroke. But since the intensity of the impulse is the force act- 

 ing upon the quiescent body at the time of the impulse, it is also 

 equal to the motion acquired by this body. Therefore, if a hard 

 ball strike another hard ball at rest, &c. 



Cor. 1. — From this proposition it is easy to determine the 

 motion and direction of a hard body striking obliquely with a 

 given momentum in a given direction on another 



hard body at rest. For if A B be the direction >^r I^ 



and momentum of the body previous to the 

 stroke, and B C the direction in which it strikes 

 the quiescent body B, produce C B to E, on 

 which demit the perpendicular A E, and draw 

 B D equal and parallel to A E, and B D will be 

 the motion and direction of A after the stroke, 

 and B C, if equal to E B, those of B. 



Cor. 2. — Hence it follows, that in any oblique collision on a 

 quiescent body, the motions of the bodies after the impact will 

 be perpendicular to each other. 



Scholium. 



I forbear to enter further into the collateral minutiae of this 

 theorem, because it would lead me too far out of my way, and I am 

 in haste to arrive at things of more importance. However, it is 

 necessary to state that 1 have chosen this indirect method of 

 demonstrating this proposition, for the sake of making it rest on 

 principles as different and as independent as possible of those of 

 a future proposition, from which it will flow as a corollary. 



Prop. IV. 



If a hard body overtake and strike another hard body, mov- 

 ing with a less velocity in the same right line, the first body 

 will, after the stroke, continue its course with the same 

 velocity which the other body had before it; and the second 

 body will acquire from the stroke a momentum equal to the 

 difference of the velocities of the bodies previous to the contact, 

 drawn into the mass of the first body ; that is, if A, B, represent 

 the two bodies, and «, b, their velocities before collision, the 

 motion of A afterwards will be A b, and that of B, B ^ + 

 (a — b) A. 



Because the bodies are both moving the same way, it is evi- 

 dent that we may conceive the second body to be at rest, and 

 the other body to strike it with a velocity equal to the difference 

 of the velocities a and b ; in which case the proposition will come 

 to the same thing as the last. Therefore (a — i) A is the 

 momentum, or force, of colHsion; and is, consequently, the 



