1822.] on Mr, Herapath's Theory. S6^ 



in the line described by the centre of gravity of the former, the 

 striking body will remain at rest after the impulse, and the other 

 will proceed in the same right line in which the former was mov- 

 ing, and with the same momentum. 



All that I require for demonstrating this Proposition is, that 

 the intensity or force of percussion he the same as, or equal to, the 

 motion generated ; and that the force of percussion he proportional 

 to the generating momentum. Without adverting to the preceding 

 propositions, each of these postulates is admitted in the quota- 

 tions I have made from the authors C. has quoted against Mr. 

 Herapath. I shall, therefore, nottrouble myself about their accu- 

 racy, which is indeed " as nearly as possible self-evident," but 

 shall proceed with the rest of the proof. Let B, W, ^eiwo per^ 

 fectly hard and equal balls at rest, and let A, A\ be any two other 

 perfectly hard balls striking respectively B, B', according to the 

 conditions of the proposition. Let also a, a', be the velocities 

 of A, A', before the strokes, so that A a = A^ a\ Then if Z> be 

 the velocity of B after the stroke, and b^ that of B', we have B b 

 = B' b\ and h = I/, Now if A move at all after the stroke, it 

 must follow the body B with an equal or less velocity than b ; 

 because it could not move the contrary way unless the force of 

 percussion was greater than the generating momentum, which is 

 impossible. The same is likewise true of the body A^. There- 

 fore, if they do not remain at rest, let them follow B and B^ with 

 the velocities p, p\ respectively. Then because the sum of the 

 momenta in each case before and after the stroke is the same 

 A« = Ap + B6, and A' a' = A' p' + B' b', and conse- 

 quently K p = A' p' ; that is, the velocities p, p', remaining to 

 A, A', after the strokes are reciprocally proportional to the bodies 

 A, A\ Suppose A = B, then p will be a certain part, for 

 instance, the wth part of 6, so that np = b. Therefore b = n p 



:= np' —- and p' = b^ -jj. Now the value of—, may be any 



thing we please, and, therefore, much greater than 7i ; in which 

 case p^ must be greater than b^ ; that is, if p has any magnitude, 

 the body A' which cannot move faster than B', because it comes 

 behind it, might nevertheless have a greater velocity in the same 

 direction, which is absurd. Therefore, p andp' must each be 

 equal to o, the only case in which the equation A p = A^ p' can 

 be universally true ; or both the bodies A, A', must remain at rest 

 after the impulses, and, consequently, the bodies B, B', proceed 

 with the momenta A «, A^ a' , respectively. 



Cor. \. — Because the intensity of the stroke is equal to the 

 momentum communicated, and this momentum is equal to the 

 momentum of the moving body before the stroke : this momen- 

 tum of the body before the stroke is equal to the intensity of per- 

 cussion ; and the whole of this intensity must be equally felt by 

 each of the bodies without any regard to their relative size. 



