366 D:s Repfy to CJs Observations [May, 



sum of the momenta of the two, or twice the momentum of 

 either one before the stroke. 



Mr. Herapath has demonstrated this theorem generally in his 

 Prop. 6, Annals for April, 1821. I have chosen this particular 

 case of it, because against this, C. has levelled his objections ; 

 and in the proof I intend to have recourse merely to what I have 

 already demonstrated from the principles admitted in the old 

 theory, and to a result whicli agrees equally well with both 

 theories. 



By the old theory, if a hard body A, having the velocity a, 

 strike directly another hard equal body A" at rest, the motion 



communicated to A^ by the impulse is — £- A = -— -^. And by 



the same theory, if the two same balls meet each other, instead 

 of one of them being at rest, with equal opposite momenta A a, 

 A' a\ the motion destroyed in either, or, which is the same, the 

 motion communicated to either is A a. But by the quotations I 

 have made from C.'s quoted authors, these communicated motions 

 are equal to the intensities of their respective strokes felt by each 

 body in the direction in which the motion is communicated. 

 Therefore, the intensity of the stroke on either body when one is 

 at rest, is half as great as when both meet with equal opposite 

 momenta. Now when one of the bodies is at rest, I have shown. 

 Cor. 1, Prop. 6, by strict mathematical reasoning from the prin- 

 ciples admitted in the old theory, that the intensity of the stroke 

 on each is equal to the momentum of the moving body ; when, 

 therefore, they are both moving with equal momenta towards 

 opposite parts, the intensity of the stroke on each is equal to 

 twice the momentum of one, or the sum of the momenta of the 

 two. 



Cor. — Hence the two equal bodies after the impulse recede 

 towards the parts whence they came with the same momenta 

 they had before they met. For the motion communicated by the 

 impulse is equal to the intensity of (he stroke on the body, and 

 this intensity is equal to 2 A « ; but at the time of the stroke, 

 the body had a momentum in an opposite direction equal to A a. 

 Therefore at the time of the contact, the body is the same as if 

 it was urged in two opposite directions by the forces A a and 

 2 A «, the former impelling it in the direction in which it was 

 moving, and the latter on the contrary ; consequently it retraces 

 its path with the momentum A a. 



This conclusion, brought out by strict mathematical induction, 

 from the principles of the old theory, coincides with Mr. Hera- 

 path's, and also with the theories of Wren, Huygens, and 

 Wallis. 



It is worthy of remark, that C. by way of mathematically refut- 

 ing this conclusion, admits the principle of Mr. H.'s proof, that 

 the intensity of the stroke is equal to the sum of the momenta, 



