1868.] Mr. R. Moon on the Impact of Comjjressible Bodies. 413 



pinging cylinder before impact, has its greatest value at the surface of col- 

 lision, and diminishes as v/e recede therefrom. 



It is clear that, in the case we are now considering, the collective mo- 

 mentum abstracted from the impinging cylinder by the collision will be 

 less, and finitely less, than that which was abstracted by the collision in 

 the former case, in which the velocity of each particle of the impinging 

 cylinder was supposed uniform and equal to V. 



Eor, if M be the momentum lost by collision when the velocity before 

 impact is uniform and equal to V, it is clear that when the velocity 

 before impact is represented by V— the quantity V^^ may be such that 

 the .momentum be/ore impact may be finitely less than M ; from which it 

 follows inevitably that the amount of momentum lost by collision in this 

 latter case must be less than M. 



Let us now vary the data by supposing that the velocity before impact 

 increases instead of diminishes as we recede from the surface of collision ; 

 so that at the moment of impact, before taking account of the effects of 

 collision, the velocity at any point of the impinging cylinder is represented 

 by V+ V, instead of V- V,. 



It is clear that the momentum abstracted by the collision in this latter 

 case will be greater, and finitely greater, than in the case where the velo- 

 city before impact is uniform and equal to V. Let the additional momen- 

 tum abstracted in this case be M^, the whole momentum so abstracted 

 being represented by M + M^. 



Let us now make a final variation in the conditions of the problem, by 

 supposing that at the moment of impact, and irrespective of the impact, a 

 velocity equal and opposite to V is communicated to each particle of the 

 impinging cylinder, so that at that instant, without taking account of 

 any action of the one cylinder upon the other, the velocities of the two 

 cylinders along their surfaces of contact will be equal, or, rather, will be 

 alike zero ; at the same time that at every other point of the impinging 

 cylinder there will be a variable velocity increasing in amount as we 

 recede from the surface of contact. 



In estimating the effect of the cylinders being in contact under the cir- 

 cumstances last described, it is clear that the abstraction from each particle 

 of the impinging body of the velocity V can only be regarded as preventing 

 the transference to the second cylinder of so much of the momentum 

 M-hM^ as that velocity, if it had constituted the entire velocity before 

 impact of the impinging body, would have given rise to, viz. M ; and that 

 the momentum whose appearance in the expression M + M^ is due to 

 the fact of the first cylinder having been originally endowed with the 

 variable velocity in addition to the constant velocity V, will still con- 

 tinue to be transmitted to the second cylinder from the first. 



We are thus led to this singular and, doubtless, pregnant conclusion, 

 that in a continuous material system in which there is neither discon- 

 tinuity of velocity nor discontinuity of density, all the consequences of 



