156 Hoy al Society. 



For, 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 — V 1? the quantity \ x may 

 be such that the momentum before 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 1 instead of V — V x . 



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 velocity before impact is uniform and equal to V. Let the 

 additional momentum abstracted in this case be M x , the whole mo- 

 mentum so abstracted being represented by M + M r 



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 vari- 

 able velocity V 1 increasing in amount as we recede from the surface 

 of contact. 



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

 circumstances 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 + Mj 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 M^ whose appearance 

 in the expression M + M 2 is due to the fact of the first cylinder 

 having been originally endowed with the variable velocity Y x in ad- 

 dition to the constant velocity V, will still continue to be transmitted 

 to the second cylinder from the first. 



We are thus led to this singular and, doubtless, pregnant conclu- 

 sion, that in a continuous material system in which there is neither 

 discontinuity of velocity nor discontinuity of density, all the con- 

 sequences of collision may occur, viz. the instantaneous transmission 

 of a finite amount of momentum from one part of the system to an- 

 other, provided we have discontinuity in the tendency to compression 

 in the different parts of the system. 



The author has endeavoured, in former communications to the 

 Royal Society, to show that when the velocity in a fluid diminishes 

 in the direction to which the motion tends, the slower particles will 

 offer a resistance to the motion of the faster particles, which the 

 received theory fails to take into account. The foregoing speculation 



