260 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



corresponding velocities for m and m after impact. Let u , v and 

 , v f be the velocities that would replace u t v and u t v if e 

 were zero. 



Now if e were zero there would be no restitution or the bodies 

 would have the same velocities in the line of centres immediately 

 after the impulse, we should therefore have 



UQ = U . 



The equation of conservation of linear momentum of the 

 system in the line of centres is in this case 



(m 4- m) u = mU+ m U , 



and the impulsive pressure, R , between the bodies is given by 

 the equation 



the sense of R being opposite to that of U. 



Now suppose that e is not zero, but the impulsive pressure 

 between the bodies is E (1 + e). We have the equations 

 mu + m u = m U + m U , 



which may be written 



- 



m m 



whence 



u-u = -e(U-U \ 



exactly as in direct impact. 



Thus, at any rate for the case of smooth spheres, Poisson s 

 hypothesis is equivalent to the following generalisation of Newton s 

 experimental result : 



The relative velocities, after and before impact, of the points of 

 two impinging bodies that come into contact, resolved along the 

 common normal to their surfaces at these points, are in the ratio 

 e : 1, where e is the coefficient of restitution. 



In what follows we shall refer to the statement just made as 



