ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 407 



Let R be the impulsive force between the bodies, measured by the momentum 

 it produces in each. 



Then, for the velocities of the centres of gravity, we have the following 

 equations : 



Z7>ffi + j|, U',= U Z -^ ...................... (62), 



with two other pairs of equations in V and W. 



The equations for the angular velocities are 



R 7? 



........... (63), 



with two other pairs of equations for q and r. 



The condition of perfect elasticity is that the whole vis viva shall be the 

 same after impact as before, which gives the equation 



Ml ( U'\ - U\) + M, ( U'\ - U\) + A, (p'\ -p\) + A, (p'\ -p\] + &c. = 0. . . . (64). 



The terms relating to the axis of x are here given ; those relating to y and 

 z may be easily written down. 



Substituting the values of these terms, as given by equations (62) and (63), 

 and dividing by R, we find 



* (U\+ U^-l t (U\ + U t ) + (y 1 n l -z 1 m 1 )(p' 1 +p 1 )-(y 3 n 1 -z,m 2 ) (p' s -hp a ) + &c. =0...(65). 



Now if v l be the velocity of the striking-point of the first body before 

 impact, resolved along the line of impact, 



v t = 1^ + (y^ - 2,771.) p, + &c. ; 



and if we put v, for the velocity of the other striking-point resolved along the 

 same line, and v\ and v\ the same quantities after impact, we may write, 

 equation (65), 



v 1 + v' l -v a -'z/ 2 = .............................. (66), 



or v l v, = v' t v\ ................................. (67), 



which shows that the velocity of separation of the striking-points resolved in 

 the line of impact is equal to that of approach. 



