Perfectly Elastic Bodies of any Form. 35 



(6.2) and (63), and dividing by R, we find 



Wi + UJ -/ 2 (U' 2 + U 2 ) + (y ini -z^) (p\ fj$ 



— {y^2 — Z 2 m 2) {P*2 +P2) + &C. = (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 i = W + (ft»i —*\ m i)Pi + &c - ■> 

 and if we pnt p a for the velocity of the other striking-point 

 resolved along the same line, and v\ and v' 2 the same quantities 

 after impact, we may write equation (65), 



v l +v' l -v 2 -v' 2 = 0, ..... (66) 

 or 



Vl — V9 -v' 2 -v' l) (67) 



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

 resolved in the line of impact is equal to that of approach. 



Substituting the values of the accented quantities in equa- 

 tion (65) by means of equations (63) and (64), and transposing 

 terms in R, we find 



2{V 1 l 1 —\J 2 l 2 +p l {y l n l -z l m l )~p 2 (y 2 n 2 -z 2 m 2 )} + &c. 



= -E{£ + £ + (y 'V m ' r + {y ^T inhY + &c - (68) 

 IM, M 2 A, A 2 v ' 



the other terms being related to y and z as these are to x. From 

 this equation we may find the value of R; and by substi- 

 tuting this in equations (63), (64), we may obtain the values of 

 all the velocities after impact. 



We may, for example, find the value of V*[ from the equation 



TV J Z i 2 j_ l * , (l/ini-^m,) 2 (y 2 n 2 -z 2 m 2 y -, \% 



^I-m^m^ — t t — + — A 2 +&C 7T 



+ 2V 2 l 2 —2p l (y l n 1 —z l m 1 )+2p 2 {y 2 n 2 — z 2 m 2 )-- &c. 



Prop. XXIII. To find the relations between the average velocities 

 of translation and rotation after many collisions among many bodies. 



Taking equation (69), which applies to an individual collision, 

 we see that V\ is expressed as a linear function of U p y^PvPa 

 &c, all of which are quantities of which the values are distributed 

 among the different particles according to the law of Prop. IV. 

 It follows from Prop. V., that if we square every term of the 

 equation, we shall have a new equation between the average 

 values of the different quantities. It is plain that, as soon as the 

 required relations have been established, they will remain the 



D2 



