426 Lord Rayleigh on Dynamical Problems in 



or 



, P + Q 2Q , 



==M+ pzrQ( w ~ v ')' • • • • (5) 



In the application which we are about to make, P will denote 

 a relatively large mass, and Q will denote the relatively small 

 mass of what for the sake of distinction we will call a pro- 

 jectile. All the projectiles are equal, and in the first instance 

 will be supposed to move in the two directions with a given 

 great velocity. After collision with a P the projectile re- 

 bounds and disappears from the field of view. Since in the 

 present problem we have nothing to do with the velocity of 

 rebound, it will be convenient to devote the undashed letter v 

 to mean the given initial velocity of a projectile. Writing 

 also q to denote the small ratio Q : P, we have 



2(7 



u' = u+- — ^(u — v) (6) 



1 — </ v v ' 



If u and v be supposed positive, this represents the case of 

 what we may call a favourable collision, in which the velocity 

 of the heavy mass is increased. If the impact of the pro- 

 jectile be in the opposite direction, the velocity u", which 

 becomes u after the collision, is given by 



2l " = u + Jl-(u + v). ..... (7) 



The symbol v thus denotes the velocity of a projectile with- 

 out regard to sign, and (7) represents the result of an un- 

 favourable collision. 



Permanent State of Free Masses under Bombardment. 



The first problem that we shall attack relates to the ultimate 

 effect upon a mass P of the bombardment of projectiles 

 striking with velocity v, and moving indifferently in the two 

 directions. It is evident of course that the ultimate state of 

 a particular mass is indefinite, and that a definite result can 

 relate only to probability or statistics. The statistical method 

 of expression being the more convenient, we will suppose that 

 a very large number of masses are undergoing bombardment 

 independently, and inquire what we are to expect as the 

 ultimate distribution of velocity among them. If the number 

 of masses for which the velocity lies between u and u + du be 



