430 Lord Rayleigh on Dynamical Problems in 



so that 



f+oo 1 (* +00 



v z e- kv2 dv=A ve-**dv. . . . (18) 



Our equation then becomes 



2kuf(u) + qf(u)=Q, 

 giving 



f{u)=Ae- M2 I« (19) 



The mean energy of the masses is \q/k, and this is now equal 

 to the mean energy of the projectiles. We see that if the 

 mean energy of the projectiles is given, their efficiency is 

 greater when the velocity is distributed according to the 

 Maxwell law than when it is uniform, and that in the former 

 case the Waterston relation is satisfied, as was to be expected 

 from investigations in the theory of gases. 



It may perhaps be objected that the law e~ kv2 is inconsistent 

 with our assumption that v is always great in comparison with 

 u. Certainly there will be a few projectiles for which the 

 assumption is violated ; but it is pretty evident that in the 

 limit when q is small enough, the effect of these will become 

 negligible. Even when the velocity of the projectiles is 

 constant, the law e~ u ' 2 ^ v2 must not be applied to values of u 

 comparable with v. 



The independence of the stationary state of conditions, 

 which at first sight would seem likely to have an influence, 

 may be illustrated by supposing that the motion of the masses 

 is constrained to take place along a straight line, but that the 

 direction of motion of the projectiles, striking always centri- 

 cally, is inclined to this line at a constant angle 6. 



If u be the velocity of the mass (unity) before impact, 



and u after impact, B the impulsive action between the mass 

 and the projectile, 



u— u' = Bcos0. 



Also, if v, V be the velocities of the projectile (q) before and 

 after impact, 



