438 Lord Rayleigh on Dynamical Problems in 



When is small, we have from (38), 



^'^TPll + " da ^ a ) «- ( *-" )2/ "'= v"r-%0) ultimately; 

 so that / >. 1 // a\ 



x«=7^ /(a ' 0) - 



Accordingly the required solution expressing the distribution 

 of velocity at t' in terms of that which obtains when t f = 0, is 



~f(u, 0= *J% * J + / a/( " ? 0) E ^{-^^- ae "^') 2 }- • (39) 



We may verify this by supposing that/(w, 0) = e~ hu % repre- 

 senting the steady state. The integration of (39) then shows 

 that 



f(u,t')=e-*« 2 , 



as of course should be. 



An example of more interest is obtained by supposing that 

 initially 



f(i h 0) = e- h,u2 ; (40) 



that is, that the velocities are in the state which would be a 

 steady state under the action of projectiles moving with an 

 energy different from the actual energy. In this case we 

 find from (32), (39), 



// 6 \ Q h ' u2 



/fcO-VWbr'K^- • • (41 > 



We will now introduce the consideration of variable velocity 

 of projectiles into the problem of the progressive state. In 

 (28) we must regard v as a function of v. If we use vdv to 

 denote the number of projectiles launched in unit of time with 

 velocities included between v and v + dv, (28) may be written 



which is of the same form as before. The only difference is 

 that we now have in place of (29), 



t f = 2q 2 l$vv 2 dv, (43) 



A= §vdv-r-q§vv 2 dv (44) 



In applying these results to particular problems, there is 

 an important distinction to be observed. By definition vdv 

 represents the number of projectiles which in the unit time 



