436 Lord Rayleigh on Dynamical Problems in 



Both in the case where the left side was omitted, and also 

 when h vanished, we found that the solution was of the form 



/=•$.«-**, (31) 



where <f> was constant, or a function of if only. We shall find 

 that the same form applies also to the more general solution. 

 The factor V$> is evidently necessary in order to make 



J + 00 

 f(u) du independent of the time. By differentiation of 



(81)* 



f/=i^-ni-2^)g, 



g=-2<^-ni-2<K), 



so that (30) is satisfied provided </> is so chosen as a function 

 of t' that 



W+% = -*& + **&, 



or 



Thus 



df 



±<pdt' - <£ 



4 ^ = jTS^I=--| 1 ^( 1 -^- 1 ) + coust. 



where, however, the constant must vanish, since <£ = c© cor- 

 responds to t r =0. Accordingly 



*-T=pa?» ( 32 > 



which with (31) completes the solution. 



If ^ is small, (32) gives <j) = l/4t f , in agreement with (24) ; 

 while if t' be great, we have c[} = h = l/qv 2 , as in (15'). 



The above solution is adapted to the case where f(u) = for 

 all finite values of u, when t' = 0. The next step' in the pro- 

 cess of generalization will be to obtain a solution applicable 

 to the initial concentration of f(u), no longer merely at zero, 

 but at any arbitrary value of u ; that is, to the case where 

 initially all the masses are moving with one constant velocity a. 



