440 Lord Bayleigh on Dynamical Problems in 



If the vibrators were left to themselves, x(r) might be chosen 

 arbitrarily, and yet the distribution of velocity, denoted by 

 f(u), would be permanent. But if the vibrators are subject 

 to bombardment, f(u) cannot be permanent, unless it be of 

 the form already determined. The problem of the permanent 

 state may thus be considered to be the determination of ^(r) 

 in (50), so as to makef(u) equal to e~ hu2 . 



We will now limit ourselves to a law of force proportional 

 to displacement, so that the vibrations are isochronous ; and 

 examine what must be the form of %(r) in (8) in order that 

 the requirements of the case may be satisfied. 



By (15'), if K be the whole number of vibrators, 



The determination of the form of % is analogous to a well- 

 known investigation in the theory of gases. We assume 



X(r) = Are~ hr2 , (52) 



where A is a constant to be determined. To integrate the 

 right-hand member of (51), we write 



■■■<t 



2 i _2 



W; • • (53) 



so that 



)u */(?-«') Jo ^ k 



Thus 



A=4AN (54) 



The distribution of the amplitudes (of velocity) is therefore 

 such that the number of amplitudes between r and r-\-dr is 



W.Mre-^dr, (55) 



while for each amplitude the phases are uniformly distributed 

 round the complete cycle. 



The argument in the preceding paragraphs depends upon 

 the assumption that a steady state exists, which would not be 

 disturbed by a suspension, or relaxation, of the bombardment. 

 Now this is a point which demands closer examination ; 

 because it is conceivable that there may be a steady state, 

 permanent so long as the bombardment itself is steady, but 

 liable to alteration when the rate of bombardment is increased 

 or diminished. And in this case we could not argue, as before, 

 that the distribution must be uniform with respect to 0. 



If x denote the displacement of a vibrator at time t, 



x=-n~ l rsin (nt — 0), dx/dt=r cos (nt — 6). 



