Systems with Propagated Coupling. 439 



If any solution of this is a pure imaginary ip, then 



and R(-m/-f K)+r(-V+|^ = 0, 



whence for the critical case at which self-maintenance just 

 begins 



— mp 2 + K - - + p<\ / ot? — — r 2 



it 

 with the condition — >?• if p is to be real, which is there- 



fore one cf the conditions for maintenance. 



Thus p 2 differs from -- by a quantity proportional to 



p which is in practice small. If the factor of p is denoted 

 by F, then 



K F 2 _, F 



m 4m 2 Im 



V m Zm 



m zm 



It should be noted that if a 2 is sufficiently small it is 

 impossible for self-maintained oscillations to exist. 



As before, to pass to the case of propagation a? is replaced 



by a 2 e 2 k x , which (if a 2 is eliminated instead off — Lp 2 + ^ J 

 as before) leads to the equation 



p sin 2qx I — Lp 2 -f ^ ) ( — mp 2 -f K) — rRp 2 J 



=p 2 cos 2qx f R( — mp 2 + K) -tr( — Lp 2 +- «v~] J. 



It would be unprofitable to consider this fully ; special 

 cases can be worked out when desired. 



(iii.) The above cases differ widely from the actual con- 

 ditions of the experiments referred to in the first paragraph. 

 They are given simply to illustrate a general mode of 

 examining the maintenance of vibrations, both with and 



2 G2 



