234 Prof. J. D. Everett on Dynamical 



which is the assumption employed above. The assumption 

 gives a choice between two values of r, one being the 

 reciprocal of the other. We must clearly choose the value 

 which gives a decreasing riot an increasing geometrical pro- 

 gression, r and its reciprocal are the roots of the equation 



r 2 -kr+l=-- 0. 

 Hence we have 



2r=/f+ N /(P-4), - = k- y(£ 2 -4), . . (15) 



where 



^-2, V(* 2 -4) =-J{<f-W- 



— = &r — V/JL — (0 s/((o' 2 — 4/i). 



Thus the resulting motion is completely determined. Suc- 

 cessive particles will be opposite in phase, and their amplitudes 

 will diminish in geometrical progression as we move away from 

 the particle which is subjected to external constraint, the 

 diminution being the more rapid as the frequency is greater. 

 § 6. The constraining force (taking the mass of the particle 

 as unity) is the excess of the actual acceleration —(o^i/q above 



the acceleration — 2fju(y — y x ) or — 2/x# (l + -J due to the 



tensions. It is therefore y ( *- w 2 + '2fi + — ). 

 But from above, 



r 

 Hence the constraining force is 



-y G> .(©*— 4fi) (16) 



It is proportional and opposite to the displacement y , and 

 therefore does equal amounts of positive and negative work. 

 These remarks apply to the permanent regime only. 



The critical value of oo which separates free from forced 

 vibrations is 2 s/\l, and this value makes the constraining 

 force zero. As a) increases from this minimum to infinity, 

 the constraining force increases from zero to infinity, and the 

 ratio r of successive amplitudes increases from unity to 

 infinity. 



The work done in the initial stage is represented by the 

 energy of the chain in the permanent state. The ratio of the 

 energy of the whole chain (extending, to infinity on both 

 sides) to the energy of the particle to which the constraint is 

 applied is 



