Illustrations of Certain Optical Phenomena. 233 



or a combination of two such undulations travelling in oppo- 

 site directions ; and this state is one which, if once started, 

 will be kept up by the internal forces of the system. 



We shall now investigate the permanent motion of the 

 chain when one of the particles is constrained to simple 

 harmonic vibration of frequency greater than the critical 

 frequency \f fijir. 



Let y be the displacement at time t of the constrained 

 particle, and y 1 , y 2 , y 3 . . . those of the consecutive particles on 

 one side of it. In any " fundamental" or " normal " mode 

 of vibration, the accelerations y 1} y 2 , &c. are proportional 

 to the displacements y^ y 2 , &c. ; so that we may write 



h = — ©Vu h = - ft)2 i/2, &c 



Hence by equation (1) we have 



and so on. 

 Hence 



— -2 = -»_& = _&_£ = & G . = k (suppose). (11) 



(12) 



if h is greater than 2, or if co 2 > 4/t, or (since g> is 2tt/T) if 

 1/T> v£/*r. 



The same result can be deduced by writing equations (11) 

 in the form 



y\ y\ y* y* 2/3 yi v y 



whence we obtain for any one of the ratios — — , — — , &c. 



f* 



y\ 



y\ 



V2 3/2 





These 



equations are satisfied by 



assuming 





y\ 



k 



#3 



1 



&c. = r, 



the value 



That is, 



yi y$ 



k — 7 — 1 = r (suppose) (14) 



k—&c. 



I l 



