OSCILLATIONS OF A SUSPENSION CHAIN. 397 



In this kind of vibration the disturbed chain is always tangential to the equilibrium 

 position of the chain at the points of suspension, but in those kinds we previously considered, 

 the inclination of the disturbed chain to its equilibrium position was a maximum at the points 

 of suspension. 



If we examine the general equation, 



dV 

 and assume for — — the following form, 



2 . A sin \/cg (pt + /3) sin (ps + a) + Ps 2 + Rs + Q, 

 A being a constant and P, R and Q functions of t, we shall find for P, R and Q {substituting 

 in (11) and equating? P «= 0, 



R + £, + -r 5 = o, 



C" 



d 2 

 — (Q + L ) = cgx2P = 2gM lt 



whence P, Q and R are given in terms of L , L x M and M v 



We shall now consider the motion of the two portions of the chain reckoning from the 

 right and the left of the origin separately, dashing the symbols which have reference to the left 

 hand, or negative part of the chain. Hence 



' c ds 2 a c ds a 



a$V IdV _,-, 



„; = o — T + - _ - + MS 



c ds 2 a c ds a 

 T d*V 



«;=o=z,' + 



ds\ 



Also u = - «„' v = v ' , 

 d 3 V d?V 



ds 3 ,, d^o 



which eight conditions are sufficient to determine L , L ly M , M 1 and £,„', L t ', J/ ', and Jf,', 

 whether the chain be symmetrical always about the vertical axis or not. Or we may consider 

 the whole length of the chain between its two ends, if a = - a, we shall then have 





and employing only one set of equations, we shall thus by these four conditions, be enabled to 



51—2 



