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OSCILLATIONS OF A SUSPENSION CHAIN. 381 



3. Let a tangent to the lowest point of the chain at rest and parallel to the horizon 

 be taken for the axis of x, a normal and vertical at that point for the axis of y. 

 Let a mass of an units length of chain = 1. 

 The area of a section of chain = 1, 



w, y co-ordinates of a point in disturbed chain 



<v, y, of same point in chain at rest 



Let x = x + u, y = y + v. 

 Let T' = tension at (x, y) in disturbed chain, 

 T = (», y) in chain at rest. 



Let T' = T + w. 



c = length of chain the weight of which equals the equilibrium tension at the lowest point 

 of the chain at rest. 



g the accelerating force of gravity, 

 s = length of chain from origin to {x, y). 

 Then dx' 2 + dy* = ds\ 



dx 2 + dy* = ds*; 



dx du dy dv . 



.•. — . — H — - -j— = 0, as tar as the first order. 

 ds ds ds ds 



This may be called the equation of continuity of the chain. As I have found that all 

 circumstances of importance are equally well developed by oscillations in which the chain is 

 always symmetrical about the vertical axis, as by oscillations in which this is not the case, I 

 shall consider only oscillations of the former description, since the number of arbitrary functions 



is diminished and the labor lessened, on this hypothesis ; which necessitates u and — beino- 



ds ° 



each = when a = 0. It may be observed that this condition does not at all affect the kind 



of analysis employed, it only shortens the arithmetic. For the equations of motion we 



shall have 



df ds \ ds) *' 



— =— (r' d -)- 



dt* " ds\ ds)' 

 d?v d f dv dy\ 



dv d f dv dy\ 



r, — r = — T-— + w — \ to first order 

 dr ds \ ds ds I 



or, 



dC ds \ ds ds I 



\ (1), 



~ (t~ 



ds \ ds 



d 2 u d f,j,dn dx\ 



df ds \ ds ds) 



dv d*V 



ds ds 3 ' 



du dy dv s <PV , . 



— = — — . — = — - remembering that 



ds dx ds c ds 6 



49 — 2 



