OSCILLATIONS OP A SUSPENSION CHAIN. 391 



and because the tension at the lowest point of the chain must be the same derived from either 

 set of equations, 



c c 



It is of course not necessary that a should = a, for the chain might have been suspended 

 from two points not in the same horizontal line. I shall however suppose that the chain 

 is kept always symmetrical, either by the passage of another similar load from the opposite 

 end of the platform, or by any other means, and that a = a, as the laws I would arrive at are 

 as I before stated, quite independent of the form of the disturbed chain being symmetrical 

 or unsymmetrical about the vertical axis. 



As a first example of the effect of a moving load, I shall suppose the bridge disturbed by 

 two equal loads moving from opposite directions with the same velocity and tending to meet 

 in the centre. I shall assume b to be >o, and shall consider only what happens till the 

 load reaches the centre. In this case it will be sufficient to employ the first of the three 

 sets of formula?. It is clear that according to these suppositions, the chain will be symme- 

 trical about the vertical axis. I shall suppose also that by means of saddles or some other 

 contrivance, the tension at the ends is kept constant, so that M = 0. 



Then we should have 



d* , . . 7r a . 8 a 8 a . tt . 



— (A) = -eg— -A, ~ng~ +ttg —. sin ~{a-pt); 



dr 4<r 7i- 7T 2 a 



d 2 . . . 7T 2 „ 8a 8a . 3tt 



7r fj.,32 (&g ir 



'•Ta t + a) + ^( Cg -fo '" , i^ (a " p0, 



dA 

 If p- be very small compared with eg, the condition that A x and may = when 



t = 0, will make h Y = ; that is to say when we put p* = in the denominator of the last term ; 



32a 3 32a 3 . tt , 



.-. A y = -m.— 7- + M-— T" sln — (a -Pt)t 

 ire ir c 2a 



32a 3 32a 3 . 3tt 



A 2 = /u. — + n — sin — (a - p t). 



81.tt 4 c 81.7r 4 c 2a v r ' 



Now when pt = a since p has been assumed to be indefinitely small ; it is clear that 

 the position of the chain ought to coincide with its statical position, on the hypothesis, that 

 while the tension at the ends is the same as before, the density has been increased throughout 

 from 1 to 1 + fx. 



And therefore the deflection in the centre as furnished by our equations ought to be the 

 same as the statical deflection, or 



M a 8 d?V nar 



, or — — m . 



2c aV„ 2c 



