80 STATICS OF SYSTEMS OF PAETICLES 



This is the Cartesian equation of the cable. It is easily seen to 



2 H 



represent a parabola of latus rectum --- Thus the cable must 



w 



hang in the form of a parabola. The greater the horizontal ten- 

 sion, the greater the latus rectum of the parabola, and therefore 

 the natter the curve of the cable. A perfectly straight cable is of 

 course an impossibility this would require infinite tension. 



57. To find the tension at any point of the cable, we square 

 equations (16) and (17), and add corresponding sides. Thus 



giving the tension at a point distant x from the center. If the 

 bridge is of length 2 a, the tension at either pier must be 



The Catenary 



58. In the problem of the suspension bridge, we neglected the 

 weight of the cable. A second problem arises when the cable is 

 supposed to be acted on by no external forces except its own 

 weight. The problem here is simply that of 

 a string of which the two ends are fastened 

 to fixed points, and which hangs freely be- 

 tween these points. 



As before, let be the lowest point, and 

 let P be any other point. The forces acting 

 on the portion OP of the string are 



(a) the tension at 0, of amount If acting 

 p horizontally ; 



Fm 40 (b) the tension at P, of amount T acting 



at an angle with the horizontal ; 



(c) the weight of OP. If we take the string to be of weight w 

 per unit length, and denote the distance OP by s, this weight is 

 ws acting vertically. 



