I27-] PARALLEL FORCES. 75. 



Let Q be the foot of the ordinate of P (Fig. 29), N the 

 intersection of the normal with the axis O'x, and draw QR 

 perpendicular to the tangent. Then PR =y sin a = s, since 

 Tsina = ws and Twy\ also QRycosa = c. Dividing, we 

 have tana=s/c; hence, differentiating, 



ds c 



_ _ 



cos 2 ** ds c da, cos 2 a 



The figure shows that the radius of curvature p is equal to the 

 length of the normal PN. 



The relation pco$ 2 a = c shows further that at the vertex 

 (a = o) the radius of curvature is p Q = c. It follows that for a 

 cord or chain suspended from two points B, C in the same hori- 

 zontal line, c (and consequently H) is large when p Q is large, i.e. 

 when the curve is flat at the vertex; in other words, when B and 

 C are far apart. 



127. Exercises. 



(1) A weightless cord ABCDEF is suspended from the fixed points 

 A, F, and carries weights at the intermediate points J3, C, Z>, 2T. Taking 

 A as origin, the axis of x horizontal, the axis of y vertically upwards, the 

 co-ordinates of the points B, C, D, , F are (2, i), (4, 1.5), 

 (^ 1.5), (8.5, i), (10, 2). If the weight at B be one pound, what 

 are the weights at C, D, ? What are the tensions of the sections of 

 the cord ? What are the reactions of the fixed points A, F? 



(2) The total weight of a suspension bridge is 2^=50 tons; the 

 span is 2/=2OO ft.; the height is /=i8ft. Find the tension of 

 the chain at the ends and in the middle, both graphically and analytically. 



(3) A uniform wire of length 2 s is stretched between two points in 

 the same horizontal line whose distance 2x is very nearly equal to 2s. 

 Find an approximate expression for the parameter c of the catenary and 

 thence for the tension of the wire. 



