S74 €n Chain Bridges. 



Having now the value of a, the stress at the lowest point o of 

 the curve ; the distance of the points of suspension ; the length, 

 weight, and swag of the chain, &c.; the tangent QC, and sub-> 

 tangent GC, become known from the equations 



a ^ ' a ' 



(see Emerson's Fhixions, page 205), whence (by Bridge's Me- 

 chanics, jiage 302), as 2CG is to CQ, so is the whole weight of 



the chain, &;c. to the pressure of the point P or Q, or, as ^^ is to 

 — ((i|+ x), so is 2z to a-\-x. Hence it appears that the stress at 



a the lowest point is to that at suspension, as a to o + x ! 



As a denotes the tension exerted at the lowest point by a 

 weight ccjuivalent to a feet in length of the chain and the weight 

 appended ; the pressure at P will be equal to a weight of the 

 bridge of a + GO in length. 



Thus we ultimately obtain the stress at o, and pressure at the 

 points of suspension, and the strength of the chain necessary to 

 support the whole bridge and its appendages, and its ability to 

 resist every strain that may occur from loaded carriages, or other 

 fortuitous causes. These, likewise, are subjects not of conjecture, 

 but of equally rigorous and difficult calculation. As the particular 

 dimensions and circumstances are not yet given, it is unnecessary 

 to introduce them on supposition. 

 Cor, 1 . The greater v is, the fewer terms will be necessary to 



find the value of a ; in practice \7jv — ' (y' + ^V) (or perhaps \vy) 



will give it sufficiently near. 

 Cor. 2. If a? be very great in comparison with ?/, the value of a. 



will be very small, as it evidently is when the chain is nearly 



in a perpendicular position: and the pressure at P is nearly 



equal the whole weight of the chain. 

 Cor. 3. If X be small in respect to 9/, the strain at a is innnense, 



and when j: is =o, innm'te ; showing that it is impossible by 



any definite force to draw a chain or cord perfectly straight. 



This is fully confirmed by common experience. 

 Cor. 4. U X and ?/ be equal, the stress at o is =fy, and at P 



ly nearly. 

 Cor. 5. When i/ is a mean proportional betwixt the length and 



swag of the chain, the stress at the lowest point is nearly equal 



to the weight of the chain. 



The superficial and practical mechanic may, perhaps, smile at 

 the preceding tedious and difficult mode of proceeding; but I as- 

 sure him, there is no easier way known of obtaining the requisite 

 information : experience, his guide, can give him |io assistance 



in 



