372 



On Chain Bridsei. 



below it; and t,o of all the rest, the la&t link being strained h^ 

 all appended fVoui it, in tlie same way as if so many equivalent 

 distinct weights were hung from it by separate strings. By 

 parity of reasoning, if the chain be moved from the vertical to a 

 leaning or curving position, similar conseciuences ensue ; but not 

 according to the same law, tlie stress varying as the links or bars 

 are inclined to each other. 



To set this matter on an incontrovertible basis, the following 

 abstract proof of the particular case under consideration, (and 

 which is the easiest that occurs,) viz. when the points of suspen- 

 sion are in the same horizontal line, and the chain of uniform 

 .construction, is adduced : If PoQ represent a curve formed by 



' ! c, 1? 



a heavy flexible chain or other thing suspended horizontally bV 

 its ends at P and Q, then it is demonstrated* that the total effort 

 exerted by the chain on the fixed points P and Q is the same as 

 if a weight W, equal to the chain, or the chain and its vviform 

 load, were suspended from the point of concourse C of the two 

 tangental springs PC and QC: and that if a tangent qnc be 

 drawn to any other point of the curve, as q, then it will be, As the 

 sine of the angle qjtC is to the sine of the angle QCW, so is the 

 pressure at P or Q to the tension at q ; or, as Gnq and QCG are 

 the supplements of the angles qnC and QCW, it will be, As sine 

 of qnG is to sine of QCG, so is the pressure at P to the tension 

 at q : but the angle quG is greater than the angle QCG (Eu- 

 clid xxxi. 1.); consecjuently the second term of the ratio being 

 less than the first, the fourth term will be less than the third, 

 (Simson's Euclid, b. v. jnob. A.) or the tension at q less than the 

 pressure at P. I have been more particular on this point than may 

 seem necessary, because an author who has some pretensions to 

 mechanical knowledge has asserted that the tension at the lowest 

 point is greater than at P ; contrary to this demonstration and 

 article 551 of MacLaurin's Fluxions, where it is also proved that 

 the tension at o is to this tension at q as the fluxion of the or- 

 dinate to the fluxion of the cueve ; consequently that at q is 



* Barratt's Meciiaiiics, p. 100 ;; Gregory's Mechanics, p. 138; Bridgaf's 

 Mechauics, 326; Emerson's Miscellanies, 163 ; Playl'air's Outlines of Phi- 

 losophy, 233. 



greate*' 



