[ 97 3 



XVL On the EquUlhrium of Arches.^ By Mr. JbHN 

 Southern, Engineer, Soho, Birmingham*. 



A 



SIB, Sobo, Birmingham, Nov. i, li^oi. 



.T the prerent moment, when the grand proje6l of 

 throwino; a fingle arch over the Thames (propofed by 

 MeflTrs. Telford and Douglas) is in contemplation, whatever 

 tends to render the theory of the equilibrium of arches more 

 eafily lo be ianderftood, will, it is reafonable to fuppofe, be 

 acceptable to many of ycnir readers ; and if you think the fol- 

 lowing paper of this defcription, yon will, by inferting it in 

 the Philofophical Magazine, obhtfe. 



Sir, 

 Your obedient fervant, 

 Mr.Tilloch. John Southern. 



1. LET the right lines, fig. i. ZBCDEF be fituate in a 

 vertical plane, and moveable about the angles B,C,D,E as 

 joints, and alfo about the points Z and F; let ZB be hori- 

 zontal, and let fuch weights be laid upon the joints BCDE 

 as will keep the whole in equilibrio, the points Z and F not 

 vieldinj : required what fuch weights are, and what the forces 

 exerted in each line. 



From any point B in the line ZB draw B/> perpendicular 

 to that line,' continue the faid line to any diftance, a, whence 

 draw the line ^M parallel to B/) ; by the equilibrium of 

 forces, the weicht (reprefented by the line) aM. laid upon 

 the joint B will be counterbalanced by a force cB in the 

 diredion ZB, and a force MB in the direftlon MB, and of 

 courfe the point B remain at reft. But aftion and rea£lion 

 being equal and contrary, the force aB will not only be ex- 

 erted againft B, but alfo be returned and aft againft the 

 point Z; alfo the force MB will be returned, and aft againft 

 the joint C. Continue BC to h, fo that bC ^ MB, and 

 from b draw the line ^N parallel to B/', meeting the line GD 

 in N ; then will the force ^C, in the direftion BC, be juft 

 balanced by a weight ^N laid upon the joint C, and a force 

 NC in the clireftion NC. In like manner continue the lines 

 CD and DE to c and d refpeftively, making the lines cD 

 = NC, and dVl — OD; and from the points c and d drawing 

 lines parallel to B/) meeting the line DE and £F in O 



• Communicated l^v the Author. 

 Vol. XT. G and 



No. 41. 

 Noi'^mber i8oi. 



