gS On the Equ'ilihriuvi of Arches. 



aiid P; fo will the weight rO laid upon the joint D, anti 

 the force OD, balance the force cD in the direftion CD ; 

 and the weight JP laid upon the joint E, and the force PE 

 balance the force JE aftinij in the direftion DE. The force 

 PE will, bv its reaction, be exerted upon the point F. 



Make BA = oB, and draw km parallel to BC; B^w will 

 therefore — cM, and Am = BM; and confequcndy the 

 lines AB, B/m, ink, will reprcfent the forces and their direc- 

 tions of aftion which keep the joint B in equilibrium. Draw 

 Am parallel to DC, then becaufe bC = MB = mk, and bN 

 is parallel to t,in (included in B/>), the triangle kinn is equal 

 and finiilar to the triangle C/^N, and confequently the lines 

 Am, mn, nk, reprcfent the forces and their dirediions of 

 aftion which keep the joint C in equilibrium. So, Ao and 

 Ap, being refpetlively drawn parallfel to DE and EF, the 

 triangle Ano =■ DcO, and Aop = EJP; and confequently 

 their fides reprcfent the forces which keep the joints D and E 

 refpeftively in equilibrium. 



2. It is obvious that the weight prefiing on anv joint is as 

 the difference of the tangents of the angles which arc formed 

 by each of the legs of that joint with the horizontal line : for 

 the weight on the joint D is as cO ~ no; but B;^ is the tan- 

 gent of the angle nAB (to the radius AB) = the angle which 

 the leg DC of the joint forms with the horizontal line AB ; 

 and ^0 is the tangent of the angle oAB (to the fame radius) 

 = the angle which the other leg DE of the joint forms with 

 the horizontal line AB; confequently, no is the difi'erence of 

 thefe tangents. 



3. It is equally obvious that the force exerted through anv 

 leer of any of the joints, and which may be called the com- 

 freffing force, is as the fecant of the angle which that leg 

 forms with the horizontal line: for the force through the 

 leg FE is as pk, the force through ED is as ok, through 

 DC as nk, throuoh CB as mk, and through BA as BA } 

 refpeftivclv the fecants of the angles, with the radius ABj, 

 ■Vvhich each leg makes with the horizontal. 



■" 4. The horizontal force, or compreHing force at the crown, 

 "(which is conrtant,) is AB — radius; becaufe the leg at the 

 crown is horizontal, and therefore the anjrie is nothing; there, 

 the fecant of which is equal to raduis. ' 



. 5. II' the weights v.'hich are difpofed upon anv nximher of 

 ^coritiguous joints h^ w\\\\<tA and laid upon one joint, and if 

 the legs which form this joint have the fame declination from 

 Ihe horizontal line, and the fame forces exerted through them 

 that the extreme legs of tlie two extreme joints of the faid 

 contiguous uuniber have, the equilibrium will Hill be exaft. 



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