A R C 



the curve a perpendicular . he lot f;ill on the longer axis, and AV be the 

 \vright of the key-stone, the weight V of any voussoir is 

 <ta W 



4. If the weights of the voussoirs are all equal, the arch of equilibrium 

 is a catenarian eurve, the same that a chain of uniform thickness would 

 assume, if hanging- freely j the horizontal distance of the points of sus- 

 pension being equal to the span of the arch, and the depth of the lowest 

 points of the chain'beiug equal to the greatest height of the arch. 



The equation to the catenary, if A' and ?/ be the corresponding coordi- 

 nates from the vertex along the axis or vertical line, is 



, a 4. .r -J_ j-2. a .r 4- r 9 



.// a .4. h. i. ~ --2 2- m 



a 



The constant quantity a may "he determined by experiment ; for the 

 <'hain being suspended ; let a tangent be drawn to any point of the curve, 

 iid produced till it meet the axis ; then as the subtangent is to the or- 

 dinate, so is the length of the chain, between the given point and the 

 vertex, to the quantity a. When a is found, the curve can be construct- 

 ed. 



5. The pressure of an arch on the piers or abutments which .support it, 

 may be estimated by considering the parts of the arch, which rest im- 

 mediately on the abutments-to a certain height, as parts of the abutments 

 themselves ; and the remainder of the arch as a wedge ; tending to sepa- 

 rate the abutments from one another. 



Thus the part ALMS (see above Fig.) which would remain in its place 

 though there were no pressure from above, may be regarded as a part <i 

 the pier, and L M E D &c., the remainder of the arch, as a wedge tend- 

 ing to overthrow the pier by its pressure on the plane M L. On these 

 suppositions the thickness of the piers, so that their weight .shall enable 

 them to resist this pressure, may be determined. 



Let the /_ .which M L makes with the vertical 0, twice the area 

 M L D E = a*, C K = #. and F C - x, then 



_?2 -\ 



k* COS.4 )' 



' -ill cos.* r*"- s \ sin. 2 "** 47^ 



lu the above demonstration, the hypothesis is that the pier A F, if th* 

 weight of the arch were too great to be sustained, would fall, by turning 

 round F as a fulcrum. Now this is not what would happen ; the part of 

 3- he HbnrmenHtehind s M wmii*! i.'-lhvu-t <>ut in the horizontal direction. 



