MECHANICS. 153 



For the properties of the catenaria, see J. BERNOUIL- 

 LI Lectiones Cole. Integ. Lect. 36. Operum, 

 torn in. p. 491. DAVID GREGORY, Phil Trans. 

 1697. COTES, Harmonia Mensurarum, p. 108., 

 and Notes, p. 115. 



The catenaria is remarkable for this mechanical pro- 

 perty, That a chain hanging in that curve, has its 

 centre of gravity lower than if it were disposed in 

 any other line, its length continuing the same, 

 and also the points from which it is suspended. 

 An arch, therefore, constructed in this form, has 

 its centre of gravity the highest possible. 



234. 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 re- 

 mainder of the arch as a wedge, tending to se- 

 parate the abutments from one another. 



Thus (fig. 19.) the parts ALMS, BONT, which 

 would remain in their places, though there were no 

 pressure from above, may be regarded as parts of 

 the piers ; and LMDNO, the remainder of the 

 arch, as a wedge tending to overthrow the piers by 

 its pressure on the planes ML and ON. On these 

 suppositions, the thickness of the piers may be de- 

 termined, so that their weight shall enable them to 

 resist this pressure. If from a point H, in the verti- 

 cal EC produced, perpendiculars UK, HX, be 

 drawn to the planes ML and ON, and if HU be 

 taken to represent the weight of the arch MLOND, 

 the parallelogram IV being described, HI and HV 



will 



