lo5 On the EquUihr'mm of Arche*, 



23. If the lop or extrados be a hocizontal ftraight line, 

 fee fig. 6y 



r- 

 a 



t = y/zax + .vvv, (i8.) therefore 



:r 



ta = A = r^a-\^2a.v + xx = ^rax X 2a + ^'. 



Example. Take the cafe of Dr. Hutton'=; laft mentioned, 

 wherein r was found equal to 56.081 ; therefore 

 rtjx = 13459, and 2a + x ^ 53. 



^13459 X 5- ^ 836.60 = A. 



23.' Tt is \vell known to mechanicians that if a heavj- body- 

 be fuitained bv two forces, their directions muft meet eilher 

 at the centre of gravity of that body, or in a vertical line 

 which paffcs throuj'^ it. Let A. fio;. 7, be the body, ^j? its 

 cei'.tre of oravitv; ea a firing bv whicli it is fufpended, which 

 aoain is fullained bv the firings a^,(7<r. flf is of necelTity 

 vertical, and, if continued, mull pafs through the centre of 

 gi-avity g of the hotly. If this firing were either longer or 

 fhorter, the point a would flill be at refl, and the firings a'^ 

 and ac under the fame tenfion. If the body were removed 

 vertically to the fituation dotted, it would be fuftained in the 

 fame poiition by the firings hf .xnA eg, which would fuffer 

 the fame tenfion as when they were united in the point a. 

 If inftead of firings, props /'i,/^/, applied in the fame direc- 

 tions on oppofite fides of the body, were fiibfiituted 5 or if, 

 inflead of thcfe jirops, others parallel to them were applied, 

 which, if continued, would meet in the vertical line, (as mn 

 and opj which meet in g, do,) the body would ftill be fup- 

 ported at refl and in the fame pofiiion, and the props would 

 fi'ffer the fame force of conipreffion in thefc latter cafes, as 

 the ftriuTs on the oppofite fides fuffered tenfion in the former. 



24. Hence the centre of gravity of the weights 

 upon the joints E,D, C, (fig. i.) niuft be in a vertical line 

 paflfing through G, becaufe they are fullained in eqiiilibriuns, 

 or at reft, by two forces in the diieftion BC and FE, which 

 lines, continued, meet in G. Alfo, the centre of gravity of all 

 the w-eiahts, fay on B, C, D, and E, mufl be in a vertical line 

 that paftes through h, becaufe thev are fuftained by two forces 

 m the direftion FE and ZB, which, continued, meet in h. 



25. Hence alfo the centre of gravitv of the materials upon 

 a curve in equilibrium will be in a vertical line that pafTes 

 through the point of interfcdion of the tangents to the ex- 

 treme points of the curve. • " 



Let 



