.6. 



in each part <if the fhord is inversely as the *i/ir of 

 its iiicl:iiiitioii to the vertical. 

 Again, we have as sin.rfC/: sin rC / :: tension rC : 



fCxsin.rC/ 

 tension dc= -. -.-TVI J bnt as r Cis inversely as 



BRIDGE. 



491 



sine r C d, therefore tension d C is as 



-. - ,' -. r^-r- 

 sin.rc x sm.flC/. 



Now, let there be an unlimited number of weights 

 hung from the chord, and indefinitely near each othrr, 

 our polygonal thread becomes a curve, Fig. 5. being 

 in fact the curve of equilibration adapted to the weight 

 which depends from it. The angles rCcl and ICd 

 become r'Crf and /' C</, which nre supplementary, and 

 have equal sines, wherefore the product of these sines 

 is the square of either. Also, as the sine of rC/or 

 r C r' is as the curvature, or reciprocally as the radius 

 of curvature, we have tension dC, or weight on C, 

 inversely as rad. curv. X sin. 1 inclination to vertical. 



This tension, in the present case, is usually produ- 

 ced by the gravity of the superincumbent materials, 

 and may be measured by the area contained between 

 two indefinitely near vertical lines, EF, e f, Fig. 5 ; 

 but while the distance EC is constant, the area F e will 

 diminish with the sine of EFc,as Ee becomes more up- 

 right. To countervail this, we must enlarge the depth 

 EF in the same proportion as sine eEF diminishes. 

 And, therefore, we have EF inversely as rad. curv. X 

 sin.' e FE. Tliat is, the height of the superincumbent 

 matter must lie inversely as the radius of curvature, 

 into tltt fit be of the tine of the inclination of the curve 

 to the i-irlic;./. 



This, then, is the leading principle of the common- 

 ly received theory of equilibration. The mode in 

 which we have derived it is concise, but we trust it 

 will not be found the less clear, or the less easily ap- 

 prehended. 



Let us proceed to apply the theory to some prac- 

 tical cases. 



If the arch be the segment of a circle, then the radius 

 of curvature is the same throughout, and the height 

 will be inversely as the cube of the sine of inclination 

 to the vertical. And from this we derive the fol- 

 lowing very simple construction, for describing the 

 equilibrating cxtrados of a circular arch, and which 

 the reader, who has examined this subject, will find 

 much easier than those commonly given. 



At any point D, draw the vertical DC/, and DF from 

 the centre C ; then laying off D equal to the thick- 

 ness at the crown, draw the perpendiculars a b, f> c, 

 cd successively, Drf is the vertical thickness at D,or 

 il is a point in the extrados. 



Foritisevident,thatDa:DA::D4:Dc:Dc:D d, 

 because of similar triangles; therefore D a : D d :: rad. 

 sec.' a D l>, or inversely as radius to cube sine ah L). 

 Now Dflis the thickness at crown, and D b is therefore 

 the thickness at D. Figure 7 is constructed in this 

 way, and may serve as a specimen of the equilibra- 

 ting extrados for a semicircular arch. By reversing 

 this operation, we may find the thickness at the crown 

 corresponding to a given thickness at any other point. 

 Acd here we may observe, that as D approaches the 

 extremity B of the semicircle, the line 1) tl rapidly in- 

 creases, until at the point B it is of an infinite length. 

 But indeed this must evidently be the case with every 



arch which sprinz at right angles with the horizon- ' '{ 

 tal line ; for the thrust of the arch should be resitted v ' 

 by a Literal pressure, and no vertical pressure can act 

 laterally on a vertical line. 



\V may also observe, that since the extrados or 

 upper outline descends first on each side of the crown, 

 and then ascends with an infinite arc, there is, for any 

 thickness of the crown, a point on each side where 

 the upper edge of the extrados is on a level with that 

 on the crown. Thus, if BD = .W, its sine is half the Fig. 6. 

 radius. D is therefore =|of DC/, so that if V u=Drt 

 be made V^ of VC the radius, we hare the point rfat 

 the same level with V. Between this point, however, 

 and the crown, there is a considerable depression, 

 which is increased if the crown be made still thinner. 

 On the other hand, if it be made thicker, the hori- 

 zontal line drawn through the crown cuts the extra- 

 dos much nearer the middle of the arch. It appears, 

 therefore, that the circle is not well adapted for the 

 purposes of a bridge, or a road, where the roadway 

 must necessarily be nearly level ; for no part of the 

 extrados of the circular arch will coincide with the 

 horizontal line. There is indeed a certain span, with a 

 corresponding thickness at the crown, where the out- 

 line differs least from the horizontal ; that is, an arch 

 of about 5 1 degrees, with a thickness at the crown 

 about ^ of the span. But that is far too great for 

 practical purposes. 



We may, however, extend the construction just 

 given, even to those arches that are formed of por- 

 tions of circles differing in curvature. For the equi- 

 librating extrados being first constructed for that 

 portion of the arch in which the crown is, as far as 

 the vertical line passing through the contact of the 

 neighbouring curves, the thickness of the crown must 

 be supposed to be enlarged, in proportion to the di- 

 minution of the radius of curvature, or the contrary, 

 and, with this", proceed as before along th succeed- 

 ing branch of the curve. This will, indeed, cause an 

 unsightly break in the extrados, for which we shall 

 not at present pretend to find any other remedy, than 

 using materials of a different specific gravity. 



Those who wish to examine this subject farther, 

 may consult Emerson's Fluxions, or Hutton's Prin- 

 ciples of Bridges. We shall only observe here, 

 that the extrados of the ellipse, and of the cycloid, 

 resemble that of the circle, having an infinite arc on 

 each side at the springing; and indeed this, as has al- 

 ready been observed, is a general rule for all those 

 curves which spring at right angles to the horizon. 

 In the parabola, the extrados is another parabola ex- 

 actly the same, only removed a little above the other. 

 In the hyperbola, the extrados is another curve, 

 which approaches the interior arch towards the spring- 

 ing. None of these curves, therefore, can, with pro- 

 priety, be employed for the arches of a bridge, though 

 there may be cases where a single arch might with 

 propriety be formed into a conic section. 



The catenaria, which has been much spoken of as 

 the best form for an arch, has an extrados, the de- 

 pression of which, below its crown, at any point, is 

 to the depression of the curve in the same vertical 

 line, in a constant ratio. This ratio is that of the 

 constant tension at the vertex, to the same tension 

 diminished by the thickness or vertical pressure m 



