And first, then, in the semicircular polygon, as it is called, Fig. 4, where weights are hung on the thread AC'CC"B, which bring it into the position ACB, we have at each angle three forces in equilibrio. Where fore, by the principles of statics, they are to one another as the sines of the opposite angles ; that is, the tension rC is to the tension IC, as sine IC\V is to sine rCW, but the tension from C to 1 is the same as from C' to r. Also sine IC\V is the same as sine r'C'NN", since these angles are supplementary, C\V, C'\V' being parallel; therefore the tension rC is to the tension r'C', as sine r'C'NN" to sine rCW. Or, the tension in each part of the chord is inversely as the sine cf its inclination to the ver tical.
Again, we have as sin. d C 1: sin r C I:: tension r C: rCxsin.rCl tension d c= sin.dC 1 ; but as r C is inversely as sine r C d, therefore tension d C is as sin.rC / sin.redxsin.dC/.
Now, let there be an unlimited number of weights hung from the chord, and indefinitely near each other, 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 r C d and / C d become r' C d and l' C d, which are supplementary, and have equal sines, wherefore the product of these sines is the square of either. Also, as the sine of r CI or r C r' is as the curvature, or reciprocally as the radius of curva ture, we have tension d C, or weight on C, inversely as rad. curv. x sin.' inclination to vertical.
This tension, in the present case, is usually produced 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 E e is constant, the area F e will diminish with the sine of EF e, as E e becomes more upright. To countervail this, we must enlarge the depth EF in the same proportion as sine e EF diminishes. And, there fore, we have EF inversely as rad. curv. x e FE. That is, the height of the superincumbent matter must be inversely as the radius of curvature, into the cube of the sine of the inclination of the curve to the vertical.
This, then, is the leading principle of the commonly 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 apprehended.
Let us proceed to apply the theory to some practical 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 following
very simple construction, for describing the equilibrating extrados 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 D d, and DF from the centre C ; Fig. 6. then laying off D a equal to the thickness at the crown, draw the perpendiculars a b, b c, c d successively, D d is the vertical thickness at D, or d is a point in the extrados.
For it isevident,thatDa:Db::D b:De:D e:D d, because of similar triangles ; therefore Da:Dd:: rad. a D b, or inversely as radius to cube sine a b D. Now D a is 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 equilibrating extrados for a semicircular arch. By reversing this operation, we may find the thickness at the crown cor responding to a given thickness at any other point. And here we may observe, That as U approaches the ex tremity B of the semicircle, the line ll d 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 springs at right angles with the horizontal lino ; fur the thrust of the arch should be resisted by a lateral pressure, and no vertical pressure can act laterally on a vertical line.
We 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=30°, its sine is half the radius. D a is therefore I) d, so that if V v=D a be made of VC the radius, we have the point d at the same level with V. Between this point, however, and the crown, there is a considerable depression, which is in creased if the crown be made still thinner. On the other hand, if it be made thicker, the horizontal line drawn through the crown cuts the extrados 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 outline differs least from the horizon tal ; that is, an arch of about 54 degrees, with a thickness at the crown about- of the span. But that is far too great for practical purposes.