Suppose the arch to descend somewhat at the crown, the stones there will hang by their upper edges, even when there is no apparent opening on the lower side of the joint. They will be pretty close for a good way on each side, so far indeed as the equilibrating super structure extends, or to about 60°. And it will then be tolerably well equilibrated, even though the superstruc ture should not be yet applied. For the arch being then at the crown, the theoretic extrados will run fur ther down on the back of the curve, ere it turns up again ; and, of course, will for a good way not differ much from the back of the archstones. But beyond this point, or about from the crown, things are not likely to be so steady. We do not say, the lower sections will slide : their friction is likely to prevent that. But the best workmanship cannot prevent them from rocking a little. At least, the sum of the motions of each joint will at length come to be something. The haunches will slip away a little, just where the equilibration ceases. The circular arch will become somewhat elliptic. The joints about that point will open behind ; and if the case be dangerous, the stones will chip away below. Some thing of this kind, indeed, goes on in the building of every arch. As the courses approach the crown, their thrust makes the lower ones recede a little from the centre. But as the process is gradual, and the finish ing courses are adapted to the shape of the opening which receives them, perhaps the only bad effect is the derangement of the crystals of Rine, which have already begun to form, while the cement fixes in the lower joints of the arch. With good workmanship, the amount of the final derangement is so small, that no joint is opened beyond the limit at which repulsion acts, especially in such great pressures ; so that every stone may be still considered as butting pretty fairly on its neighbours.
Having now exhibited the effects that may be ex pected from the friction of the parts of an arch, one thing only remains to be considered in this department of our subject, which is, the lateral pressure likely to arise on the back of the arch, from the materials em ployed to raise the structure to the horizontal line.
If the materials employed here be only a solid mass of masonry, it is not easy to see, every thing being steady, how it can act in any other way than in the ver tical direction. If, however, a motion takes place in the arch, the mass of materials lying nearly over the springing, when the arch is not very different from a semicircle, will have such an enormous friction, if well built and bonded together, as would appear equal to the resistance of any pressure that is likely to be opposed to it. And when the arch is a segment much smaller than a semicircle, the rules we have already given for its equilibration must be considered. But, instead of solid courses of masonry, the haunches of arches arc often filled up with coarse gravel or shiver, and some times with mere earth or sand. Materials of this des cription does by no means act by mere dead weight. It has a tendency to slide down towards a horizontal position ; and, of course, possesses, in some slight degree, the quaquaversuin pressure of a fluid. This may act on
our arch in a manner altogether new, and produce strains for which hitherto we have made no provision. \Ve shall first consider the back of the arch as filled up with a fluid substance, as water. The pressure in part will be in a direction perpendicular to the curve, and will be proportional to the depth. A pressure per pendicular to the curve will be equivalent, in effect, to a vertical pressure, which exceeds it in the ratio of the secant of the inclination to the vertical. Of course, the pressure at the springing, when all is equilibrated, must be equal to the horizontal thrust in a semicircular arch. Take the thickness of matter at the crown of radius, the weight of one degree =k, then the horizontal thrust will be and the height of fluid necessary for this will be 574- times the thickness at vertex, provided the specific gravity of the fluid be the same with that of the arch. But if not, let f= the gravity of the fluid, and that of the arch at vertex, then 571 S R will be f nc the height required. Suppose the arch made of brick, which is about double the specific gravity of water and we have, for water filling up the flanks, till just covering the crown of the arch, a depth at the spring ing nearly equal to the radius ; and, of course, the thick ness at crown should be about R, or of the spar., when in equilibration at the springing. NVe take no notice of the effect of the arch in assisting this. \Va ter, therefore, is much too light for equilibrating an arch at the springing, in any moderate thickness of crown. It might, however, be so employed. The quantity requisite is always finite, even at the vertical spring courses ; and by expanding the arch. or otherwise em ploying its hydrostatical properties, the requisite weight of fluid could without doubt be obtained in any case. But it is unnecessary to pursue this speculation farther than merely to observe, that its weight on the arch, where a variation is requisite, might be adjusted, by attending to the modes of altering the density which we have noticed, when speaking of filling up the arch by masonry alone.
Though the action of sand, gravel, or mould, in situa tions such as this, be not exactly the same with that of water, in following the laws of hydrostatical pressure ; yet these materials resemble water, and may be conceived to hold the middle place between the fluid and the solid back ing. In some respects they arc more advantageous than the fluid. They arc stiffer, so to speak, affording a lateral abutment to the arch, if it is likely to yield ; and as the parts have a great friction among themselves, it will require a much greater pressure acting horizontally, to make the matter rise, than in the case of a fluid. We must not, however, be too confident. Materials of this kind arc compressible ; and we have already seen, that very slight shifts are attended with dangerous con sequences. At the same time, we need not be much afraid of a trivial departure from exact equilibration ; for it is not likely that materials of this kind will act with the powerful effort of hydrostatical pressure.