I. Let it be required to determine the position of the joints in an arch, when each section is just prevented from sliding outwards by the friction at its lower sur face.
Let the arch, Plate LXXXI, Fig. 4, spring from a hori zontal joint, as Nn, where, of course, the friction acting in YN, is just equal to the horizontal thrust, and must therefore have to Tn' or vN the weight of the semi arch, the ratio which friction has to the incumbent pressure, say 11. TN is the direction of the absolute pressure at the abutment Nn. Take NM the weight of the section N, TM is the pressure on the joint of M, and making arm similar to NTil, aim will also represent the extreme friction in that joint, and TM its load, and so on succes sively. Wherefore, if Tm, Ti, be found, the joints of the arch may be drawn at right angles to these lines respectively, and every stone will be exactly in the pre dicament of N, that is, just kept by its friction from slid ing away.
The positions of TM, Ti, may be readily discovered; for the angle n'Tni must be equal to NTM. If, therefore, we make Ta equal to TN, draw the tangent aw, and mak ing ab=1431, and joining Tb, we have aTb=NTM. And, in this manner, taking ab, bc, &c. for the weights of the successive sections from the scale, and drawing lines from T, the joints may be formed perpendicular to the lines thus drawn.
Upon the same principles, we readily find a construc tion for the extreme weights of the sections, when the positions of the abutments, Ste. are given. This is so evident, that we shall not stop to point it out.
But a more convenient construction perhaps would be, to take the horizontal thrust, or quantity of friction in the vertical line Cd, Plate LX XXI, Fig. 5. Lay off the weight of the semi-arch da, draw Ca, make Cx equal to it, also xz, mark off the weight of the sec tions along xz, and through the divisions draw lines from the centre ; the joints required are parallel to these lines.
II. Let it be required, in the next place, to determine the other limit to the position of the joints, or that in which each section is just prevented from sliding in, by the friction on its lower bed.
Here it is evident, that as the friction acts precisely opposite to its direction in the former case, the joints may have, on the opposite side, exactly the same de gree of obliquity to the position of equilibrium. Draw, therefore, the tangent vy parallel to or, cut it with Cv equal to aC, lay of the weights of the sections along vy, and draw lines from C; these lines will exhibit the po sitions of the joints, which of course may be drawn par allel to them. We have marked these two limits of po
sition in three joints of the half arch above the same fi gure, assuming the friction at one-third, and taking the first section of 30° as equal to the thrust : and any other arch might have been introduced as well as the circular. Any of the lines in the triangle Cda makes, with the cor responding line in Cyv, or in Czx, an angle equal to aCx, that is, when the friction is one-third of the pressure, equal to 18° 26'; and when the friction is one-half, this angle is 26° 34'. The position of any joint, therefore, may vary in the former 18° 26', and in the latter case 26° 34', on either side of the position of equilibrium, before any sliding can take place among the sections. Nay, the friction of polished freestone is even more than one half, perhaps it is two-thirds of the pressure, which would give SS° 4'. And it is proper to observe, that this is not confined to the annulus of archstones, but holds equally with whatever weight the sections may be load ed. We may observe then, that in any arch, the position of the joints may be varied about perhaps SO° from that of equilibrium, before any derangement can arise from the sliding of the archstones.
This is a most important conclusion, and leads to ex tensive practical consequences. It affords a true ex planation of the facility with which arches are every where constructed, even by the common country ma son. The equilibration theory has shown us, that by adjusting the inclination of the joint to the weight of in cumbent matter, we may suit an arch to any given cir cumstances; and we here find in the friction of the parts a powerful addition to its stability. We trust, there fore, that the reader now sees the propriety of the ob servation, which we made above respecting the inutili ty of searching very minutely into the exact position of these joints. It is in common cases scarcely possible to go wrong. But it must be observed, that the varia tion of position above mentioned, is to be reckoned from the position of equilibration, not from the common joints radiating all from one centre, or perpendicular to the curve, unless where such an arch is equilibrated by the superincumbent weight. For in an annulus of arch stones, with radiating joints, which is the most common mode of construction, those towards the vertex can be drawn only a very little lower, and those towards the springing only a very little higher than the original centre, though either of them admits of a considerable variation in the opposite direction.