` 13.—Let the arch spring from a horizontal joint, ass n, where, of course, the friction acting in v N, is just equal to the horizontal thrust, and must therefore have to T a' or v s the weight of the semi-arch, the ratio which friction has to the incumbent pressure, say 1. T s is the direction of the absolute pressure at the abutment v n. Take m m the weight of the section N, T M is the pressure on the joint of M, and making T m similar to x T 11', M 7/t will also represent the extreme friction in that joint, and T ?it its load, and so on successively. Wherefore, it' T T 1, &c. 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 predicament of N, that is, just kept by its friction from sliding away.
" The positions of T m, T 1 may be readily discovered ; for the angle iv T m must be equal to v T m. if therefore, we make T a equal to T s, draw the tangent a ?v, and making a b=s m, and joining T b, we have a T b=x T M. And, in this manner, taking a b, b c, &e. for the weights of the succes sive sections from the scale, and drawing lines from T, the joints may be formed perpendicular to the lines thus "Figure 14.—But a more convenient construction per haps would be, to take the horizontal thrust, or quantity of friction in the vertical line c d, lay off the weight of the semi arch d a, draw c a, make c x equal to it, also x z, mark off the weight of the sections along x z, 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 ease, the joints may have on the opposite side, exactly the same degree of obli quity to the position of equilibrium. Draw, therefore, the tangent v y parallel to a c, cut it with c v equal to a c, lay off the weights of the sections along v y, and draw lines from c ; these lines will exhibit the positions of the joints, which of course may be drawn parallel to them. We have marked these two limits of position in three joints of the half-arch above the same figure, 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 c d a, makes with the corres ponding line in c y v, or in c z x, an angle equal to a c x, that is, when the friction is one-third of the pressure, equal to 18° 20'; and when the friction is one-half, this angle is 26034'. The position of any joint, therefore, may vary, in the former 1S° 26', and in the latter case 20° 34', on either side of the position of equilibrium, before any sliding can take place among the sections. Nay, the friction of polished free stone is even more than one-half, perhaps it is two-thirds, of the pressure, which would give 33° 4'. And it is proper to observe, that this is not confined to the annulus of arch-stones, but holds equally with whatever weight the sections may be loaded. We may observe, then, that, intny arch, the position of the joints may be varied about perhaps from that of equilibrium, before any derangement can arise from the sliding of the arch-stones.
" This is a most important conclusion, and leads to exten sive practical consequences. It affords a true explanation of the facility with which arches are everywhere constructed, even by the common country mason.
"For this reason, therefore, we approve highly of the practice, which we believe is very general among artificers, we mean that of backing up the arch with solid masonry, for several courses above the springing. If great security is thought necessary, cement, being a compressible substance, ought to be sparingly employed in the vertical joints at the back of the arch-stones.
"The friction of the sections of the arch, as it permits a considerable variation to take place in the position of the joints, will also admit of a considerable deviation from the load which is necessary for equilibrium over any point of the curve.
"It would not be difficult to investigate the extent to which this variation of weight might be carried. But we shall at present only remind the reader, that, as we find a variation of 200 practicable in the position of the joints, he may conclude that each section will admit of its load being altered to that which would suit a point in the curve on either side of it.
"One thing only remains to be considered in this depart ment of our subject, which is, the lateral pressure likely to arise on the back of the arch, from the materials employed to raise the structure to the horizontal line.