=125. Old and New Theories of the Arch.=—In the older theories of the arch, considered as a series of blocks simply abutting against each other, the resultant loading on each block was assumed to be vertical. In the modern theories, on the other hand, the resultant loading on any block is taken precisely as it is, either vertical or inclined, as the case may be. Many arches are loaded with earth over their arch-rings. This earth loading produces a horizontal pressure against each of the stones, as well as a vertical loading due to its own weight. In such cases it is necessary to recognize this horizontal or lateral pressure of the earth, as it is called, as a part of the arch loading. It is known from the theory of earth pressure that the amount of that pressure per square foot or any other square unit may vary between rather wide limits, the upper of which is called the abutting power of earth, and the latter the conjugate pressure due to its own weight only. If _w_ is the weight per cubic unit of earth and _x_ the depth considered, and if φ be the angle of repose of the earth, the abutting power per square unit will have the value: 1 + sin φ _p_ = _wx_ ---------. (64) 1 - sin φ while the horizontal or conjugate pressure due to the weight of earth only will be: 1 - sin φ _pʹ_ = _wx_ ---------. (65) 1 + sin φ The use of these formulæ will be illustrated by actual arch computations. [Illustration: FIG. 36.] [Illustration: FIG. 37.] Fig. 36 is supposed to show a set of ring-stones for an arch of any curvature whatever. The joints _LM_ and _ON_ represent the skew-backs or springing joints, while _R_ and _R₁_ represent the supporting forces or reactions with centres of action at _aʹ_ and _a₁_. The ring is divided into blocks or pieces by the joints at _a_, _b_, _c_, _d_, and _e_, the resultant loading or force on each block being given by the lines with arrow-heads and numbered 1, 2, 3, 4, 5, 6, and 7. Fig. 37 represents a force polygon constructed in the ordinary manner by laying off carefully to scale the two reactions _R_ and _R₁_, together with the loads or forces numbered 1 to 7, inclusive. By constructing the so-called polygonal frame in the ring-stones of Fig. 36 in the usual manner with its lines or sides parallel to the radiating lines in Fig. 37, as shown by the broken lines, the points _a_, _b_, _c_, etc., are found where the resultant forces cut each joint. The line drawn through those points thus determined is called the line of resistance of the arch. Obviously, if that line of resistance be determined, the complete stability or instability of the arch, as the case may be, will be established. Furthermore, the complete determination of the force polygon in Fig. 37, and the corresponding polygonal frame drawn in the arch-ring, constitute all the computations involved in the design of an arch.