The horizontal line drawn to the left from _f_ gives the vertical projection of that part of the extrados between _d_ and _f_, and the horizontal earth pressure on _df_ will be precisely the same in amount as that on the vertical projection of _df_, as just found. In the same manner the horizontal earth pressure on that part of the extrados between any two adjacent joints may be found. The mid-depths of these vertical projections below the line _E′O_ are to be carefully measured by scale and then used for the values of _x_ in equations (64) and (65), which now become equations (66) and (67), as the angle of repose φ is taken to correspond to a slope of earth surface of 1 vertical on 1½ horizontal. _p = 3.51wx._ (66) _pʹ = 0.285wx._ (67) The horizontal earth pressures thus found are as follows: _h₁_ = { 101,500 pounds; _h₃_ = { 30,625 pounds; { 8,700 ” { 2,625 ” _h₂_ = { 59,500 ” _h₄_ = { 9,800 ” { 5,100 ” { 840 ” These quantities _h₁_, etc., are found by multiplying the two intensities _p_ and _p′_ by the vertical projections of the surface on which they act. The larger values are found by equation (66) and represent the abutting power of the earth, while the smaller values are found by equation (67), and represent the horizontal or conjugate pressure of the earth due to its own weight only. The actual horizontal earth pressure against the arch-ring may lie anywhere between these limits. The weights of the moving load, earth, and ring-stones between each pair of vertical lines and radial joints shown in Fig. 38 are next to be determined, and they are as follows: _W₁_ = 27,300 pounds; _W₆_ = 12,300 pounds; _W₂_ = 27,900 ” _W₇_ = 15,550 ” _W₃_ = 24,500 ” _W₈_ = 19,500 ” _W₄_ = 21,300 ” _W₉_ = 19,400 ” _W₅_ = 18,300 ” _W₁₀_ = 24,300 ” The centres of gravity of these various vertical forces are shown in Fig. 38 at the points _W₁_, _W₂_, etc. The triangles of forces shown in that figure and composed, each one, of a vertical and horizontal force as described, are laid down in actual position on the arch-ring, as shown. All data are thus secured for completing the force polygon and polygonal frame or line of resistance. It will be assumed that the reactions _R_ and _R′_ cut the springing joints at _c_ and _a_, respectively, one third of the width of the joint from the soffit, and it will further be assumed that _b_, the mid-point of the joint at the crown, is also in the line of resistance. The assumption of the location of these three points is made for the reason, as is well known, that with a given system of forces a polygonal frame may be found which will pass through any three points in the ring. [Illustration: FIG. 39.]