PLAN, NO. IV. Estimate of the expense of raising the valley forty feet, cutting twenty-seven feet from the summit at Foster's Booth, and reducing the inclinations from 1 in 14 and 15 to 1 in 30 on the west side of Lovell's Hill. To 104,671 cubic yards of earth-work, at 1s. To 12,440 cubic yards from back cut ting, at 9d. · To 63,931 cubic yards from back cutting, at 7d. To 2611 cubic yards of earth from back cutting, at 6d. To three acres of land to be purchased for supplying earth for embankment, at 100%. To twelve acres of land to be purchased for the site of embankments, and for the cutting at Foster's Booth and Lovell's Hill, at 1007. To 616 yards of fences over the embankment, at 10s. To 1870 yards of post and rail fence, at 6s. To 2486 yards of road-making, at 15s. To field-gates, &c. To toll house and gates Contingencies £ S. d. PLAN, NO. V. Estimate for the expense of raising the valley forty feet, cutting twenty-seven feet from the summit at Foster's Booth, and fourteen from the summit at Stowe Hill. To 198,200 yards of earth-work in embankments taken from the cuttings, at 1s. To 23 acres of land for cuttings and embankments, at 100%. To 682 yards of fences on the embankments, at 10s. To 3374 yards of post and rail fences, at 6s. To 4055 yards of road materials, at 15s. £ S. d. 9,910 0 0 2,350 0 0 NOTE B. Page 65. THE resistance produced by collision is seldom a constant retarding force; loose stones, or hard substances, are sometimes met with, and will give a sudden check to the horses, according to the height of the obstacle: the momentum thus destroyed is often very considerable. The power required to draw a wheel over a stone or other obstruction may be thus determined:-Suppose ABD to represent a carriage wheel 52 inches in diameter, the axis 2.5 inches in diameter, the weight of the wheel 200 lbs., and the load on the axle 300 lbs. Let a stone or other obstacle four inches high be represented as at S; the power necessary to be applied to the axle to draw the wheel over the stone is thus found:Suppose P the power which is just sufficient to keep the wheel balanced, or in equilibrio, when acting from the centre C in the direction CP. The force acting against this power is gravity, and is equal to the weight of the wheel and load on the axle, acting from the centre C in the direction CB. These forces act together against the point D in the direction CD. Gravity acts in the direction CB with the energy or length of lever D B, and the power acts in the direction C P, with the leverage BC; and the equation of equilibrium will be W DB PxCB. In this equation, CB the radius of the wheel diminished by the height of the obstacle, and BD equals DC-BC2; hence the Wx/DC-BC2 power P = DC-DS -: in the present example, W=500; DC= DC=52=52–26; BC=DC—DS=26−4=22; and ✓DC-BC2=13.85; the power, therefore, which is necessary to keep the whole in equilibrium, or resting off the ground, supported at the point D, will be equal to 314.3 lbs. 500 x 13.85 22 The pressure at the point D is equal to the joint A action of the power and weight, as before stated, this in the present instance is represented by the radius of the wheel: hence CB CD::500: = CD x 500 26 × 500 =591 lbs. nearly. The injury which a road sustains by this pressure acting on a small point, and in an oblique direction, is very great but it is not alone in this that the road suffers; the force with which the wheel strikes the surface, in its descent from the top of the stone, is considerable, and would soon wear a hole in the hardest road. But it must be observed, that a carriage mounted on proper springs will be drawn over an obstacle of this kind with much less power than if the carriage had no springs; for the springs allow the wheels to mount over the obstacle without raising the body of the carriage, and its load along with it, to the same height: upon this principle alone it is that carriages mounted on proper springs are easier moved than those without springs; and, for the same reason, springs are more necessary on rough and uneven roads than on smooth ones; and in proportion to the roughness of the roads should the springs be free and elastic; and it is to the improvement in the roads, of late years, that the rigid elliptic springs on carriages have been substituted for the C springs formerly in use; for, when ruts were very numerous, and the surface of the roads rough and uneven, the springs that are now used would have been of very little use, as their vertical motion is so limited, besides having no lateral play. By enlarging the diameter of the wheel, the power required to draw it over an obstacle will be diminished; and, should the weight of the wheel remain the same, the power will decrease nearly as the diameter of the wheel increases: we have seen that, when the wheel was 52 inches in diameter (which is the general size of the front wheels of waggons), it required about 314 lbs. to keep the wheel in equilibrium when resting on a point four inches above the general surface. Suppose, now, that the hind wheel (the diameter of which is sixty-four inches) is to be pulled over the same obstacle, the power will be found to be only 305 lbs., although the weight is increased 70 lbs. by enlarging the wheel; Wx✓DC - BC2_570 x 15.5 for P DC-DS 28 =305. In this investigation we have supposed the power to act directly from the centre, and parallel to the horizon, which supposition is sufficient for practical purposes; but, if great accuracy be required, it will be necessary to allow for the thickness of the axletree, and to diminish the length of the lever by which the power and weight acts; for suppose the power to balance the weight, the |