. Railroad construction. Theory and practice . parabola passing through the point n (Fig. 36)may be written 2/2 + 2/„2 = 2v{x + 2:,,) 5 •j/2 y jjfrom which x„ =— h -r— — x, * 2p 2p ^ - When X = xj y = yj^ and we have 64 RAILROAD CONSTRUCTION. § 57, The general equation of a tangent passing through the point A may be written yijA = ■ P(x + x^y from which X = vva When X = x^, y = 1/s[ = = 2/, 1, and we haveVnyA sn = ^n . 8n = yA^ + yn^- 2p (yA-ynY 2pyA -Ae^A eh A 2 —rAm eh -„. Ae^ ■^ynVA Am^2p This proves the general proposition that if secants are drawn parallel tothe axis of x, intersecting a
. Railroad construction. Theory and practice . parabola passing through the point n (Fig. 36)may be written 2/2 + 2/„2 = 2v{x + 2:,,) 5 •j/2 y jjfrom which x„ =— h -r— — x, * 2p 2p ^ - When X = xj y = yj^ and we have 64 RAILROAD CONSTRUCTION. § 57, The general equation of a tangent passing through the point A may be written yijA = ■ P(x + x^y from which X = vva When X = x^, y = 1/s[ = = 2/, 1, and we haveVnyA sn = ^n . 8n = yA^ + yn^- 2p (yA-ynY 2pyA -Ae^A eh A 2 —rAm eh -„. Ae^ ■^ynVA Am^2p This proves the general proposition that if secants are drawn parallel tothe axis of x, intersecting a parabola and a tangent to it, the intercepts be-tween the tangent and the parabola are pioportional to the square of thedistances (measured parallel to y) from the tangent point. CHAPTER III. EARTHWORK. FORM OF EXCAVATIONS AND EMBANKMENTS. 58. Usual form of cross-section in cut or fill. The normalform of cross-section in cut is as shown in Fig. 37, in whiche . . gr represents the natural surface of the ground, no matter. how irregular; ah represents the position and width of the re-quired roadbed; ac and hcl represent the ^side slopes whichbegin at a and h and which intersect the natural surface at such
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