. Railroad construction : theory and practice : a textbook for the use of students in colleges and technical schools . 20, + () = DEMONSTRATION OF EQ. 44. The general equation of a parabola passing through the point n (Fig. 36)may be written 2/2 + 2/,,2 = 2pix + x,,h V- y n from which a-„ = —— + — x. 2p 2p When X ^^ Xj^ j/ — y .^ and we have The general equation of a tangent passing through the point A may bewritten from which x = Xa When X =- a;^, 2/ = 2/g[= 2/,^]. and we have ^5 VuVaV - ^A. Va^ yn- ^ynyj •^s 2p (UA-yny2p Am2p 2p yA ^A Ae eh 8n —rAm Tiiis proves the general
. Railroad construction : theory and practice : a textbook for the use of students in colleges and technical schools . 20, + () = DEMONSTRATION OF EQ. 44. The general equation of a parabola passing through the point n (Fig. 36)may be written 2/2 + 2/,,2 = 2pix + x,,h V- y n from which a-„ = —— + — x. 2p 2p When X ^^ Xj^ j/ — y .^ and we have The general equation of a tangent passing through the point A may bewritten from which x = Xa When X =- a;^, 2/ = 2/g[= 2/,^]. and we have ^5 VuVaV - ^A. Va^ yn- ^ynyj •^s 2p (UA-yny2p Am2p 2p yA ^A Ae eh 8n —rAm Tiiis proves the general proposi*^ion 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 proportional to the square of thedistances (measured parallel to y) from the tangent point. CHAPTER in. 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 , . .g represents the natural surface of the ground, no matter. how irregular; ah represents the position and width of the re-quired roadbed; ac and bd represent the side slopes whichbegin at a and h and which intersect the natural surface at such
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