. Field-book for railroad engineers. Containing formulas for laying out curves, determining frog angles, levelling, calculating earth-work, etc., etc., together with tables of radii, ordinates deflections, long chords, magnetic variation, logarithms, logarithmic and natural sines, tangents, etc., etc . — C2 n fi — c2 n [n -3) d^ n2 [n -5) f/2 n2 [n_ -7) «2 d^ &c. It will be seen, that each of these values is formed from the preceding,by adding the same quantity —-—, and subtracting ^ multiphed inSTiccess-lorj hr w — 1, n - Z -n - 5, v^ ^flaking: ^> ~ i, we have RADIUS OF CURVATUKb. ra c^^ =
. Field-book for railroad engineers. Containing formulas for laying out curves, determining frog angles, levelling, calculating earth-work, etc., etc., together with tables of radii, ordinates deflections, long chords, magnetic variation, logarithms, logarithmic and natural sines, tangents, etc., etc . — C2 n fi — c2 n [n -3) d^ n2 [n -5) f/2 n2 [n_ -7) «2 d^ &c. It will be seen, that each of these values is formed from the preceding,by adding the same quantity —-—, and subtracting ^ multiphed inSTiccess-lorj hr w — 1, n - Z -n - 5, v^ ^flaking: ^> ~ i, we have RADIUS OF CURVATUKb. ra c^^ = c^ 4-^(r2_c«)-igcr, C2 = c,^-hUT-c-)-ud\ Ca^ = c i-\- ^ {T^ - c) + ud. the quantities, wliicb enter in* j tlic expressions for the radii, arenow known, and the radii may, therefore, be determined. The samemethod will apply to the other half of the parabola. The manner of obtaining the preceding formulte is as follows. Theradius of curvature at any given point on a parabola is, by the Differ-ential Calculus. R = 2^i^^.3 E ^ which p represents the parameter ofI lie parabola for rectangular coordinates, and E the angle made witha diameter by a tangent to the curve at the given point. First, let themiddle station E (fig. 42) be the given point. Then the angle E is the Fig. 42. angle made with E Dhy n tangent at E, or since A B is parallel tothe tangent at E (§ 84, IV.), sin. E = sin. ADE = sin. BDE. Letp be the parameter for the diameter E D. Then, by Analytical Ge ometry, f p E2 E ^c3 p sin 2 E. Therefore, at this point R = 2 E ~2sihE■ ^^^P-^^ = Vd Therefore, R = j^ --= . = — : since A ^^ cd sin. E (Tab. X. 17).c d sin. E A ^ ^ Next, to find 7?i, or the radius of curvature at H, the first stationfrom E. Through ff draw EG parallel to CD, and from Fdraw thetangent EK. Join A K, cutting C Dm L. Then from what has justbeen pioved for the radius of curvature at E, we have for the radius of curvature at //. A, = a F K ^^^^ A G • A L = A F: A C
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Keywords: ., bookcentury1800, bookdecade1870, booksubjectrailroadengineering
