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VIII <br />TABLE II. - Radii, Ordinates and Deflections. Chord =100 ft. <br />Deg: <br />Radius. <br />Mid.' <br />Ord. <br />Tnn. <br />Dist. <br />Def. <br />Dist. <br />Def. <br />1 Ft. <br />Deg. <br />Radius <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />Def. <br />Dist. <br />Def. <br />IfFt. <br />2° �7 <br />ft. <br />t <br />ft. <br />ft. <br />eft. <br />1°59' <br />2°25' <br />it. <br />ft, <br />ft. <br />34° <br />0'10' <br />34377. <br />.036 <br />.145 <br />.291 <br />0.05 <br />7' <br />810.0 <br />1.528 <br />6.105 <br />12.21 <br />2.10 <br />20 <br />17159. <br />:073 <br />.291 <br />.582 <br />0.10 <br />20' <br />781.8 <br />1.600 <br />6.395 <br />12.79 <br />2.20 <br />30 <br />11459. <br />.109 <br />.430 <br />.873 <br />0.15 <br />30 <br />764.5 <br />1.637 <br />6.540 <br />13.08 <br />2.25 <br />40 <br />8594.4 <br />.145 <br />.682 <br />1.164 <br />0.20 <br />40 <br />747.9 <br />1.673 <br />6.685 <br />13.37 <br />2.30 <br />50 <br />6875.5 <br />.182 <br />.727 <br />1.454 <br />0.25 <br />8 <br />716.8 <br />1.746 <br />6.976 <br />13.95 <br />2.40 <br />1 • <br />5729.6 <br />.218 <br />.573 <br />1.745 <br />0.30 <br />20 <br />688.2 <br />1.819 <br />7.2GG <br />14.53 <br />2.50 <br />10 <br />4911.2 <br />.255 <br />1.018 <br />2.036 <br />0.35 <br />30 <br />674.7 <br />1.855 <br />7.411 <br />14.82 <br />2.55 <br />20 <br />4297.3 <br />.291 <br />1.164 <br />2.327 <br />0.40 <br />40 <br />661.7 <br />1.892 <br />7.556 <br />15.11 <br />2.60 <br />30 <br />3819.8 <br />.327 <br />1.309 <br />2.618 <br />0.45 <br />9 <br />037.3 <br />1.96.5 <br />•7.846 <br />15.69 <br />2.70 <br />40 <br />3437.9 <br />.364 <br />1.454 <br />2.909 <br />0.50 <br />20 <br />014.6 <br />2.037 <br />8.136 <br />16.27 <br />2.80 <br />50 <br />3125.4 <br />.400 <br />1.600 <br />3:200 <br />0.55 <br />30 <br />603.8 <br />2.074 <br />8.28L <br />16.56 <br />2.85 <br />2 <br />2864.9 <br />.436 <br />1:745 <br />3.490 <br />0.60 <br />40 <br />593.4 <br />2.110 <br />8.426 <br />16.83 <br />2.90 <br />10 <br />2644.6 <br />.473 <br />1.591 <br />3.781 <br />0.65 <br />10 <br />573.7 <br />2.153 <br />8.716 <br />17.43 <br />3.00 <br />20 <br />2455.7 <br />.509 <br />2.036 <br />4.072 <br />0.70 <br />•30 <br />546.4 <br />2.292 <br />0.150 <br />18.30 <br />3.15 <br />30 <br />2292.0 <br />.545 <br />2.181 <br />4.363 <br />0.75 <br />11 <br />521.7 <br />2.403 <br />D. r,95 <br />19.16 <br />3.30 <br />40 <br />2148.8 <br />.582 <br />2.327 <br />4.654 <br />0.80 <br />30 <br />499.1 <br />2.511 <br />10.02 <br />20.04 <br />3.45 <br />50 <br />2022.4 <br />.618 <br />2.472 <br />4.945 <br />0.85 <br />12 <br />478.3 <br />2.620 <br />10.45 <br />20.91 <br />3.60 <br />8 <br />1910.1 <br />.655 <br />2:618 <br />5.235 <br />0.90 <br />30 <br />459.3 <br />2.750 <br />10.89 <br />21.77 <br />3.75 <br />10 <br />IS09.6 <br />.691 <br />2.763 <br />5.526 <br />0.95 <br />13 <br />441.7 <br />2.839 <br />11.32 <br />22.6.4 <br />3.90 <br />20 <br />1719.1 <br />.727 <br />2.903; <br />5.817 <br />1.00 <br />30 <br />425.4 <br />2.949 <br />11.75 <br />23.51 <br />4.05 <br />30 <br />1637.3 <br />.764 <br />3.054 <br />6_.108 <br />1.05 <br />14 <br />410.3 <br />3.058 <br />12.18 <br />24.37 <br />4.20 <br />40 <br />1562.9 <br />.800 <br />3.199 <br />6.398 <br />1.10 <br />30 <br />396.2 <br />3.168 <br />12.G2 <br />25.24 <br />4.35 <br />50 <br />1495.0 <br />.836 <br />3.345 <br />6.659 <br />1.15 <br />15 <br />383.1 <br />3.277 <br />1.3.0:5 <br />26.11 <br />4.50 <br />U <br />1432.7 <br />.873 <br />3.490 <br />6.950 <br />1.20 <br />30 <br />370.8 <br />3.387 <br />13.49' <br />26.97 <br />4.65 <br />'10 <br />137.5.4 <br />.909 <br />3.635 <br />7.271 <br />1.25 <br />16 <br />359.3 <br />3.496.13.92 <br />27.84 <br />4.50 <br />20 <br />1322.5 <br />.945 <br />3.718', <br />7.561 <br />1.30 <br />30 <br />348:5 <br />3.606 <br />14.35 <br />28.70 <br />4.95 <br />30 <br />1273.6 <br />.082 <br />3.026 <br />7.852 <br />1.35 <br />17 <br />338.3 <br />3.716 <br />14.78 <br />29.56 <br />5.10 <br />40 <br />1228.1 <br />1.01S-4.071 <br />8.143 <br />1.40 <br />18 <br />319.6 <br />3.935 <br />15.64 <br />3t.29 <br />5.40 <br />'50 <br />1185.8 <br />1.055 <br />4.217 <br />S.433 <br />1.45 <br />19 <br />302.9 <br />4.155 <br />16.51 <br />33.01 <br />5.70 <br />5 <br />1146.3 <br />1.091 <br />4.362 <br />5.724 <br />20 <br />257.9 <br />4.374 <br />17.37 <br />34.73 <br />G.00 <br />10 <br />1109.3 <br />1.127 <br />4.507 <br />9.014 <br />_1.50 <br />1.55 <br />21 <br />274.4 <br />4.504 <br />15.22 <br />36.44 <br />6.30 <br />20 <br />.1074.7 <br />1.164 <br />4.653 <br />9.305 <br />1.60 <br />22 <br />262.0 <br />4.814 <br />19.08 <br />33.16 <br />6.60 <br />-30 <br />1042.1 <br />1.200 <br />4.79 8 <br />9.596 <br />I.G5 <br />23. <br />250.S <br />5.035 <br />19.94 <br />39.87 <br />6. 90 <br />:40 <br />1011.5 <br />1.237 <br />4.943 <br />9.886 <br />1.70 <br />24 <br />240.5 <br />5.255 <br />20.79 <br />41.53 <br />7.20 <br />-50 <br />982.6 <br />1.273 <br />5.OSS <br />10.18 <br />1.75 <br />25 <br />231.0 <br />5.476 <br />21.64 <br />43.23 <br />7.50 <br />6- . <br />055.4 <br />1.309 <br />5.234 <br />10.47 <br />1.80 <br />26 <br />222.3 <br />5.697 <br />22.50 <br />44.99 <br />7.50 <br />10 <br />029.6 <br />1:346 <br />5.379 <br />10.76 <br />1.55 <br />27 <br />214.2 <br />5.918 <br />23.35' <br />46.69 <br />S.10 <br />20 <br />905.1 <br />1.382 <br />5.524.11.05 <br />1.90 <br />28 <br />206.7 <br />6.139 <br />24.19 <br />48.35 <br />8.40 <br />30 <br />881.9 <br />1.418 <br />5.669 <br />11.34 <br />1".95 <br />29 <br />199.7 <br />6.360 <br />25.04 <br />50.07 <br />8.70 <br />40 <br />859.9 <br />1.455 <br />5.814 <br />11.63 <br />2.00 <br />30 <br />193.2 <br />6.583 <br />25.88 <br />51.76 <br />9.00 <br />The middle ordinate in inches for any cord of length (C) is equal to0012 C' <br />multiplied by the middle ordinate taken from the above table. Thus, if it <br />desired'to bend a 30 ft. rail to fit a 10 degree curve, its middle ordinate should <br />be .0012X90OX2.183 or 2.30 inches. <br />TABLE III. Deflections for Sub Chords for Short Radius Curves. <br />Degree <br />Of <br />Curve <br />Radius <br />50 <br />A sub chord <br />R = sin of 1 def. annle <br />hi <br />of arc <br />for 100 ft. <br />sin. l def, ang.. <br />12.5 Ft. <br />15 Ft. <br />20 Ft. <br />25 Ft. <br />30° <br />_193..18 <br />1'51' <br />2° �7 <br />2°.58' <br />3° 43' <br />101.15 <br />320 <br />181.39 <br />1°59' <br />2°25' <br />3° 10' <br />3°58' <br />101.33 <br />34° <br />171.01 <br />2006' <br />2°33' <br />3°21' <br />4°12' <br />101.48 <br />36° <br />i6 Ao <br />2° 13' <br />°41' <br />2,41 , <br />' 3° 33' <br />4° 26' <br />101.66 <br />fi°. <br />1:53 58. <br />2° 20' <br />2° 49' <br />. 3° 44' <br />4° 40' <br />101.85 <br />.3 <br />40° <br />146. i9 <br />2° 27' " <br />2° 57' <br />3° 55•' <br />4° 54" <br />102.06 <br />42° <br />139.52- <br />2° 34' <br />3° 05' <br />4° 07' <br />5° o8'. <br />102.29 <br />44° <br />133.47 <br />2° 41'. <br />3° 13'. <br />4'18' <br />S°22' <br />102.53 <br />460 <br />127.97 <br />2° 48 <br />3` 21' <br />4° 29' <br />5° 36' <br />102.76 <br />48°. <br />122..92 <br />2° 55' <br />3°29' <br />4°40' <br />5°50' <br />103.00 <br />50° <br />118.31 <br />3° 02' <br />3° 38'. <br />4° 51' <br />6-04 <br />103.24 <br />52° <br />114.06 <br />3° 09' <br />30 46' <br />5° 02' <br />6-17 <br />103•'54 <br />56° <br />l0 <br />30 16' <br />3o 54 <br />50 13/ 1 <br />60 31 <br />02' <br />103.84 <br />29 <br />106. <br />39 <br />4 <br />5 23 <br />6 44 <br />104.14 <br />g8° <br />103-4 <br />3° 29' <br />4° 10 <br />50 34' <br />6° 57' <br />104.43 <br />6o°100.00 <br />1 .6667 11 <br />3o 35 <br />4° 18' <br />S° 44 <br />f 11' <br />104.72 v <br />IX <br />CURVE FORMULAS <br />T= R tan's I R= T cot. Z I chords <br />50 tan J I Chord def. = <br />T 50 R <br />R Sin. J D = <br />Sin. D = 50 Sin. Z D No. chords = I <br />R E= R ex. see I I D - <br />Sin. D = 50 tan I E = T tan I Tan. def. =3:i chard def. <br />The square of any distance, divided by twice the radius, will equal <br />the distance from tangent to curve, very nearly. <br />To find angle for a given distance and deflection. <br />Rule 1. Multiply the given distance by .01745 (def, for 1° for I ft. <br />see Table II.), and divide given deflection by the product. <br />Rule 2. Multiply given deflection by 57.3, and divide the product by <br />the given distance. <br />To find deflection fora given angle and distance. Multiply the angle <br />by .01745, and the product by the distance. <br />GENERAL DATA <br />RIGHT ANGLE TRIANGLES. Square the altitude, divide by twice the <br />base. Add quotient to base for hypotenuse. <br />Given Base loo, Alt. 10.102=200=.5. 100-}-.5=100.5 hyp. <br />Given Hyp. ioo,Alt. 25.252:-200=3.125.100-3.125=96.875=Base. <br />Error in first example, .002; in last, .045. <br />To find Tons of Rail in one mile of track: multiply weight per yard <br />by I1, and'divide by 7. <br />LEVELING. The correction for curvature and refraction, in feet <br />and decimals of feetis equal to 0.674d2, where d is the distance in miles. <br />The correction for curvature alone is closely, id 2. The combined cor- <br />rection is negative. <br />PROBABLE ERROR. If dl, d, da, etc. are the discrepancies of various <br />results from the mean, and 1f :_d2=the sum of the squares of these differ- <br />ences and n=the number of observations, then the probable error of the <br />mean=} 0.6745 Ede <br />1 11(n-1) <br />SOLAR ErnrMERIS. Attention is called to the Solar Ephemeris for <br />the current year, published by Keuffel & Esser Co., and furnished free of <br />charge upon request, which is 31x5s in., with about 90 pages of data very <br />useful to the Surveyor; such as the adjustments of transits, levels and <br />solar attachments; directions and tables for determining the meridian <br />and the latitude from observations on the sun and Polaris; stadia meas- <br />urements; magnetic declination; arithmetic constants; English and Metric <br />conversions; trigonometric formulas; Natural andLogarithmic Functions; <br />and Logarithms of Numbers. <br />TABLE IV. - Minutes in Decimals of a Degree. ` <br />1' <br />.0167 <br />11' <br />.1833 <br />211 <br />.3500 <br />l' <br />.5167 <br />41' <br />.6333 <br />51' <br />.8500 <br />2 <br />.0333 <br />12 <br />.2000 <br />22 <br />.3667 <br />32 <br />.5333 <br />42 <br />.7000 <br />52 <br />.8667 <br />3 <br />.0500 <br />13 <br />.2167 <br />23 <br />.3833 <br />33 <br />.5500 <br />43 <br />.7167 <br />. 53 <br />.8833 <br />4 <br />XG67 <br />1.4 <br />.2333 <br />24 <br />.4000 <br />34 <br />.5667 <br />44 <br />.7333 <br />54 <br />.9000 <br />5 <br />.0833 <br />15 <br />.2500 <br />25 <br />.4167 <br />35 <br />.5833 <br />45 <br />.7500 <br />55 <br />.9167 <br />6 <br />.1000 <br />16 <br />.2667 <br />26 <br />.4333 <br />36 <br />.6000 <br />4G <br />.7667 <br />66 <br />.9333 <br />7 <br />.1167 <br />1.7 <br />.2833 <br />27 <br />.4500 <br />37 <br />.6167 <br />47 <br />.7833 <br />57 <br />..9500 <br />8 <br />.1333 <br />18 <br />.3000. <br />28 <br />.4667 <br />33 <br />:6333 <br />48 <br />.5000 <br />58 <br />.9667 <br />9 <br />..1500 <br />19 <br />.3167 <br />29 <br />.4833 <br />39 <br />.6500 <br />49 <br />8167 <br />59 <br />.9533 <br />10 <br />1 .1667 11 <br />20 <br />1 .3333 1130 <br />".5000 11 <br />40 <br />1 .6667 11 <br />50 <br />1 .8333 11 <br />60 <br />11.0000 <br />W, <br />TAnLr V. -'Inches in Decimals of a Foot. <br />1-1G <br />3-32 <br />1 <br />3-16 <br />% <br />5-16 <br />? y <br />?� <br />0052 <br />.0078 <br />.0101 <br />.0156 <br />.0208 <br />.0260 <br />.0313 <br />.0117 <br />.0521 <br />.0625 <br />'.0729 <br />1 <br />2 <br />3 <br />4 <br />5 <br />6 <br />7 <br />8 <br />9 <br />10 <br />11 <br />.0833 <br />.1667 <br />.2500 <br />.3333 <br />.4167 <br />.5000 <br />.5533 <br />.6667 <br />.7500 <br />.5333 <br />1 .9167 <br />W, <br />