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VIII <br />TABLE II. -- Radii, Ordinates and Deflections. Chord =100 ft. <br />Deg. <br />Radius <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />Def. <br />Dist: <br />Def.Mid. <br />1for <br />Ft.rd. <br />Deg. <br />Radius <br />O <br />Tan <br />Dist. <br />Def. <br />Dist. <br />Def. <br />1 Ft. <br />2° 17' <br />ft. <br />IL <br />it: <br />ft. <br />! <br />1°59' <br />ff. <br />ft. <br />it. <br />It. <br />i <br />0°10' <br />34377. <br />.036 <br />.145 <br />:291 <br />0.05 <br />7° <br />819.0 <br />1.523 <br />6.105 <br />12.21 <br />2.10 <br />20 <br />17189. <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 />.436 <br />.573 <br />0.15 <br />30.764.5 <br />4° 07' <br />1.637 <br />6.540 <br />13.08 <br />2.25 <br />40 <br />8594.4 <br />.145 <br />.5S2 <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 />.873 <br />1.745 <br />0.30 <br />20 <br />688.2 <br />1.819 <br />7.266 <br />14.53 <br />2.50 <br />10 <br />4911.2 <br />_ <br />.255 <br />1.018 <br />2.03G <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 />637.3 <br />1:965 <br />7.84G <br />15.69 <br />2.70 <br />40' <br />3437.9 <br />.304 <br />1.454 <br />2.909 <br />0.50, <br />20 <br />614.G <br />2.037 <br />8.136 <br />16.27 <br />2.80 <br />60 <br />3125.4 <br />.400 <br />1.600 <br />3.200 <br />0.55 <br />30- <br />603.8 <br />2:074 <br />8.281 <br />16.56 <br />2.85 <br />2 <br />2864.9 <br />.436 <br />1.745 <br />3.490 <br />0.60 <br />140 <br />593.4 <br />2.110 <br />8.426 <br />16.85 <br />2.90 <br />10 <br />2644.6 <br />.473 <br />1.591 <br />3.7SI <br />O.G5 <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 />2.030 <br />14.072 <br />0.70 <br />30 <br />546.4 <br />2.292 <br />9.150 <br />18.30 <br />3.15 <br />30 <br />2292.0 <br />.545 <br />2.181 <br />4.3G3 <br />0.75 <br />tt <br />521.7 <br />2.402 <br />9.585 <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 />3 <br />1910.1 <br />..655 <br />2.618 <br />5.235 <br />0.90, <br />30. <br />459.3 <br />2:730 <br />10:59 <br />21.77 <br />3.75 <br />10 <br />1809.6 <br />.691 <br />2.763 <br />5.526 <br />0.95 <br />13 <br />441..7 <br />2.839 <br />11.32 " <br />22. 64 <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 />.500 <br />3.199 <br />6.39S <br />1.10 <br />. 30 <br />396.2 <br />3:163 <br />12.62 <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 />353.1 <br />3.277 <br />13:05 <br />26.11 <br />4.50 <br />A. <br />1432:7 <br />.873 <br />3."490 <br />6.980 <br />1.20 <br />30 <br />370.8 <br />3.357 <br />13'.49 <br />20.97 <br />4.65 <br />.10 <br />1375.4 <br />.909 <br />3.635 <br />7.271 <br />1.25 <br />1G <br />359.3 <br />3.496 <br />13.92 <br />27:84"4.80 <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 />29.70 <br />4.95 <br />30 <br />1273.6 <br />.9S2 <br />3.920 <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 />3228.1 <br />1.018 <br />4.071' <br />8.143 <br />1.40 <br />1S <br />319.6 <br />3.935 <br />15.64 <br />31.29 <br />5.40 <br />50 <br />1185.8 <br />1.055 <br />4.217 <br />8.433 <br />1.45 <br />19 <br />302.9 <br />4.155 <br />16.51. <br />33.01'5.70 <br />6 <br />1146.3 <br />1.091 <br />4.362 <br />S.724 <br />1.50 <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 />1.55 <br />21 <br />274.4 <br />4.594 <br />18.22 <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 />38.16 <br />6.60 <br />30 <br />1042.1 <br />1.200 <br />4.795 <br />9.596 <br />1.65 <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.SS6 <br />1.70 <br />24 <br />240.5 <br />5.255 <br />20.79 <br />41.58 <br />7.20 <br />50 <br />982.6 <br />1.273 <br />5.058 <br />10.18 <br />1.75 <br />25, <br />231.0 <br />5.476 <br />21.64 <br />43.28 <br />7.50 <br />6 <br />055.4 <br />1.309 <br />5.234'10.47 <br />1.80 <br />20 <br />222.3 <br />5.697 <br />22.50 .44.99 <br />7.50 <br />10 <br />929.6 <br />1.346 <br />5.379 <br />10.76 <br />1.S5 <br />27 <br />214.2 <br />5.018 <br />23.35 <br />46.69 <br />8.10 <br />20 <br />905.1 <br />1.382 <br />5.524 <br />11.05 <br />1.00 <br />28 <br />20G.7 <br />6.139 <br />24.19 <br />48.38 <br />8.40 <br />30 <br />881.0 <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 />1 859.9 <br />1.455 <br />5.814 <br />11.63 <br />2.00 <br />f 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 to .0012 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 .0012X900X2.183 or 2.36 inches. <br />TABLE III. Deflections for Sub Chords for Short Radius Curves. <br />Degree <br />of <br />Curve <br />Radius <br />56 <br />A subchord <br />R =sin of A def. angle <br />Length <br />of arc <br />for 100 ft. <br />sin. i 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° 17' <br />2° 58' <br />3° 43 <br />101.15 <br />32° <br />181.39 <br />1°59' <br />2025' <br />30 1o' <br />30 58' <br />101.33 <br />34° <br />171.01 <br />2° 06' <br />2° 33' <br />3° 21' <br />4° 12' <br />101.48 , <br />360 <br />-16i.8o <br />2°'13' <br />2° 41'' <br />3° 33' <br />4° 26' <br />1o1.66 <br />38° <br />1J3.58 <br />2* 26' <br />z° 49' <br />3°44' <br />°' <br />440 <br />1o1.85 <br />qo° <br />146.19 <br />2° 21' <br />2° 57' <br />3 55 <br />4° 54' <br />162.06 <br />42° <br />139.52 <br />2° 34' <br />3° 05' <br />4° 07' <br />5° 08' <br />102.29 <br />44° <br />133.47 <br />2° 41' <br />•3° 13' <br />40 18' <br />5° 22' <br />102.53 <br />46° <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°09'3°46' <br />.4667 <br />0°02' <br />6°17' <br />103.54 <br />54° ' <br />110-11 <br />3° 16' <br />3° 54' <br />5° 13' <br />6° 31' <br />103.84 <br />56° <br />io6.5o <br />3° 22' <br />4° 02' <br />5° 23' <br />60441 <br />104.14 <br />58° <br />-103.14 <br />3° 29'4° <br />10' <br />5° 34' <br />6° 57' <br />104.43 <br />60° <br />100.00 <br />30 35' <br />4° 18' <br />5° 44' <br />7° 11', <br />104.72 <br />CURVE FORMULAS IX. <br />T= R tan � I R= T cot. ; I ch0rd� <br />5o tan z I Chord def. R <br />T, - Sin. J D 12 = Sin. , D <br />5° 50 <br />Sin. D = No. chords = I <br />R E= Rex. sec J I D <br />Sin. JI) = 5o tan i I E, =4 tan J Tan. def. = a chord 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 .01945 (def. far I° for lift. <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 for a 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 zoo, Alt. 10.102-200=.5. 1oo+.5=100.5 hyp. <br />'Given Hyp. Ioo, Alt. 25.25$=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 ii, and divide by 7. <br />LEVELING. The correction for curvature and refraction, in feet <br />and'decimals of feet is equal to 0.574ds, where d is the distance in miles. <br />The correction for curvature alone is closely, Jd2. The combined cor- <br />rection is negative. <br />PROBABLE ERROR. If tl , d2, d8i etc. are the discrepancies. of various <br />results from the mean, and if Xds=the sum of the squares of these differ- <br />ences and n=the number of observations, then the probable error of the <br />mean= f 0.6745 , n n-1) <br />SOLAR EPHEMERIS. 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 3}x58 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 and Logarithmic Functions; <br />and Logarithms of Numbers. <br />TABi..E IV. - Minutes in Decimals of a Deuce. <br />1' <br />.0167 <br />11' <br />1833 <br />21' <br />.3500 <br />31' <br />.5167 <br />41' <br />.6833 <br />51' <br />.8500 <br />2 <br />.0333 <br />12 <br />.2000 <br />22.3G67 <br />.0078 <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 />.0667 <br />14 <br />.2333 <br />24 <br />.4000• <br />34 <br />.5667 <br />44 <br />.7333 <br />54 <br />.9000 <br />5.0833 <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 />.2067 <br />26 <br />.4333 <br />36 <br />.6000 <br />46 <br />.7667 <br />56 <br />.9333 <br />7 <br />.1167 <br />17 <br />.2533 <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 />38 <br />.6333 <br />4S <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 />.9833 <br />10 <br />1 .1667 11 <br />20 <br />1 .3333 1130 <br />1 .5000 <br />11 40 <br />1 .616167 11 <br />50 <br />1 5383 11 <br />60 <br />11.0000 <br />TABLE V. - <br />Inches in DLcimals of a Foot. <br />1-16 <br />3-32 <br />3-16 <br />YJ <br />5 -IG <br />'/a <br />1 <br />Ma <br />1 <br />0052 <br />.0078 <br />.0104 <br />.0156 <br />.0208 <br />.0260 <br />.0313 <br />.0417 .0521 <br />.0625 <br />.0729 <br />1 <br />2 <br />3 <br />4 <br />5 <br />6 <br />7 <br />8 9 <br />10 <br />11 <br />.0533 <br />.1667 <br />.2500 <br />1 .3333 <br />.4167 1 <br />.5000 <br />.5833 <br />.G667 .7500 <br />.8333 <br />.9167 <br />