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VIII <br />TABLE II. - Radii, Ordinates and Deflections. Chord =100 ft. <br />Deg. <br />Rad018 <br />Mid.,Tan. <br />Ord. <br />Dtat. <br />Def. <br />Dist. <br />Def. <br />for ' <br />1 Ft. <br />Deg. <br />Radius <br />Mid. <br />Ord. <br />Tan <br />Dist. <br />Def. <br />Dist. <br />Def. <br />for <br />1 Ft. <br />2° 17' <br />It. <br />It. <br />ft. <br />it. <br />181.39 <br />1059' <br />It. <br />it. <br />It, <br />ft. <br />34° <br />0`10' <br />34377. <br />030 <br />.145 <br />.291 <br />0.05 <br />7° <br />819.0 <br />1.528 <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 />751.8 <br />1.600 <br />6.395 <br />12.79 <br />2.20 <br />30 <br />'11459. <br />109 <br />:43G <br />.873 <br />0.15 <br />30 <br />764.5 <br />1.637 <br />6.540 <br />13.OS <br />2.25 <br />40 <br />8594.4 <br />.145 <br />.582 <br />1.164 <br />0.20 <br />40 <br />747:9 <br />1.673 <br />6.655 <br />13.37 <br />2.30 <br />50 <br />687.5.5 <br />.182 <br />.727 <br />1.454 <br />0.25 <br />8 <br />716.8 <br />1.746 <br />6.976 <br />13.05 <br />2.40 <br />1 <br />5729.6 <br />,218 <br />.S73 <br />1.745 <br />0.30 <br />20 <br />GSS.2 <br />1.819 <br />7.266 <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 />1.555 <br />7.411 <br />14.52 <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 />1.592 <br />7.556 <br />15.13 <br />2.60 <br />30 <br />3819.8 <br />'.327 <br />1.309 <br />2.018 <br />0.45 <br />9 <br />G37.3 <br />1.965 <br />7.846 <br />15.69 <br />2.70 <br />40 <br />3437.0 <br />.364 <br />1.454 <br />2.909 <br />0.50 <br />20 <br />614.6 <br />2.037 <br />8.136 <br />16.27 <br />2.80 <br />50 <br />3125.4 <br />.4UO <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 />O.GO <br />40 <br />593.4 <br />2.110 <br />5.426 <br />16.85 <br />2.90 <br />10 <br />2644.6 <br />.473 <br />1.891 <br />3.781 <br />0.65 <br />10 <br />573.7 <br />2.183 <br />8.716 <br />17.43 <br />3.00 <br />20 <br />2455.7 <br />.509 <br />2.03G <br />4.072 <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 />2.151 <br />4.363 <br />0.75 <br />11 <br />521.7 <br />2.402 <br />9.585 <br />19.16 <br />3.30 <br />40 <br />214S.8 <br />.582 <br />2.327 <br />4.G54 <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 />5.235.0.90 <br />30 <br />459.3 <br />2.730 <br />10.89 <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 />22.64'3.90 <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 />6.393 <br />1.10 <br />30 <br />396.2 <br />3.168 <br />12.62 <br />25.24 <br />4.35 <br />50 <br />1495.0 <br />.836 <br />3.345 <br />6.689 <br />1.15 <br />15 <br />383.1 <br />3.277 <br />13.05 <br />26.11 <br />3.50 <br />4 <br />1432.7 <br />.873 <br />3.490 <br />6.980 <br />1.20 <br />30 <br />370.8 <br />3.387 <br />13.49 <br />26.97 <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 <br />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 />2S.70 <br />4.95 <br />30 <br />1273.6 <br />.082 <br />3.92G <br />7.852 <br />1.35 <br />17 <br />33S.3 <br />3.716 <br />14.73 <br />29.56 <br />5.10 <br />40 <br />1228:1 <br />1.015 <br />4.071. <br />8.143 <br />1.40 <br />18 <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 <br />5.70 <br />S <br />1146.3 <br />1.091 <br />4.362 <br />8.724 <br />1.50 <br />20 <br />257.9 <br />4.374 <br />17.87 <br />34.73 <br />6.00 <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 />3G.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 />G.60 <br />�30 <br />1042.1 <br />1.200 <br />4.79S <br />9.596 <br />1.65 <br />23 <br />250.8 <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 />5.058 <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 />935.4 <br />1.309 <br />5.234 <br />10.47 <br />1.80 <br />26 <br />222.3 <br />5.697 <br />22.50 <br />44.997.80 <br />'10 <br />929.6 <br />1.346 <br />5.379 <br />10.76 <br />1.85 <br />27 _ <br />214.2 <br />5.918 <br />23.35 <br />46.69 <br />8.10 <br />20 <br />905.1 <br />1.382 <br />,5.524 <br />11.05 <br />1.90 <br />28 . <br />206.7 <br />6.139 <br />24.10 <br />48.38 <br />8.40 <br />30 <br />881.0 <br />1.418 <br />5.669 <br />.11.34 <br />1.05 <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 1 <br />.1.314 <br />11.63 <br />2.00 1 <br />30 <br />193.2 <br />6.583 <br />25.88 <br />51.76 <br />9.00 <br />`.rhe 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 />ilius. <br />sub chord =sin of 1 deE.angle <br />R <br />Length <br />of arc <br />for 100 ft. <br />sin. ldcf. 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°5s' <br />3°43' <br />101:15 <br />320 <br />181.39 <br />1059' <br />2° 25' <br />3° 10' <br />3° 58' <br />101.33 <br />34° <br />171.01 <br />2° o6' <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' I <br />to 26' <br />101.66 <br />38° <br />i53:58 <br />2° 20' <br />. 2° 49` <br />3° 44' <br />4° 40' <br />101:.85 <br />40° <br />146. ig <br />2° 27' <br />2° 57 <br />3° 55' <br />4° 54', <br />102.06 <br />42' <br />139.52 <br />-° 34' <br />3° 05'4° <br />07' <br />S° 08' <br />102.29. <br />440 <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 />4029 , <br />°29' <br />5° 36'• <br />102.76 <br />48°' <br />122.92 <br />2° 55' <br />3° 29'4° <br />40' <br />S° 50' <br />103.00 <br />Soo <br />118.31 <br />30 02' <br />3° 38' <br />4° 51' <br />6° 04' <br />103.24 <br />52° <br />114-o6 <br />3° 09' <br />3° 46' <br />5° 02' <br />60 17' <br />1.03.54 <br />54° <br />110.11 <br />3° 16'3° <br />54' <br />5° 13' <br />6° 31' <br />103.84 <br />560 <br />Io6.5o <br />3° 22' <br />4° 02' <br />S° 23' <br />6° 44' <br />104.14 <br />58° <br />103.14 <br />3°29' <br />4°1O' <br />5°34' <br />6°57 <br />104.43 <br />60° <br />loo.00 <br />3°35 <br />4°f8' <br />S°44' <br />7° II' <br />104.72. <br />IX <br />CURVE FORMULAS <br />T = R tan i I R = T cot. 3 I. Chord def. = chords <br />Z .- 5o tan 2 I R <br />Sin. B.D R = 50 <br />Sin. §. D = 5o Sin. 2 D No. chords = I <br />.- I R. E =1� c::. sce I D <br />Sia. --J D = 50 t In ; I E ='I' tan J I Tan. def. =1 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 I. Multiply the given distance by .01745 (clef. for 0 for I ft..\ <br />see Table II.), and divide given deflection by the product. ' <br />Rule 2. 11'Iultiply 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 AIrGLE TRIANGLES. Square the altitude, divide by twice the <br />base.' Add quotient to base for hypotenuse. - <br />Given Base 100, Alt. Io.1o2=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; 1n last,, .045. <br />To find Tons of Rail in one mile of track: multiply weight per yard <br />by i I, and divide by 7- <br />LrvEI.IN6. The correction for curvature and refraction, in feet <br />and decimals of feet is equal to 0.574d2, where d is the distance in miles. <br />The correction for curvature alone is closely, Bd2. The combined cor- <br />rection is negative. <br />P11011AaLE ERROR. If d3, d_, ds, etc. are the discrepancies of various <br />results from the mean, and if %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 EdL <br />Vi <br />(n-1) <br />SOLAR EPHEMERIS. Attention is called to the Solar Ephemeris for <br />4e 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 determi6ing:.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 andLagarithmic Functions; <br />and Logarithms of Numbers. <br />TABLE IV. - Minutes in Decimals of a Degree. <br />1" <br />,0167 <br />11' <br />1533 <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 <br />.3667 <br />32 <br />.5333 <br />42 <br />.7000 <br />5'L <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 />13 <br />.2333 <br />24 <br />.4000 <br />34 <br />.5667 <br />44 <br />.7333 <br />54 <br />.0000 <br />5 <br />.0833 <br />'15 <br />.2500 <br />25 <br />.4167 <br />35 <br />.5833 <br />45 <br />.7500 <br />55 <br />.0167 <br />6 <br />16 <br />.2667 <br />26 <br />.4333 <br />36 <br />.6000 <br />4G <br />.7667 <br />5G <br />.0333 <br />7 <br />..1000 <br />.1167 <br />17 <br />.2833 <br />27 <br />.4500 <br />37 <br />G167 <br />47 <br />.7533 <br />57 <br />.9500 <br />8 <br />.1333 <br />13 <br />.3000 <br />28 <br />.4667 <br />38 <br />G333 <br />48 <br />.1000 <br />58 <br />.9667 <br />9 <br />.1500 <br />19 <br />.3167 <br />29 <br />.4833 <br />39 <br />.0500 <br />49 <br />.9167 <br />59 <br />.9833 <br />10 <br />1 .1667 11 <br />20 <br />1 .3333 1130 <br />1 .5000 11 <br />40 <br />1 .6067 <br />SO <br />.5333 <br />60 <br />1.0000 <br />TArLr V. - <br />Inches in Decimals of a hoot. <br />1-1G <br />0052 <br />3-32 <br />.0078 <br />% <br />.0101 <br />3-10 <br />.0156 <br />% <br />.0203 <br />5-16 <br />.0260 <br />N <br />.0318 <br />?, r' <br />A1. .0521 <br />.0625 <br />.0729 <br />1 <br />.0833 <br />2 <br />.1667 <br />3 <br />.2500 <br />4 <br />.3333 <br />5 <br />.4167 <br />.506 00 <br />7 <br />.5833 <br />8 9 <br />.6667 .7500 <br />10 <br />..8333 <br />it <br />.9167 <br />