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VIII IX <br />TABLE II. - Radii, Ordinates and Deflections. Chord =100 ft. CURVE FORAiULAS - <br />Deg. <br />Radius <br />1'fid.Tun. <br />Dist. <br />Def. <br />Dist. <br />Def <br />for <br />1 or <br />Deg. <br />Radios <br />Mid. <br />Ord. <br />Tau. <br />Dist. <br />Def. <br />Dist. <br />Def. <br />1 Ft. <br />20 17' <br />ft. <br />ft. <br />tt. <br />t. <br />' <br />1°J9'2'25' <br />ft. <br />ft. <br />ft. <br />ft. <br />34° <br />0"10' <br />34377. <br />.036 <br />.145 <br />.291 <br />0.05 <br />7' <br />519.0 <br />1.528 <br />6:105 <br />12.21 <br />2.10 <br />20' <br />17189. <br />.073 <br />.291 <br />.552 <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 />.873 <br />0.15 <br />30 <br />764.5 <br />1.637 <br />6.540 <br />13.08 <br />2.25 <br />40 <br />8594.4.145 <br />4° i8' <br />.552 <br />1.164 <br />0.20 <br />40 <br />747.9 <br />1.673 <br />G.GS5 <br />13.37 <br />2.30 <br />' 50 <br />6875.5 <br />.182 <br />.727 <br />1.454 <br />0.25 <br />S <br />716.8 <br />1.746 <br />6.976 <br />13.95 <br />2.40 <br />1 <br />5729.6 <br />.215 <br />.573 <br />1.745 <br />0.30 <br />20 <br />G88.2 <br />1.519 <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 />667.7'1.892 <br />3° 35' <br />7.556 <br />15.11 <br />2.60 <br />30 <br />3S19.8 <br />.327 <br />1.309 <br />2.618 <br />0.45 <br />9 <br />637.3 <br />1.965 <br />7.546 <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 />614.6 <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 />8.251 <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 />16.55 <br />3.90 <br />10 <br />2644.6 <br />_'.473 <br />1.891 <br />3.751 <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 />2.030 <br />4.072 <br />0.70 <br />30 <br />546.4 <br />2.292' <br />9.150 <br />15.30 <br />3.15 <br />30 <br />2292.0 <br />.545 <br />2.181 <br />4.363 <br />0.75 <br />11 . <br />521.7.2.402 <br />9.;,S5 <br />19.16 <br />3.30 <br />40 <br />2148.8 <br />.582 <br />2.327 <br />4.654 <br />O.SO <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 />^0.91 <br />3.60 <br />8 <br />1910.1 <br />.655 <br />2.61S <br />5.235 <br />0.90 <br />30 <br />459.3 <br />2.730 <br />IO.S9 <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 />2.908 <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 />3.055 <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 />356.2 <br />3!168 <br />12.62 <br />25.24 <br />4.35 <br />50 <br />1495.0 <br />.836 <br />3.345 <br />6.659 <br />1.15 <br />15 <br />383.1 <br />3'.277 <br />13.05 <br />26.11 <br />4.50 <br />4 <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 />1375:4 <br />.909 <br />3.635 <br />7.271 <br />1.25 <br />16 <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 />25.70 <br />4.95 <br />30 <br />1273.6 <br />.982 <br />3.926 <br />7.S52 <br />1.35 <br />17 <br />338.3 <br />3.776, <br />14.78 <br />?9.56 <br />5.10 <br />40 <br />1228.1. <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 />1155.8 <br />1.055 <br />4.217 <br />8.433 <br />1.45 <br />19 <br />302.9 <br />4.15.5 <br />16.51 <br />33.01 <br />".'0 <br />S <br />1146.3 <br />1.091 <br />4.362 <br />8.724 <br />1.50 <br />20 <br />237.9 <br />4.374 <br />17.37 <br />34.73 <br />6.60 <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 />G.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 />4.798 <br />9.596 <br />1.65 <br />23 <br />250.8 <br />5.035 <br />19.94 <br />39.57 <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 />5.2:5.5 <br />20.79 <br />41.55 <br />7.20 <br />50 <br />952.6 <br />1.273 <br />5.OSS <br />10.18 <br />1.75 <br />25 <br />231.0 <br />5.476 <br />21..64 <br />43.28 <br />7.50 <br />8 ' <br />955.4 <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 />929.G <br />1.34G <br />5.379 <br />10.76 <br />1.55 <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 />24.19 <br />48.35 <br />S.40 <br />30 <br />581.9 <br />1.415 <br />5.669 <br />11.34 <br />1.95 <br />29 <br />199.7 <br />6.360 <br />25.04 <br />50.07 <br />S.70 <br />40 1 <br />859.9 <br />1.455 <br />5.814 <br />11.63 <br />2.00 <br />30 <br />.193.2 <br />6.583 <br />25.85 <br />51.76 <br />9.00 <br />The middle ordinate in inches for any cord of length (C) is equal to .0012 C2 <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 shouhl <br />be .0012X90OX2.183 or 2.36 inches. <br />TABLE III. Deflections for Sub Chords for Short Radiu's Curves. <br />Degree <br />of <br />Curve <br />Radius <br />A sub chord _sin of a def. angle <br />R <br />.Length <br />of arc <br />for 100 R. <br />sin's def. ang. <br />12.5 Ft. <br />15 Ft. <br />20 Ft. <br />25 Ft.' <br />. 300 <br />193.18 <br />1° 51' <br />20 17' <br />'° 58' <br />3° 43' <br />101.Ij <br />32 <br />181.39 <br />1°J9'2'25' <br />.5333 <br />3° 10' <br />3° 58' <br />101.33 <br />34° <br />171.01 <br />2° 06' <br />2° 33'3° <br />2t' <br />4° 12' <br />101.48 <br />360 <br />161.80 <br />2°13' <br />2°41' <br />3°33' <br />4°26' <br />101.66 <br />38° <br />153.58 <br />2° 20' <br />2° 49' <br />3° 44' <br />4 ° 40' . <br />1:01.85 <br />40° <br />146.19 <br />2°27' <br />2°57' <br />30 55' <br />4°54' <br />102.o6 <br />420 <br />139.52 <br />20 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 />4° i8' <br />5° 22' <br />102.53 <br />46° <br />1'7.97 <br />2° 48' <br />3° 21' <br />4° 29' <br />5° 36' <br />1o2.76 <br />48° <br />122.92 <br />2° 55' <br />3° 29 <br />4° 40' <br />5° So' <br />103.00 <br />50° <br />118.31 <br />3,021 <br />3° 38' <br />4° jI' <br />6° 04' <br />103.24 <br />52° <br />I14 -o6 <br />3° 09' <br />3° 46' <br />5° O2' <br />6° 17' <br />103.54 <br />54° <br />II0.11 <br />3°16' <br />3°54' <br />S°13' <br />6°31' <br />Io3.84 <br />56 <br />io6.5o <br />3°22' <br />4°02' <br />5°23' <br />6°44 <br />104.14 <br />58° <br />103.14 <br />3° 29' <br />4° f0' <br />5° 34' <br />60 57' <br />104.43 <br />6o° <br />100.00 <br />3° 35' <br />4° 18' <br />1 5044 <br />7° 11' <br />104.72 <br />T _ tan 1 <br />5 tan k I R= T cot. l I Chord def. - chord2 <br />T Sin. J D R 50 R <br />= <br />Sin. D = 50 Sm. '' D N o. chords = D <br />R E = R es. sec ; I <br />So tan a I ; <br />Sin. � D = .1 E = T tan } I 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 r. Dlultiply the given distance by :01745 (def. for 1° for r 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 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 loo, Alt. Io.102=200=.5. 1oo+.5=1oo.5 hyp. <br />Given Hyp. loo,Alt. 25.252=Zoo=3.125.100-3.125=96.875=Base. <br />Error iri first example, .002; in last, .045• <br />To find Tons of Rail in one mile of .track: multiply weight per yard <br />by 11, and divide by 7. <br />LEVELING. The correction for curvature and refraction, in feet <br />and.decimals of feet is equal to 0.574d -,where d, is the distance in miles. <br />The correction for curvature alone is closely, 3d2. The combined cor- <br />rection is negative. <br />PROBABLE ERROR. If d„ d,, da, etc. are the discrepancies of various <br />results from the mean, and if 2d =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 � a <br />1 n (n-1) <br />SOLAR EPHEMERIS. Attention is called to the Solar Ephemeris for <br />the current year, published by Keuftel & Esser Co., and furnished free of <br />charge upon request, which is 341-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 />TABLE IV. - Minutes in Decimakoi a Degree. <br />1' <br />.0167 <br />11' <br />.1533 <br />21' <br />.3500 <br />31' <br />.5167 <br />4Y <br />GS.33 <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 />.3333 <br />33 <br />.5500 <br />43 <br />.71657 <br />53 <br />SS33 <br />4 <br />.0667 <br />14 <br />.2333 <br />24 <br />.4000 <br />34 <br />.5667 <br />44 <br />.7333 <br />54 <br />.0000 <br />5 <br />OS33 <br />15 <br />.2500 <br />25 <br />.4167 <br />35 <br />.5S33 <br />45 <br />.7500 <br />55 <br />.9167 <br />6 <br />.1000 <br />1G <br />.2667 <br />26 <br />.4333 <br />36 <br />.6000 <br />4G <br />.7667 <br />56 <br />.9333 <br />7 <br />.1167 <br />17 <br />.2833 <br />27 <br />.4;500 <br />37 <br />.61G7 <br />117 <br />.7833 <br />57 <br />.9500 <br />8t <br />.1333 <br />18 <br />.3000 <br />28 <br />.4667 <br />38 <br />.0333 <br />48 <br />5000 <br />58 <br />.0667 <br />9 <br />.1500 <br />19 <br />.3167 <br />29 <br />.4833 <br />39 <br />.6500 <br />t9 <br />S167 <br />59 <br />.9533 <br />16 <br />1 .1667 11 <br />20 <br />1 .3333 1130 <br />1 .5000 11 <br />40 <br />1 .6067 11 <br />50 <br />1 5333 <br />60 <br />1.0000 <br />TABLE V. - <br />Inches in Decimals of a Foot. <br />1-]G <br />3-32 <br />,a <br />3-16 <br />% <br />5 -IG <br />00.2 <br />.0075 <br />.0504 <br />.0156 <br />.0208 <br />.0260 <br />.0313.0521 <br />.0625 <br />.07291 <br />2 <br />3 <br />4 <br />5 <br />6 <br />79 <br />�11 <br />10 <br />1OS33 <br />.7667 <br />.2500 <br />.3333 <br />.4167 <br />.5000 <br />.SS33.7500 <br />.5333 <br />.916,7 <br />