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VIII <br />TABLE II. -Radii, Ordinates and Deflections. -Chord= 100 ft. <br />Deg. <br />Itndius <br />Mid. <br />O: d: <br />iru. <br />bis*. <br />Def. <br />i Dist. <br />lief. <br />1 or <br />'Dcg. <br />Radius <br />Dlid. <br />Ord. <br />Txa <br />Dist. <br />Def. <br />Dist. <br />lief. <br />1.rt I <br />'° 17 <br />It. <br />it. <br />ft. <br />ft, <br />ISI .,39 <br />i° J9' <br />ft: <br />ft. <br />ft. <br />ft. <br />34° <br />0'10' <br />34377. <br />.036 <br />.14;5 <br />•.291 <br />0.03 <br />7° ' <br />810.0 <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 />781.8 <br />1.600 <br />6.395 <br />12.70 <br />2.20 <br />30 <br />11459. <br />.109 <br />.436 <br />S73 <br />0.15 <br />30 <br />764.5 <br />1.637 <br />6.540 <br />13.05 <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 />6.655 <br />13.37 <br />2.30 <br />50 <br />GS75.5 <br />.152 <br />.727 <br />1.454 <br />0.25 <br />8 <br />716.8 <br />1.746 <br />0.976 <br />13.95 <br />2.40 <br />1 <br />5729.6 <br />.21S <br />.573 <br />1.745 <br />0.30 <br />20 <br />GS8.2 <br />1.819 <br />7.266 <br />14.53 <br />2.50 <br />10. <br />4911.2 <br />.255 <br />1.018 <br />2.0360.35 <br />1o6.5:1 <br />30 <br />G74.7 <br />1.855 <br />7.411 <br />14.82 <br />2.55 <br />20 <br />4297.3 <br />.^-91 <br />1.164 <br />2.327 <br />0.40 <br />40 <br />661.7 <br />1.592 <br />7.556 <br />15.11 <br />2.60 <br />30 <br />3319.8 <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 />G14.G <br />2.037 <br />8.136 <br />10.27 <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 />8.281 <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 />503.4 <br />2.110 <br />S.426 <br />16.85 <br />2.00 <br />10 <br />2644.6 <br />.473 <br />1.591 <br />3.751 <br />0.63 <br />10' <br />573.7 <br />2.183 <br />5.716 <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 />2:2'92 <br />9.150 <br />18.30 <br />3.16 <br />30 <br />2292.0 <br />.54:5 <br />2.181 <br />4.3G3 <br />0.75 <br />11 <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 />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 />O.S5. <br />12 <br />473.3 <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 />10.89 <br />21.77 <br />3.75 <br />10 <br />1809.6 <br />.691 <br />2.763 <br />5.526 <br />0.95--13 <br />441.7 <br />2.839 <br />11.32 <br />22.64 <br />3.90 <br />20 <br />1719..1 <br />.727 <br />2.008 <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 />356.2 <br />3.168 <br />12.62 <br />25.24 <br />4.35 <br />50 <br />1495.0 <br />.MG <br />3.345 <br />6.GS9 <br />1.15 <br />15 <br />333.1 <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 />3.357 <br />13.49 <br />26.97 <br />4.05 <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 />.94.5 <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 />.9S2 <br />3.926 <br />7.852 <br />1.35 <br />17 <br />338.3 <br />3.710 <br />14.78 <br />29.56 <br />5.10 <br />40 <br />1225.1 <br />1.018 <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 />1135.8 <br />1.055 <br />4.217 <br />8.433 <br />1.45 <br />19 <br />4.155 <br />16.51 <br />33.01 <br />5.70 <br />6 <br />1146.3 <br />1'.091 <br />4.362 <br />8.724 <br />1.50 <br />20 <br />.302.9 <br />237.9 <br />4.374 <br />17.37 <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 />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 />10.08 <br />35.16 <br />3.60 <br />30 <br />1042.1 <br />1.200 <br />4.798 <br />9.596 <br />1.65 <br />23 <br />250.8 <br />5.03.5 <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 />9S2.6 <br />1.273 <br />5.OSS <br />10.13 <br />1.75 <br />25 <br />231.0 <br />5.476 <br />21.64 <br />43.25 <br />7.50 <br />6 <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 />929.6 <br />1.346 <br />5.379 <br />10.76 <br />1.35 <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 />1.1.05 <br />1.90 <br />28 <br />206.7 <br />6.139 <br />24.19 <br />48.33 <br />8.40 <br />30 <br />881.9 <br />1.418 <br />6.669 <br />11.34 <br />1.95 <br />29 <br />199.7 <br />6.360 <br />2.5.04 <br />50.07 <br />S.70 <br />40 <br />1 S59.9 <br />1.455 <br />5.814 <br />11.63 <br />12.00 <br />1 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 C2 <br />mnitiplied 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 .00l2X900X2.1II3 or 2.36 inches. <br />TABLE III. Deflections for Sub Chords for Short Radius Curves. <br />Degree <br />of <br />Curve <br />Radius <br />50 <br />sub chord <br />T( =sin of .'_. clef. angle <br />f cr Length . <br />for 100 ft. <br />sin. r def. ang. <br />12,5. Ft. <br />15 Ft. <br />20 Ft. '- <br />25 Ft. <br />30° <br />193-18 <br />1° 51' <br />'° 17 <br />2° 58' <br />3° 43' <br />161.1.5] <br />32° <br />ISI .,39 <br />i° J9' <br />2° 25 <br />3° lo' <br />3° 58' <br />101.33 . <br />34° <br />171.01 <br />^° 06' <br />'° 33' <br />3°21 <br />}° 12' <br />I01.48 <br />360 <br />161.80 <br />2° 13' <br />2°41': <br />3 °33' <br />4°26' <br />Io1.6G <br />38° <br />153.53 <br />2' 20'c <br />2' =19' <br />3° 44' <br />4° 40' <br />101.85 <br />40° <br />f_}6.19 <br />2° 27' <br />2° 57' <br />3° 55' <br />4° 54� <br />102.06 <br />42' <br />139.5'- <br />'-°34' <br />3°05 <br />4°07o8' <br />43 <br />102.29 <br />44° <br />133.47 <br />2°41' <br />3°13' <br />} IS' <br />j°22'' <br />102.53 <br />46° <br />127•.97 <br />.10;4s,: <br />3° 21' <br />4° 29' <br />50 36' <br />102.76 <br />.48° <br />122'. 92 <br />° 55 <br />3° 29' <br />4° 4c' <br />5° 5o' <br />103.00 <br />500 <br />118.3f.' <br />3°02' <br />3°38' <br />4°-1' <br />6°04' <br />103.24 <br />5z°, <br />114..c6 ... <br />3° 09.' <br />'3° 46' <br />5° 02' <br />6° 17' <br />103.54 <br />54° <br />IIo. It" <br />3° 16' <br />3° 54' <br />1° 13' <br />6-31, <br />103.84 <br />56° <br />1o6.5:1 <br />3° 22` <br />4° 02' <br />5° 23' <br />6° 41' <br />104.14 <br />S8° <br />103..1 4 <br />.3°•29' <br />4° 10, <br />5° 34' <br />6° 57' <br />104.43 <br />Go° <br />100.00" <br />3° 35' <br />. 4° 18' <br />5' 44' <br />7° if' <br />104.72 <br />IX <br />CURVE FORIlIULAS <br />T= R tan � I R T cot. 3,I chord2 <br />I = 50 tangy I Chord def. = R <br />Sin. } D R = 50 <br />.Sin. i D =' 1o Sin. 2 D No. chords = I <br />R E= R ex. scc z I D <br />Sin. j = 5o tT ' I E =T tan} I Tan. def. = I 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 I° 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 for a given angle and distance. Multiply the angle <br />by .or745, and the product by the distance. <br />GENERAL DATA <br />RIGIIT ANGLE TRIANGLES. Square the altitude, divide by twice the <br />base. Add quotient to base for hypotenuse. <br />Given Base loo, Alt. Io.io2=2oo=.5. IoO+-5 =ioo.5 hyp. <br />Given Hyp. Ioo,Alt. 25.252_200=3.125.100-3.125=96.875=Base. <br />Error in first example, .oO2; in last, .045.' <br />To find Tons of Rail in one mile of track: multiply weight per yard <br />by I I, and divide by �. <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, J-1 -. The combined cor- <br />rection is negative. <br />PROBABLE ERROR. If'd, , d3, d3, etc. are the discrepancies of various <br />results from the mean, and if 7'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= y 0.6745= n (n-1) <br />SOLAR EPHEMERIS. Attention is called to the Solar Ephemeris for <br />the current year, published by KeuiTel & Esser Co., and furnished free of <br />charge upon request, which is 3.zx5j 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 andLogarithnlic Functions; <br />and Logarithms of Numbers'. <br />TABLE IV. - T-Winutes in Decimals of a Degree. <br />1' <br />.0167 <br />11' <br />.1833 <br />2t' <br />1:3500 <br />31' ' <br />.5167 <br />41' <br />6533 <br />54' <br />Ewe <br />2 <br />.0333 <br />12 <br />.2000 <br />22 <br />.3667. <br />3.2 <br />..5333 <br />42 <br />OOU <br />52 <br />.8667 <br />3.0500 <br />._0.729 <br />13 <br />.2167 <br />23 <br />.3833' <br />33� <br />.5500 <br />43 <br />.7167 <br />5:. <br />5833. <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 <br />.0633 <br />15 <br />.2500 <br />25 <br />.4167 <br />35 <br />.5533 <br />43 <br />.7500 <br />5.5 <br />.9167 <br />6 <br />.1000 <br />16 <br />.2667 <br />26 <br />.4333. <br />36 <br />.G000 <br />4n <br />.7007 <br />56 <br />.9333 <br />7 <br />.1167 <br />17 <br />.2533 <br />27 <br />.4500 <br />37 <br />.G167 <br />41 <br />.7333 <br />57 <br />.9500 <br />8.1333 <br />1S <br />.3000 <br />2S <br />.4667 <br />38 <br />.6333 <br />48' <br />0000 <br />58 <br />.9667 <br />9 <br />.1500 <br />1.9 <br />.3167 <br />29 <br />.4533 <br />39 <br />6500 <br />49. <br />SIG7 <br />59 <br />.9833 <br />10 <br />.1Gf,7 If <br />20 <br />.3333 1130 <br />1 .5000 11 <br />40 ' <br />.66567 <br />50 <br />I. ."333 <br />1 GO <br />11.0000 <br />TABLE V. -.Inches <br />in Decimals of a Foot. <br />_ <br />1-16 <br />•^-32 <br />% - <br />'-16 <br />% <br />-1G <br />.0052 <br />.0078 <br />.0104 <br />.0156 <br />.0203 <br />.502150 <br />.021.3 <br />.0417 .0621 <br />.Ofl3 <br />._0.729 <br />1 <br />2 <br />3 <br />4 <br />S <br />6 <br />7 <br />8 0 <br />10 <br />]1 <br />.0833 <br />.1667 <br />.2500 <br />.3333 <br />.4167 <br />.5000 <br />.5833 <br />:6667 .7500 <br />.5333 <br />.0167 <br />