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TABLE II. -Radii, Ordinates and Deflections. Chord 100 ft. <br />Dom' <br />Radius <br />50 <br />Mid. <br />"Ord. <br />Tan. <br />Dist. <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 />51' <br />t. <br />t. <br />ft. <br />ft. <br />7 <br />43' <br />ft. <br />ft. <br />ft. <br />ft. <br />i <br />0"-10' <br />34377. <br />.036 <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 />17169. <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 />'•.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 />.582 <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 />.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 />80 <br />3819.8 <br />•.327 <br />1.309 <br />"2.618 <br />0.45. <br />9 <br />637.3 <br />1.965 <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 />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 />8.281 <br />16.56 <br />2.85 <br />8 <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 />16.85 <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 />2.036 <br />4:072 <br />0.70 <br />30 <br />546.4 <br />2.292 <br />9.150 <br />18.30 <br />3.15 <br />80 <br />2292.0 <br />.545 <br />2.181 <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 />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 />20.0.1 <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 />!! <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 <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 />_80 <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 />24.37.4.20 <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 />12.62 <br />25.2.4 <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 />4.50 <br />L <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 />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 />28.70 <br />4.95 <br />30 <br />,1273.6 <br />.982 <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 />1.018 <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 />60 <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 />6 <br />1146.3 <br />1.091 <br />4.362 <br />8.724 <br />1.50 <br />20' <br />287.9 <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 />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 />4.798 <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 />41.58 <br />7.20 <br />50. <br />982.6 <br />1.273 <br />5.088 <br />10.18 <br />1.75 <br />25' <br />231.0 <br />5.476 <br />21.64 <br />43.28 <br />7.50 <br />0 <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 />7.80 <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 />24.19 <br />48.38 <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 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 />D ofee <br />Curve <br />Radius <br />50 <br />sub chord <br />R =sin of z def. angle <br />Lenth <br />of <br />for 100 ft. <br />sin. z def. ang. <br />12.5 Ft. <br />15 Ft. <br />20 Ft. <br />25 Ft. <br />30° <br />193.18 <br />1° <br />51' <br />2° <br />17' <br />2° <br />58' <br />3° <br />43' <br />101.15 <br />32° " <br />181.39 <br />1 ° <br />59 <br />2 o <br />25 <br />° <br />3 <br />10 <br />3 0 <br />58 , <br />101.33 <br />340 <br />171.0I <br />20o6 <br />1 <br />2° <br />33' <br />3° <br />21' <br />4° <br />12' <br />10I.48 <br />360 <br />161.80 <br />2° <br />13' <br />2° <br />41' <br />3°'33' <br />54 <br />4° <br />26' <br />io1.66 <br />380 <br />153.58. <br />2° <br />20' <br />. 2' <br />_49' <br />3° <br />44 <br />4° <br />40 <br />ioi . 85 <br />40° <br />146-.19 <br />2° <br />27'; <br />2° <br />57' <br />3° <br />55' <br />4° <br />54' <br />io2.o6 ". <br />42° <br />139.52 <br />2° <br />34' <br />3° <br />05' <br />4° <br />07, <br />50 <br />08' <br />102.29 <br />44°" <br />133-47- <br />2° <br />41' <br />3° <br />13'. <br />4° <br />18 <br />S° <br />22' <br />102.53 <br />46° <br />127:97 <br />2° <br />48' <br />3° <br />21' <br />-4° <br />29' <br />5° <br />36' <br />102.76. <br />480 <br />122.92 <br />2° <br />55' <br />30 <br />29� <br />40 <br />40' <br />50 <br />50' <br />103.00 <br />500 <br />I18.31 <br />3° <br />02' <br />3°38' <br />4°51' <br />6°04' <br />60 <br />103.24 <br />520 . <br />114.06 <br />3°•09' <br />3° <br />46' <br />S° <br />oz' <br />17' <br />103.54 <br />54° <br />.110:I1 <br />3° <br />16' <br />3° <br />54 <br />5° <br />13' <br />6° <br />3f' <br />103.84 <br />56° <br />106.50 <br />30 <br />22' <br />4002, <br />°23' <br />5023 <br />, <br />60 <br />44' <br />104.14 <br />58° <br />103.14 <br />3° <br />29' <br />4° <br />10' <br />50341 <br />60 <br />57' <br />Io4,43 <br />60° <br />100.00 <br />3035, <br />40 <br />18' <br />S° <br />44' <br />7° <br />II' <br />I04.72 <br />CURVE FORMULAS <br />T o tan � I R= T cot. a I Chord def. = chord9 <br />T = 50 R <br />Sin. 8 D R = <br />Sin. J D = 50 Sin. $ D No. chords = I <br />R E= R ex. sec -1I D <br />50 tan J I <br />Sin:" b D = ,1, E = T tan f I Tan. def. = z 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 <br />eflection.Rule I. 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 for a given angle and distance. Multiply the angle <br />by .61745, 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. lo.102+200=.5. 100-}-.5=100.5 hyp. <br />Given Hyp. loo, Alt'. z5.z5z=too=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 I I, and divide by 7. <br />'LEVELING. The correction for: curvature and refraction, in.feet <br />and decimals of feet is equal to 0.574 de, where d is the distance in miles. <br />The correction for curvature alone is closely, ids. The combined cor- <br />rection is negative. <br />PROBABLE ERROR: If d�, de, da, etc. are the discrepancies of various <br />results from the mean, and if Y:da=the sum of the.squares of these dif;oit <br />ences and n=the number of observations, then the probable error of the. <br />mean= + 0.6746 n�n 1) <br />SOLAR EPHEMERIS. Httention is called to the Solar Ephemeris far <br />the current year, published by Keuffel & Esser Co., and furnished free of <br />charge upon request, which is 3}x51 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 f ormulas; Natural and Logarithmic Functions; <br />and Logarithms of Numbers. <br />TABLE IV. - Minutes <br />TABLE V. - Inches in Decimals of a'Foot. <br />in Decimals of a De ee.: <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 <br />.3667 <br />32 <br />.5333 <br />42 <br />.7000 <br />52 <br />.8667 <br />it <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 <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 />46 <br />.7667 <br />56 <br />:9333 <br />7 <br />.1167 <br />17 <br />.2833 <br />27 <br />.4500 <br />37 <br />.6167 <br />47- <br />.7833 <br />57 <br />.9500 <br />8.1333 <br />18 <br />.3000 <br />28 <br />.4667 <br />3S <br />.6333 <br />48 <br />.8000 <br />58 <br />.9667 <br />9 <br />.1500 <br />19 <br />'.3167 <br />29 <br />.4533 <br />39 <br />.6500 <br />49 <br />.8167 <br />59 <br />.9833 <br />10 <br />1 .166711 <br />20 <br />1 .3333 1 <br />30 <br />1 .5000 <br />1140 <br />1 .6667 11 <br />50 <br />1 .833311 <br />60 <br />11.0000 <br />TABLE V. - Inches in Decimals of a'Foot. <br />1-16 3-32 <br />4 <br />3-16 <br />5-16 <br />%?� <br />.0052 :.0078 <br />.0104 <br />.0156 <br />.0208 <br />.0260 <br />.0313 <br />:0417: ..(521 <br />625 <br />.0729 <br />1 2 <br />3 <br />4 <br />5 <br />6 <br />7 T <br />8 1 9 1 <br />1011 <br />.0833 .1667 <br />.2500 <br />.3333 <br />.4167 <br />.5000 <br />.5833- <br />.6667 .7500 1 <br />.8333 <br />.9167 . <br />