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M <br />TABLn IL - Radii, Ordinates and'Deflections. Chord=100-ft. <br />Deg. <br />Radius, <br />Mid. <br />Ord. <br />Tan. - <br />Dist. <br />Def. <br />Dist. <br />Dor. <br />1 or <br />Deg. <br />Radius <br />Mid. <br />Ord. <br />Tan <br />Dist. <br />Def. <br />Dist. <br />Def <br />1 Ft. <br />2° 17'... <br />ft. <br />ft. <br />ft. <br />ft. <br />a <br />1° 59' <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 />819.6 <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 />12.79 <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 />30 <br />3819.8 <br />.327 <br />1.309 <br />2.618 <br />0.45. <br />'9 <br />637.3 <br />1.965 <br />7.846 <br />15.69 <br />2.70 <br />40 <br />3437.9 <br />.364 <br />1.454 <br />2.909 <br />0.50 <br />20'614.6 <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 />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 />16.85 <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 />30 <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 />20.04 <br />3.45 <br />'60 <br />2022.4 <br />:018 <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 />8 <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 />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 />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 />4.50 <br />6 <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 />3.926 <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 />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 />a <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 />5D <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 />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 />7.80 <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 />6.300 <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 />Degree <br />of <br />Curve <br />Radius <br />50 <br />34 sub chord <br />R = sin of } def. angle <br />I:ength <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 />?° 58' <br />3° 43' <br />_Io1. i5 .. <br />'320 <br />181.39 <br />1° 59' <br />i° 25' <br />3o..1O, <br />30 58' <br />101.33 <br />34° <br />171.0.1 <br />2° o6' <br />2° 331 <br />.3° 21' <br />4° 12' <br />101.48 <br />360 <br />161.80 <br />2° 13' <br />2° 41`. <br />3° 33' <br />38° <br />--4° 26' <br />ioi.66 <br />53 <br />153.58 <br />2° 20' <br />-' <br />2°.49' <br />3° 44' <br />. 4° 40-... <br />1 <br />. >oi.85 <br />o-' <br />40 <br />146.19 <br />a0 2'j1 <br />0 z 571 <br />' 30 551 <br />0- <br />-4 .54 <br />102.06 <br />42° - <br />139.52 <br />2° 34' <br />3° 05' <br />4° 0 1 <br />5° 08' <br />102.29 . <br />44°133.47 <br />45 <br />2° 41t <br />3° 13' <br />40 18' <br />-5022 ' <br />102.53 <br />460. <br />127-97 <br />2° 48, . <br />3° 21' <br />40 29' <br />5036 1 .. <br />102.76 <br />48o <br />122.92. <br />20 55' <br />30 29'. <br />40 40' <br />5'_50' <br />103.00 <br />50° <br />118.31. <br />3' 02' <br />3° 38' <br />4° Si'. <br />6° 04' <br />j03.24 <br />52° <br />114.°6 <br />3° 09' <br />3-461 <br />5° 02' <br />6°•17' <br />103.54' <br />54° <br />IIo. II <br />-3° 16' .. <br />30 541 <br />50 131 .." <br />60 311. <br />103.84 <br />$60 . <br />106.5° <br />. 3o z2� <br />•40 02' <br />5o z3� <br />60 44� :. <br />104.14 <br />g8° <br />- 103.14 <br />30 z9� <br />4° ion <br />50 34� <br />6° 57� <br />104•:43 <br />60 <br />100.00 <br />3 35 <br />4018 <br />5 44 <br />7"1I1 <br />104.72 <br />r <br />CURVE FORMULAS - <br />T= R tan j I R= T cot. a I chord$ <br />5o tan I 2 Chord def. = R <br />_ T Sin. } D R = 50 <br />Sin. JD = 5° Sin. � D No. chords = I <br />R E = R ex. see D <br />= 5° tT I <br />Sn. 0 E = T tan j 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 <br />early:To find angle for a given distance and deflection. <br />Rule I. Multiply the given distance by .01745 (def, 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 />thegiven distance. <br />To find deflection for a given angle and'distance: Multiply the angle <br />by .oi745, 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. 10.102_200=.5. 100-}-.5=100.5 hyp. a <br />Given 'Hyp: zoo, Alt. 25.252+200=3.I25-100-3-125-96.-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'.1I, and divide by 7. <br />LEVELING. The correction for curvature and ;refraction, -in feet <br />and decimals of feet is equal to 0.574 da, where d is the distance -in miles. <br />The correction for curvature alone is closely, fd2. The combined cor- <br />rection is negative. <br />PROBABLE ERROR. If dl, d2, da, etc. are the "discrepancies of various <br />results from the mean, and if Eda=the sum of the squares of these diflor- <br />ences and n -the number of observations; then the.probable error of, the, <br />mean= FITS - <br />t 0.674 n n 1) <br />SOLAR EPHEMERIS. Attention is called to the Solar- Ephemefiif& <br />the current year, published by Keuffel & Esser Co., and furnished free of <br />charge upon request; which is 3ix5j 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 <br />in Decimals of a Degree. <br />1' <br />0167' <br />11' <br />.18 33 <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 />;8 <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 />14000, <br />34 <br />.5667. <br />44 <br />.7333 <br />54 <br />.9000 <br />b <br />.0833'-15 <br />5 6 <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 <br />.1333 <br />18 <br />.3000. <br />28 <br />:4667• <br />38 <br />.6333 <br />48 <br />.8000 <br />58 <br />.9667 <br />0, <br />:1500 <br />19 <br />.3167 <br />29 <br />.4833' <br />39 <br />49 <br />59 <br />.9833 <br />:'10' <br />.1667•:20. <br />.3333 <br />30 <br />-.5000,11-40, <br />.6500 <br />1 .666711 <br />50 <br />.8167 <br />1 -.833311 <br />60 <br />11.0000 <br />i <br />- <br />-TasLE V. -' <br />Inclies in Decimals -of -a -Foot. <br />7 <br />1 -le <br />3 82 <br />3j 3-16 <br />YA 5-16 <br />% <br />Ma <br />%1.8333 <br />v <br />0052 <br />.0078 <br />.0104 .0166' <br />:0208 .0260 <br />.0313 <br />.0417 <br />.0521.0 <br />1 <br />2 <br />3 4 <br />5 6 <br />7 <br />8 <br />911 <br />.083.4,...1667- <br />.2500 .3333 <br />.4167. .5000 <br />..5833 -.6667. <br />..7500.9167 <br />