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VIII <br />TABLE II. - Radii, Ordinates and Deflections. Chord =100 ft. <br />Deg. <br />Radius - <br />milt <br />Ord. <br />Tan. <br />Dist. <br />Def. <br />Dist. <br />Def. <br />for <br />1 Ft <br />- <br />Deg.' <br />Radius <br />Mid. <br />Ord_, <br />Tan.' <br />Dist. <br />Def. <br />Dist. <br />Def. <br />fa <br />I r <br />Ft <br />2P �7' <br />ft. <br />ft. <br />ft. <br />ft. <br />! <br />1° 59' <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 />819.0 <br />1.528 <br />6.105 <br />12.21 <br />2.10 <br />20 <br />17189. <br />.073.291 <br />2° 20 <br />.5S2 <br />0.10 <br />20' <br />781.8 <br />1.600 <br />6.395 <br />12.79 <br />2.20 <br />.30.11459. <br />40 54' <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 />6.635 <br />13.37 <br />2.30 <br />•50 1 <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 />7.260 <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 />674.7.1.855 <br />5° 23� <br />7.411 <br />14.52 <br />2.55 <br />20 <br />4297.3 <br />.291 <br />1.1G4 <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 />2.613 <br />0.45 <br />9 <br />637.3 <br />1.065 <br />7.846 <br />15.69 <br />2.70 <br />40 <br />3437.9 <br />.304 <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..5.; <br />30 <br />G03.8 <br />2:074 <br />8.281 <br />16.56 <br />2.85 <br />2 <br />2864.9 <br />`.430 <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 />1.591 <br />3.781 <br />0:GS <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 />1.654 <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 />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 />2.903 <br />5.817 <br />1.00 <br />130 <br />425.4 <br />2.049 <br />11.7.5 <br />23.51. <br />4.05 <br />30 <br />1637.3 <br />.76-1 <br />3.054 <br />6.105 <br />1.05 <br />14 <br />410.3 <br />3:055 <br />12.18 <br />21.37 <br />4.20 <br />40 <br />1562.9 <br />.800 <br />3.199 <br />6.393 <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 />6.689 <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 />G.080 <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 />..909.3.635 <br />,7.271 <br />1.25 <br />16 <br />359.3 <br />3.496 <br />13.92 <br />27.84 <br />4.S0 <br />20 <br />'1322.5 <br />.945 <br />3.715 <br />7.561 <br />1.30 <br />30 <br />348.5 <br />3.606 <br />14.35,•,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 />3.710 <br />14.78 <br />29.5G <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 />4.15516.51' <br />33.01 <br />5.70 <br />.5 <br />1146.3 <br />1.091 <br />4.362 <br />8.724 <br />1.50 <br />20 <br />2S7.9 <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 />36.4.1 <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 />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 />39.57 <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 />952.6 <br />1.273 <br />5.083 <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 />9:15.4 <br />1.309 <br />5.234 <br />10.47 <br />1.80 <br />2G <br />222.3 <br />5.697 <br />22.50, <br />44.99 <br />7.S0 <br />10 <br />029.6 <br />1.346 <br />5.379 <br />10.76 <br />1.55 <br />21 <br />214.2 <br />5.918 <br />23.35 <br />46.69 <br />S.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 />48.33 <br />S.40 <br />30 <br />881.9 <br />1.418 <br />5.669 <br />11.34 <br />1.95 <br />29 <br />199.7.6.360 <br />25.04 <br />50.07 <br />S.70 <br />40 <br />859.0 <br />1.4.55 <br />5.814 <br />11.63 <br />2.00 <br />30 <br />103.2 <br />6.583 <br />25.83 <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 />Degred <br />of . <br />Curve <br />Radius <br />50 <br />sub chord <br />It =sin of 1 def. angle <br />Length <br />of arc <br />for 100 ft. <br />sin.1 def. ang.12.5 <br />Ft. <br />IS Ft. <br />20 Ft. <br />25 Ft. <br />•30' - <br />193.18- <br />10 51' <br />2P �7' <br />2° 58 <br />30 43' <br />101.15 <br />320 <br />181.39 <br />1° 59' <br />2° 25' <br />3° 1o' <br />3° 58' <br />101.33 <br />34° <br />171.01 <br />-.2' -06' <br />2° 33' <br />30 21' <br />4° 12' <br />I01.48 <br />36° <br />161.80 <br />2° 13 <br />2' 41' <br />3° 33' <br />4° z6' <br />1o1.66 <br />3g° <br />153';58- <br />2° 20 <br />2° 49�- <br />3°,44' <br />4° 40, <br />101.85 <br />40° <br />146.19 <br />2° 27'. <br />2° 57`� <br />3° 55' <br />40 54' <br />IO2.o6 <br />42' <br />139.52 <br />2° 34' <br />3° 05' <br />4° 07= <br />5° 08' <br />102.29 <br />440 <br />133.47 <br />2041 ' <br />3° 13' <br />4° 18' <br />5° 22' <br />102.53 <br />460 <br />127:97 <br />2° 48' <br />3° 21' <br />4° 29' <br />S° 36' <br />102.76 <br />.48° <br />1222.92 <br />2° 55'. <br />3° 29' <br />4° 40' <br />5° 50, <br />103.00 <br />50°. <br />118.31 <br />3° oz'. <br />3° 38 <br />4° 51' <br />6° 04' <br />103.24 <br />52° <br />114.o6 <br />3° o9' <br />3° 46' <br />S° 02' <br />6°.i7' <br />103.54 <br />54° <br />IIo. I I <br />3° 16 <br />3° 54 <br />5° 13 <br />6031f <br />103.84 <br />56° <br />1o6.50 <br />3° 22' <br />4002 1 <br />5° 23� <br />60 44' <br />104.. 14 <br />58° <br />103.14 <br />3° z9' <br />4° 10' <br />5° 34 <br />6° 57' <br />104.43 <br />6o° <br />Ioo.,00 <br />3° 35' .. <br />4°,.18' <br />. 50 44' <br />79 11'L124.72 <br />IX <br />CURVE- FORMULAS <br />T= R tan I R= T cot. ; Ichord2 <br />50 tan 8 I 2 . Chord def. =. <br />T _Sin..} D R 50 R <br />= <br />50 Sin. D... I <br />Sin. D No. chords = - <br />R E=Rex,scc I D <br />Sin. D = 50 tT I r -T tan 8 I' . 'Tan. def. _ Ie 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 .04745 (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 .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 too, Alt. Io.102-200=.5: 100+.5=ioo.5 hyp. <br />Given Hyp. Ioo,Alt.25.252=200=3.125.ICb-3.125=96.875=Base. <br />1 Error in first example, .002; in Iast, .o45. <br />To find Tons of Rail in one mile of track:. multiply weight per yard <br />by I I, and divide by 7. <br />LLVELING. The correction for curvature and refraction, is 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, id'. 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 fd2=the sum of the squares of these differ- <br />ences and li=the number of observations, then the probable error of tho <br />mean= + 0.6745 fid' <br />11(n-1) <br />SOLAR EPHEMERIS. Attention is called to the Solar Ephemeris for <br />the current year,, published by Keuffel & Esser Co., and furnished free of <br />charge upon request, which is 31x58 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 Deeirnal.3 of a Dezree. <br />1' <br />.0167 <br />11' <br />.1833' <br />21' <br />.3500 <br />31' <br />.51G7 <br />41' <br />.6833 <br />51' <br />8500 <br />2 <br />.0333 <br />12.2000 <br />22 <br />.3607 <br />32 <br />.5333 <br />42 <br />.7000 <br />52 <br />SG67 <br />G <br />.0500 <br />13 <br />.2167 <br />23 <br />.3333 <br />33 <br />.5501D <br />43 <br />.7IG7 <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 />.9000 <br />5 <br />.0833 <br />16 <br />.2500 <br />25 <br />.4167 <br />35 <br />.55:53 <br />45 <br />.7500 <br />55 <br />.9167 <br />8 <br />.1000 <br />111 <br />2667 <br />211 <br />4333. <br />30 <br />6000411 <br />7667 <br />611 <br />9333 <br />7 <br />.1167 <br />17 <br />.2833 <br />27 <br />.4500' <br />37 <br />G1 r7 <br />47 <br />.7833 <br />57 <br />.9500 <br />S <br />.1333 <br />1S <br />3000 <br />28 <br />.4667 <br />38 <br />.6333 <br />4S <br />S000 <br />58 <br />.9667 <br />9 <br />.1500 <br />19 <br />.8167 <br />20 <br />.4833 <br />30 <br />.6500 <br />49 <br />.8167 <br />59 <br />.9833 <br />10 <br />.1661 <br />20,k <br />3333 <br />130 <br />500HO <br />1 40 <br />1 .6Gt7 11 <br />50 <br />1 .833311 <br />GO <br />11.0000 <br />TABLE V. = <br />Inches iwDeeimals of a Foot. <br />1-111 <br />3 32 . <br />Y. <br />3 -1G <br />'; <br />5-16 <br />% <br />i9 <br />4 <br />0032 <br />.0078 <br />.0101 <br />.0156 <br />.0208 <br />.0260 <br />.0318 <br />.0117 .0521 <br />.0625 <br />.0729 <br />1 <br />2 <br />3. <br />4b <br />G <br />? <br />S 9 <br />10 <br />11 <br />.0833 <br />.18117 <br />.2500 <br />.3333 <br />.4167 <br />.5000 <br />.5833 <br />.8867 .7000 <br />.5333 <br />.91167 <br />