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7M ; gg <br />TABLE]- IL -- Radii, Ordinates and Deflections. Chord=100 ft. CURVE FORMULAS <br />Deg' <br />Radius <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 I <br />Dist. <br />Def. <br />Dist. <br />Def. <br />for <br />1 rt. <br />193.18 <br />ft. <br />ft. <br />ft. <br />ft.. <br />1 <br />32° <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 />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 />.552 <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 <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 />2864.9.436 <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 />13.781 <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 />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 .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 />1562.9.800 <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 />b <br />1432.7 <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 />_.873 <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"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 />1.200.4.798 <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 />1.2735.088 <br />10.18 <br />1.75 <br />25 <br />231.0 <br />5.476 <br />21 .64 <br />8 <br />43.27.50 <br />4 " <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 />44.92 <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 />1 <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 egnal 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 IIL - Deflections for Sub Chords for Short Radius Curves. <br />Degree <br />of <br />Radius <br />50 <br />sub chord = sin of I def. angle <br />R <br />Length <br />of arc <br />Curve <br />sin. l def. ang• <br />12.5 Ft. <br />15 Ft. <br />20 Ft. <br />25 Ft. <br />for 100 ft. <br />..300.., <br />193.18 <br />,I° 51' <br />z°-17' <br />z° 58.1- <br />3° 43' - <br />I01.15 <br />32° <br />181.39 <br />1°59' <br />2°25' <br />3° 10' <br />3058' <br />101.33 <br />34° <br />I'71.OI <br />2° 06' <br />2° 33' <br />3° 2I' <br />4° I2' <br />101.48 <br />360 <br />161.8o <br />2° �3' . <br />2° 41'"• <br />3° 33'. <br />4° 26' <br />ioI.66 <br />380 <br />153.58. <br />2° 20' <br />z° 49' <br />3° 44'- <br />4° 40' • <br />101.85 <br />40° <br />146.19 <br />2° 27' <br />-z° 57' <br />3° 55' <br />4° 54' <br />102. o6 <br />4z' <br />'139.- 52 <br />'-° 34' <br />3° 05' <br />4° 07' <br />5° 08' <br />102.29 <br />44° <br />133.47 <br />2°.41' <br />30 13' <br />4° 18' <br />5°'22' <br />102.53 <br />46° <br />127.97" <br />2° 48' <br />3° 21' <br />4°.29 <br />5° 36' <br />102.76 <br />48° <br />122.92 <br />2°.55, <br />3029, <br />4°40, <br />5°50' <br />103.00 <br />50° <br />I1S.31 <br />3° 02 <br />3° 38' <br />4° 51 . <br />60 04' <br />103.24 <br />52° ' <br />114.06. <br />3° 09' <br />3° 46' <br />S° 02' <br />6° 17 <br />103.54 <br />540 <br />"110.11 <br />3° 16' <br />3' 54 <br />5° 13 <br />6031 <br />.103:84 <br />.560 <br />1o6.5o <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°Io' <br />5° 34' <br />6°57' <br />104.43 <br />6o° <br />Ioo.00 <br />3°35' <br />4°18' <br />5.°44' <br />7°ii'- <br />1 104.72 <br />T= R tan � I R= T cot. 2 I chord$ <br />_ 50 tan i I Chord def. _ <br />T Sin. J ll 50 R <br />R = Sin. a <br />Sin. } D = 5o 2 D No. chords'= I <br />R E= R ex. sec a I D <br />Sin. J D = 5o tan a I E = T tan J I Tan. def.= ; 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 (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 />the given distance., <br />To find deflection for a given angle and distance. Multiply the angle <br />by .o1745; grid the product by the distance. <br />GENERAL DATA <br />RIGHT ANGLE TRIANGLES. Square the altittide., divide by twice •the <br />base. Add quotient to base for hypotenuse. . <br />Given Base ioo,Alt:16.102-'200=.5. 100+-5=100.5 hyp. <br />Given Hyp: Ioo,Alt.25.252_200=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 d2, where d is the distance in miles. <br />The: correction for curvature alone is closely, Jd?. The combined cor- <br />rection is negative. <br />PkOBABLE ERROR. If dt, d,, da, etc. are the discrepancies of various <br />results from the mean, and.lf 7-d2=the sum of the squares of these difor- <br />ences and it=the number•of observations, then the probable error of the. <br />mean= 0.67401-Vn (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 3}x58 in., with about 90 pages of datawery <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 andLogarithmic"Functions; <br />and Logarithms of Numbers. <br />TABLE IV. - Minutes in Decimals of a DP.PrP.{P- <br />1' <br />.0167 <br />11' <br />.1833 <br />21' <br />.3500 <br />31' <br />.5167 <br />41' <br />.6833 <br />51' <br />5-16 <br />2 <br />.0333 <br />12 <br />.2000 <br />22 <br />.3667 <br />32 <br />.5333 <br />42 <br />.7000 <br />52 <br />.8500 <br />8 <br />.0500 <br />13 ..2167 <br />2 <br />23 <br />.3833 <br />33 <br />.5500 <br />43 - <br />.7167 <br />53 <br />.8667 <br />4 <br />.0667 <br />14 <br />.2333 <br />24 <br />.4000 <br />34 <br />.5667 <br />44 <br />.7333 <br />54 <br />.8833 <br />5 <br />.0833 <br />15 <br />.2500 <br />25 <br />.4167 <br />35 <br />.5833 <br />45 <br />.7500 <br />55 <br />.9000 <br />.9167 <br />01000 <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 />1S <br />.3000 <br />28 <br />.4667 <br />38 <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 />.1667 11 <br />20 <br />1 .3333 1130 <br />1 .5000 <br />11 40 <br />1 .6667 11 <br />50 <br />1 .833311 <br />60 <br />11.0000 <br />TABLE V. - <br />Inches in Decimals of a Foot. <br />t <br />1-16 <br />3-82 <br />Y <br />3-16 <br />Y <br />5-16 <br />'/s <br />1 <br />.0052 <br />.0078 <br />.0104 <br />.0156 <br />.0208 <br />.0260 <br />.0313 <br />.0417 A521 1 <br />.0625 <br />0729 <br />1 <br />2 <br />3 <br />4 <br />5 <br />6 <br />7 <br />8 9 <br />10 <br />11 <br />.0833 <br />.1667 <br />.2500 <br />..3333 <br />.4167 1 <br />.5000 <br />.5833 <br />.6667 .7500 <br />.8333 1 <br />.9167 <br />