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r <br />VIII <br />TABLE II. - Radii, Ordinates and Deflections: Chord =100 ft. <br />Deg. <br />Radius <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />Dd. <br />Dist. <br />lief, <br />for <br />I Ft. <br />Deo. <br />Radius <br />Mid. <br />0-d. <br />Tan <br />' D st: <br />Def. <br />Dist. <br />Def. <br />for <br />7 Ft. <br />2° 17' <br />it. <br />ft. <br />ft. <br />ft. <br />I <br />1° 59' <br />ft. <br />ft. <br />ft, <br />it. <br />34° <br />0'10' <br />34377. <br />.036 <br />.145 <br />.291 <br />0.05 <br />7' <br />319.0 <br />1.52S <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 />73t.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 />7frt.5 <br />1.637 <br />6.540 <br />13.08 <br />2.25 <br />40 <br />8504.4 <br />.145 <br />.5S2 <br />1.164 <br />0.20 <br />40 <br />747.9 <br />t.673 <br />6.655 <br />13.37 <br />2.30 <br />50 <br />GS75.5 <br />.1S2 <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 />638.2 <br />1.819 <br />7.266 <br />14.53 <br />2.50 <br />10 <br />4911.2 <br />.255 <br />1.018 <br />2.03G <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 />3S19.8 <br />.327 <br />1.309 <br />2.613 <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 />614.0 <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 />16.8.5 <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 />2.153 <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 />51G.4 <br />2.292 <br />9.150 <br />18.30 <br />3.15 <br />30 <br />2292.0 <br />.54:5 <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 />19. <br />478.3 <br />2:620 <br />10.45 <br />20.91.3.60 <br />3 <br />1910.1 <br />.655 <br />2. GIS <br />5.235 <br />0.90 <br />30 <br />459.3 <br />2.730 <br />10.59 <br />21.77 <br />3.75 <br />10 <br />1809.6 <br />.691 <br />2.763 <br />5.526 <br />0.93 <br />13 <br />441.7 <br />3.539 <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 />.7G4 <br />3.054 <br />6.10S <br />1.05 <br />14 <br />410.3 <br />3.058 <br />12.19 <br />24.37 <br />4.20 <br />40 <br />1662.9 <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 />6.659 <br />1.15 <br />15 <br />333.1 <br />3.277 <br />13.05 <br />265.11 <br />4.50 <br />4 <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 />26.07 <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 />3.490 <br />13.92 <br />2'7.84 <br />4.80 <br />20 <br />1322.5 <br />.045 <br />3.718 <br />7.5651 <br />1.."-.0 <br />30 <br />'348.5 <br />3.606 <br />14.35 <br />28.70 <br />4.95 <br />30 <br />1273.6 <br />.952 <br />3.926 <br />7.852 <br />].3 <br />17 <br />338.3 <br />3.716 <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 />3L.29 <br />5.40 <br />50 <br />1155.8 <br />1.055 <br />4.217 <br />8.433.1.45 <br />19 <br />302.9 <br />4.15.5 <br />76.51 <br />33.01 <br />5.70 <br />S <br />1146,3 <br />L 091 <br />4.362 <br />8.724 <br />1.50 <br />20 <br />287.9 <br />4.374 <br />17.37 <br />34.78 <br />6.00 <br />10 <br />1109.3 <br />1.127 <br />4.507 <br />9.014 <br />1.5.5 <br />21 <br />274.4 <br />4.594 <br />18.22 <br />06.44 <br />0.30 <br />20 <br />1074.7 <br />1.161 <br />4.653 <br />9.305 <br />1.60 <br />2? <br />262.0 <br />4.814 <br />19.03 <br />38.16 <br />6:60 <br />30 <br />1042.1 <br />1.200 <br />4.793 <br />9.596 <br />1.6:; <br />23 <br />250.8 <br />5.035 <br />19.94 <br />39.87 <br />6.00 <br />40 <br />1011.5 <br />1.237 <br />4.943 <br />9.83G'1.70 <br />24 <br />240.5 <br />0.255 <br />20.79 <br />41.53 <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 />43.25 <br />7.50 <br />G <br />935.4 <br />1.309 <br />5.234 <br />10.47 <br />1.80 <br />26 <br />222.3 <br />5:697 <br />22.50 <br />44.09 <br />7.80 <br />10 <br />929A <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 />1.332 <br />5.524 <br />11.05 <br />1.90 <br />2S <br />206.7 <br />6.139 <br />24.19 <br />48.35 <br />8.40 <br />30 <br />831.9 <br />1.415 <br />5.669 <br />11.34 <br />1.95 <br />2-3 <br />199.7 <br />6.360 <br />25.04 <br />50.07 <br />8.70 <br />40 <br />1 559.9 <br />1.455 <br />5.814 <br />11.63 <br />12.00 <br />1 30 <br />111)3.216.583125.88 <br />151.7619.00 <br />The middle ordinate in inches for any cord of length (C) is equal to .0012 U' <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 />�4, sub chord - sin of I def. angle <br />R <br />Length <br />of a <br />for 100 it. <br />sin. ldef. 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 />101.15 <br />32° <br />131.39 <br />1° 59' <br />20 �5' <br />3' Io' <br />3° 58' <br />101.33 <br />34° <br />I7I.0I <br />2° 06 <br />3°33 <br />3° 2I' <br />Ao 12' <br />I0I.4$ <br />36° <br />161.8o <br />z° 13' <br />2° 41' <br />3° 33'}° <br />26' <br />ioi.66 <br />38' <br />153.58 <br />2° 20' <br />2° 49' <br />3° 44' <br />4° 40' <br />101.85 <br />40° <br />146: i9 <br />2° 27 <br />2' 57' <br />3° 55' <br />4' 54' <br />102.o6 <br />42 <br />139.52 <br />20 34' <br />3' 05 <br />4°'07' <br />5° 08 <br />102.29 <br />440 <br />133.47 <br />2° 41' <br />3° 13' <br />4' 18' <br />5° 22' <br />102.53 <br />46° <br />127.97 <br />2° 48 <br />3° 21' <br />4o 29' <br />5° 36' <br />102.76 <br />48' <br />122.92 <br />2' 55' <br />3° 29' <br />4° 4c' <br />5° 50' <br />103.00 <br />50°118.31 <br />47 <br />3° 02' <br />3' 38' <br />40 51' <br />6° 04' <br />103.24 <br />52° <br />114. cG <br />3° 09' <br />3° 4G' <br />5° 02' <br />6° 17 <br />103.54 <br />54° <br />IIO. If <br />30 16', <br />3° 54' <br />S° 13'6° <br />31' <br />I03.84 <br />56° <br />1o6.5o <br />3° 22' <br />4° 02' <br />5° 23' <br />6° 47 <br />104-14 <br />58° <br />103-14 <br />3° 29' <br />4' I0' <br />5' 34' <br />6°5i' <br />104.43 <br />.60°• <br />100.00 <br />3°35' <br />4°18 <br />5°44' <br />7°11 <br />104.72 <br />Ili <br />CURVE FORMULAS <br />T _ 5o tan � I R = T cot. b I Chord def. =chord$ <br />T Sin. D R = 50 R <br />Sin.4 i D - 50 Sin.: D No. chords = I <br />R E= R ex. sec l I D <br />_ 5o tan ; I <br />Sin. � D - E =T tan f 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 .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 -Ioo,Alt."IO.Io'-_200=.5.,I0O+-5=100.5 hyp. <br />Given Hyp.'ioo,Alt.25.25'=zoo=3.I25, 100-3.125=96.875=ease. <br />Error in first example, .002; to last, .045. <br />To find Tons of Rail in one mile of track: multiply weight per yard" <br />by it, and divide by 7. <br />LEVELING. The correction for curvature and refraction, in feet <br />and decimals of feetis equal to 0.574da, where d is the distance in miles. <br />The correction for curvature alone is closely, 3d°. The combined cor- <br />rection is negative. <br />PROBABLE ERROR. 'If d , d2, do, 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= 0 G745' oda <br />1 n (n-1) , <br />SOLAR EPHEDIERIS. 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 -q1 -X5-.1 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 Ineas- <br />urements; magnetic declination; arithmetic constants; English and Metric <br />conversions; trigonometric f or-nulas; Natural and Logarithmic Functions; <br />and Logarithms of Numbers. <br />TABLE IV. - TWinutes'in. Decimals of a Deg-ree. <br />1' <br />0167 <br />11' <br />1833 <br />21' <br />.3500 <br />31' <br />.5167 <br />41' <br />.6533 <br />51' <br />3500 <br />2 <br />.0333 <br />12 <br />.2000 <br />22 <br />.3667 <br />32 <br />.5333 <br />42 <br />.7000 <br />52 <br />.8067 <br />3 <br />.0500 <br />13 <br />.2167 <br />23 <br />.3333 <br />33 <br />.5500 <br />43 <br />.7167 <br />53 <br />.8533 <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 />.5sm <br />45 <br />.7500 <br />55 <br />9167 <br />6 <br />.1000 <br />16 . <br />.2667 <br />265 <br />.4333 <br />36 <br />.6000 <br />46 <br />.7667 <br />56 <br />.9333 <br />7 <br />.1167 <br />17 <br />.2333 <br />27 <br />.4500 <br />137 <br />.6167 <br />47 <br />.7833 <br />57 <br />.9500 <br />8 <br />.1333 <br />18 <br />.3000 <br />28 <br />.46167 <br />38 <br />.6333 <br />48 <br />.6000 <br />68 <br />.9607 <br />9 <br />.1500 <br />19 <br />.3167 <br />29 <br />.4533 <br />39 <br />.6500 <br />49 <br />.3167 <br />59 <br />.9833 <br />10 <br />.1607 1 <br />, 20 <br />1 .3333 <br />130 <br />1 .5000 <br />11 40 <br />1 .6667 11 <br />50 <br />1 .8333 11 <br />60 <br />1.0000 <br />TABLE V. - <br />Inches in Deciinals of a Foot. <br />1-1G <br />% <br />.0051 <br />.007$ .0104 <br />.0156 <br />.0205 <br />.0260 <br />.0313 <br />.0417 <br />.0:,21_ <br />.0625 <br />.0729 <br />1 <br />3 <br />4 <br />5 <br />G <br />7 <br />8 <br />9 <br />10 <br />11 <br />0833..:1667 <br />.2500 <br />.3333 <br />.4107 <br />.5000 <br />.5333 <br />.6667 <br />.7500 <br />.8333 <br />.E1GT <br />