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VIII <br />TABLE IL - Radii, Ordiniites and Deflections. Chord =100 ft. <br />Deg. <br />Radius . <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />Def. <br />Dist. <br />I lief' <br />T <br />Deg. <br />Radius <br />Mid.t. <br />Ord. <br />Tan ' <br />Dist. <br />Def. <br />Dist. <br />Def* <br />for <br />l Ft. <br />2 � -17' <br />t. <br />ft. <br />t. <br />ft. <br />r <br />1° 59' <br />ft. <br />I 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 <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 />.43G <br />.873 <br />0.15 <br />30 <br />764.5 <br />1.637 <br />6.540 <br />13.03 <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 <br />6875.5 <br />.182 <br />.727 <br />1.454 <br />0.25 <br />8 <br />716.8 <br />1.740 <br />6.976 <br />13.95 <br />2.40 <br />1 <br />5720.6 <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 />1.013 <br />2.036 <br />0.35 <br />30 <br />674.7.1.855 <br />5° 23' <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 />601.7 <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 />0.4.5 <br />9 <br />637.3 <br />1.965 <br />7.846 <br />15.69 <br />2.70 <br />40 <br />3437.0 <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 />22864.9 <br />.436 <br />1.745 <br />3.490 <br />0. GO <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 />3.751 <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 />4.303 <br />0.75 <br />11 <br />521.7 <br />2.402 <br />.9.585 <br />19.10 <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 />20.01 <br />3.45 <br />50 <br />2022.4 <br />.615 <br />2.472 <br />4.945 <br />0.65 <br />1; <br />478.3 <br />2.620 <br />10.45' <br />20.913.60 <br />8 • <br />1910.1 <br />.G55 <br />2.615 <br />5.235 <br />0.00 <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 />0.95 <br />13 <br />441.7 <br />2.839 <br />11.32 <br />22.6.1 <br />3.90 <br />20 <br />1719.1 <br />.727 <br />2.903 <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 />6.393 <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 />6.659 <br />1.15 <br />15 <br />383.1 <br />3.277 <br />13.05 <br />26.11 <br />4.50 <br />a <br />'1432.7 <br />.S73 <br />3.490 <br />6.950 <br />1.20 <br />30 <br />370.8 <br />3.357 <br />13.49 <br />20.97 <br />4.65 <br />10 <br />1375.4 <br />.909 <br />3.G35 <br />7.271 <br />1.25 <br />16 <br />359.3 <br />3.406 <br />13.92 <br />27.84 <br />4.80 <br />20 <br />1322.5 <br />.945 <br />3.719 <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 />.082 <br />3.9265 <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 />3.035 <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 />61146.3 <br />1.091 <br />4.362 <br />' 8.724 <br />1.50 <br />20 <br />287.9 <br />4.374 <br />17.37 <br />3-4.73 <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 />33.16 <br />6.60 <br />30 <br />1042.1 <br />1.200 <br />4.793 <br />9.596 <br />1.65 <br />23 <br />250.8 <br />5.035 <br />19.94 <br />39.87 <br />G.90 <br />40 <br />1011.5 <br />1.237 <br />4.043 <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 />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 />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 />4G. G9 <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 />4S.33 <br />8.40 <br />30 <br />881.9 <br />1.418 <br />5.669 <br />11.34 <br />1.95 <br />29 <br />109.7 <br />6.360 <br />25.04 <br />50.078.70 <br />40 <br />1 859.9 <br />1.455 <br />5.814 <br />11.63 <br />2.00 <br />30 <br />193.2 <br />6.553 <br />25.58 <br />51.76 <br />9.00 <br />Tho 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 .0012X90OX2.183 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 subchord <br />R =sin of 's def. angle <br />of Length <br />arc <br />for 100 ft. <br />sin. l 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 />2' 58' <br />3' 43'- <br />to f .15 <br />32° <br />181.39 <br />1° 59' <br />2° 25' <br />3° 10' <br />30 58' <br />101.33 <br />34' <br />171.01 <br />2° 06', <br />2° 33'3' <br />21' <br />4' 12' <br />f0I.48 <br />-•36° <br />161.8o <br />2° 13' <br />20 41' <br />3' 33' <br />4' 26' <br />io f . 66 <br />38° <br />153.58 <br />2° 20' <br />20 49' <br />3° 44' <br />4' 40'101.85 <br />base. Add quotient to base for hypotenuse. <br />400 <br />146.ig <br />20 27' <br />2° 57' <br />30 55' <br />4' 541 <br />102.o6 <br />42' <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 />3' 13' <br />4' 18' <br />5' 22' <br />.102.53 <br />46° <br />127.97 <br />2° 48' <br />3° 21' <br />40 29' <br />50 36' <br />102.76 <br />48° <br />122.92 <br />z° 55' <br />3' 29' <br />4' 40' <br />5' 50' <br />103.00 <br />50° <br />118.31 <br />3°02' <br />3°38 <br />4°51' <br />6°04' <br />1o3.24 <br />52° <br />114. o6 <br />3°09 <br />3046' <br />5002' <br />6°i7' <br />103.54 <br />540 <br />IIo.II <br />3° 16' <br />3° 54' <br />5' 13' <br />60311 <br />103.84 <br />56° <br />1o6.5o <br />3° 22' <br />40 02' <br />5° 23' <br />6°44' <br />I04.14 <br />58° <br />103.14 <br />3' 29' <br />4° 10' <br />5° 34' <br />6' 57' <br />104.43 <br />60° <br />Ioo.00 <br />-3' 35 <br />4' 18' <br />5° 44" <br />70 11' <br />104.72 <br />\b1 <br />a (p4 CURVE FORMULA -S IX <br />T= R tan 1 <br />50 tan I <br />. <br />R= T cot ' I <br />chord2 <br />Chord def. = <br />.1833 <br />,I, _ <br />Sin. J D <br />R = 50 <br />12 <br />.51G7 <br />Sin. J' D = 5o <br />1 <br />Sin. , D <br />No. chords = I <br />.8500 <br />R <br />E= R ex. sec 12 I <br />D <br />.2000 <br />,Sia. JD = 5o t1n a I <br />E = T tan j I <br />Tan. def. _ ; chord def. <br />.5333 <br />The'square of any distance, divided by twice <br />the radius, will equal <br />52 <br />the distance from tangent to curve, very nearly. <br />3 <br />- To find angle for a given distance and deflection. <br />13 <br />Rule I.. Multiply the given distance by .01745 (def. for I0 for f ft. <br />23 <br />see Table II.), and divide given deflection by the product. <br />33 <br />Rule 2. Multiply given deflection by 57.3, and divide the product by <br />43 <br />the given distance. <br />53 <br />To find deflection for a given angle and distance. Multiply the angle <br />4 <br />by .01745, and the product by the distance. <br />14 <br />GENERAL DATA <br />24 <br />• RIGHT ANGLE TRIANGLES. Square the al bt'ude, divide by twice the <br />34 <br />base. Add quotient to base for hypotenuse. <br />44 <br />Given Base ioo, Alt. 10.102_200=.5. 100-}-.5=1oo.5 hyp• <br />.54 <br />Given Hyp., ioo, Alt. 25.252-200=3.125. 100-3.125=96.875 =Base. <br />5 <br />Error in first example, .cot; In last, .045- <br />15 <br />To find Tons of Rail in one mile of track: multiply tiveight per yard <br />25 <br />by I I, and divide by 7. <br />35 <br />LEVELING. The correction' for curvature and refraction, in feet <br />45 <br />and decimals of feet is equal to 0.574 da, when d is the distance in miles. <br />55 <br />The correction for curvature alone is closely, jd2. The combined cor- <br />6 <br />rection is negative. I - <br />t <br />PROBABLE ERROR. If d,, da, da, etc. are the discrepancies of various <br />, <br />results from the mean, and if 2:d2=the sum 0f the squares of these differ- <br />rr <br />ences and n=the number of observations then therobable error of the <br />p <br />(\ <br />�0 1, <br />a <br />mean= X0.6746 <br />56.9333 <br />n�n 1) <br />�~f'�� <br />SOLAR EPHEMERIS. Attention is'ealled to the Solar Ephemeris for <br />° <br />the current year, published by Kcuffel & Esser Co., and furnished free of <br />1 f; <br />charge upon request, which is 3}x5j- in., with about 90 pages of data very <br />J <br />useful to the Surveyor; such as the adjustments of transits, levels and <br />47 <br />solar attachments; directions and tables for determining the meridian <br />the latitude from <br />57 <br />and observations on the sun and Polaris; stadia meas- <br />n <br />urements; magnetic declination; arithmetic constants; English and Metrie <br />18 <br />conversions; trigonometric formulas; Naturall and Logarithmic Functions; <br />G b' <br />and Logarithms of Numbers. <br />39 <br />TABLE IV. - Minutes in Decimals of a Degree. . <br />48 <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 <br />.2000 <br />22 <br />.3667 <br />32 <br />.5333 <br />42 <br />.7000 <br />52 <br />.8667 <br />3 <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 />.0333 <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 />is <br />.2667 <br />26 <br />.4333 <br />39 <br />.6000 <br />46 <br />.7667 <br />56.9333 <br />7 <br />.1167 <br />17 <br />.2833 <br />27 <br />.4500 <br />37 <br />.,6167 <br />47 <br />.7S33 <br />57 <br />.0500 <br />8 <br />.1333 <br />18 <br />.3000 <br />28 <br />.4667 <br />39 <br />.,6333 <br />48 <br />.8000 <br />58.9667 <br />9 <br />.1500 <br />19 <br />.3167 <br />29 <br />.4833 <br />39 <br />.6500 <br />49 <br />.5167 <br />59 <br />.9833 <br />]0 <br />.1667 <br />20 <br />.3333 <br />30' <br />1 .5000 11 <br />40 • <br />.566711 <br />50 <br />1 .833311 <br />GO <br />1.0000 <br />TABLE V. - Inches in Decimals of a Foot. <br />1.16 <br />3-32 <br />Y8 <br />3-16 <br />Y, <br />5-16 <br />", j <br />.0052 <br />.0078 <br />.0104 <br />.0156 <br />.0203 <br />.0_GO <br />.0313 <br />.0417 .0521 <br />.0625 <br />.0729 <br />1 <br />2 <br />' 3 <br />4 <br />b <br />1 6 <br />7 <br />8 9 <br />10 <br />11 <br />.0833 <br />.1667 <br />.2500 <br />.3333 <br />.4167 <br />.5000 <br />.5933 <br />.6667 .7500 <br />.8333 <br />.9167 <br />