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�-�VIII IX <br />Tnnit•; II. - Radii, Ordinates and Deflections. Chord =100 ft. CURVE FORMULAS <br />Deg. <br />Radius <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />Def. <br />I Dist. <br />Def. <br />for <br />1 Ft. <br />Deg. <br />Radius <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />. Def. <br />Dist. <br />Def. <br />for <br />1 Ft. <br />I° <br />t. <br />ft. <br />ft. <br />ft. <br />r <br />3° <br />ft.ft. <br />101.15 <br />ft, <br />ft. <br />1° <br />10' <br />34877. <br />.036 <br />:145 <br />.201 <br />0.05 <br />7° <br />819.0 <br />1.52S <br />6.105 <br />12.21 <br />2.10 <br />20 <br />17159. <br />.073 <br />.291 <br />.582 <br />0.10 <br />20' <br />781.8 <br />.1.000 <br />6.395 <br />12.79 <br />2.20 <br />30 <br />11450. <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 />6.685 <br />13.37 <br />2.30 <br />50 <br />6875.5 <br />.182 <br />.727 <br />1.454 <br />0.25 <br />8 <br />716.5 <br />1.746 <br />6.97G <br />13.95 <br />2.40 <br />1 <br />5729.6 <br />.218 <br />.873 <br />1.745 <br />0.30 <br />20 <br />G3S.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 <br />1.855 <br />7.411 <br />14.82 <br />2.55 <br />20 <br />4397.3 <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 />2.613 <br />0.45 <br />9 <br />G37.3 <br />1'.9G5 <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 />G14.6 <br />2.037 <br />8.136 <br />16.27 <br />2.50 <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 />8.231 <br />16.56 <br />2.85 <br />2 <br />2864:0 <br />.436 <br />1.745 <br />3.490 <br />0.60 <br />40. <br />593.4 <br />2.110 <br />8.426 <br />16. S5 <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 />546.4 <br />2.292 <br />9.150 <br />13.30 <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 />20.01 <br />3.45 <br />• 50 <br />2022.4 <br />.618 <br />2.472 <br />4.045 <br />0.8:, <br />12 <br />478.3 <br />2.620 <br />10.45 <br />20.91 <br />3.610 <br />8 <br />1910.1 <br />.655 <br />2.613 <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 />3.00 <br />20 <br />i719.1 <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 />6.103 <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 />6.39S <br />1.10 <br />30 <br />396.2 <br />3.163 <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 />6. <br />1432.7 <br />.873 <br />3.490 <br />G.9S0 <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 />3:496'13,92 <br />27.84 <br />4.S0 <br />20 <br />1322.5 <br />.945 <br />3.713' <br />7.561 <br />1.30• <br />30 <br />348.5' <br />3.606'14.35 <br />28.70 <br />4.95 <br />30 <br />1273.6 <br />.982 <br />3.92G <br />7.852 <br />1.35 <br />17 <br />338.3 <br />3.716 <br />14.78 <br />20.56 <br />5.10 <br />40 <br />1225.1 <br />1.018 <br />4.071 <br />8.143 <br />1.40 <br />1S <br />319.6 <br />3.935 <br />15.64 <br />31.29 <br />5.40 <br />50 <br />1155.8 <br />1.055 <br />4.217 <br />5.433 <br />1.45 <br />19 <br />302.9 <br />4.155 <br />16.51 <br />33.01 <br />5.'r0 <br />e <br />1146.3 <br />1.001 <br />4.362 <br />3.724 <br />1.50 <br />20 <br />257.9 <br />4.374 <br />17.37 <br />34.73 <br />G.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 />13.22. <br />35.44 <br />6.30 <br />20 <br />1074.`7 <br />1.164 <br />4.653 <br />0.305 <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 />4.79, <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 />24 0. 5 <br />.5.255 <br />20.79 <br />41.58 <br />7.20 <br />50 <br />032.6 <br />1.273 <br />5.088.10.18 <br />1.75 <br />25 <br />231.0 <br />5.476 <br />21.64 <br />43.23 <br />7.50 <br />6 <br />0+55.4 <br />1.309 <br />5.234 <br />10.47 <br />1.80 <br />26 <br />222.3 <br />5.607 <br />22.50 <br />44.09 <br />7.30 <br />10 <br />929.6 <br />1.346 <br />5.379 <br />10.76 <br />1.85 <br />27 <br />214.2 <br />.5.913 <br />23.35 <br />46.69 <br />S.Io <br />20 <br />905.1 <br />1.352 <br />5.524 <br />11..05 <br />1.00 <br />23 <br />206.7 <br />6.139 <br />24.19 <br />48.33 <br />SAO <br />30 <br />831.9 <br />1.41.8 <br />5.669 <br />11.34 <br />1.05 <br />29 <br />190.7 <br />6.360 <br />25.04 <br />50.07 <br />8.70 <br />40 <br />850.9 <br />1.455 <br />5.814 <br />11.63 <br />2.00 <br />30 <br />193.2 <br />0.533 <br />25.88 <br />51.76 <br />9.00 <br />The middle ordinate in inches for anv 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 bond a 30 ft. rail to fit a 10 degree curve, its middle ordinate should <br />be .001'2X900X2.183 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 />Yj sub chord = <br />It <br />sin of 'I def. angle <br />° g <br />Length <br />of arc <br />for 100 ft. <br />sin.'J def. ang. <br />12.5 Ft. <br />15 Ft. <br />20 Ft. <br />25 Ft. <br />3o° <br />193.18. <br />I° <br />51' <br />2° <br />17' <br />2° <br />5S' <br />3° <br />43' <br />101.15 <br />320 <br />181.39 <br />1° <br />59' <br />2°25' <br />13 <br />3°l0' <br />23 <br />3° <br />58' <br />101.33 <br />340 <br />171.01 <br />2° <br />o6' <br />z° <br />33' <br />3° <br />21 <br />4° <br />I2' <br />I01.43 <br />36° <br />161.8o <br />2° <br />13' <br />2° <br />41' <br />30 <br />33' <br />4° <br />26' <br />Ioi.66 <br />38° <br />153.58 <br />2° <br />20' <br />2° <br />49' <br />3° <br />44' <br />4° <br />40' <br />I01 . X15 <br />40° <br />146. i9 <br />z° <br />27' <br />2° <br />57 <br />3' <br />55 <br />4° <br />54' <br />IO2. o6 <br />4z' <br />139.52 <br />2° <br />34' <br />3° <br />05' <br />4° <br />07' <br />5° <br />08 <br />102.29 <br />44° <br />133.47 <br />2° <br />41' <br />3° <br />13' <br />4° <br />18' <br />5° <br />22' <br />102.53 <br />46° <br />127.97 <br />2° <br />48' <br />3° <br />21 <br />4° <br />29' <br />S° <br />36' <br />102.76 <br />4S° <br />122.92 <br />z° <br />55' <br />3' <br />29' <br />4° <br />40' <br />5° <br />5o' <br />103.00 <br />50° <br />118.31 <br />3° <br />02' <br />3° <br />38' <br />4° <br />51' <br />60 <br />04' <br />103.24 <br />52° <br />I14 -o6 <br />3009 <br />1 <br />3° <br />46' <br />5° <br />O2' <br />6° <br />17' <br />103.54 <br />54' <br />110.11 <br />3° <br />16'30 <br />54' <br />5° <br />13' <br />6° <br />31 <br />103.84 <br />56° <br />Io6.5o <br />3° <br />22' <br />4° <br />O2' <br />50 <br />23' <br />6' <br />44' <br />104.14 <br />580 <br />103.14 <br />3029' <br />4° <br />10' <br />5° <br />34 <br />6' <br />Si' <br />I04.43 <br />60° <br />I00.00 <br />3° <br />35' <br />4° <br />18' <br />5° <br />44' <br />7° <br />11' <br />104.72 <br />T= R tan I R. = T cot. <br />,I, - 50 -tan I' Chord def. = R <br />Sin. J DR = 2 <br />Sin. D = R Sin' 2 D )\ 0. chords = I. <br />E= R ex'. sec 1 I D <br />5o tan <br />_ a I <br />Sin. D - 7 T 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 fora given distance and deflection. <br />Rule I. Multiply the given distance by .01i45 (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 loo, Alt. 10.102-200=.5. IOO+-5=100.5 hyp• <br />Given Hyp. loo, 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 II,and divide by 7. <br />LEVELING. The correction for curvature and refraction, in feet <br />and decimals of feetis equal to 0.574d2,where d is the distance in miles. <br />The correction for curvature alone is closely, ,d2. 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 Xd2=the sum of the squares of these differ- <br />ences and n=tlle number of observations, then the probable error of tho <br />mean= + 0.6745 Ede <br />SOLAR EPIIEil1ERIS. Attention is called to the Solar Ephemeris for <br />the c irrent year, published by Keuffel & Esser Co., and furnished free of <br />charge upon request, which is 3.}x58 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 andLogarithmie Functions; <br />and Logarithms of Numbers. <br />n TABrx IV. - Minutes in Decimals of a Degree. <br />1' <br />.0167 <br />11' <br />.1833 <br />21' <br />.3500 <br />31' <br />. 5167 <br />41' <br />.6533 <br />51' <br />- .8500 <br />2 <br />.0333 <br />1. <br />.2000 <br />22 <br />.3667 <br />32 <br />.5333 <br />42 <br />.7000 <br />52 <br />.8667 <br />a <br />.0500 <br />13 <br />.2167 <br />23 <br />.3833 <br />33 <br />.5500 <br />43 <br />.7167 <br />53 <br />.8533 <br />4 <br />.0667 <br />14 <br />.2333 <br />'4 <br />.4000 <br />34 <br />.5667 <br />44 <br />.7333 <br />54 <br />..0000 <br />5 <br />.0333 <br />15 <br />.2500 <br />25 <br />.4167 <br />35 <br />.5333 <br />45 <br />.7500 <br />55 <br />.9167 <br />0 <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 />MG7 <br />17 <br />.2S33 <br />27 <br />.4500 <br />37 <br />.6167 <br />47 <br />.7533 <br />57 <br />.9500 <br />8 <br />.1333 <br />1S <br />.3000 <br />28 <br />.46G7 <br />33 <br />.6333 <br />48 <br />.8000 <br />53 <br />.9667 <br />9 <br />.1500 <br />19 <br />.3167 <br />29 <br />.4S33 <br />39 <br />.6500 <br />49 <br />.8107 <br />59 <br />.9833 <br />10 <br />.1667 il <br />20 <br />1 .3333 1130 <br />.5000 <br />40 <br />.6667 <br />V 50 <br />S333 <br />60 <br />1.0000 <br />TABL>; V. - Inches in Decimals of a Foot. <br />1.16 <br />3-32 <br />3-10 <br />% <br />5-1G <br />s <br />0052 <br />.0073 <br />.0104 <br />.0156 <br />.0208 <br />.0260 <br />A- <br />.0117 .0521 <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 />.33.^,3 <br />.4167 <br />.000 <br />.5833 <br />.6667 .7500 <br />.5333 <br />.9167 <br />