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L' <br />VIII <br />TAnL>; IL - Radii, Ordinates and Deflections. Chord =100 ft. <br />Deg. <br />Padius, <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />Def. <br />Dist. <br />Dcf. <br />I Ft <br />D°2' <br />Rai"ie <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />Def. ' <br />Dist. <br />Def. <br />1 Ft. <br />1° <br />ft, <br />ft. <br />it. <br />ft. <br />a <br />3° <br />it. <br />ft. <br />it. <br />ft. <br />i°59' <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 />.41,82 <br />0.10 <br />20' <br />7S1.8 <br />1.600 <br />6.395 <br />12.79 <br />2.20 <br />30 <br />11459. <br />.109 <br />.436 <br />.373 <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 />4.673 <br />6.685 <br />13.37 <br />3.30 <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 />.213 <br />.873 <br />1.745 <br />0.30 <br />20 <br />GS8.2 <br />1.819 <br />7.266 <br />14.53 <br />2.50 <br />10 <br />4011.2 <br />.255 <br />1.015 <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.S92 <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 />1.065 <br />7.846 <br />15:69 <br />2.70 <br />40 <br />3437.9 <br />.361 <br />1.454 <br />2.909 <br />0.50 <br />20 <br />G14.G <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 />16.85 <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 />2:133 <br />8.716 <br />17.43 <br />3.00 <br />20 <br />2155.7 <br />.509 <br />2.036 <br />4.072 <br />0.70 <br />30 <br />54G.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 />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 />202'3.4 <br />..582 <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 />1SOO.G <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 />223.51 <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 />3.05S <br />12.18 <br />24,37 <br />4.20 <br />40 <br />1562.9 <br />.500 <br />3.199 <br />6.398 <br />1.10 <br />30 <br />390.2 <br />3.165 <br />12.62 <br />25.24 <br />4.35 <br />50 <br />1495.0 <br />.836 <br />3.345 <br />6.689.1.15 <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 />6.980 <br />1.20 <br />30 <br />370.8 <br />3.3S7 <br />,13.49 <br />26.97 <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 <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 />318.5 <br />3.606 <br />14.35 <br />2S.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 />29256 <br />5.10 <br />40 <br />1228.1 <br />1.018 <br />4.071 <br />8.143 <br />1.40 <br />18 <br />319.G <br />3.935 <br />15.G4 <br />31.29 <br />5.40 <br />60 <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 />G <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 />G.00 <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 />3G.44 <br />6.30 <br />_10 <br />20 <br />1074.7 <br />1.164 <br />4.653 <br />9.305 <br />1.60 <br />22 <br />202.0 <br />4.814 <br />19.08 <br />33:16 <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 />39.87, <br />6.90 <br />40 <br />1011.5 <br />1.237 <br />4.943 <br />9.586 <br />1.70 <br />24 <br />240.5 <br />5.111-121.79 <br />41.5S <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 />43.28 <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 />5.91S <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 />531.9 <br />1.418 <br />5.669 <br />11.34 <br />1.95 <br />29 <br />199.7 <br />G.360 <br />25.04 <br />50.07 <br />8.70 <br />40 <br />859.9 <br />1.455 <br />5.514 <br />11.G3 <br />2.00 <br />30 <br />193.2 <br />6.553 <br />25.88 <br />51.76 <br />9.00 <br />The middlo ordinate iu inebes for any cord of length (C) is equal to .0012 C' <br />multiplied by the middle ordinato taken from the above table. 'Thus, if it <br />desired to bend a 110 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 />50of <br />11' <br />34 sub chord= sin of I def. angle <br />1115 <br />Length <br />arc <br />for 100 ft. <br />sin. u def. ang. <br />12.5 Ft. <br />Ft. <br />20 Ft. <br />25 Ft. <br />30° <br />193.18 <br />1° <br />51, <br />° <br />17' <br />2° <br />58' <br />3° <br />43 <br />IOI.I5 <br />32° <br />181.39 <br />i°59' <br />3 <br />20 <br />25' <br />3°10' <br />23 <br />3°58' <br />33 <br />101.33 <br />34° <br />171.OI <br />2° <br />06' <br />z° <br />33' <br />3° <br />21' <br />4° <br />12' <br />I0I.48 <br />360 <br />161.8o <br />20 <br />13' <br />2° <br />41' <br />3° <br />33' <br />4°.26' <br />25 <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 />IOI.85 <br />40° <br />146.Ig <br />20 <br />_ <br />27' <br />2° <br />57' <br />3° <br />55' <br />4° <br />54' <br />fo2.o6 <br />420 <br />139.52-° <br />37 <br />34' <br />3° <br />05' <br />4° <br />07 1 <br />5° <br />o8' <br />102.29 <br />44° <br />133-47 <br />2° <br />41' <br />3° <br />13' <br />40 <br />18' <br />S° <br />22' <br />102:53 <br />46° <br />127.97 <br />2°48' <br />.4833 <br />3°21' <br />.6500 <br />4°29' <br />-.5167 <br />S°36' <br />.9833 <br />102:76 <br />48° <br />122.92 <br />20 <br />55 <br />3° <br />29' <br />40 <br />40 <br />j° <br />50' <br />103.00 <br />50° <br />118.31 <br />3° <br />02' <br />3°4-51,° <br />38' <br />6 <br />04' <br />103.24 <br />52° <br />114.06 <br />3° <br />09' <br />3° <br />46' <br />° <br />5 <br />oz ' <br />60 <br />17 <br />103.54 <br />54° <br />110.11 <br />3° <br />16' <br />30 <br />54' - <br />5° <br />13' <br />6° <br />31' <br />103.84 <br />56° <br />106.50 <br />3° <br />2s' <br />40 <br />oz' <br />S° <br />23' <br />6° <br />44' <br />104.14 <br />58° <br />103.14 <br />3° <br />29' <br />4° <br />10' <br />50 <br />34' <br />60 <br />57' <br />104.43 <br />600 <br />100.00 <br />3°J5' <br />4° <br />18' <br />5°44' <br />7° <br />11' <br />104.72 <br />jX <br />CURVE FORMULAS <br />T= R tan I P. = T cot. I chord2 <br />jo tan a I Chord def. _ <br />T - Sin. ,} ll 50 R <br />R = � <br />i D 5o Sin. 2 D No. chords = I <br />R E= R ex. sec 3 I D <br />j ; I <br />Sin. ID =o tan r E = T tan 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 O 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 />ho 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. 1oo+.5 =1oo.5 hyp. <br />Given. Hyp. ioo, Alt. 25.252=200=3.125. loo-3.125=96.87J=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 1r, 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, jd2. The combined cor- <br />rection is negative. <br />Pr.OnAULE ERROR. If d, , d°, da, etc. are the discrepancies of various <br />results from the mean, and if Ede=the sum of the squares of these differ- <br />ences and n=the number of observations, then the probable error of the, <br />mean -.-O.6745 FY -d7 - <br />n (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 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 andLogarithmic Functions; <br />and Logarithms of Numbers. <br />TABLE 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 />GS33 <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 />.3333 <br />33 <br />.5500 <br />43 <br />.7167 <br />53 <br />.&533 <br />4 <br />.0667 <br />14 <br />.2333 <br />24 <br />.4000 <br />34 <br />.5667 <br />44 <br />.7333 <br />51 <br />'.9000 <br />5 <br />OS33 <br />15 <br />.2500 <br />25 <br />.4167 <br />35 <br />.5333 <br />'45 <br />.7500 <br />55 <br />.9167 <br />6 <br />.1000 <br />16 . <br />567 <br />26 <br />.4333 <br />36 <br />GOOO <br />46 <br />.7667 <br />Sri <br />.9333 <br />7 <br />.1167 <br />17 <br />.2333 <br />27 <br />.4500 <br />37 <br />G1G7 <br />47 <br />.7533 <br />57 <br />.9500 <br />8 <br />.1333 <br />18 <br />.3000 <br />28 <br />.4667 <br />38 <br />G333 <br />48 <br />.5000 <br />fig <br />.96617 <br />9 <br />.1500 <br />19 <br />.3167 <br />19 <br />.4833 <br />39 <br />.6500 <br />49 <br />-.5167 <br />59 <br />.9833 <br />10 <br />.1667 <br />20 <br />1 .3333 1130 <br />1 .5000 11 <br />49 <br />1 .GG67 11 <br />40 <br />1 .8333 11 <br />CO <br />1.0000 <br />TABLE V. - <br />Inches in Decimals of a Foot. <br />1.16 <br />3-32 <br />3-16 <br />% <br />5-16 <br />Y8 <br />.0052 <br />.0075 <br />.0101 I <br />.0156 <br />.0203 <br />.0260 <br />.0313.0117 <br />1 <br />.0521 <br />.0625 <br />.0729 <br />1 <br />2 <br />3 <br />4 <br />5 <br />6 <br />7 <br />1 3 <br />1 9 <br />10 <br />11 <br />.0833 <br />.lfiC,7 <br />.2500 <br />.3333 <br />.4167 <br />.5600 <br />.5333 <br />.6667 <br />.7500 <br />.5333 <br />.9167 <br />