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m 1z <br />TAB <br />mo IL - Radii, Ordinates and Deflections. Chord =100 ft. CURVE FORMULAS - <br />Dom'' <br />Radius <br />50 <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. <br />Dist. <br />Def. <br />Dist. <br />Def. <br />for <br />1 Ft. <br />2° 17- <br />ft. <br />t. <br />ft. <br />ft. <br />eft. <br />1°59' <br />2°25' <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 />.145.582 <br />5° 22' <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 />Il <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 />120 <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 <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 />1 <br />3.780.65 <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 <br />21.77 <br />3.75 <br />10'.11809.6 <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 />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 />6.408 <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 />6.398 <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 />8- • <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 />26.97 <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 />30.348.5 <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 <br />5.70 <br />[d <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 <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 />9.886 <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 />1.273 <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 />8 <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 />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 />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 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 .0012X900X2.183 or 2.36 inches. <br />TABLE III. ' Deflections for Sub Chords for Short Radius Curves. - <br />Degree <br />Of <br />Cure <br />Radius <br />50 <br />Ma sub chord <br />R- = sin of $ def. angle <br />Length <br />of arc <br />for 100 ft. <br />sin. 12 def. ang• <br />12.5 Ft. <br />15 Ft. <br />20 Ft. <br />25 Ft. <br />_30° <br />193.18 <br />-i-° 5?' _ <br />2° 17- <br />2° 58' _ <br />._ _3� 43' .. <br />_ 101-15 <br />320 <br />181.39 <br />1°59' <br />2°25' <br />3°l0' <br />3° 58' <br />101.33 <br />34 <br />171.01 <br />2° O6' <br />2° 33' <br />3° 21' <br />4° 12' <br />101.48 <br />36° <br />i61.8o <br />2°13' <br />2° 41': <br />_ 3°33' <br />`4°26'. <br />ioi.66 <br />3$° <br />153.58_ <br />2°20' <br />2`..49'._ <br />.3°.441 <br />4°40'- <br />Ioi.85 <br />40° <br />146.19 <br />2°27' <br />2° 57' <br />3° 55' <br />4° 54' <br />102.o6 <br />4z° <br />139.52 <br />20 34' <br />3°. 05' <br />4° 0T <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 />4° 29' <br />5° 36' <br />162.76 <br />48° <br />122.92 <br />z° 551 <br />3° z9' <br />4° 40 <br />5° 50' <br />103.00 <br />..50° <br />118.31. <br />3°0', <br />3°38' <br />4°51' <br />6°04' <br />103.24 <br />52° <br />114. o6 <br />3° 09' <br />3° 46' <br />5° 0z', <br />6° 17 <br />103.54 <br />54° <br />I IO. I I- <br />3° 16' <br />_ • 3° 54' <br />5° 13' <br />6° 31 <br />103.84 <br />56° <br />io6.5q <br />3° 22' <br />4° 02' <br />5° 23' - <br />6° 44' <br />104.14 <br />8 0 <br />60° <br />4 <br />3° 29� <br />40 1O, <br />5a 34� <br />60 57' <br />104.43 <br />.3333' <br />100.0 <br />3 35 <br />4 18 <br />5 44 . <br />7 .11 <br />104.72 <br />T = R tan 8 I R = T cot. ' I Chord def. = chords <br />50 tan I I • R <br />T Sin. 8 D R = 60 - <br />Sin. I D = 5o Sin. ' D No. chords= I <br />R E.= R ex. sec z I D <br />5o tan a I - <br />Sin. 8 D= h E T tan I 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 .oiM (def. for i°for ; 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.102=2oo=.5: ioo-}-:5=ioo.5 hyp- <br />Given Hyp. Ioo,Alt:25.252=200=3.125. Ioo-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; feet is 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 />-P ROBABLE ERROR. If d,, da, da, etc. are the discrepancies of various <br />results from the mean, and if Y-da=the sura of the squares of these diflcr- <br />ences and n=the number of observations; then the probable error of the, <br />mean Xd2. <br />± 0.6745 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 31x58 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 f ormulas; Natural andLogarithmie Ftinctions; <br />and Logarithms of Numbers.- <br />TABLE <br />umbers: <br />TABLE 1V. = Minutes <br />in Decimals of a Degree. <br />1' <br />.0167 <br />11' <br />.1833 <br />21' <br />:3500 <br />31' <br />.5167 <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 />$ <br />.05Q0. <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 />.0833 <br />15 <br />.2500 <br />25 <br />.4167 <br />.35 <br />.5833 <br />45 <br />.7500 <br />55 <br />.9167 <br />1.6 <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 />-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 />18 <br />.3000 <br />28 <br />.4667 <br />38 <br />.6333 <br />48 <br />.8000 <br />58' <br />.9667 <br />.1500 <br />19. <br />3167. <br />29 <br />.4833 <br />39 <br />.6500 <br />49 <br />:8167 <br />59 <br />'.9833 <br />L9 <br />10 <br />1667 <br />20 <br />.3333' <br />30 <br />.5000 <br />40 <br />.6667 <br />50 <br />8333 <br />60 <br />1.0000 <br />TABLE V. - Inches in Decimals of a Foot. - <br />- <br />146 <br />3-32 <br />% <br />3=165-16 <br />.0052 <br />.0078 <br />.0104 <br />.0156 <br />.0208 <br />.0260 <br />.0313 <br />.0417' .0521 <br />.0625 <br />.0729 <br />12 <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 <br />..5000 <br />1 .5833 <br />1 .6667 .7500. <br />.8333 <br />.9167, <br />