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VIII <br />TABLE II. - Radii, Ordinates and Deflections. Chord =100 ft. <br />Deg. <br />Radius <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />' Def.' <br />Dist. <br />Def. <br />I1fur <br />Deg. <br />Radius <br />Mid. <br />O:d. <br />Tan. <br />P•iat. <br />Def. <br />Dist. <br />Def. <br />lfF t. <br />2° 17' <br />ft. <br />fi. <br />ft. <br />ft. <br />! <br />1° 59' <br />ft, <br />ft. <br />ft: <br />ft. <br />34° <br />0'10 <br />34377. <br />'.0330 <br />.145 <br />.291 <br />0.05 <br />7. <br />519.0 <br />1.525 <br />. <br />6.105 <br />12.21 <br />2.10 <br />20. <br />17159. <br />.073 <br />.291 <br />:552 <br />0.1 0 <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 />-764.6. <br />1.63T <br />6:540 <br />13.05 <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 />6.655 <br />13.37 <br />2.30 <br />50 <br />'6545.5 <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 />.218 <br />.573 <br />1.745 <br />0.30 <br />20 <br />G98.2 <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 />1.555 <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 />7.550 <br />15.11 <br />2.00 <br />30 <br />• 3819.8 <br />.327 <br />1.309 <br />2.618 <br />0.45 <br />0 <br />637.3 <br />1.965 <br />7.846 <br />15.69 <br />2.70 <br />40 <br />3437.9 <br />.364.1.454 <br />2.909 <br />0.50 <br />20 <br />614.6 <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 />8.281 <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 />3.781 <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 />2.03G <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 />22 2.0..54:; <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 />2145.8 <br />'.582 <br />2.327' <br />4.65.1 <br />0.£0 <br />30 <br />499.1 <br />2.511 <br />10.02 <br />20.01 <br />3.45 <br />50 <br />2022.4 <br />.613 <br />2.472 <br />4.945 <br />0.85 <br />"3- <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 />1809.0 <br />.691 <br />2.763 <br />5:526 <br />0.93 <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 />2.1103 <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 />3.0.54', <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 />1562.0 <br />.800 <br />3.199' <br />6.393 <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 />.836.1 <br />3.345! <br />6.689 <br />1.15. <br />15 ' <br />383.1.3.277 <br />13.051 <br />26.11 <br />4.50 <br />4 <br />1432.7 <br />.873 <br />3.490' <br />6.950 <br />1.20, <br />'30 <br />370.8 <br />3.357 <br />13.49 <br />26.97 <br />4.65 <br />10. <br />1375.4.909 <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 />345.5 <br />3.606 <br />14.35. <br />28.70 <br />4.95 <br />30 <br />1273.6 <br />.982 <br />3.92G <br />7.S52 <br />1.35 <br />17 <br />338.3 <br />3.710 <br />14.78 <br />29.56 <br />5.10 <br />i40 <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 />1185.5 <br />1.055 <br />4.217 <br />5.433 <br />1.4:5 <br />10 <br />302.9.4.155 <br />16.51 <br />33,Q1 <br />5.70 <br />5 <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 />107417 <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 />"1042.11 <br />1.200 <br />4.705 <br />9.506 <br />1.65 <br />23 <br />250.8 <br />5.035 <br />19.94 <br />39.57 <br />6.90 <br />40 <br />1011.5 <br />1.237 <br />4.913 <br />9.886 <br />1.70 <br />24 <br />240.5 <br />5.255 <br />20.79' <br />41.5S <br />7.20 <br />50 <br />992.6 <br />1.273 <br />6.088 <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 />'055.4 <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 />1.3.46 <br />5.379 <br />10.76 <br />1.85 <br />27 <br />214.2 <br />5.918 <br />23.35 <br />46.69 <br />5.10 <br />20 <br />905.1 <br />1.352 <br />5.52411.05 <br />1.90 <br />28, <br />206.7 <br />6.139 <br />24.19 <br />4S.38 <br />S.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 />S170 <br />40 <br />1 559.9 <br />1.455 <br />5.S14 <br />11.63 <br />2.00 <br />1 30 <br />193.2 <br />6.5S3 <br />25.88 <br />151.76 <br />0.00 <br />The middle ordinate in inches for any cord of length (C) is equal to .0012C2 <br />multiplied by the middle ordinate taken from the above table. Thus, it it <br />desired to bend i:00 ft, rail to fit a IU degree curve, its middle ordinate should <br />be .0012X90UX2.183 or 2.36 inches. <br />TABLE IIT. 'Deflections for Sub Chords for Short Radius Curves. <br />Degree <br />- of <br />Curve <br />Radius <br />50 <br />sub chord <br />1 t = sin of r def. angle <br />pei'a tch <br />for 100 ft. <br />sin.1 def, ane. <br />12.5 'Ft. <br />15 Ft. <br />20 Ft. <br />25 Ft. <br />30° <br />193. i8 <br />1° 51' <br />2° 17' <br />2' 58' <br />3° 43'" <br />IOI.15 <br />320 <br />181.39 <br />1° 59' <br />2° 25' <br />3° 10 <br />3058 <br />101.33 <br />34° <br />171.01 <br />2° 06' <br />2° 33' <br />3° 21' <br />4° 12' <br />101.48 <br />36° <br />161.80 <br />2°13' <br />2° 41'3° <br />33' <br />4° 20' <br />io1.66 <br />380 <br />153.58 <br />2°20' <br />2°49' <br />3044' <br />4° 40' <br />101.85 <br />40° <br />146.19 <br />2° 27' <br />2° 57' <br />3° 55' <br />4° 54' <br />102.o6 <br />42' <br />139.52 <br />2° 34' <br />3° 05' <br />4° 07`S,08, <br />45 <br />102.29 <br />44° <br />133.47 <br />2° 41' . <br />30 13' <br />4°18' <br />5° 22' <br />102.53 <br />460 <br />127-97 <br />2' 48' <br />3° 21' <br />4° 29' <br />5° 36' <br />102.76 <br />48° <br />122.92 <br />2° 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 />103.24 <br />520 <br />114.06 <br />3°09' <br />3°46' <br />5°o2' <br />6°17' <br />103.54 <br />54° <br />110.11 <br />3° 16' <br />3° 54' <br />S° 13' <br />6° 31' <br />103.8:} <br />56° <br />1o6. 5o <br />3° 22 <br />4° 02' <br />S° 23' <br />6° 44' <br />104-14 <br />580 <br />103.14 <br />3° 29' <br />4° lo' <br />S° 34' <br />6° 57' <br />.104.43 <br />6o° <br />100.00 <br />3°3J' <br />4°18' <br />5°44' <br />7°I1' <br />104.72 <br />IX <br />CURVE FORMULAS <br />T= R tan 2 I Ttchorda <br />R = T.co.. s I Chord def. <br />'_ 5o tan. I . � .. _ `k <br />T*_ -Sin. J D R = 50 <br />Sin. } D = 5o Sin. J D No. chords = I- <br />R E= R ex. sec jI U <br />5o tan <br />Sin. g D = E. = T tan 4 I ; 'Tan. def. = a 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•giSen distance and deflection.- <br />Rule <br />eflection: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.io2=Zoo=.5. 100•}-.5=Ioo.5 hyp. <br />Given Hyp. loo, Alt. 25.252=200=3.125. I00 -3.I25=96 -875=P -ase - <br />Error in first example, :oo2; in last, .045. <br />-To find Tons of Rail i,i_one mile of track: multiply weight per yard <br />by ii, and divide by 7. . <br />Li-wELING. The correction for curvature and refraction, in feet <br />and decimals of feet is equal to 0.574 d2, where d is the distance in'miles. <br />The correction for curvature alone is closely, Jd2. The combined cor- <br />rection -is negative. <br />"PROBABLE ERROR. If d,, d„ d3, etc. are the discrepancies of various <br />results from the mean, and if 7-d3=the sum of the squares of these differ- <br />ences and 11=the number of observations, then the probable error of the <br />mean= + 0.6745Ed' <br />�n (r.-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•.1,x5$ 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 and Logarithmic Functions; <br />'and Logarithms of Numbers. <br />TABLE IV. - Minutes in Decimals of a Degree. <br />1' <br />.01(37 <br />11' <br />.1833 <br />21' <br />.3500 <br />31' <br />.51157 <br />41' <br />.6S33 <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 />S <br />.0500 <br />13 <br />.2167 <br />23 <br />.3833 <br />33 <br />.5500 <br />43 <br />.7167 <br />53 <br />SS33 <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 />6 <br />..0833 <br />15 <br />.2500 <br />25 <br />.41G7 <br />35 <br />.5833 <br />45 <br />.7500 <br />55 <br />.9167 <br />6 <br />.1000 <br />16 <br />.2667 <br />26 <br />.4333 <br />36 <br />G000 <br />40 <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 />.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 />58 <br />.9667 <br />0 <br />.1500 <br />19 <br />.3167 <br />29 <br />.4533 <br />39 <br />.6500 <br />49 <br />.8167 <br />59 <br />.9533 <br />10 <br />.1667 11 <br />20 <br />1 .3333 <br />30 <br />J .5000 <br />n 40 <br />.6667 1, <br />50 <br />.5333 <br />60 <br />1.0000 <br />TABLE V. - <br />Inches in Decimals of a Foot. <br />1-115 <br />1 3-323-16 <br />5-1G <br />% <br />1 <br />.0052 <br />.007$ <br />.010-1 <br />.0156 <br />02% <br />.03 <br />.0260 <br />.0313 <br />.0417 1 .0521 <br />.0625 <br />.0729 <br />1 <br />2 <br />3 <br />4 <br />5-1 <br />6 <br />7 <br />8 9 <br />10 <br />11 <br />0333 <br />.1607 <br />.2500 <br />.3333 <br />.4167 <br />.5000 <br />.5833 <br />.6067 .7500 <br />.5333 <br />.9167 <br />