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VIII - - Y IX <br />TAar.E II.-- Radii, Ordinates and Deflections. Chord= 100 ft.CURVE . FORMULAS <br />s <br />_ <br />Radius' <br />Mid. <br />Ord. <br />Ord. <br />Tan, <br />Dist. <br />Def. <br />Dist.' <br />Def. <br />for <br />I Ft. <br />'Deg. <br />Radius <br />`fill. <br />Ord.' <br />Tan. <br />' Dist. <br />.Def. <br />Dist. <br />-0 -1f- <br />for <br />1 Ft. <br />2. 17' <br />t, <br />t. <br />ft. <br />fc. <br />Iit. <br />1° 59' <br />2° 25 <br />ft._ <br />ft, <br />it. <br />1 <br />6' W <br />34377..030 <br />2°33 <br />.145 <br />-.291 <br />0.05 <br />7° <br />819.0 <br />1.528 <br />6.105 <br />12.21 <br />4° 26' <br />20 <br />17189{' <br />.073 <br />.291 <br />.582 <br />0.10 <br />20' <br />781.8 <br />1:600 <br />61395 <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 -OS <br />2,2.- <br />40 <br />559.7.4 <br />=.145 <br />.582 <br />1.164 <br />0.20 <br />40 <br />747.9 <br />1.673 <br />G. 610 <br />13.37 <br />2.30' <br />50 <br />GS75.5 <br />.IS2 <br />.727 <br />1.454 <br />0.25 <br />8 <br />716.8 <br />1.74G <br />6.976 <br />13.95 <br />2.40 <br />1 <br />5729.6 <br />.21S <br />' .873 <br />1.745 <br />0.30 <br />20 <br />GSS.2 <br />1.819 <br />.7.266 <br />14.53 <br />2.50 <br />IO <br />4911.2 <br />.255 <br />1.018 <br />2.030 <br />0.35 <br />'30G74.7 <br />4° 02' <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 />G61.7 <br />4.592 <br />7.556 <br />15:11 <br />2.60 <br />30. <br />3819.5 <br />.327.1.309 <br />2.618 <br />•2.909 <br />0.4, <br />9 <br />637.3 <br />1.905 <br />7.846 <br />15.69 <br />2.70 <br />40 <br />3437.9 <br />.364 <br />1.454 <br />0.50 <br />20 <br />614.G <br />2.037 <br />8.136 <br />16.27 <br />.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 />.930 <br />1.745 <br />3.490 <br />0.00 <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 />O.G5 <br />:10 <br />573.7 <br />211S3 <br />8.716 <br />17.43 <br />3.00 <br />20 <br />2455.7 <br />.509 <br />2.030 <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 />2.151 <br />4.3G3 <br />0.75 <br />11 <br />521.7 <br />2.402 <br />9.555 <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 />2.511'10.02 <br />20.04 <br />3.45 <br />50 <br />2022.4 <br />.618 <br />2.472 <br />4.94.5 <br />0.85 <br />12 <br />478.3 <br />2.620 <br />10.45 <br />20.91 <br />3.60 <br />3'. <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 />1809.6 <br />.691 <br />2.763 <br />5.526 <br />0.95 <br />13 <br />441.7 <br />2.S39 <br />11.32 <br />22.64 <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 />6.103 <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 />.600 <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 />25 <br />383.1 <br />3.277 <br />13.05 <br />26.11 <br />4.50 <br />1432.7 <br />.673 <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 <br />.909 <br />3.635 <br />7.271 <br />1.25 <br />10 <br />359.3 <br />3:496 <br />13.92 <br />27.54 <br />4.80 <br />20 <br />1322.5 <br />.945 <br />3.718 <br />7.561 <br />1.30 <br />30 <br />348.5 <br />3.606 <br />14.35 <br />23.70 <br />4.95 <br />30 <br />1273.6 <br />.982 <br />3 MG <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 />5.90 <br />50 <br />1185.5 <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 />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 />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 />18.22 <br />36.44 <br />G.30 <br />20 <br />1074.7 <br />1.164 <br />4.653 <br />9:305 <br />1.60 <br />22 <br />262.0 <br />4.514 <br />19.08 <br />38.10 <br />6.60 <br />30 <br />1042.1 <br />1.200 <br />4.70S <br />9.596 <br />1.65 <br />23 <br />250.8 <br />5.035 <br />19.04 <br />39.87 <br />G.90 <br />40 <br />1011.3 <br />1.237 <br />4.943 <br />9.SS6 <br />1.70 <br />24 <br />240.5 <br />5.255 <br />20.79 <br />41.58 <br />7.20 <br />5.0 <br />982.6 <br />1.273 <br />5.088 <br />W A S <br />1.75 <br />25 <br />231.0 <br />5.476 <br />21.64 <br />43.23 <br />7.50 <br />0 " - <br />955.4 <br />1.309 <br />5.234 <br />10.47 <br />1.50 <br />20 <br />222.3 <br />5.697 <br />22.50 <br />44.99 <br />7.50 <br />10 <br />929.6 <br />1.346 <br />5.370 <br />10:76 <br />1.55 <br />27 <br />214.2 <br />5.915 <br />23.35 <br />40.59 <br />8.10 <br />20 <br />905.1 <br />1.352 <br />5.524 <br />11.05 <br />1.90 <br />28 <br />200.7 <br />6.139 <br />24.19 <br />48.38 <br />8.40 <br />30 <br />851.9 <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 />5.514 <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 />Curve <br />Radius' <br />50 <br />M sub chord = sin of Idef. angle <br />P. ' - <br />Length <br />of arc <br />for 1011 ft. <br />sin. l def. ang. <br />12.5 Ft. <br />15 Ft. <br />20 Ft. <br />25 Ft. <br />39°193.15 <br />51' <br />10 51 <br />2. 17' <br />r° 53' <br />'.3° 43' <br />101- 15 <br />32° <br />181.39 <br />1° 59' <br />2° 25 <br />3° 10 <br />3° 58' <br />101.33 <br />34° <br />171.01 <br />2° o6' <br />2°33 <br />3° z1' <br />4° Iz' <br />I0I.0- <br />36° <br />161.86 <br />20 13' <br />2° 411 <br />3°.33' <br />4° 26' <br />1oi'.66 <br />38°153.58 <br />. <br />20 20' <br />20 49' <br />3° 44' <br />_4° 40' <br />-101.95 <br />40° <br />146. i9 <br />2° 27'. <br />'-° 57' , <br />3° 55' <br />4° 54' <br />Io2.o6 <br />42' <br />139.52 <br />z° 34' <br />3° 05' <br />40 07` <br />5° 0S' <br />102.29 <br />.44 <br />133.4-7 <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 />102.76 <br />490 <br />122.92 <br />2' 55' <br />3° �9' <br />4° 4c, <br />•5° 50' <br />103.00 <br />50° <br />115.31 <br />3° 02'. <br />3° 3S' <br />4° 51 <br />6° o4f <br />I03.24 <br />52° <br />114-o6 <br />3009, <br />3° 46/. <br />5° 02' <br />6°-17' <br />103-54 <br />540 <br />1 ro. 11 <br />3° 16 <br />3° 54' <br />�5° 13' <br />6° 31' <br />Lo3.84 <br />56° <br />Io6.30 <br />3° 22' <br />4° 02' <br />S°.23' <br />6° 44' <br />104.14 <br />580- <br />103. 14 <br />30 29' <br />4° 10' <br />50 34.' <br />6° 57' <br />104.43 <br />60° <br />loo.co <br />3035 <br />4° 16' <br />5°'44' <br />70 11' <br />104.72 <br />T= R tan o I R= T cot. I = chord2 <br />5o.tan.f I Chord de.{R <br />R <br />Sin..J D .. 50 <br />-= � <br />Sin. J D= i0 Sin. 2 D No. chords = I <br />R. E = R' ex. sec i I D <br />Sin. D = 50 tT I .. E = T tan. l I Tan. def. =,21 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 I° for f 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. 100+.5 = 100-5 hyp. <br />,Given Hyp. ioo, Alt. 25.252=200=3.125. Ioo-3.I25=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 11, and divide by 7. <br />LEVELING. The correction for curvature and refraction, in feet <br />and decimals of feet is equal to 0.574 de, 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=the number of observations, then the probable error of the <br />mean= + 0.6745J 7-d2 <br />V 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 31x5) in., with about 190 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 i\ietric <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 />^I' <br />.5167 <br />41' <br />.6833 <br />51' <br />.8500 <br />2 <br />.0333 <br />12.2000 <br />.0052 <br />22 <br />.3667 <br />32 <br />.5333 <br />42 <br />.7000 <br />52 <br />.SG67 <br />3 <br />.0500 <br />13 <br />2167 <br />23 <br />.3533 <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 />31 <br />.5GG7 <br />44 <br />'.7333 <br />54 <br />.9000 <br />.5 <br />.0833 <br />15 <br />.2500 <br />25 <br />.4167 <br />35 <br />.5S33 <br />45 <br />.7560 <br />55 <br />.9167 <br />6 <br />.1000 <br />16 <br />.2007 <br />2r, <br />.4.',3.', <br />3G <br />.6000 <br />'46' <br />.7667 <br />66 <br />.0333 <br />7.1167 <br />17 <br />.2533 <br />27 <br />.4500 <br />37 <br />G167 <br />47 <br />.7333 <br />57 <br />.0500 <br />S <br />.1333 <br />18 <br />.3000 <br />23 <br />.4607 <br />38 <br />.0333 <br />43 <br />.6000 <br />58 <br />.9667 <br />9 <br />.1500 <br />19 <br />.3167 <br />29 <br />.48.33 <br />39 <br />.6500 <br />'49 <br />..'107 <br />59 <br />:9833 <br />10.1667 <br />20 <br />.3333 <br />30 <br />.5000 <br />40 <br />.6667 1150 <br />.8333 11 <br />CO <br />11.0000 <br />TABLE V. - <br />Inches in Decimals of a Foot. - <br />1 -16 <br />3-32 1 is <br />3-1G <br />r} <br />5-1G <br />% <br />1 <br />ii <br />.0052 <br />.0078 .0101 <br />.0156 <br />:0208 <br />.0200 <br />.0:313 <br />.11 7 <br />.0521 1 <br />.0625 <br />.0729_ <br />.1 <br />3- <br />4 <br />S <br />•G <br />7 <br />3 <br />9 <br />10 <br />11 <br />.0833 <br />.1667 .2500. <br />.3333 <br />.4167 <br />.5000 <br />.5833 <br />.6667. <br />.7500 <br />.5333 <br />.fl167 <br />