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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 />lliat. <br />Def. <br />Dist. <br />Def' <br />I Ft <br />Deg. <br />Radius <br />Mid. <br />Ord. <br />Taa <br />Dist. <br />Def. <br />Dist. <br />Def. <br />fcr <br />1 Ft. <br />2° 17' <br />ft. <br />t <br />ft. <br />ft. <br />f <br />1° 59' <br />it. <br />ft., <br />ft. <br />it. <br />340 <br />0"10' <br />34377. <br />036 <br />. 145 <br />.291 <br />0.05 <br />7° <br />519.0 <br />1.525. <br />6.105 <br />12.21 <br />2.10 <br />20 <br />17159. <br />.073 <br />.291 <br />.5S2 <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 />109.436 <br />42° <br />S73 <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 />1.1G4 <br />0.20 <br />40 <br />747.9 <br />1.G73. <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 />1'• <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 />.2.03G <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.G18 <br />0.45 <br />9 <br />637.3 <br />1.965 <br />7.546 <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 />614.6 <br />2.037 <br />5.136 <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 />G03.8 <br />2.074 <br />8.251 <br />1G.56 <br />2.55 <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 />8.42G <br />1G.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 />2.183. <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 />2.40 <br />9.555 <br />19.16 <br />3.30 <br />40 <br />2148.8 <br />'.5S2 <br />2.327 <br />4.654 <br />0.80 <br />30 <br />499.1 <br />2.511 <br />10.02 <br />20.0.1 <br />3.45 <br />50 <br />'2022:4 <br />.618 <br />2.472 <br />4.945 <br />0.85 <br />12 <br />475.3 <br />2.620 <br />10.45 <br />20.91 <br />3.60 <br />8. <br />1910.1 <br />.655 <br />2.615. <br />5.235 <br />0.90 <br />. 30 <br />459.3 <br />2.730 <br />10.59 <br />21.77 <br />3.75 <br />10 <br />1809.6 <br />.691 <br />2.763 <br />5.52G <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 />23.51 <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 />3.058 <br />12.18 <br />21:37 <br />4.20 <br />40 <br />1562.9 <br />.800 <br />3.199 <br />6.395 <br />1.10 <br />30 <br />'96.2 <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 />4 <br />1432.7 <br />.873 <br />3.490 <br />6.980 <br />1.20 <br />30 <br />370.8 <br />3.387 <br />13.42 <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 />27.54 <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 />25.70 <br />4,05 <br />30 <br />1273.6 <br />.982 <br />3.026 <br />7.852 <br />1.35 <br />17 ° <br />338.3 <br />3.716 <br />14.75 <br />29.56 <br />5.10 <br />.40 <br />1228.1 <br />.1.018 <br />4.071 <br />5.143 <br />1.40 <br />18 <br />319.6 <br />3.935 <br />15.G4 <br />31.20 <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 />5 <br />1146.3 <br />1.091 <br />4.362 <br />5.724 <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 />38.1G <br />61.119 <br />-30 <br />1042.1 <br />1.200 <br />4.798 <br />9.596 <br />1.65 <br />Z3 <br />250.8 <br />5.035 <br />19.94 <br />39.87 <br />61.90 <br />40 <br />-1011.5 <br />1.237 <br />4.943 <br />9.586 <br />1.70 <br />24 <br />240.5 <br />5.255 <br />20.79 <br />41.55 <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 />43.23 <br />7.50 <br />6 <br />955.4 <br />1'.309.5.234 <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 />929.0 <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 />190.7 <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 />6.553 <br />25.85 <br />51.76 <br />9.00 <br />The middle ordinate in inches for anv cord of length (C) is equal to .0012 C2 <br />multiplied by the middle ordinate taken from the above table. Three, if it <br />dedred'to bend a 30 ft. rail to fit 10 degree curve, its middle ordinate should <br />be M12X900X2.183 or 2.3G inches. <br />TABLE III. Deflections for Sub Chords for Short Radius Curves. <br />Degree <br />of <br />Cun,c _ <br />Radius, <br />50 <br />3/2 sub chord =sin of 1 def., angle <br />r <br />Length <br />of arc <br />for 100 ft. <br />sin. i def. ang. <br />12.5 Ft. <br />IS Fc. <br />20 Ft. <br />25 Ft. <br />.30 °. <br />193.18 <br />1° 51' <br />2° 17' <br />2° 58' ' <br />-3° 43' . <br />101.15 <br />32° <br />181.39 <br />1° 59' <br />2° 25' <br />3° 10' <br />3° 5S' <br />101.33 <br />340 <br />171.01 <br />2° o6' <br />2°33' <br />3° 21' <br />4° 12' <br />101.48 <br />36° <br />16i.8o <br />2013' <br />2° 41' <br />3° 33' <br />4° 26' <br />ioi.66 <br />38. <br />153.58.. <br />.2° 20' <br />2° 49' <br />3°44 <br />4° 40' <br />101.85 <br />40° <br />146. i9 <br />2° 27' <br />2° 57' <br />3° 55 <br />4° 54' <br />102.o6 <br />42° <br />139.52 <br />20 34' <br />3° 05' <br />4° 07' <br />50 o8' <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 />30 21 <br />4° 29' . <br />5° 36' <br />102.76 <br />48° <br />122.92 <br />'° 55' <br />30 ''-9' <br />4° 40'' <br />5° 50' <br />103.00 <br />5o° <br />113.31 <br />3002/, <br />3° 38' <br />4° 51' <br />6004 1 <br />103.24 <br />52° <br />114.06 <br />3° 09' <br />3°46' <br />5°02' <br />60 17' <br />103.54 <br />54° <br />110.11 <br />3° 16' <br />3 ° 54' <br />5° 13' <br />6031 1 <br />103:84 <br />56° <br />io6.50 <br />30 22' <br />40 02' <br />50 23' <br />60 44' <br />104..14 <br />580 <br />103.14 <br />3°29' <br />4010' <br />5034' <br />60 57' <br />104.43 <br />600 <br />100.00 <br />3°35' <br />4° 18 <br />5°44' <br />7°11' <br />104-72 <br />CURVE FOR'i4IULAS <br />IX <br />T=Rtan hI s <br />R = T.cot. At I chord <br />,I. _ 5o tan I Chord def. = R <br />Sin. J DR = 50 <br />Si..4 D _ 50 Sin. � D No. chords = I <br />R, E = Res. secs 1 D <br />Sin. I D = 50 tan ; I 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 for a given distance and deflection. <br />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 .oi745, 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=206=.5. IDo+.5=100.5 hyp. <br />Given Hyp. ioo, Alt. 25.252=200=3.125. 100-3.125=96.875=Base. <br />Error in first example, .002; In last, <br />To find Tons of Rail in one mile of track: multiply weight per yard <br />by I I, and divide by 7. <br />LFVELING. 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, #d2. The combined cor- <br />rection is negative. <br />•PRODASLE ERROR. If d,, d , da, etc. are the discrepancies of various <br />results from the mean, and 1f fd2=the sum of the squares of these differ= <br />ences and n=the number of observations, then the probable error of the <br />mean= =0.6745oda <br />1 n (n-1) <br />SOLAR EP13EUERIs. Attention is called to' the Solar Epbemeris'for <br />the current year, published by Keuffel & Esser Co., and furnished free of <br />charge upon request, which is 3.;x55 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; magneticdeclination; arithmetic constants; English and Metric <br />conversions; trigonometric formulas; Natural andLogarithmic Functions; <br />and Logarithms of Numbers. <br />TABLE IV. - 1\1inutes in Decimals of a Degree. <br />1' <br />.0167 <br />11' <br />1533 <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 />52.8C,67 <br />.0625 <br />8 <br />.0500 <br />13 <br />.2167 <br />23 <br />.3833 <br />33 <br />.5500 <br />43 <br />.7167 <br />53 <br />.58.33 <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 />.0533 <br />15 <br />.2500 <br />25 <br />.4167 <br />35 <br />.5833 <br />45 <br />.7500 <br />55 <br />.9167 <br />G <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 />.2333 <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 />.6333 <br />48 <br />.9000 <br />58 <br />.9667 <br />9 <br />.1500 <br />19 <br />.3167 <br />29 <br />.4633 <br />39 <br />.6500 <br />49 <br />.8167 <br />59 <br />.9833 <br />10 <br />.1667 <br />1 20 <br />.3333 11 <br />30 <br />.5000 11 <br />40 <br />1 .6667 11 <br />50 <br />1 .8333 it <br />60 <br />11.0000 <br />TABLE V. - Inches in Decimals of a Foot. <br />1-1G <br />3-32 <br />3-16 <br />Y45 <br />-1G <br />% <br />%a <br />.0052 <br />.on78 <br />.0104 <br />.0156 <br />.0208 <br />.0260 <br />.0313 <br />.0117 .0521 <br />.0625 <br />.0729 <br />1 <br />2 <br />3 <br />4 <br />5 <br />G <br />7 <br />8 9 <br />10 <br />11 <br />...0533 <br />.1667 <br />.2500 <br />.3333 <br />.4187 <br />.5000 <br />.5533 <br />.6667 .7500 <br />.5333 <br />.9167 <br />