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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 />1)eL <br />for <br />I Ft <br />- Deg. <br />Radius <br />Mid. <br />Ord. <br />Tan <br />Dist. <br />Def. <br />Dist. <br />Det. <br />for <br />1 Ft. <br />2° 17' <br />it, <br />ft. <br />ft, <br />ft. <br />a <br />1°59' <br />It.. <br />it. <br />IL <br />It. <br />34° <br />0°10' <br />34377. <br />.036 <br />.145 <br />.291 <br />0.05 <br />7. <br />810.0 <br />1.528 <br />6.105 <br />12.21 <br />2.10 <br />20 <br />17159.. <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 />.573 <br />0.1.5 <br />30 <br />764. F) <br />1.6137 <br />6.540 <br />13.03 <br />2.25 <br />40 <br />8594.4 <br />+.145 <br />.552 <br />1.164 <br />0.20 <br />40 <br />747.9 <br />1.673 <br />6.635 <br />13.37 <br />2.30 <br />50 <br />6875.5 <br />.182 <br />.727 <br />1.454 <br />0.25 <br />8 <br />71G.S <br />1. 746 <br />.6.976 <br />13.95 <br />2,40 <br />1 ' <br />5729.6 <br />.218 <br />.S73 <br />1.745 <br />0.30 <br />'20 <br />638.2 <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 />.201 <br />1.164 <br />2.327 <br />0.40 <br />40 <br />661.7 <br />1.592 <br />'7.556 <br />15.11 <br />2.60 <br />30 <br />3819.8 <br />.327 <br />1.309 <br />2.615 <br />0.45 <br />9 <br />637.3 <br />1.965 <br />7.84E <br />15.69 <br />2.70 <br />40 <br />,3437.9 <br />.304 <br />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 />8.231 <br />16.56 <br />2.35 <br />2 <br />.2864:9 <br />.436 <br />1.745 <br />3.490.0:60 <br />40 <br />593.4 <br />2.110 <br />8.426 <br />16.85 <br />2.J0 <br />10 <br />2644.6 <br />.473 <br />1.591 <br />3.751 <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 />.500 <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 />2.181 <br />4.363 <br />0.75 <br />11 <br />521.7 <br />2.403 <br />9.535 <br />19.16 <br />3.30 <br />40 <br />2148.8 <br />.592 <br />2.327 <br />4.654 <br />0.80 <br />30 <br />499.1 <br />2.511 <br />10.02. <br />20.01 <br />3.45 <br />'50 <br />2022.4 <br />.618 <br />2.472 <br />1.945 <br />0.85 <br />12 <br />•175.3 <br />2.620 <br />10.45 <br />20.91 <br />3.60 <br />3' <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 />5.526 <br />0.95 <br />13 <br />•1.11.7 <br />2.839 <br />11.32 <br />22.64 <br />3.00 <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 />•1•.05 <br />30 <br />1637.3. <br />.764.3.054 <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 />.800 <br />3.190 <br />6.395 <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 <br />3.345 <br />6.659 <br />1.15 <br />15 <br />333.1 <br />3.277 <br />13.05 <br />26.11 <br />4.50 <br />4 ' <br />1432.7 <br />.873 <br />3.400 <br />6.080 <br />1.20 <br />30 <br />370.3 <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 />350.3 <br />3.496 <br />13:92 <br />27. S4 <br />4.80 <br />20 <br />1322.5 <br />.945 <br />3.718 <br />7.561 <br />1.30 <br />130 <br />343.5 <br />3.606 <br />14.35 <br />23.70 <br />4.95 <br />30 <br />1273.0 <br />.982 <br />3.923 <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.G4 <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 />163.51 <br />33.01 <br />5.70 <br />6 <br />1146.3 <br />1.091 <br />4.362 <br />8.724 <br />1.50 <br />'20 . <br />237.9 <br />4.374 <br />17.37 <br />34.73 <br />6.00 <br />10 <br />1109.3.1:127 <br />4.507 <br />9.014 <br />1.55 <br />21 <br />274.4 <br />4.594 <br />19.22 <br />36.4.4 <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 />19.03 <br />3S.1G <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.S7 <br />6.90 <br />40 <br />1011.5 <br />1.237 <br />4.943 <br />9.856 <br />1.70 <br />24 ' <br />240.5 <br />5.255 <br />20.79 <br />41.53 <br />7.20 <br />50 <br />932:6 <br />1.273 <br />5.068 <br />10.15 <br />1.75 <br />25 <br />231.0 <br />5.476 <br />21.64 <br />'3.25 <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 />214.2.5.918 <br />23.35 <br />46.69 <br />5.10 <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 />48.33 <br />8.40 <br />30 <br />681.9 <br />1.418 <br />5.669 <br />11.34 <br />1.95 <br />29 <br />199.7 <br />6.360 <br />2.5.04 <br />50.07 <br />S.70 <br />40 <br />659.0 <br />1.455 <br />5.514 <br />11.613 <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 fita 10 degree curve, its middle ordinate should <br />be .0012X900X2.183 or 2.36 inches. <br />TABLE III. Deflections for Sub Chords for Siiort Radius Curves. <br />Degree <br />of <br />Curve <br />Radius <br />so <br />Y2 sub chord <br />R =sin of 4 def. angle <br />Length <br />of are <br />for 100 ft. <br />sin.1 def. ang. <br />12.5 Ft. <br />15 "Ft. <br />20 Ft.. <br />25 Ft. <br />30° <br />193. i8 <br />r° Si'- <br />2° 17' <br />-'° 58' <br />3° 43` <br />IOI . i 5 <br />32° <br />181.39 <br />1°59' <br />2°25'3°10' <br />.0208 <br />3° 58' <br />101.33 ' <br />34° <br />171.01 <br />2° 06 <br />2° 33 <br />3° 21' <br />12' <br />I01.48 <br />360 <br />i6r.8o <br />2° 13' <br />2° 41' <br />3° 33' <br />4° 26'." <br />ioi.66 <br />38° <br />15J.58 <br />2° 20' <br />2° 49, <br />3° 44 <br />40 40 <br />.6667 .7500 <br />40° <br />146. i9 <br />2° 27' . <br />2° 57' <br />3° 55' <br />4° 54' <br />IO2.o6 <br />.42' <br />139.52 <br />2° 341 <br />3° 05' <br />4° 07' <br />5° 08" <br />102.29 <br />440 <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 />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 />600,111 <br />103.24 <br />520 <br />.114.06 <br />3°'09'. <br />3-46' <br />5° oz' <br />.6° 17' <br />103.54 <br />54° <br />110.11 <br />3° 16' <br />3' 54' <br />S° 13' <br />6-31, <br />103.84 <br />56° <br />io6.5o <br />3° 22' <br />4° 02' <br />S° 23' <br />6° 44' <br />104. 14 <br />58° <br />103. I.} <br />3° 29' <br />4° 10' <br />50 34' <br />60 57' <br />104.43 <br />6o° <br />100.00 <br />3° 35' <br />4° 18' <br />5° 44, <br />7° 11' <br />I04.72 <br />IX <br />CURVE FORMULAS <br />T= R tan A I R.= T cot. z I Chf= chord2 <br />_ 5o tan . I orde. R <br />T - Sin. D R = 50 <br />Sin. }'D = 50 ' Sin. 4' D No. chords = I <br />R E =Res. sec 2 I D <br />.' So tan -'II <br />SIn a <br />j D , T . E = T tan t 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 near]}-. <br />To find angle fora given distance and deflection. <br />Rule i. Multiply the given distance by -01745 (def. for f° for. I ft. <br />see .Table II.), and divide given deflection by the product. <br />Rule 2. Multiply given deflection by j7.3, and divide the product by <br />the given distance. <br />To find deflection for a given angle and distance. Multioly the angle <br />by .01745, and the product by the distance. <br />GENERAL DATA <br />RIGHTANGLE TRIANGLES. Square the altitude, divide by twice the <br />base. Add quotient to base for hypotenuse. <br />Given Base loo, Alt. 1o.Io2=200=.5. Ioo-I-.5=10o.5 hyp• <br />Given Hyp. ioo; A1t.25.25=200=3.125. IOO-3.125=96.875=13ase. <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 I I, and divide by 7. <br />LEVELING. The correction for curvature and refraction, in feet <br />and decimals of feet is equal to 0.574 d', where d is the distance in miles. <br />The correction for curvature alone is closely, J14. The combined cor- <br />rection is negative. ' <br />PROBABLE ERROR. If d„ d2, da, etc. are the discrepancies of various <br />results from the mean, and if 7-d' =the sum of the squares of these differ- <br />ences and n=the number of observations, then the probable error of the <br />mean= + 0.674qid' <br />SOLAR EprimaERI9. Attention is called to the Solar Ephemeris for <br />the current year, published by Iteuffel & Esser Co., and furnished free of <br />charge upon request, which is 341x58 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;.trigonometrid f ormulas; Natural andLogarithmic Functions; <br />and Logarithms' of Numbers. <br />TABLE IV. = Xfinutes in Decimals of . Degree. <br />1' .0167 11' .1833 21' .3500 31' .5167 41' .6333 51' .5500 <br />2 .0333 12 .2000 22 .3667 32 ..1333 49 .7000 63 .8667 <br />3 .0500 ]3 .2167 23 .3333' 33 .6500 43 .7167 63 .5533 <br />4 :0667 31 .^333 24 .4000 34 .5667 44 .7333 54 .9000 <br />5 .0833 15 .2500 25 .4167 35 .5333 45 .7500 55 .916 <br />TABLE V. -'Inches <br />in Decimals Of a l+oot. <br />1-16 <br />3-32 <br />? e <br />3-lG <br />% <br />5-163 <br />?'s <br />'L4 11 <br />�+ <br />.0052 <br />.0078 <br />.0101 <br />.0156 <br />.0208 <br />.0260 <br />.G>13 <br />D11.7 .6521 <br />0625 <br />.0729 <br />1 <br />2 <br />3 <br />4 <br />6 <br />G <br />7 <br />s 9 <br />10 <br />11 <br />0833 <br />.166'7 <br />2500 <br />.3333 <br />.4167 <br />.5000 <br />.5833 <br />.6667 .7500 <br />.8333 <br />.9167 <br />7 <br />6 <br />.1000 <br />16 <br />26.4333 <br />363 <br />.6000 <br />40 <br />.7667 <br />56 <br />.9333 <br />7 <br />17 <br />.2667 <br />27 . <br />37 <br />Gltii <br />47 <br />.7833 <br />57 <br />.0500 <br />8 <br />.I1G7 <br />is <br />.2833 <br />3000 <br />28 <br />.4500 <br />.4667 <br />3S <br />G333 <br />48 <br />;600 <br />58 <br />.9667 <br />9 <br />.1'3:;3 <br />.1500 <br />19 <br />1 <br />20 <br />.4533 <br />39 <br />.650, <br />49 <br />;1617 <br />59 <br />.9333 <br />10 <br />.1667 <br />20 <br />.3167 <br />.3333 <br />30 <br />..1 <br />40 <br />.6667 <br />50 <br />1 .8333 <br />GO <br />1.0000 <br />TABLE V. -'Inches <br />in Decimals Of a l+oot. <br />1-16 <br />3-32 <br />? e <br />3-lG <br />% <br />5-163 <br />?'s <br />'L4 11 <br />�+ <br />.0052 <br />.0078 <br />.0101 <br />.0156 <br />.0208 <br />.0260 <br />.G>13 <br />D11.7 .6521 <br />0625 <br />.0729 <br />1 <br />2 <br />3 <br />4 <br />6 <br />G <br />7 <br />s 9 <br />10 <br />11 <br />0833 <br />.166'7 <br />2500 <br />.3333 <br />.4167 <br />.5000 <br />.5833 <br />.6667 .7500 <br />.8333 <br />.9167 <br />