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VIII <br />TAnLn IL - Radii, Ordinates and Deflections. Chord =100 ft. <br />Deg. - <br />Kj <br />�Y <br />Radius <br />Mid. <br />Ord. <br />Tan. <br />Dist. <br />Def. <br />Dist. <br />'ef.IMid <br />for <br />1 Ft. <br />Deg. <br />Radius <br />Ord: <br />Tan <br />Dist. <br />Def. <br />Dist. <br />Def. <br />for <br />1 Ft. <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 />G 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 />C% 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 />8594.4.145 <br />.102.06 <br />.582 <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 />�1 <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 />48' <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 />30 <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 />c 30 <br />3819..8 <br />.327 <br />1.300 <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 />20 <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 />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 />I0' <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 />2.402 <br />9.585 <br />19.16 <br />3.30 <br />40' <br />2148.8 <br />.582 <br />2.a27 <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 />9 <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 />2.839 <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 />2.949-11.75 <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 />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 />4 <br />1432.7.873 <br />3.490 <br />6.980 <br />1.20 <br />30 <br />370.8 <br />3.387 <br />13.49 <br />26.07 <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. <br />348.5 <br />.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 />.155 <br />16.51' <br />33.01 <br />5.70 <br />S ' <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 />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 />_ <br />22.50 <br />44.99'7.80 <br />10 <br />929.6 <br />1.346 <br />5.379 <br />10.76 <br />1.85 <br />27 <br />214.2 <br />.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 />b.669 <br />11.34 <br />1.95 <br />29 <br />199.7:6.360 <br />25.04 <br />50.07 <br />8.70 <br />40 <br />859.9 <br />1.455 <br />b.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 'n inches for any cord of length (0) is equal to .0012 C' <br />multiplied by the 111 Ne 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 />TAME III. Deflections for Sub Chords for Short Radius Curves. <br />Degree f <br />of <br />Curve <br />R 50us <br />ARchord =sin of I def. angle <br />Length. <br />of arc <br />for 100 ft. <br />sin. a def. ang. <br />12.5 Ft. <br />15 Ft: <br />20 Ft. ' <br />25 Ft. <br />30° <br />i93.18 <br />1° 5f' <br />2° 171 <br />2° 58• <br />3° 43' <br />I0I.15 <br />°- - <br />X32 <br />IHI.39.... <br />- 1 <br />1o 59 <br />..20 251 - <br />o <br />...3 tor <br />.. <br />o 1 :. <br />3 58 <br />101.33" <br />34° <br />171.01 <br />2°'06' <br />20 33' <br />3° 21' <br />4° 12' <br />I01.48 <br />36° <br />161-. 8o <br />2° 13' <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 />IOI.85 <br />- o <br />-40 <br />..' <br />146.19" <br />o r- <br />2, 27 <br />o r <br />2 57 <br />o 1- <br />3 55 <br />o <br />4. 54. <br />.102.06 <br />. 42° <br />139.52 <br />2° 341. <br />3o O51 <br />4'67' <br />5o'o8 <br />i02.29 <br />• 44° <br />133.47: <br />2°..41' <br />3° 131 <br />4.0 18' <br />5°. 22' <br />102.53 <br />46° 1 <br />127.97 <br />2°"48' <br />39-21'. <br />4° 29'. <br />5° 36' " <br />' -102..76` <br />48' <br />122.92 <br />2°55' <br />3°29' <br />4°4O' <br />Soso, <br />163.00 <br />• 5o° ; <br />118.31•:... <br />3° 02' <br />30 381 <br />.4o 51:• ' <br />. 6° 04-1 <br />103.24 <br />520 _I <br />_ 114.06, <br />3° 09, <br />30 46' <br />5° 62' <br />6°..17' <br />103.54 <br />:54°� <br />IIO.I1 <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 />5° 23' :. <br />6° 44' <br />104. 14. <br />.58°i <br />' "103.14" <br />3° 29'. <br />4° lo' <br />5° 341 <br />.6057' <br />10443 <br />60° • <br />100.00 <br />3° 35' <br />4° i8! <br />5° 44� <br />7°• II' <br />104.72 <br />IX <br />CURVE FORMULAS <br />.. b <br />T = R tan 2 I R= T cot. z I chord' <br />5o tan ;:I Chord def. = <br />T= Sin. zll R= 50 R <br />3 <br />Sin. s D = R Sin. 2_D No. chords = I <br />E = R ex. sec I D <br />Sin. I D =- . t T 2 I .E = T tan I I Tan. def. = z 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 .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. IO.IO`2o0=.5. 100-1-.5=100.5 hyp. <br />Given Hyp. loo, Alt. 25.252=too=3.125. 100-3.125=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 I I, and divide by 7. <br />LEVELING. The correction for curvature ,and refraction, in feet + <br />and decimals of feet is equal to 0.574d', 'where d is the distance in. miles. <br />The correction for curvature alone is closely,:2sd2. The combined cor- <br />rection is negative.- <br />PROBABLE <br />egative.PROBABLE ERROR. If di, d2, d3, etc. are the discrepancies of various <br />results from the mean, and if Ede=the sum of the squares.of these differ- <br />ences'and n=the number'of observations, then the.probable error of the <br />mean= Y_d2: <br />10.6745 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 upon <br />trequest. This handy booklet, 3-gx6 in., has about 190 pages of data very <br />'useful to the Surveyor; such as the adjustments of transits, levels and solar <br />attachments; directions and tables for determining the meridian and the <br />latitude from observations on- the 'sun and Polaris; stadia measurements; <br />magnetic declination; arithmetic constants, etc. <br />TABLE IV. =Minutes in Decimals of a Degree. <br />1t :0167 11' .1833 21' .3500 311 .5167 41' .6833 511 .8500 <br />2 • .0333 12 :2000 22 .3667 32 .5333 42 .7000 52 .8667 <br />�3 .0500 13 .2167 23 .3833 33 .5500 43 .7167 53 .8833' <br />4 ..0667 14 .2333 24 .4000 34 .5667 44 .7333 54 .9000 <br />5 .0833 15 .2500 25 .4167 35 .5833 45 .7500 55 .9167 <br />- 6 .1000 16 .2667 26 .4333 36 .6000 46 .7667 56 .9333 <br />71167 17 .2833 27 - .4500 37 .6167 47 .7833 57 .9500 <br />8 .1333 18 .3000 28 .4667 38 .6333 48 .8000 58 .9667 <br />9 .1500 19 .3167 29 .4833 39 .6500 49 :8167 59 .9833 <br />10" .1667 20 .3333 30 .5000 40 .6667 50 .8333 60 1.0000 <br />TABLE V. -Inches in Decimals of a Foot. <br />1-16 3-32 Y 3-16% 5-16 % A I 1.0729 .0052 .0078 .0104 .0156 I .0208 I .0260 .0313 I .0417 I .0521 .0625 <br />1 2 3 4 5 6 7 8 9 to 11 <br />.0833 .1667 .2500 .3333 .4167 :5000 .5833 .6667 .7500 .8333 .9167 <br />