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VIII 1X <br />TABLE IL --Radii, Ordinates and Deflections. Chord 100 ft. CURVE FORMULAS <br />Deg. <br />Radius <br />Mid. <br />Ord <br />Tan. <br />Dist. <br />Def. <br />Dist. <br />Def.Mid <br />for <br />1 Ft. <br />Deg. <br />Radius <br />Ord. <br />Tan. <br />Diet: <br />Def. <br />Dist' <br />Def. <br />for <br />1 Ft. <br />2° 171 <br />2° 58' <br />it. <br />101.15 <br />32° <br />181.39 <br />1° 59' <br />t. <br />t. <br />t. <br />t. <br />34° <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 />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 />.30 <br />11459. <br />.109 <br />.436 <br />.873 <br />0.15 <br />30 <br />764.5 <br />1.637 <br />6.54013.08 <br />144° <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 />6.685 <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 />2.036 <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 />30 <br />-3819.8 <br />.327 <br />1.309 <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 />3137.9 <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 />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 />2155.7 <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 />2.327 <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 />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 />2.839 <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 />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 />1195.0 <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 />! <br />1132.7 <br />.873 <br />3.490 <br />6.980 <br />1.20 <br />30 <br />370.8 <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 />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 />3 0 <br />348.5 <br />3.606 <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 />1.018.4.071 <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 />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 />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 />39.87'6.90 <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 />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 <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 />199.7 <br />6.360 <br />25.04 <br />50.07 <br />8.70 <br />40 <br />859.9 <br />1.455 <br />5.814 <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 in inches for any cord of length (C) is equal to .0012 C2 <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 />R 50adius/2 <br />sub chord _sin of def. angle <br />R <br />Length <br />of arc <br />for 100 ft. <br />sin, c def. ang. <br />12.5 Ft. <br />15 Ft. <br />20 Ft. <br />25 Ft. <br />30° <br />193.18 <br />10 51' <br />2° 171 <br />2° 58' <br />3° 43 <br />101.15 <br />32° <br />181.39 <br />1° 59' <br />2° 25' <br />3° lo' <br />3° 58' <br />101.33. <br />34° <br />171.01 <br />2° 06' <br />2° 33' <br />3° 21' <br />4° 12' <br />101.48 <br />360 <br />161.8o <br />2° 13'_ <br />2°41'' <br />3°33' <br />4°26' <br />101.66 <br />380 <br />153.58 <br />2° 20' <br />2°49' <br />3° 44' <br />4° 40' <br />101.85 <br />400 <br />146.19 <br />202 7' <br />z° 57 <br />30 ;55' <br />4° 54' <br />102. o6 <br />42° <br />139.'52 <br />2°34 <br />3° 05' <br />4° 07 <br />5° o8 <br />102.29 <br />144° <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 />4o29 <br />50361 <br />102.76 <br />480 <br />122.92 <br />2° 55'3° <br />29", <br />4° 40 <br />5° 50' <br />103.00 <br />500 <br />118.31. <br />3° 02 <br />3° 38' <br />4° 5i'- <br />60 04' <br />103.24 <br />5z° <br />i14._o6 <br />3° 09, <br />3° 46/5° <br />O2' <br />6° 17 <br />103.54 <br />540 <br />1lO:rl <br />1110-.Il <br />3° 16' <br />3° 54' <br />5° 13' <br />6°.31' <br />103.84 <br />56° <br />.4833 <br />3° 22' <br />4° 02' <br />50 23' <br />6044 <br />104-14 <br />590 <br />'103.14 <br />3° 29 <br />4° 1O. <br />5° 34'' <br />6° 57' <br />104.43 <br />60° <br />Ioo.00 <br />3° 35' <br />4° 18' <br />5° 44' <br />7° 11' <br />104.72 <br />T R tan <br />Z I R= T cot. 2 1 chord= <br />_ Chord def. = <br />T 5o tan Sin. 2 D R 50 R <br />Sin. 2 3 D = 50 Sin. 2 D No. chords = I <br />R E= R ex. sec 2 I D <br />Sin. 12 D 5o,t T 2 I E= .T tan t I Tan. def. = 2 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 1 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.1o2=200=,5. loo+: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, .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 0:574d2, where d is the distance in miles. <br />The correction for. curvature alone is closely, ;d2. The combined cor- <br />rection is negative. <br />PROBABLE ERROR. If dt, d2, ds, etc. are the discrepa6cies.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 />rnean FYd2, <br />-0.6745 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 />request. This handy booklet, 36x6 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 />11 <br />.0167 <br />116 <br />.1833 <br />211 <br />.3500 <br />311 <br />.5167 <br />41" <br />.6833 <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 />3 <br />.0500 <br />13 <br />.2167 <br />23 <br />.3833 <br />33 <br />.5500 <br />43 <br />.7167 <br />53 <br />.8833 <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 />.0833 <br />15 <br />.2500 <br />25 <br />.4167 <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 />.6000 <br />46 <br />.766756 <br />.9333 <br />7 <br />.1167 <br />17 <br />.2833 <br />27 <br />.4500 <br />37 <br />.6167, <br />47 <br />.7833. <br />57 <br />:9500 <br />8 <br />.1333 <br />18 <br />.3000 <br />28 <br />.4667 <br />38 <br />.6333 <br />48 <br />.8000 <br />58 <br />.9667 <br />9 <br />.1500 <br />19 <br />.3167 <br />29 <br />.4833 <br />39 <br />.6500 <br />49 <br />.8167 <br />59 <br />.9833 <br />10 <br />-3667 <br />1 20 <br />1 .3333 <br />1 30 <br />1 .5000 11 <br />40 <br />1 .6667 11 <br />50 <br />1 .8333 11 <br />60. <br />1.0000 <br />TABLE V.=Inches in Decimals of a Foot. <br />1-16 3-32 3-16 5-16 % Y % <br />0052 .0078 .0104 .0156 .0208 .0260 .0313 .0417 .0521 .0625 .0729 <br />.0833 .16;67 .2500 .3333 I .4167 .5000 .5833 .6667 .7500 .8333 .9167 <br />