CURVE FORMULAS T = R tan 1/2 I T = 50 tan 1/2 I / Sin. 1/2 D Sin. 1/2 D = 50 / R Sin. 1/2 D = 50 tan 1/2 I / T R = T cot. 1/2 I R = 50 / Sin. 1/2 D E = R ex. sec 1/2 I E = T tan 1/4 I Chord def. = chord^2 / R No. chords = I / D Tan. def. = 1/2 chord def. The square of any distance, divided by twice the radius, will equal the distance from tangent to curve. very nearly. To find angle for a given distance and deflection. Rule 1. Multiply the given distance by .01745 (def. for 1° for 1 ft.) and divide given deflection by the product. Rule 2. Multiply given deflection by 57.3, and divide the product by the given distance. To find deflection for a given angle and distance. Multiply the angle by .01745, and the product by the distance. GENERAL DATA RIGHT ANGLE TRIANGLES. Square the altitude, divide by twice the base. Add quotient to base for hypotenuse. Given Base 100, Alt. 10.10^2 ÷ 200 = .5. 100 + .5 = 100.5 hyp. Given Hyp. 100, Alt. 25.25^2 ÷ 200 = 3.125. 100 - 3.125 = 96.875 = Base. Error in first example, .002; in last, .045. To find Tons of Rail in one mile of track: multiply weight per yard by 11, and divide by 7. LEVELING. The correction for curvature and refraction, in feet and decimals of feet is equal to 0.574 d^2, where d is the distance in miles. The correction for curvature alone is closely, 3/8 d^2. The combined correction is negative. PROBABLE ERROR. If d1, d2, d3, etc. are the discrepancies of various results from the mean, and if Σd^2 = the sum of the squares of these differences and n = the number of observations, then the probable error of the mean = ± 0.6745 √(Σd^2 / n(n-1)) MINUTES IN DECIMALS OF A DEGREE | 1' | .0167 | 11' | .1833 | 21' | .3500 | 31' | .5167 | 41' | .6833 | 51' | .8500 | | 2 | .0333 | 12 | .2000 | 22 | .3667 | 32 | .5333 | 42 | .7000 | 52 | .8667 | | 3 | .0500 | 13 | .2167 | 23 | .3833 | 33 | .5500 | 43 | .7167 | 53 | .8833 | | 4 | .0667 | 14 | .2333 | 24 | .4000 | 34 | .5667 | 44 | .7333 | 54 | .9000 | | 5 | .0833 | 15 | .2500 | 25 | .4167 | 35 | .5833 | 45 | .7500 | 55 | .9167 | | 6 | .1000 | 16 | .2667 | 26 | .4333 | 36 | .6000 | 46 | .7667 | 56 | .9333 | | 7 | .1167 | 17 | .2833 | 27 | .4500 | 37 | .6167 | 47 | .7833 | 57 | .9500 | | 8 | .1333 | 18 | .3000 | 28 | .4667 | 38 | .6333 | 48 | .8000 | 58 | .9667 | | 9 | .1500 | 19 | .3167 | 29 | .4833 | 39 | .6500 | 49 | .8167 | 59 | .9833 | | 10 | .1667 | 20 | .3333 | 30 | .5000 | 40 | .6667 | 50 | .8333 | 60 | 1.0000 | INCHES IN DECIMALS OF A FOOT | 1-16 | 3-32 | 1/8 | 3-16 | 1/4 | 5-16 | 3/8 | 1/2 | 5/8 | 3/4 | 7/8 | |.0052 | .0078 | .0104 | .0156 | .0208 | .0260 | .0313 | .0417 | .0521 | .0625 | .0729 | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |.0833 | .1667 | .2500 | .3333 | .4167 | .5000 | .5833 | .6667 | .7500 | .8333 | .9167