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VIII z✓ 3. . TABLE IL-Radii, Ordinates and Deflections. Chord=100 f t. L.-1Ix CURVE FORMULAS --� Mid.' T. Def. Mid. Tim Def. ' T = R tan I ' R"d11° Ord. Diet. Dist. 1 F!. Deg• Radme Ord. Dist. Dist. i Ft 5o tan}I R = T cot. J I Chord def. a chord t t t. t. T Sin.j D R = 60 R tt. ft. ft. ft. � •"10' 34377. .036 .145 .291 0.05 7" 819.0 1.528 6.105 12.21 2.10 20 17189. 073 .291 .582 0.10 20' 781.8 1.800 6.395 12.79 2.20 Sin. J D =So tan}I E-Rln. jex D I No.chords=D 30 11459. .109 .436 .873 0.15 30 764.5 1.637 6.540 13.08 2.25 R j' 40 8594.4 .145 .582 1.164 0.2o 40 747.9 1.673 6.685 13.37 2.30 50 6875.5 .182 .727 1.454 0.25 8 716.8 1.746 6.976 13.95 2.40Sin. D - E=T tan I Tan.def._}chord d�? 6 1 5729.6 .218 .873 1.745 0.30 20 688.2 1.819 7.266 14.53 2.50 T 10 4911.2 .255 1.018 2.036 ,0.35 30 674.7 1.855 7.411 14.82 2.55 The square of any distance, divided by twice the radius will equal 20 4297.3 .291 1.164 2.327 0.40 40 661.7 1.892 7.556 15.11 2:60 30 3819.8 .327 1.309 2.618 0.45 9 637.3 1.965 7.846 15.69 2.70 the distance from tangent to curve,very nearly. //f7 40 3437.9 .364 1.454 2.909 0.50 20 614.6 2.037 8.136 16.27 2.80• To find angle for a given distance and deflection. ° 50 3125.4 .400 1.600 3.200 0.55 30 603.8 2.074 8.281 16.56 2.85 Rule I. Multiply the veri distance b OI ° 2 2864.9 .436 1.745 3.490 0.60 40 593.4 2.110 8.426 16.85 2.90 p O y 74$ (def. I for I ft 10 2644.6 .473 1.891 3.781 0.65 10 573.7 2.183 8.716 17.43 3.00 see Table IL),and divide given deflection by the product. 20 2455.7 .509 2.036 4.072 0.70 30 546.4 2.292 9.150 18.30 3.15 Rule 2. Multiply given deflection by 37.3,and divide the product by 30 2292.0 .545 2.181 4.363 0.75 11 521.7 2.402 9.585 19.16 3.30 thei0ven distance. 40 2148.8 .582 2.327 4.654 0.80 30 499.1 2.511 10.02 20.04 3.45find deflection for a i 50 2022.4 .618 2.472 4.945 0.85 12 478.3 2.620 10.45 20.91 3.60 given angle and distance. Multiply the angle ! 1910.1 .655 2.618 5.235 0.90 30 459.3 2.730 10.89 21.77 3.75 by•01745.and the product by the distance, i 10 1809.6 .691 2.763 5.526 0.95 13 441.7 2.839 11.32 22.64 3.90 4, f 20 1719.1 .727 2.903 5.817 1.00 30 425.4 2.949 11.75 23.51 4.05 GENERAL DATA 3,-�. 30 1837.3 .764 3.054 6.108 1.03 14 410.3 3.058 12.18 24.37 4.20 RIGHT ANGLE TRIANGLES. Square the altitude, divide b�t�viCe the 40 1562.9 .800 3.199 6.393 1.10 30 396.2 3.168 12.62 25.24 4.35 base. Add quotient to base for hypotenuse. �/ / 50 1495.0 .836 3.345 6.689 1.15 15 383.1 3.277 13.05 26.11 4.60 Given Base I00 Alt.IO.IO�=200=. I00 l 1432.7 .873 3.490 6.980 1.20 30 370.8 3.387 13.49 26.97 4.85 r 5 -F.5-100.5 hyp. 10 1375.4 .909 3.635 7.271 1.25 16 359.3 3.496 13.92 27.84 4.80 Given Hyp. Ioo,Alt.25.252+200=3.125. 100-3.125=96.875=Base. 20 1322.5 .945 3.718 7.561 1.30 30 348.5 3.606 14.35 28.70 4.95 Error in first example, in last, 30 1273.6 .982 3.921 7.852 1.35 17 338.3 3.716 14.78 29.56 5. 0 .043• 40 1228.1 1.018 4..071 8.143 1.40 18 319.8 3.935 15.64 31.29 5.440 To find Tons of Rail in one mile of track: multiply weight per yard . 50 1185.8 1.055 4.217 8.433 1.45 19 302.9 4.155 16.51 33.01 5.70 by II,and divide by 7. i_ 1146.3 1.091 4.362 8.724 1.50 20 287.9 4.374 17.37 34.73 6.00 LEVELING. The correction for curvature and refraction ii 10 1109.3 1.127 4.507 9.014 1.55 21 274.4 4.594 18.22 36.44 6.30. 20 1074.7 1.164 4.653 9.305 1.60 22 262.0 4.814 19.08 38.16 6.60 and decimals of feet is equal to 0.574 d2,where d is the distance in feet in miles. 30 1042.1 1.200 4.798 9.596 1.65 23 250.8 5.035 19.94 39.87 63.90 _` The correction for curvature alone is closely,Jds. The combined cod- 7 /o 40 1011.5 1.237 4.943 9.886 1.70 24 240.5 5.255 20.79 41.58 7.20 rection is negative. v 7 50 982.6 1.273 5.088 10.18 1.75 25 231.0 5.476 21.64 43.28 7.50 y/ 6 955.4 1.309 5.234 10.47 1.80 26 222.3 5.697 22.50 44.99 7.80 PROBABLE ERROR. If d4,ds,d„etc.are the discrepancies of various 10 929.6 1.346 5.379 10.76 1.85 27 214.2 5.918 23.35 46.69 3.10 results from the mean,and 1f Fd'=the sum of the squares of these differ- 20 905.1 1.382 5.524 11.05 1.90 28 206.7 6.139 24.19 48.33 8.40 ences and n=the number Of observations,then the probable error of the 30 881.9 1.418 5.669 11.34 1.95 29 199.7 6.360 25.04 50.07 8.70 40 859.9 1.455 5.814 11.63 2.00 1 30 193.2 6.583 25.88 51.76 9.00 mean=f OX74 The middle ordinate in inches for any cord of length(C)is equal to .0012 C' n(n-1f1 `y o multiplied by the middle ordinate taken from the above table. Thus,if it SOLAR EPHEMERIS.Attent10 is called to the Solar Ephemeris fir' b.8.00d to bend 130 ft.rail to fit a 10 degree curve,its middle ordinate should the current year,published by Keuffel&Esser Co.,and furnished free of •7 S be.om2xsoox2.1s3 or 2.:.6 inches. charge upon request,which is 3x5}in.,with about 90 pages of TABLE III. Deflections for Sub Chords for Short Radius Curves. useful to the Surveyor; such as the adjustments data very of transits, levels and Degree Radius 34 sub chord Length solar attachments;directions and tables for determinin the meridian of 50 R =sin of}def•angle and the latitude from observations on the sun and Polaris;stadia meas- Curve sin.l def.ang. 12.5 Ft. 15 Ft. 20 Ft. of arc 25 Ft. for 100 ft tlrements;magnetic declination;arithmetic constants;English and Metric 30° 193.18 I°51' 2° 11 2°58' 3°43' 101.13 conversions;tngonometricformulaa;NaturalandLogarithmicFunetion8; 32° 181.39 1°59' -o z5' 3° 1O' 3°58' IOI.33 and Logarithms of Numbers. 34° 171.01 2°o6' z°33' 3°21' 4° Iz' IOI.48 TABLE IV.-Minutes in Decimals of a Degree. 36° 161.8o 2° 13' 2°41' 3°33' 4°26' Ioi.66 11 .0167 lY .1833 21' 3500 31' 5167 411 8833 SI' 8500 38° 153.58 2°20' 2°49' 3°44' 4°40' 101.83 0333 12 .2000 22 .3667 32 .5333 42 .7000 b2 .8667 40° 146.19 2°27' 2°57' 3°5$' 4°54' 102.06 6 •0`00 13 .2167 23 .3833 33 .5500 43 .7167 53 .8833 2° 1 0 o i o 1 o 4 .0667 14 .2333 24 .4000 i4 .5667 44 .7333 i4 .9000 4 39.52 2 34 3 05 4 07 5 08' 102.29 a 0833 15 .2500 25 .4167 35 .5933 46 .7500 d5 .9167 44° 133.47 2041 1 3° 13' 4° 18' 5°22' 102.53 6 .1000 10 .2667 26 .4333 ae .6000 46 .7667 66 .9333 46° 127.97 2°48' 3°21' 4°29' 5°36' 102.76 7 .1167 17 .2833 27 .4500 37 .0167 47 .7833 57 .9500 480 122.92 2°55' 3°2 ' ° O' ° o' IO 00 8 - .1333 18 .3000 28 .4667 38 G= 4s .8000 58 .9667 9 4 4 5 5 3• 9 .1500 18 .3187 29 .4833 39 .6500 49 .9167 69 .9833 50 I18.31 3°02' 3°38' 4°51' 6004' 103.24, 19 .1667 20 .3333 30 .5000 49 .6667 50 .8333 N 1.00°0 520 114.06 3°09' 3°46' 5°oz' 6°17' 103.54 _ TABLE V.-Inches in Decimals of a Foot. 54 II0.II 3° 16' 3°54' S° 13' 6°31' 103.84 560373-32 -16 0 106.50 30 22 40 02�. 50 23� 60 44� 104.14 s 0052 0078 l0 64 .0156 .020826 X13 .01'17 0521 as o7 $8 103.14 3 z9 4 Io 5 34 6 57 104.43 1 2 3 4 b 8 7 8 9 10 11 60° 100.00 3°35' 4°18' 5°44� 7°11' 104.72 _ 0633 .1067 .2500 .3333 .4107 .5000 .5833 .6667 .7500 .8333 .9167