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TABLE IL - Radii, Ordinates and Deflections. Chord =100 f t. CURVE FORMULAS
<br />Dom''
<br />Radius,
<br />Mid,
<br />Ord.
<br />Tan.
<br />Dist,
<br />Def.
<br />Dist.
<br />Def.
<br />for
<br />1 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 />t.
<br />ft.
<br />ft.
<br />ft.
<br />i
<br />I°59'
<br />ft.
<br />ft.
<br />ft,
<br />ft.
<br />i
<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 />6.540
<br />13.08
<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 />Il
<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 />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 />$
<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 />3.7810.65
<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 />2.327
<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 .478.3
<br />2.620
<br />10.45
<br />20.91
<br />3.60
<br />8..
<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 />SO
<br />1495.06
<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 />1432.7
<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 />30
<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
<br />4.071
<br />8.143
<br />1,40
<br />IS
<br />319.6
<br />3.935
<br />15.64
<br />31.29
<br />5,40
<br />50
<br />1185.5
<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 />8
<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 />0"'.
<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 />1 859.9
<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 C'
<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 />Radius,
<br />A sub chord = sin of $ def. angle
<br />R
<br />Length
<br />of arc
<br />for 100 ft.
<br />sin. z def. ang.
<br />12.5 Ft.
<br />15 Ft.
<br />20 Ft.
<br />25 Ft.
<br />..,30193,.18
<br />..
<br />51'
<br />i° 51'
<br />2° 17'
<br />2° 58' .
<br />3' 43'
<br />101.15
<br />32°
<br />181.39
<br />I°59'
<br />2°25'
<br />3°Io'
<br />3° 58'
<br />101.33
<br />34'
<br />I'7 I , 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 />ioi .66
<br />38'
<br />153,58 _
<br />2020 1
<br />2° 49'
<br />3° 44'
<br />4° 40'
<br />IoI.85
<br />40°
<br />146. i9
<br />2° 27'
<br />2° 57 1
<br />30 55'
<br />° 4 54'
<br />102.o6
<br />42'
<br />139.52
<br />2° 34'
<br />3° 05'
<br />4° 07'
<br />S° 08'
<br />102.29
<br />44°
<br />133.47
<br />20 41'
<br />3° 13'
<br />4° 18"
<br />5° 22'
<br />102.53
<br />46°
<br />-127.97
<br />2' 48'3°
<br />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 />6° o4'
<br />103.24
<br />52°
<br />114,06
<br />3° 09'
<br />3° 46'
<br />5° 02'
<br />6° 17'
<br />103.54
<br />54°
<br />IIO.II_
<br />3° 16'
<br />3' 54'
<br />8'•13'
<br />6° 31'
<br />Io3.84
<br />56°
<br />166.50
<br />3022
<br />4' O2'
<br />5° 23'
<br />6°44'
<br />104.14
<br />58°
<br />103.14
<br />3° 29
<br />4' Io'
<br />S° 34'
<br />6° 57'
<br />104.43
<br />60°
<br />lbe. 00
<br />3° 3S'
<br />4° 18'
<br />S° 44'
<br />7' 11'
<br />104.72
<br />T R tan 8 I R= T cot. 2 I chorda
<br />50 tan I Chord def. = R
<br />T Sin. B D R = 50
<br />Sin. D = 50 Sin. D No. chords
<br />D
<br />R' E = R ex. sec ;,I
<br />a I
<br />Sin. D -_'5o tan , r 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 .ioo,Alt. 1o.102=200=.5. IoO+-5=ioo.5 hyp.
<br />Given Hyp. ioo, Alt. 25.252 +200=3.125. 100-3.125 =96.875 =Base.
<br />Error in first example, .002; 1n last, .045.
<br />To find Tons of Rail in one mile of track: multiply weight per yard
<br />by iI, and divide by 7.
<br />LEVELING. The correction for curvature and refraction, in feet
<br />and decimals of feet is equal to 0.574 da, where d is the distance in miles.
<br />The. correction for curvature alone is closely, Jd2. The. combined cor-
<br />rection is. negative.
<br />PROBABLE ERROR. If d; d., da, etc. are the discrepancies of various
<br />results from the mean, and if 7_&=the sum of the squares of these dif�cr-
<br />ences and n=the number of observations, then the probable error of the,
<br />mean=+0.6745 n(d
<br />SOLAR EPHEMERIS. Attention is called to the Solar Ephemeris for'
<br />the current year, published by Keuffel & Esser Co., and furnished free "of
<br />charge upon request, which is 3;x58 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-, trigonometric formulas; Natural andLogarithmic Functions;
<br />and Logarithms of Numbers.
<br />TABLE IV. - Minutes.in Decimals of a Deeree.
<br />1'
<br />.0167
<br />11'
<br />.1833
<br />21'_ ,
<br />.3500
<br />31'
<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 />S
<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 />.7667
<br />56
<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 />9500
<br />'19
<br />.3167
<br />29
<br />.4833
<br />39
<br />,6500
<br />49 ,
<br />.8167
<br />b9
<br />.9833
<br />10
<br />.1667
<br />20
<br />.3333
<br />30
<br />.5000
<br />40
<br />.6667
<br />50
<br />.8333
<br />60
<br />1,0000
<br />TABLE V. - Inches in Decimals of a Foot.
<br />1-16
<br />3-32
<br />3-16
<br />5-16
<br />Y
<br />%
<br />?�
<br />.0052
<br />.0078
<br />.0104
<br />,0156
<br />:0208
<br />,0260
<br />.0313 .0417 .0521
<br />.0625
<br />.0729
<br />1
<br />2
<br />3
<br />45
<br />6
<br />7 8 9
<br />10
<br />11,
<br />0833
<br />.1667
<br />.2500
<br />.3333
<br />.4167
<br />.5000
<br />.5833 ..6667 .7500
<br />1 .8333
<br />.9167
<br />
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