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VIII
<br />TABLE II. - Radii, Ordinates and Deflections. Chord =100 f t.
<br />Deg. Radius
<br />Md.,
<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 />It.
<br />ft
<br />t.
<br />ft.
<br />/
<br />320
<br />ft.
<br />ft.
<br />ft,
<br />ft.
<br />3° 58'
<br />0110' 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 17189.
<br />.073.291
<br />4° 26'-'
<br />.582
<br />0.10
<br />20'
<br />781.8
<br />1.600
<br />6.395
<br />12.79
<br />2.20
<br />30 11459.
<br />.109
<br />.436
<br />S73
<br />0.15
<br />30
<br />764.5
<br />1.637
<br />6.540
<br />13.08
<br />2.25
<br />40 8594.4
<br />.145.582
<br />102.29
<br />1.164
<br />0.20
<br />40
<br />747.9
<br />1.673
<br />6.GS5
<br />13.37
<br />2.30
<br />50 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 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 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 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 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 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 3125.4
<br />.400
<br />1.600
<br />3,200
<br />0.55
<br />30
<br />603.S
<br />2.074
<br />8.281
<br />16.56
<br />2.85
<br />3 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 2644.6
<br />.473
<br />1.891
<br />3.781
<br />0.65
<br />10
<br />573.7
<br />2.183
<br />17.43
<br />3.00
<br />20 2455.7
<br />.509
<br />2.030
<br />4.072
<br />0.70
<br />30
<br />546.4
<br />2.292
<br />.8.716
<br />9.150
<br />18.30
<br />3.15
<br />30 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 />8.30
<br />40 2148.8
<br />.582
<br />2.327
<br />4.654
<br />0.80
<br />30
<br />499.1
<br />2.511
<br />16.02
<br />20.04
<br />3.45
<br />AO 2022.4
<br />.618
<br />2.472
<br />4.945
<br />0.85
<br />13
<br />478.3
<br />2.620
<br />10.45
<br />20.91
<br />3.60
<br />a 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 1809.6
<br />.691
<br />2.763
<br />5.52G
<br />0.95
<br />13
<br />441.7
<br />2.839
<br />11.32
<br />22.64
<br />3.90
<br />.20 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 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 />24.87
<br />4.2U
<br />40 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 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 />a 1432.7
<br />.873
<br />3.490
<br />6.9S0
<br />1.20
<br />30
<br />370.8
<br />3.387
<br />13.49
<br />26.97
<br />4.65
<br />10 1375.4
<br />.009
<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 1322'.5
<br />.045
<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 1273.6
<br />.982
<br />3.92G
<br />7.552
<br />1.35
<br />17
<br />338.3
<br />3.716
<br />14.78
<br />29.56
<br />5.10
<br />40 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 11.85.8
<br />1.055
<br />4.217
<br />8.433
<br />1.45
<br />19
<br />303.9
<br />4.155
<br />1G.51
<br />33.01
<br />5.70
<br />6 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 1109_3
<br />1.127
<br />4.507
<br />9.014
<br />1.55
<br />21
<br />274.44.594
<br />18.22
<br />36.44
<br />6.30
<br />20 1074.7
<br />1.164
<br />4.653
<br />9.305
<br />1.60
<br />22
<br />263.0
<br />4.814
<br />19.08
<br />83.16
<br />6.60
<br />30 1042.1
<br />1.200
<br />4.798
<br />9.596
<br />1,65
<br />23
<br />250.5
<br />5.035
<br />19.94
<br />39.87
<br />6.90
<br />40 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 />50 982.6
<br />1.273
<br />5.08S
<br />10.18
<br />1.75
<br />25
<br />231.0
<br />5.476
<br />21.64
<br />43.28
<br />7.50
<br />6 955.4
<br />1.309
<br />5.234
<br />10.47
<br />1.80
<br />26
<br />222.3
<br />5.607
<br />22.50
<br />44.99
<br />7.80
<br />10 929.6
<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 />20 905.1
<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 />30 881.9
<br />1.418
<br />5.669
<br />11.34
<br />1.95
<br />20 .
<br />190.7
<br />6.360
<br />25.04
<br />50.07
<br />8.70
<br />40 859.9
<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 e nal to .G012 C1
<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 />50
<br />A sub chordsLength
<br />It = in of 1 def. angle
<br />of arc
<br />for 100 ft.
<br />sin. 22 def. ang.
<br />12.5 Ft.
<br />15 Ft.
<br />20 Ft.
<br />25 Ft.
<br />300193.18
<br />1°51.
<br />2°11'
<br />2°58'
<br />3°43'
<br />1:01.15
<br />320
<br />181.39
<br />10 59'.
<br />2° 25'
<br />3 10'
<br />3° 58'
<br />101.33
<br />34°
<br />171.01
<br />2° 06
<br />2° 33'
<br />3°2I'
<br />4° 12'
<br />101.48
<br />36°
<br />161,80
<br />2° 13
<br />2041'
<br />30 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 />2° 27'
<br />2° 57'
<br />3° 55
<br />4° 54`
<br />102.06
<br />42'
<br />139.52
<br />2° 34'
<br />3° o5'
<br />4° 07'
<br />S° 08
<br />102.29
<br />44'
<br />133.47
<br />2° 41'
<br />3° 13'
<br />4° 1S'
<br />5° 22'
<br />102.53
<br />460
<br />727.97
<br />z° 48'
<br />3° 21'
<br />4° 29'
<br />S° 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 />so.
<br />118.31
<br />3° 02'
<br />3,38 1
<br />4° 51'
<br />6004 1
<br />103.24
<br />520
<br />I14.o6
<br />3° o9'
<br />3° 46'
<br />5° 02'
<br />6° 17'
<br />103.54
<br />54
<br />110.11 ._
<br />` 3° 16'
<br />3° 54'
<br />5° 13'
<br />6° 31'
<br />103. 84
<br />56°
<br />ig6,56
<br />3° 22'
<br />4° 02'
<br />5° 23'
<br />6° 44'
<br />104.14
<br />58° -
<br />" 103.14
<br />3,29?
<br />4° Io'
<br />5,34 1
<br />6° 57'
<br />10443
<br />60°
<br />100.00
<br />3° 35'
<br />4° 18'
<br />5° 44`
<br />1 7° 11'
<br />104.72
<br />CURVE FORMULAS
<br />T = R tan 211 R = T cot. Z I Chord def. = chord2
<br />T _ 5o tan 1 I R
<br />Sin, J D R = 50
<br />Z D
<br />Sin. } D = 5� Sin. 2 No. chords =
<br />R E= R es. sec Z 1 D
<br />So tan a I
<br />Sin. } D = ,h E = T tan I I Tan. def. = 12 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 />RIGUT ANGLE TwANGLns. Square the altitude, divide by twice the
<br />base. Add quotient to base for hypotenuse.
<br />Given Base loo, Alt. 10.1o2-,200-,5. 1oc,+.5 � 1oo•5 hyp•
<br />Given Hyp. loo, 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 11, and divide by 7.
<br />Ly-vEi.i c.. 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, 8d2. The combined cor-
<br />rection N negative.
<br />PROBABLE ERROR. If d1, d:i, d„ etc. are the discrepancies of various
<br />results from the mean, and if Y-d2=the sum of the squares of these differ-
<br />ences and n=the number of observations, then the probable error of the
<br />mean -0.6745 Id'
<br />n (n-1}
<br />SOLAR EP11EmERIS. Attention is called to the Solar Ephemeris for
<br />the current year, published by Kcuffel & Esser Co., and furnished free of
<br />charge upon request, which is 3,1-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 and Logarithmic Functions;
<br />and Logarithms of Numbers.
<br />TABLE IV. - Minutes
<br />in
<br />Decimals of n Degree.
<br />P
<br />.0167
<br />11'
<br />.1833
<br />f 21'
<br />.3500
<br />31'-11157
<br />41'
<br />6833
<br />51'
<br />.8500
<br />2
<br />.0333
<br />12
<br />.2000
<br />.0104
<br />.3667
<br />32
<br />.5333
<br />42
<br />.7000
<br />52
<br />.8667
<br />3
<br />.0500
<br />13
<br />.2167
<br />122
<br />23
<br />.3833
<br />33
<br />.5500
<br />43
<br />.7167
<br />53
<br />.8833
<br />4
<br />.1667
<br />14
<br />.2333
<br />24
<br />.4000
<br />34
<br />.5667
<br />44
<br />.7333
<br />54
<br />.9000
<br />6
<br />.0667
<br />15
<br />.2500
<br />25
<br />.4167
<br />35
<br />.5533
<br />45
<br />.7500
<br />55
<br />,9167
<br />6
<br />.0833
<br />.1000
<br />16
<br />.2667
<br />20
<br />.4333
<br />36
<br />.6000
<br />46
<br />.7667
<br />56
<br />.0333
<br />7
<br />.1167
<br />17
<br />.2833
<br />27
<br />.4500
<br />37
<br />.6167
<br />47
<br />.7833
<br />57
<br />.0500
<br />8
<br />18
<br />.3000
<br />2S
<br />.4667
<br />3S
<br />.6333
<br />48
<br />.8000
<br />58
<br />.9657
<br />9
<br />.1333
<br />19
<br />.3167
<br />29
<br />.4833
<br />39
<br />.6500
<br />49
<br />.5167
<br />59
<br />.9833
<br />10
<br />.1500
<br />.166,7
<br />20
<br />.3333
<br />30
<br />.5000
<br />40
<br />.6667
<br />50
<br />S333
<br />60
<br />1.0000
<br />TABLE V.
<br />Inches in Decimals of a Foot.
<br />1-1G
<br />3-32
<br />5-1G
<br />%
<br />f
<br />l
<br />.0052
<br />.0078
<br />.0104
<br />.OI56
<br />.0203
<br />.02GD
<br />.0313
<br />.0417
<br />.0521
<br />.0625
<br />.0729
<br />1
<br />2
<br />3
<br />4
<br />5
<br />6
<br />7
<br />8�
<br />9--
<br />10
<br />J0-
<br />11^
<br />0833
<br />.1667
<br />.2300
<br />.3333
<br />.4167
<br />.5000
<br />5832
<br />.5333
<br />.6667
<br />.7500
<br />.5333
<br />.9167
<br />
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