|
"VIII lig
<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 />Dor
<br />1 Ft
<br />Deg.
<br />Radius
<br />Mid.
<br />Ord.
<br />Tan
<br />Dist.
<br />Def.
<br />Dist.
<br />Dor'
<br />I Ft.
<br />.6833
<br />t.
<br />ft.
<br />t.
<br />ft.
<br />7
<br />43,
<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 />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 />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 />a
<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 />104.43
<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'
<br />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 .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 />.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 />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 1
<br />359.3
<br />3.496
<br />13.92.
<br />27.84
<br />4.80
<br />20
<br />A322.5
<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'`.982
<br />3.926
<br />.7.852
<br />1.35
<br />AT
<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 />'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 />6
<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 />6 •
<br />; 955.4
<br />1.309
<br />5.234
<br />10.47:
<br />1.80_
<br />20
<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 1859.9
<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 ,0012X90OX2.*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 />j, sub chord
<br />R =sin of i def. angle
<br />Length
<br />of arc
<br />for 100 ft.
<br />sin. J def. ang.
<br />12.5 Ft.
<br />15 Ft.
<br />20 Ft.
<br />25 Ft.
<br />30°
<br />193.18
<br />.1°.51'
<br />.6833
<br />,°
<br />17'2°
<br />2
<br />58,:
<br />3°
<br />43,
<br />I01.15
<br />32°
<br />181.39
<br />1°
<br />59'
<br />2°25'
<br />52
<br />3°.I0'
<br />8
<br />3°58'
<br />13
<br />I0I.33
<br />340
<br />171.01
<br />2°
<br />o6'
<br />2°33'
<br />.7167
<br />3°21'
<br />'.8833.
<br />4°12'
<br />.5000
<br />101.48
<br />360
<br />161.8o
<br />2°13'
<br />34
<br />2`41'
<br />44
<br />3°33':
<br />54
<br />4°26'.
<br />5"
<br />'loI.66
<br />380
<br />153.58
<br />2°20'
<br />.4167
<br />2°49'
<br />.5833
<br />3°44'
<br />.7500
<br />4°40'
<br />.9167
<br />IOI.85
<br />o.
<br />40
<br />146.19
<br />o
<br />2
<br />r
<br />.27
<br />o
<br />2
<br />,
<br />57
<br />o
<br />355
<br />i
<br />o
<br />4
<br />r
<br />54
<br />102.o6
<br />42°.
<br />139.52
<br />2°
<br />34'
<br />3°05'
<br />.4500
<br />4,07'.
<br />.6167
<br />5°08'
<br />.7833
<br />102.29
<br />44°'
<br />133.47
<br />z°
<br />41'
<br />3°
<br />13'
<br />4°
<br />i8'
<br />5°
<br />2z'
<br />102.53
<br />460
<br />127:97
<br />2°48'
<br />.1500
<br />3°2I'
<br />.3167
<br />4°29'
<br />.4833
<br />S°36'
<br />.6500
<br />IO2.76
<br />48°
<br />I22.92
<br />2°
<br />55'
<br />3°
<br />29'
<br />4°
<br />40'
<br />50
<br />50'
<br />103.00
<br />50°
<br />118.31
<br />3°
<br />02'
<br />3¢
<br />38'
<br />4°
<br />51'
<br />60
<br />o4'
<br />103.24
<br />52°
<br />I14.o6
<br />3°09'
<br />3°46'
<br />5°02'
<br />.6°i7'
<br />103.54
<br />54°
<br />IIo.II
<br />3°
<br />16'
<br />3°
<br />54'
<br />5°
<br />13'
<br />60
<br />31'
<br />103.84
<br />560
<br />106.50
<br />3°
<br />22'
<br />4°
<br />oz'
<br />5°
<br />23
<br />6°
<br />44.
<br />104.14
<br />580
<br />103.14
<br />3°
<br />29'
<br />4°
<br />1O'
<br />5°
<br />34'
<br />6°
<br />57'
<br />104.43
<br />600
<br />100.00
<br />3°
<br />35'
<br />4°
<br />18'
<br />5°
<br />44'
<br />7°
<br />il'
<br />104.72
<br />T R50 tan I R= T cot. 1 I Chord def. "-chords
<br />T - Sin. J D R =
<br />b0 R
<br />a
<br />Sin. I D = 50 Sin: 2 D No. chords = I
<br />R E= R ex. sec I. I D
<br />Sin. J, D _ 5o tan J I ' ,I, E =T tan 1 I . Tan. def. = s a
<br />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 .6i745,'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. 10.102-200=.5. Ioo+.5=roo.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 I I, and divide by 7.
<br />LEVELING. The correction for curvature and refraction, in 'feet
<br />and decimals of feet is equal to 0.574 da, where d is the distance in miles.
<br />The correction for curvature alone is closely, Jd2. The combinedcor-
<br />rection is negative. .
<br />PROBABLE ERROR. If d�, de, da, etc. are the discrepancies of various
<br />results from the mean, and if Zd2=the sum of the squares of these dif 6i-
<br />ences and n=the number of observations, then the probable error 0f the.
<br />mean= =b 0.6745 n -n 1)
<br />SOLAR EPHEMERIS. Attention'is called to the Solar Ephemeris f&
<br />the current year, published by Keuffel & Esser Co.; and furnished free of
<br />charge upon request, which is 3'Jx5ji in., with about 90 pages of datawery
<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 f ormulas,• Natural andLogarithmic Functions;
<br />and Logarithms of Numbers...
<br />TABLE IV. - Minutes
<br />in Decimals of a Degree.
<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 />8
<br />.0500
<br />13
<br />.2167
<br />23
<br />.3833
<br />33
<br />.5500
<br />43
<br />.7167
<br />53
<br />'.8833.
<br />4.0667
<br />.5000
<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 />23
<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 />.1667
<br />20
<br />.3333'
<br />30
<br />1 .5000
<br />11 40
<br />1 .60711
<br />50
<br />1 .83333,
<br />60
<br />1.0000
<br />TABLE V. - inches in Decimals of a root.
<br />1-16
<br />3-82
<br />3-16
<br />5=16
<br />%
<br />�2 %
<br />°/a
<br />.0052
<br />.0078
<br />.0104
<br />.0156
<br />.0208
<br />.0260
<br />.0313
<br />.0417 .521
<br />. 625
<br />0729
<br />1
<br />2
<br />3
<br />4
<br />b
<br />6
<br />7
<br />8 9
<br />10
<br />11
<br />.0833
<br />.1667
<br />.2500
<br />.3333
<br />.4167
<br />.5000
<br />.5833
<br />.6687• .7500
<br />.8333
<br />.9167
<br />
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