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M
<br />TABLz IL - Radii, Ordinates and Deflections. Chord =100 ft.
<br />.-Deg.
<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 />Or.
<br />Tan.
<br />Dist,
<br />Def.
<br />Dist.
<br />Def.
<br />for
<br />1 Ft.
<br />2° 17'
<br />t.
<br />ft.
<br />ft.
<br />ft. ,
<br />/
<br />1059'
<br />ft,
<br />ft.
<br />it.
<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 />$
<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 />2
<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 />3.781.0,65
<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 />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 />6
<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 .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 />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 />a
<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 />8t'1.
<br />955.4
<br />1.309
<br />5.234
<br />10.47
<br />1,86
<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 />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 />50
<br />34 sub chord _sin of def. angle
<br />R
<br />Length
<br />of arc
<br />for 100 ft.
<br />sin, j def, ang,
<br />12.5 Ft.
<br />I5 Ft,
<br />20 Ft.
<br />25 Ft.
<br />31'.
<br />1.93.18
<br />1° 51'
<br />2° 17'
<br />2°.58'
<br />3' 43'_
<br />161.15
<br />_2390
<br />32°
<br />181,39 -
<br />1059'
<br />2° 25'
<br />3° lo'
<br />3058'
<br />101.33
<br />34°
<br />171,01
<br />2°o6'
<br />2°33'
<br />3°21'
<br />4012'
<br />101,48
<br />360
<br />16 r-: 8o
<br />z 13'.
<br />2° 41'
<br />3°33°
<br />4° 26'
<br />io1.66
<br />38°-
<br />153.58
<br />2°.20'
<br />2° 49`
<br />3° 44'
<br />4° 40'
<br />1o1:85
<br />40°
<br />146. i9
<br />2° 27
<br />2°57'
<br />3° 55':
<br />4° 54'
<br />102. o6
<br />42°
<br />139.51
<br />20 34
<br />3° 05'
<br />4° 07'_
<br />50 08'
<br />102.29
<br />44°
<br />133.47
<br />2° 41'
<br />3° 13'
<br />4°18'
<br />50 22'
<br />102.53
<br />460
<br />127,97
<br />2°48'
<br />3°21'
<br />4°29'
<br />5°36'
<br />102,76
<br />480
<br />122.92
<br />2° 551
<br />30 29'
<br />4°401
<br />5° 50'
<br />103.00
<br />500
<br />118.31
<br />3°02'
<br />3°38'
<br />4°51'
<br />6°04'
<br />103.24
<br />52°
<br />114,06
<br />3° 09'
<br />3° 46'5°
<br />02'
<br />60 17'
<br />103,54
<br />540
<br />110.11
<br />3° 16'
<br />30 54'
<br />5° 13'
<br />6031 1
<br />103.84
<br />56°
<br />1o6.50
<br />3° 22'
<br />4° 02'
<br />5° 231'-
<br />60 44'.
<br />104.14
<br />580
<br />103.14
<br />3°29'
<br />4°10�
<br />5034'
<br />6°57'
<br />104.43
<br />6o°
<br />Ioo. oo
<br />3° 35'
<br />-40 18'-
<br />50 44'
<br />70 11'
<br />104.72
<br />i
<br />CURVE FOI 'MULLS
<br />T= R tan J I R= T cot. I chorda
<br />5o tali I Chord def. = R
<br />T Sin.,} D 60
<br />R =
<br />Sin. D - 50 Sin. # D No. chords = I
<br />• R E=Rex. sec ,1 D
<br />Sin. ,} D = 5o tan ; I E = T tan I I Tan. def.= ; 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 />{he 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 loo, Alt. 10.102=200 =.5. 100+.5 =100.5. hyp.
<br />Given Hyp. loo, Alt. 25.252=200=3.125. 100-3.125=96.875=Base.
<br />Error infirst 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 0.574d,', 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,., d2, da, etc. are the discrepancies of various
<br />results froln•the mean, and if 2:d2=the sum of the squares of these dlfior-
<br />ences and n --the number of observations' then the probable error of the,
<br />mean= + 0.6745
<br />SOLAR EPHEMERIS. Attention is called to the Solar Ephemeris far,
<br />the current year, published by Keuffel & Esser Co., and furnished free of
<br />charge upon request, which is 3.1x58. 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
<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 />a
<br />.0500
<br />13
<br />.2167
<br />23
<br />.3833
<br />33
<br />.5500
<br />43
<br />.7167
<br />53
<br />.8833
<br />L-
<br />.0667
<br />14
<br />.2333
<br />24
<br />.4000
<br />34
<br />.5667
<br />44
<br />.7333
<br />54
<br />.9000
<br />5.0833
<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 />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 1130
<br />1 .5000
<br />11 40
<br />1 .6667 11
<br />50
<br />1 .8333 jj
<br />60
<br />1.0000
<br />-
<br />TABLE V. -
<br />Inches in Decimals of a Foot.
<br />1-16
<br />3-32
<br />%
<br />3-16
<br />Y
<br />5-16
<br />Y.
<br />8
<br />Yl
<br />X
<br />.0052
<br />.0078
<br />.0104
<br />.0156
<br />:0208
<br />.0260
<br />.0313
<br />.0417 ,0521
<br />.0625
<br />.0729
<br />1
<br />L
<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 />.6667 .7500
<br />.8333 .9167
<br />
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