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VIII
<br />TABLE II. - Radii, Ordinates and Deflections. Chord =100 ft.
<br />Deg. I
<br />Radius
<br />Mid.
<br />Ord. (Dist.
<br />Tan.
<br />`Dist.
<br />Def.
<br />lief.
<br />for
<br />Deg.
<br />Radius
<br />Mid.
<br />Ord.
<br />Tan.
<br />•Dist.
<br />Def.
<br />Dist.
<br />Def.
<br />for
<br />1 Ft.
<br />2° 17'1.2°
<br />t.
<br />ft.
<br />ft.
<br />it.
<br />'
<br />1' 59'
<br />ft,
<br />ft.
<br />ft.
<br />ft.
<br />34°171.01
<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 />17159.
<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 />.43G
<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 />6.970
<br />13.95
<br />2.40
<br />1
<br />5729.6
<br />.218
<br />.573
<br />1.745
<br />0.30'-
<br />20
<br />GS8.2
<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 />074.7
<br />1.555
<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 />0,37.3
<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 />S.13G
<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 />3.281
<br />16.56
<br />2.85
<br />2
<br />2864.9
<br />.43G
<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 />1.591
<br />3.751
<br />0.65
<br />10
<br />573.7
<br />2.153
<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 />9.555
<br />19.16
<br />3.30
<br />40
<br />214S.8
<br />.582
<br />2.327
<br />4.654
<br />0.80
<br />30
<br />499.1.2.511
<br />10.02
<br />20.04
<br />3.45
<br />50
<br />2022.4
<br />.G18
<br />2.472
<br />4.945
<br />O.S.,
<br />12
<br />478.3
<br />2,620
<br />10.45
<br />20.91
<br />3.G0
<br />8
<br />1910.1
<br />.655
<br />2.613
<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 />2.903
<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 />.S00
<br />3.199
<br />6.393
<br />1.10
<br />30
<br />396.2
<br />3.165
<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 />L
<br />.1432.7
<br />.873
<br />3.490
<br />6.9SO
<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 />.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
<br />1273.6
<br />.082
<br />3.926,
<br />7:552
<br />1.3.5
<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 />1S
<br />319.6
<br />3,935
<br />15.64
<br />31.29
<br />5.40
<br />50
<br />'1185.8
<br />1.055
<br />4.2171
<br />8.433
<br />1.45
<br />19
<br />302.9
<br />4,155
<br />16.51
<br />33.01
<br />5.70
<br />S
<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 />:;6.44
<br />6.30
<br />20
<br />1074.7
<br />1.164
<br />4.653
<br />0.305
<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 />4.708
<br />0.596
<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 />0.386
<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 />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 />10.70
<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 />1.00
<br />28
<br />206.7
<br />6.139
<br />24.19
<br />48.38
<br />8.40
<br />30
<br />SS1.9
<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 />1 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 />Y/ sub chord J sin of } def. angle
<br />R
<br />Length
<br />of arc
<br />for 100 ft. .
<br />sin. i def. ang.
<br />.12.5 Ft.
<br />15 Ft.
<br />20 Ft.
<br />25 Ft.
<br />300
<br />193.18
<br />to 51'
<br />2° 17'1.2°
<br />58'
<br />3° 43'
<br />101.15
<br />32° .
<br />181.39
<br />1' 59'
<br />2° 25'
<br />3° 1o'
<br />3° 58'
<br />101.33
<br />34°171.01
<br />3
<br />2° 06.
<br />20 33'
<br />- 3' 21'
<br />4° 12'
<br />101.48
<br />36°
<br />161 .8o
<br />2° 13' '
<br />2' 41'
<br />3° 33'
<br />a° 26'
<br />Ioi.66
<br />38°
<br />153:58
<br />2°20'
<br />2°49'
<br />3' 44
<br />4° 40
<br />101.85
<br />40°
<br />146.19
<br />2° z7'
<br />2' 57'
<br />3' 55'
<br />4° 54
<br />102.o6
<br />42'
<br />139.52
<br />'-° 34'
<br />3' 05'
<br />4° 07'
<br />5' 08
<br />102.29
<br />44°
<br />133.47
<br />2° 41'
<br />3° 13'
<br />4° 18'
<br />S°'22'
<br />102.53
<br />46°
<br />127.97
<br />2° 48'
<br />3° 21'
<br />4° 29
<br />5° 36'
<br />102.76
<br />4R'
<br />122.• 92
<br />2° 55'
<br />3' 29'
<br />4° 40
<br />5° 50'
<br />103.00
<br />500
<br />118.31
<br />3°02'
<br />3°38
<br />4°51'
<br />6004'
<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 />110.11
<br />3° 16'
<br />3' 54'
<br />S' 13'
<br />6° 31'
<br />103.84
<br />56°
<br />Io6.5o _
<br />3° 22'
<br />4° 02'
<br />5'23!
<br />6? 44
<br />104-14
<br />58°
<br />103.14
<br />3'29'
<br />4'10'
<br />5'34'
<br />6'57'
<br />104.43
<br />6o°
<br />100.00 • _
<br />3' 35
<br />4° 18'
<br />5' 44
<br />7° 12'
<br />104.72
<br />IR
<br />CURVE FORMULAS
<br />T = R tan J I It = T cot. ' I chords
<br />5o tan's- Chord def. _
<br />T Sin. J D R= 50 R
<br />Si.. i D = 5o Sin. , D No. chords = I
<br />R E = R ex. sec 12I
<br />50 tan a I
<br />Sin. D = r 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 />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 loo, Aft. 10.10x_200=.5. 100-x-.5=100.5 hyp.
<br />Given Hyp. loo, Alt. 25.252+200=3.I25.'100-3-125=96.875 =Basc.
<br />Error in first example, .002; in last, .o45.
<br />To find Tons of Rail in one mile of track: multiply weight per yard
<br />by 11, and divide by 7.
<br />Lr;vnLING. 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, d„ etc. are the discrepancies of various
<br />results from the mean, and if Ida=the sum of the squares of these differ-
<br />ences and n=the number of observations, then the probable error of tho
<br />mean= 4-0.6745 n n a-1)
<br />SOLAR EPHI IlMERIS. 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:}x5• in., with about 90 pages of data very
<br />useful to Vie 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 andLogarithmie Functions;
<br />and Logarithms of Lumbers.
<br />TABLE IV. - Alinutes 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 />2.0333
<br />12
<br />.2000
<br />22
<br />.3667
<br />32
<br />.5333
<br />42
<br />.7000
<br />52
<br />.8667
<br />3
<br />.0500
<br />l3
<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 />21
<br />.4000
<br />34
<br />.5667
<br />44
<br />.7333
<br />54
<br />.9000
<br />5
<br />.0333
<br />15
<br />,2500
<br />25
<br />.4167
<br />35
<br />.5533
<br />45
<br />.7500
<br />55
<br />.9167
<br />6
<br />.1000
<br />16
<br />.2667
<br />20
<br />.4333
<br />36
<br />.6000
<br />46
<br />.7667
<br />50,
<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 />IS
<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 />SW7
<br />59
<br />.9833
<br />10
<br />.1667
<br />.20
<br />.3333
<br />30
<br />.5000
<br />40
<br />1 .6667 1
<br />50
<br />1 .8333
<br />60
<br />1.0000
<br />TADLt V. -
<br />Inches in Decimals of a Foot.
<br />1-16
<br />3-32
<br />3-16
<br />5-16
<br />MY4
<br />7
<br />.0052
<br />.0078
<br />.6104
<br />.0156
<br />.0208
<br />.0260
<br />.0313
<br />.0417 .0521
<br />.06251.0729'
<br />1_
<br />2
<br />3
<br />'4
<br />b
<br />6
<br />7
<br />8 9
<br />10
<br />11
<br />1
<br />.0833
<br />.1667
<br />.2500
<br />.3333
<br />.4167
<br />.5000
<br />.5833
<br />:6667 .7500
<br />1 .8333
<br />.9167.
<br />
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