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TABLE -IL - Radii, Ordinates and Deflections. Chord =100 ft.
<br />Deg•
<br />Radius
<br />Md'
<br />U: d.
<br />Tan.
<br />Vs.,.Aiat..�'
<br />DJ. -Dvf-
<br />for
<br />1 Ft
<br />Dcg '
<br />itadiva
<br />Mid.
<br />Ord.
<br />Tan.
<br />List.
<br />Def.
<br />hist.
<br />fcr
<br />1 Ft.
<br />2' 17'
<br />ft.
<br />It.
<br />- -1t.
<br />ft.
<br />!
<br />°
<br />1 59
<br />it.
<br />ft.
<br />it.
<br />, ft.,
<br />34°
<br />0'10'
<br />34377.
<br />.036
<br />:145`
<br />.291
<br />0.05
<br />7"
<br />819.0
<br />1.528
<br />0.105
<br />12.21
<br />2.10
<br />20
<br />17189.
<br />.073
<br />..291,
<br />-.552
<br />0.10
<br />20'
<br />781.8
<br />1.600
<br />6.395
<br />12.70
<br />2.20
<br />. 30,
<br />114,9.
<br />.109
<br />.43T
<br />•873
<br />0.15
<br />30
<br />764.5
<br />1.637
<br />6.540
<br />13.03
<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 />6.635
<br />13.37
<br />2,30
<br />50
<br />6875.5
<br />.182
<br />.72T
<br />1.454
<br />0.25
<br />8
<br />716.8
<br />1.746
<br />6,076
<br />13.05
<br />2.40
<br />1
<br />5729.6
<br />.218
<br />.573
<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 />2.03G
<br />U.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 />0.60
<br />40'
<br />661.7
<br />1.892
<br />7.530
<br />15.11
<br />2.60
<br />30
<br />3819.8
<br />-.327
<br />1.309
<br />2.61S
<br />0.45
<br />9'
<br />637.3
<br />1.905
<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 />014,6
<br />3.037
<br />8.136
<br />16.27
<br />'2.80
<br />.50
<br />3125.4
<br />.400
<br />1.600
<br />3.200
<br />0.53
<br />30
<br />003.8
<br />3.074
<br />8,231
<br />16.56
<br />2.85
<br />2
<br />2864.9
<br />.430
<br />1.745
<br />3.490
<br />0.60
<br />40
<br />593.4
<br />2.110
<br />8.426
<br />16.85
<br />2.00
<br />10
<br />2644.6
<br />.473
<br />1,891
<br />3.761
<br />0.63',
<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 />19.1G.
<br />3.30
<br />40
<br />2148.8
<br />.582
<br />2.327
<br />4.654
<br />0,80
<br />30
<br />199.1
<br />2.511
<br />10.02
<br />20.01
<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 />3.50
<br />3'
<br />1.910.1
<br />•655
<br />2.618
<br />5:235
<br />0.90
<br />30
<br />459'.3
<br />3:730.10.89
<br />21.77
<br />3.75
<br />10
<br />1809.6
<br />,.691.2.763.-5:526
<br />0.93
<br />13 :
<br />441.7
<br />3.839
<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 />'.761
<br />3.054
<br />6.1081.05
<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 />3.109
<br />6:395
<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 />.830
<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 />4
<br />1432:7
<br />.573
<br />3,490
<br />6.080
<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 />.500
<br />3.635
<br />7.271
<br />1.25
<br />16
<br />359.3
<br />3,496
<br />13.92
<br />2784
<br />4.80
<br />20'
<br />1322.5
<br />.0453.718
<br />7.501
<br />1.30
<br />30
<br />348.5
<br />3.606
<br />14.35
<br />28.70,4.95
<br />30
<br />1273.6
<br />.982
<br />3.926
<br />7.853
<br />1.35
<br />17
<br />338.3
<br />3.71G'14.78
<br />29.56
<br />5.10
<br />40
<br />1225.1
<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 />1155.8
<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 />b
<br />1146.3
<br />1.091
<br />4.862
<br />8.724
<br />1.50
<br />20,
<br />287.9
<br />4.374
<br />1.7.37
<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 />36.4:1
<br />6.30
<br />20
<br />1074.7
<br />1.164
<br />4.653
<br />0.305
<br />1.60
<br />2,�
<br />262.0
<br />4.814
<br />19[08
<br />38.16
<br />0:60
<br />30•
<br />1042.1
<br />1.200
<br />4.798
<br />0.596
<br />1.65
<br />23
<br />250.8
<br />5.035
<br />19:04
<br />"9.87
<br />6.90
<br />40
<br />1011.45
<br />1.237
<br />4.943
<br />9.886
<br />1.70
<br />24 .
<br />240.5
<br />:1.255
<br />20.79 -
<br />41.53
<br />7.20
<br />50
<br />982.0
<br />1.273
<br />5.088
<br />10.18
<br />1.75
<br />25
<br />231.0
<br />5:476
<br />21.0443.23
<br />7.50
<br />6
<br />055.4
<br />1.309
<br />5.234
<br />10.4-7
<br />'1,80
<br />26
<br />222,3
<br />5.697
<br />22.50
<br />44.99
<br />7.80
<br />10
<br />920.6
<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 />°.10
<br />20
<br />05.1
<br />1.383
<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.GG9
<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 />G:583
<br />25-88
<br />,51.76
<br />0.00
<br />The niiddlo ordinate in inches for any cord of length (G) is Pilnal to .0012 C'
<br />multiplied by the middle ordinate taken from the above table. Thus, it it
<br />desired to bend a.90 ft. rail to fit a 10 degree curve, its middle ordinate should
<br />be .0012X900X2.187 or 2.36 inches.
<br />TABLE III. . Deflections for Sub Chords for Short Radius Curves.
<br />Degree
<br />Curve
<br />Radius
<br />50
<br />A sub chord "
<br />1 = sin of 1 dci'. angle
<br />Lengthof
<br />of arc
<br />for 100 ft.
<br />sin. E def. ane.
<br />_ 12.5 rt.
<br />15 IIt.
<br />20 Ft.
<br />25 Ft.
<br />300
<br />193- 18
<br />1° 51'
<br />2' 17'
<br />2' 58'
<br />3° 43'.
<br />1()[.15
<br />°
<br />32
<br />131.39
<br />°
<br />1 59
<br />° 25
<br />° .
<br />3 to
<br />° _ ,
<br />�8
<br />'101 -33
<br />34°
<br />171.01
<br />2' 0('
<br />-° 33'
<br />3° 2I'
<br />"1 12'
<br />101.48
<br />360
<br />16f. 8o
<br />2° 13'
<br />^_' 41'
<br />3°33'
<br />4°2G'
<br />IOI.66
<br />38°
<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° .74.
<br />I62.o6
<br />42'
<br />139-52
<br />__
<br />2° 34'
<br />3 05'
<br />4° 07'
<br />5° C8•
<br />102.29
<br />44'
<br />133.47.
<br />20 41'
<br />3° 13'
<br />4° 13'
<br />5° 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 />4Ro
<br />122.92
<br />26 5J,
<br />3° 29'
<br />4° 40
<br />5° 50'
<br />.103. o0
<br />50
<br />118.31
<br />3 O2
<br />3° 38'
<br />4' 51'
<br />6° c4'
<br />103.2.4
<br />1R
<br />114- GG
<br />° O9'
<br />3 �
<br />6
<br />,�° 4'
<br />S ° C2'
<br />6° I '
<br />7
<br />IO 3.54
<br />54°
<br />110. 11
<br />3° IG'
<br />3°j%'
<br />•.1500
<br />6° 31`
<br />I03.84
<br />56°
<br />- Io6. so
<br />3° 22'
<br />4° 03
<br />S° 23'.
<br />6° 4.''
<br />I04- 14
<br />53°
<br />103,14
<br />3° 29'
<br />4° 10'
<br />S° 34'
<br />6",97 '
<br />104.43
<br />60°
<br />ICO. oa
<br />3° 3J'
<br />4° 18'
<br />59 44'
<br />7' 11'
<br />104.72
<br />Ix
<br />CURVE FORMULAS
<br />T= R tan I R= T cat.
<br />5o tan I Chord
<br />`I - Sin. I D R = 50 R
<br />Sin. a D = - Sin. : L? No, chords = I
<br />R >✓ = R.ex. 5ec I 1 D
<br />D =. 0° tT2 I E = T tan t I Tan. def. = z chord def.
<br />The square of any distance, divided" by twice the radius, - will cqual
<br />the distance_ from tangent to curve, very nearly.
<br />To find angle for a given distance and deflection.
<br />Rule >. Multiply the given distance by .or745 (def, for 1° for a 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 />Uy •o1745, 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. IOAO' . 206 =.5. I00+.5=Ioo•5 hYp•
<br />Given Hyp. loo,Alt. 25.2�a=2oo-3.r25.loo-3:125=96.873=Bise.
<br />Error in first example, .002; In last, .04.5.
<br />To find Tons of Rail in one mile of track: multiply weight per yard
<br />by i i,and divide by ,7. =
<br />L1;vaLING. The correction for curvature and refraction, in feet
<br />and decimals of feet is equal to 0.57442, where d is the distance in miles.
<br />The correction for curvature alone is closely, 5da. The combined cor-
<br />rection is negative. -
<br />PROBABLE ERROR. If d,, d;; d,, etc. are the discrepancies of various
<br />results from the mean, and if 202 'the sum of the squares of these differ-
<br />ences and n=the number of observations, then the probable error of the
<br />mean= FXd''
<br />SOLAR E1?HEP1Ei1IS. Attention is called to the Solar Ephemeris for
<br />the current year, published by E cuffel & Fsser Co., and furnished free of
<br />charge upon request, which is 3.}x51, 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 mcas-
<br />urernents; magnetic declination; arithmetic constants; English and Metric
<br />conversions; trigonometric formulas; Natural and Logarithmic Functions;
<br />and Logarithms of Numbers.
<br />T9SLr IV. -Minutes 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 />1;3
<br />•2000
<br />22
<br />.3667
<br />32
<br />.5333
<br />47
<br />.7000
<br />52
<br />.86§7
<br />3
<br />.0500
<br />13
<br />.2167
<br />23
<br />.3833
<br />33
<br />.5500
<br />43
<br />.7167
<br />53
<br />.3833
<br />4
<br />.0667
<br />1.4
<br />'.2333
<br />.2k
<br />.4000
<br />34
<br />.5667
<br />44
<br />.7333
<br />54
<br />.9000
<br />5
<br />.0833
<br />•15 -
<br />.2506
<br />25
<br />.41 G7
<br />35
<br />5233
<br />45
<br />.7503
<br />55
<br />.9167
<br />6
<br />.1000
<br />1G
<br />.2GG7
<br />23
<br />.4333
<br />36
<br />.6000
<br />46
<br />.7607
<br />56
<br />.9333
<br />7
<br />.1167
<br />17
<br />.2833
<br />27
<br />.4500
<br />37
<br />.0167
<br />47
<br />.7033
<br />57
<br />.9500
<br />6
<br />.1333
<br />1R
<br />.3000
<br />28
<br />.4667
<br />3S
<br />.G333
<br />49
<br />.1000
<br />58
<br />.9667
<br />9
<br />•.1500
<br />19
<br />.3107
<br />29
<br />.4833
<br />30
<br />.6500
<br />49
<br />.3167
<br />59
<br />.9833
<br />10
<br />.1667 1
<br />20
<br />•3333
<br />30
<br />.5000
<br />11 40
<br />.6067
<br />56
<br />1 .9'MI 11
<br />CO
<br />I 1.0000
<br />TAnLE V. -
<br />Inches in Decmlals of a Foot.
<br />-
<br />1.1G
<br />3-32
<br />3,J
<br />3-16
<br />14 5-16
<br />1.3
<br />'�
<br />%
<br />:i i
<br />.0052
<br />.0073
<br />.010.1
<br />.0156
<br />.0205 ,.0260
<br />•0313
<br />.0417
<br />.0521
<br />.0625 .0729
<br />1
<br />2
<br />3
<br />4
<br />5 67
<br />S
<br />0
<br />10 11
<br />.0833
<br />.1667
<br />.2300
<br />.3333
<br />.4167 .5000
<br />.5833
<br />.6667
<br />.7500
<br />.5333 1-9107
<br />
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