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TAaLE.VI:-CORRF_CTIONS FOR SUB -CHORDS AND LONG CIPORDS.
<br />.FOR SUB -CHORDS ADD
<br />Excess
<br />oP'are
<br />LONG CHORDS
<br />A,. B, b
<br />sin. A=a, cos. B=a, b= (c+a) (c a)
<br />a; b
<br />A, B, c
<br />tan. A= -°b, cot. Bib; c--
<br />"A, a
<br />B, b; a
<br />B=90° -A, b=a cot. A, sin. A*
<br />A, b
<br />A, c
<br />B, a, c
<br />B, a, b
<br />B=90° -A, a=b tan. A, C=_ -Cos. t'
<br />B=90° -A, a=c sin. A, b=c cos. A
<br />Given
<br />Bought.
<br />Oblique triangles. See 8g.' (b)
<br />A B, a
<br />D
<br />10
<br />20
<br />30
<br />40 150
<br />a, b, C
<br />60
<br />70
<br />80
<br />90
<br />,IC, ft.
<br />D
<br />20C
<br />300
<br />400
<br />500
<br />40
<br />.00
<br />.00
<br />.01
<br />.01.01
<br />area
<br />.01
<br />.01
<br />.01
<br />.00
<br />.02
<br />1
<br />199.99
<br />299.97
<br />399.92
<br />499.85
<br />6
<br />.00
<br />.01
<br />.01
<br />.02
<br />.02
<br />.02
<br />.02
<br />.01
<br />.01
<br />.05
<br />2
<br />199.97
<br />299.88
<br />399.70
<br />499.39
<br />8
<br />.01
<br />.02
<br />.02
<br />.03
<br />.03
<br />.03
<br />.03
<br />.02
<br />.01
<br />.08
<br />3
<br />199.93
<br />299.73
<br />399.32
<br />498.63
<br />10
<br />.01
<br />'.02
<br />-.03
<br />.04
<br />:05
<br />.05
<br />.05
<br />.04
<br />.02
<br />.13
<br />4
<br />199.88
<br />299.51
<br />398.78
<br />497.57
<br />12
<br />:02
<br />.04
<br />.05
<br />.06
<br />.07
<br />.07
<br />.07
<br />.05
<br />.03
<br />.18
<br />'5
<br />199.81
<br />299.24
<br />398.10
<br />496.20
<br />14
<br />.02
<br />.05
<br />.07..08
<br />'.09
<br />'.10
<br />.07
<br />.04
<br />.25
<br />6
<br />199.73
<br />298.90
<br />397.26
<br />494.53
<br />16
<br />.03
<br />.06
<br />.09
<br />.11
<br />.12
<br />.12
<br />..09
<br />.12
<br />.09
<br />.05
<br />.33
<br />7
<br />199.63
<br />298.51
<br />396.28
<br />492.57
<br />18'
<br />.04
<br />.08
<br />.11
<br />.14
<br />.15
<br />.16
<br />.15
<br />.12
<br />.07
<br />.41
<br />8
<br />199.51
<br />298.05
<br />395.14
<br />490.31
<br />20
<br />.05
<br />.10
<br />.14
<br />.17
<br />.19
<br />.20
<br />.18
<br />.15
<br />.09
<br />.51
<br />9
<br />19938
<br />297.54
<br />393.86
<br />487.75
<br />22
<br />.06
<br />.12
<br />.23
<br />.24
<br />.22
<br />.18
<br />.10
<br />.62
<br />10
<br />199.24
<br />296.96
<br />392.42
<br />484.90
<br />24
<br />:07
<br />".14
<br />.17
<br />.20
<br />..21
<br />.25
<br />..28
<br />.28
<br />.26
<br />.21
<br />.12
<br />.74 ,
<br />12
<br />198.90
<br />295.63
<br />389.12
<br />478.34
<br />26
<br />.09
<br />.17
<br />.24
<br />.29
<br />'.32
<br />.33
<br />.31
<br />.25
<br />.15
<br />.86
<br />14
<br />198.51
<br />294.06
<br />385.22
<br />470.65
<br />28
<br />.10
<br />.19
<br />.27
<br />.34
<br />.37
<br />.38
<br />.36
<br />.29
<br />.17
<br />1.00
<br />16
<br />198.05
<br />292.25
<br />380.76
<br />461.86
<br />30
<br />.11
<br />22
<br />.31
<br />:39
<br />'.43
<br />-..44
<br />.41
<br />.33
<br />.19
<br />1.15
<br />18,
<br />197.54
<br />290.21
<br />375.74
<br />452.02
<br />32E•
<br />.13-
<br />.25
<br />.36
<br />:44
<br />..49
<br />r
<br />.50
<br />...47
<br />.38
<br />.22
<br />1.31
<br />20
<br />196.90
<br />287.94
<br />370.17
<br />441.15'
<br />34.
<br />.15
<br />:28
<br />'.40
<br />.50
<br />.55
<br />.57
<br />-.53
<br />,.43
<br />.25
<br />1.48
<br />22
<br />196.32
<br />285.44
<br />364.06
<br />429.30.
<br />36
<br />.17
<br />.32
<br />.45
<br />.56
<br />.62'
<br />.64
<br />.59
<br />.48
<br />.28
<br />1.66
<br />24
<br />195.63
<br />282.71
<br />357.43
<br />416.53
<br />381
<br />.18
<br />.36
<br />.51
<br />...62
<br />'.70
<br />.71
<br />'.66
<br />-:53
<br />.31
<br />.1.86
<br />26
<br />194.87
<br />279.76
<br />350.30
<br />402.89
<br />40:
<br />.21
<br />'40
<br />.56
<br />..69
<br />.77
<br />.79
<br />.73
<br />.59
<br />.35.
<br />2.06
<br />28
<br />194.06
<br />276.59
<br />342.69
<br />388.42
<br />42
<br />.23
<br />.44
<br />.62
<br />.76
<br />.85
<br />.87
<br />.81
<br />:65
<br />.38
<br />2.28
<br />30
<br />193.18
<br />273.20
<br />334.61
<br />373.20
<br />44
<br />25
<br />.48
<br />.68
<br />.84
<br />.94
<br />.96
<br />.89
<br />.72
<br />.42
<br />2.50
<br />32
<br />192.25
<br />269.61
<br />326.08
<br />357.28
<br />46.
<br />.27
<br />,52
<br />:75
<br />..92
<br />1.02
<br />1.05
<br />.98
<br />.78
<br />.46
<br />2.74,
<br />34
<br />191.26
<br />265.81
<br />317.12
<br />340.73
<br />48
<br />30
<br />.57
<br />.81
<br />100
<br />1.12
<br />1.14
<br />1.06
<br />.86
<br />.50
<br />: 2.99
<br />36.
<br />190.21
<br />261.80
<br />307.77
<br />323.61
<br />50'
<br />.32
<br />.62
<br />:89
<br />1.09
<br />1.21
<br />1.24
<br />1.15
<br />.93
<br />.55
<br />3.24
<br />38
<br />189.10
<br />257.60
<br />298.03
<br />305.99
<br />62
<br />.35
<br />.67
<br />.96
<br />1.18
<br />1.31
<br />1.35
<br />1.25
<br />1.01
<br />.59
<br />3.52
<br />40
<br />187.94
<br />253.21
<br />287.94
<br />287.94
<br />54
<br />.38
<br />`.73
<br />1.04
<br />1:28
<br />1.42
<br />1.46
<br />1.35
<br />1.09
<br />.64
<br />3.80
<br />42
<br />,186.72
<br />248.63
<br />277.51
<br />269.54
<br />56
<br />.41
<br />.78
<br />1.12
<br />1.38
<br />1.53
<br />1.57
<br />1.46
<br />1.17
<br />.69
<br />4.09
<br />44
<br />185.44
<br />243.87
<br />266.78
<br />250.85
<br />58
<br />.44
<br />.84
<br />1.20
<br />1.48
<br />1.65
<br />1.69
<br />1.57
<br />1.20
<br />.74
<br />4.40
<br />46
<br />184.10
<br />239.93
<br />255.78
<br />231.95
<br />60
<br />.47
<br />.91
<br />1.29
<br />1.59
<br />1.76
<br />1.81
<br />1.68
<br />1.35
<br />.80
<br />4.72
<br />48
<br />182.71
<br />233.83
<br />244.51
<br />212.92
<br />Noes. -When a chord of less than 106 ft. is used the corrections given in the above
<br />table should be added to the nominal length of chord to get the length which should
<br />be used- in order that the 100 ft. points will check with those obtained by using the
<br />standard I06 ft. chard. Thus in locating a 140 curve by 25 ft: chords measure 251.08
<br />for each chord. _ Long.chords are useful in passing obstacles;
<br />TABLE VII. -MIDDLE ORDINATES FOR RAILS`IN FEET,
<br />Deg LENGTH _OF RAILS
<br />.
<br />Deg
<br />LENGTH OF RAILS
<br />of
<br />of
<br />Curve ,32 30 28 26 24 22 20
<br />Curve
<br />32,
<br />30
<br />28
<br />- 26
<br />24
<br />22
<br />20
<br />10 `.022
<br />.020
<br />.016 .613 .011
<br />.009
<br />.008
<br />'166
<br />.356 .313
<br />.213
<br />:236
<br />.200
<br />.170
<br />.139
<br />2 .045
<br />.038
<br />.034 .029 -.025
<br />.021
<br />:017
<br />17
<br />.378 .333
<br />.290
<br />.252
<br />.213
<br />.180
<br />.148
<br />-3
<br />'.051 .044 .037
<br />.031
<br />.026.
<br />18
<br />.400 .351
<br />.306
<br />..265
<br />.225
<br />.190
<br />;156
<br />.067
<br />:089
<br />.058
<br />.079
<br />.0% .060 '•050
<br />.042
<br />.035
<br />19 '•
<br />.423 '.371
<br />.324
<br />.280
<br />.238
<br />.201
<br />4
<br />'.165
<br />5 .112
<br />.099
<br />.086 .074 .063
<br />.053
<br />.044
<br />' 20
<br />.445 .392
<br />..341
<br />.296
<br />.250
<br />.212
<br />.174
<br />'6 .134
<br />.117
<br />.102 .088 .076
<br />:064
<br />.052
<br />" 21
<br />.466 .410
<br />•.357
<br />.309
<br />.262
<br />.222
<br />.182
<br />7 .156
<br />.137
<br />.120 .104 .088
<br />.074
<br />.061
<br />'•22
<br />.487 ..430
<br />.375
<br />.325
<br />.275
<br />.233
<br />•191
<br />8 .179
<br />:158
<br />,.137 :119 .100
<br />.085
<br />.070
<br />23
<br />.509 .450
<br />.390
<br />•338
<br />.287
<br />.243
<br />.199
<br />9 .201
<br />.175
<br />.153.133 .112
<br />.095
<br />.078
<br />24
<br />.531 .469
<br />.408
<br />.354
<br />.299
<br />.253
<br />.208
<br />10 ,223
<br />.196
<br />.171 .148 .125
<br />.106
<br />.087
<br />25
<br />552 .486
<br />.424
<br />.367
<br />.311
<br />.263
<br />.216
<br />11 .245
<br />:216
<br />.188 .163 .139
<br />.117
<br />.096
<br />'' 28
<br />.573 .506
<br />.441
<br />.382
<br />.323
<br />.274
<br />.225
<br />12 .'268
<br />.236
<br />.206 .179 .151
<br />.128
<br />.105
<br />27
<br />.594 .524
<br />.457
<br />.396
<br />.335
<br />.284'
<br />.233
<br />13 .290
<br />.254
<br />.222 ,192 .163
<br />.136
<br />.113
<br />' 28
<br />.618 .545
<br />.475
<br />.411
<br />.348
<br />.294
<br />.242
<br />14 .312
<br />.275
<br />.239 .207 .175
<br />.148
<br />.122
<br />29.
<br />.638 .564
<br />:491
<br />'.424
<br />.361
<br />.303
<br />.250
<br />15 .334
<br />.295
<br />.257 .223 .188
<br />.159
<br />.131.
<br />30
<br />.660 .583
<br />.508
<br />.438
<br />.374
<br />.313
<br />.259
<br />IX
<br />SLOPE REDUCTIONS.
<br />When distances are measured on a .slope .that may be reduced to
<br />the equivalent horizontal distance by the following approximate rule: -
<br />subtract from the slope distance the square of the rise divided by twice
<br />the slope distance. Thus for a slope distance of 250.3 ft. and a rise
<br />of 15 ft. correction=151=2X250.3=.45 (by slide rule) or horizontal
<br />distance=250.3-.45=249.85. When vertical angle=V. A. is'measured
<br />horizontal distance lope distance -slope distance (1 -Cos. V. A.).
<br />Thus for slope distance of 248.7 ft. and V. A. of 4° 20' from Table VIII
<br />Cos=.99714 and correction=1-.99714=.00286 per foot or total of .286 X
<br />2%'(near enough)=.57 and horizontal distance=248.7-.57=248.13 ft.
<br />' . see t. (a). TRIGONOMETRICAL FORMULAS.
<br />sin. A=a b P
<br />cos. A
<br />tan. A`a (a) c a c a
<br />cot. A= -Q
<br />sec. A=b ao
<br />cosec. Ac A. b C A b C
<br />F'onmuLA FOR SOLVING TRIANGLES.
<br />Given
<br />Sought.
<br />Right triangles. Bee ng. (a).
<br />a, c
<br />A,. B, b
<br />sin. A=a, cos. B=a, b= (c+a) (c a)
<br />a; b
<br />A, B, c
<br />tan. A= -°b, cot. Bib; c--
<br />"A, a
<br />B, b; a
<br />B=90° -A, b=a cot. A, sin. A*
<br />A, b
<br />A, c
<br />B, a, c
<br />B, a, b
<br />B=90° -A, a=b tan. A, C=_ -Cos. t'
<br />B=90° -A, a=c sin. A, b=c cos. A
<br />Given
<br />Bought.
<br />Oblique triangles. See 8g.' (b)
<br />A B, a
<br />b
<br />b -a sin. B
<br />61n. A
<br />A,. a, b `
<br />'R
<br />sin. B_b em. A
<br />a
<br />a, b, C
<br />A - B
<br />tan. Y2 (A -B) -a -b tam b (A +B)
<br />C, b, c
<br />A
<br />If s=% (a+b+c); sin. 34.A=J(s-b)
<br />Dc
<br />(d -a)
<br />cos. A=�a b ,tan. 3/ A=Ve (s -a) ,
<br />l
<br />Q sin. A=2 4(8--a) (3-0 (sem) s
<br />ba
<br />A, B, C, a
<br />area
<br />area -a1 alt. B sln. C
<br />2 sin. A
<br />A, b, c
<br />area
<br />area=% b c sin. A
<br />a, b, c
<br />area
<br />s=A (a+b+c),
<br />
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