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VIII
<br />TABLE VI.-CORRFCTIONS FOR SUB -CHORDS AND LONG CI10116S.
<br />FOR SUB -CHORDS ADD
<br />Excess
<br />of arc
<br />LONG CHORDS
<br />sought.
<br />Right triangles. See 0g. (a).
<br />a, c
<br />A., B, b
<br />sin. A- cos.
<br />a, b
<br />A, B, c
<br />tan, A=!, cot. B=-6, c= n' +e'•
<br />A, a
<br />B, b, c
<br />B=90° -A, b=a. cot. A, mein aA°
<br />A, b
<br />B, a, c
<br />B=90° -A, a==b tan. A,
<br />D
<br />10
<br />20
<br />30
<br />40
<br />50
<br />60,
<br />70
<br />80
<br />90
<br />100 ft.
<br />D
<br />2001
<br />300
<br />400 500
<br />1
<br />40
<br />.00
<br />.00.01
<br />.01
<br />.01
<br />.01
<br />•.01
<br />.01
<br />00
<br />.02
<br />1
<br />199.99
<br />299.97
<br />399.92 499.85
<br />6
<br />.00
<br />.01
<br />.01
<br />.02
<br />.02
<br />.02
<br />.02
<br />.01
<br />.O1
<br />.05
<br />2
<br />199.97
<br />299.88
<br />399.70 499.39
<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 />399.32 498.63
<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 />398.78 497.57
<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 />398.10 496.20
<br />14
<br />.02
<br />.05
<br />.07
<br />.08
<br />.09
<br />.10
<br />.09
<br />.07
<br />.04
<br />.25
<br />8
<br />199.73
<br />298.90
<br />397.26 494.53
<br />16
<br />.03
<br />.06
<br />.09
<br />..11
<br />.12
<br />.12
<br />.12
<br />..09
<br />.05
<br />.33
<br />7
<br />199.63
<br />298.51
<br />396.28 492.57
<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 />395.14 490.31
<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 />199.38
<br />297.54
<br />393.86 487.75
<br />22
<br />.06
<br />.12
<br />.17
<br />.21
<br />.23
<br />.24
<br />.22
<br />.18
<br />.10
<br />.62
<br />10
<br />199.24
<br />296.96
<br />392.42 484.90
<br />24
<br />.07
<br />.14
<br />.20
<br />.25
<br />.28
<br />.28
<br />.26
<br />.21
<br />.12
<br />.74
<br />12
<br />198.90
<br />295.63
<br />389.12 478.34
<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 />385.22 470.65
<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 />380.76 461.86
<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 />375.74 452.02
<br />32
<br />.13
<br />.25
<br />.36
<br />.44
<br />.49
<br />'.50
<br />.47
<br />.38
<br />.22
<br />1.31
<br />20
<br />196.90
<br />287.94
<br />370.17 441.15
<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 />364.06 429.30
<br />36 ..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 />357.43 416.53
<br />38
<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 />350.30 402.89
<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 />342.69 388.42
<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 />334.61 373.20
<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 />326.08 357.28
<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 />317.12 340.73
<br />48
<br />.30
<br />.57
<br />.81
<br />1.00
<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 />307.77 323.61
<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 />298.03 305.99
<br />52
<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 />287.94 287.94
<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 />277.51 269.54
<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 />Cog
<br />44
<br />185.44
<br />243.87
<br />266.78 250.85
<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 />255.78 231.95
<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 />244.51 212.92
<br />Noce. -When a chord of less than loo 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 loo ft. points will check with those obtained by using the
<br />standard loo ft.�chord. Thus in locating a 140 curve by 25 ft. chords measure 25f.06
<br />for each chord. Long chords are useful in passing obstacles.
<br />TABLE VII. -MIDDLE ORDINATES FOR RAILS IN FEET.
<br />Deg.
<br />LENGTH OF RAILS
<br />Deg
<br />LENGTH OF RAILS
<br />of
<br />of
<br />Curve
<br />32
<br />30
<br />28
<br />26
<br />24
<br />22
<br />20
<br />Curve
<br />32
<br />30.
<br />28
<br />26
<br />24
<br />22
<br />20
<br />10
<br />.022
<br />.020
<br />.016
<br />.013
<br />.011
<br />.009
<br />.008
<br />16°
<br />.356
<br />.313
<br />.273
<br />.236
<br />.200
<br />.170
<br />.139
<br />2
<br />.045
<br />.038
<br />.034
<br />.029
<br />.025
<br />.021
<br />.017
<br />17
<br />.378
<br />.333
<br />.290
<br />.252
<br />.213
<br />.180
<br />.148
<br />3
<br />.067
<br />.058
<br />.051
<br />.044
<br />.037
<br />.031
<br />.026
<br />18
<br />.400
<br />.351
<br />.306
<br />.265
<br />.225
<br />.190
<br />.156
<br />4
<br />.089
<br />.079
<br />.069
<br />.060
<br />.050
<br />.042
<br />.035
<br />19
<br />.423
<br />.371
<br />.324
<br />.280
<br />.238
<br />.201
<br />,.165
<br />5
<br />.112
<br />.099
<br />.086
<br />.074
<br />.063
<br />.053
<br />.044
<br />20
<br />.445
<br />.392
<br />.341
<br />.296
<br />.250
<br />.212
<br />.174
<br />6
<br />.134
<br />.117
<br />.102
<br />.088
<br />.076
<br />.064
<br />.052
<br />21
<br />.466
<br />.410
<br />.357
<br />.309
<br />.262
<br />.222
<br />.182
<br />7
<br />.156
<br />.137
<br />.120
<br />.104
<br />.088
<br />.074
<br />.061
<br />22
<br />.487
<br />.430
<br />.375
<br />.325
<br />.275
<br />.233
<br />.191
<br />8
<br />.179
<br />.158
<br />.137
<br />.119
<br />.100
<br />.085
<br />.070
<br />23
<br />.509
<br />.450
<br />.390
<br />.338
<br />.287
<br />.243
<br />.199
<br />9
<br />.201
<br />.175
<br />.153
<br />.133
<br />.112
<br />.095
<br />.078
<br />24
<br />.531
<br />.469
<br />.408
<br />.354
<br />.299
<br />.253
<br />.208
<br />10
<br />.223
<br />.196
<br />.171
<br />.148
<br />.125
<br />.106
<br />.087
<br />25
<br />.552
<br />.48U.491
<br />.367
<br />.311
<br />.263
<br />.216
<br />11
<br />.245
<br />.216
<br />.188
<br />.163
<br />.139
<br />.117
<br />.096
<br />21
<br />.573
<br />.50.382
<br />.323
<br />.274
<br />.225
<br />12
<br />.268
<br />.236
<br />.206
<br />.179
<br />.151
<br />.128
<br />.105
<br />27
<br />.594
<br />.524.396
<br />.335
<br />.284
<br />.233
<br />13
<br />.290
<br />.254
<br />.222
<br />.192
<br />.163
<br />.138
<br />.113
<br />28
<br />.618
<br />.545.411
<br />.348
<br />.294
<br />.242
<br />14
<br />.312
<br />.275
<br />.239
<br />.207
<br />.175
<br />.148
<br />.122
<br />29
<br />.638
<br />.56.424
<br />.361
<br />.303
<br />.250
<br />15
<br />.334
<br />.295
<br />.257
<br />.223
<br />.188
<br />.159.131
<br />30
<br />.660
<br />.58.438
<br />.374
<br />.313
<br />.259
<br />11
<br />NEW
<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=l-.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 fig. (a). TRIGONOMETRICAL FORMULAS.
<br />sin. A=c B B
<br />cos.. Ate°
<br />c
<br />tan. Awa (a). a a (b) a a.
<br />cot. A=! -
<br />sec.. A- °P
<br />cosec. A= a A b C A b C
<br />FORMULA
<br />FOR SOLVING TRIANGLES.
<br />Given
<br />sought.
<br />Right triangles. See 0g. (a).
<br />a, c
<br />A., B, b
<br />sin. A- cos.
<br />a, b
<br />A, B, c
<br />tan, A=!, cot. B=-6, c= n' +e'•
<br />A, a
<br />B, b, c
<br />B=90° -A, b=a. cot. A, mein aA°
<br />A, b
<br />B, a, c
<br />B=90° -A, a==b tan. A,
<br />A; c
<br />B, a, b
<br />B=90° -A, . a=te sin. A, • b=c cos. A
<br />Given
<br />Sought.
<br />Oblique triangles. See Ag. (b) '
<br />A, B, a
<br />b ,
<br />b=a sin, e
<br />sin. A
<br />A,.a, b
<br />B
<br />sin. B -b sin. A
<br />a
<br />a, b, C
<br />A - B
<br />Jan. -Y2 (A -B) -a btan. a+ D (A +B)
<br />D
<br />c, b, c
<br />A
<br />If s=112(a+b+c), sin. M A=, J(8 -b)
<br />be
<br />cos. A= Is (& , tan. I/ A=�(d,
<br />sin. A-2 Ds -a)
<br />i
<br />be
<br />A, B, C, a
<br />area
<br />ea_ a= sia B sin. C
<br />2 sin. A
<br />A, b, c
<br />area
<br />area= ---M b c sin. A
<br />a, b, c
<br />area
<br />s ---A (a+b+c), area= s (s-a)(s-b)(s c)
<br />
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