Fremling Field Notebook 1968 - 1969

star

thTANCES FROM S1Ie STAas FOR CO s-SEcToNNo SLOPE I TO 1. ROADWAY Of ANT WsAIDTH

.5 0.50 1.50 2.50 8450 4.50

.6 0.60 1.60 2.0 3460 4.60

.7 0.70 1.70 2.70 3.70 4.70

5.30 5.40 5.50 5.20 6.20 76.10 6.30 8 .40 6.50 . ,203 8.60 8.40 10.60 10.40 51.3 10.10 10.20 11.60 .1 1.20 11.30 11.40 3.1 .20130 1.30 12.40 12.50 3 1 .20 1.30 13.40 13.50 14.40 14.60 14.10 14.20 14.80 15.10 15.20 15.30 15.40 15.60 16.50 16.10 16.20 16.30 16.40 17.40 17.50 17.10 17.20 17.30 .10 1.20 1.30 18.40 18.50

5.60

5.70 .70

0 0 0.00 1.00 1 2 2.00 148 140 4.00 S .00

.1 .2 0.10 0.20 1.10 1.20 2.10 52.20 1.10 3.20 4.20 4.10

7

7.00

4

3.00

410

10 11 12

8.10

1~0 37.

12.

.

183 14 14.

15 15.00 .00 16 S4.0 100.

S

3 300

30

3

20 .20 21.0 22.20 23.20 24.20 25.20 28.20

20.30

21.30 22.0 24.30 24.30 25.30 28.0

20.40 20.50 21.50 21.40 22.40 22.50 24.50 28.40 49.40i .60 25.40 25.50 2.40 26.60

32.2 0 21 332 30.10.10 30.20 31.20

32.30 330 80.30 81.30

83.40 80.40

7.20

3830

45. 10 4.00 .00 21.10 22 200 22.10 38.00 3.10 2 29 4.00 24.10 23 28.00 26 10 200 261 2 S

0 0 33 0.00

.4 0.40 1.40 2.40 3.40 4.40

.3 0.30 1.30 2.30 1.80 4.30

.90 2.90 22.90 24.90 29.90 28.90 28.90

1.90

9

41 22

24 24 25 28

82 32.70 2.80 32.9 3,90 33.80 3 3.70 303 30.70 080 30.90 .80 1.90 3 31.70

4040 4.50 4340 48.60 50 140 .40 1.6

40.60 43.60 4.050

40.70 43.70 4.70

4.80 4380 180

490 4 0 1.80

42.40

42.60

42.7

4280

42 9042 44 44,9 .0 47 46 45.90

43.

.10 43.10 41.10

4020 4320 4 .0

41.0o 4330 4.30

42

42.0

42.10

4220

42 30

44 47 45 48 4

44.00 47.00 45.00 48.0 48.00

44 20 44.10 47.20 47.10 48.20 48.10 .0 01 4820 48.20 48.10

.

20.70 21.70 22.70 24.70 24.70 25.70 28.70

81 12 13 14 16 16 17 1

7.908 39.90 I8 3 180 4.90 8.900 39908

31.40 3.40 38.40 38.40 39.40

i

20.80 21.60 22.60 24.60 24.60 25.80 28.80

1.90 12.90 13.90 14.90 16.90 16.90 17.00

3.89 38.80 35.80 39.80

3580 39.30

39 39.

8 6

38.60 36.70 8 4.0 .8.70 31.70 38.60 38.60 35.70 39.60 39.70

36.20 320

0 .3

10.60 11.60 12.60 13.60 14.60 15.60 16.60 17.60 18.60

20.80 21.80 22.80 48.80 4.80 3.80 28.80

0 1 2 14

6.90

4.0 0 16.90

32.0 32.60 3.60 30.503.0 0.0 1.40o 16.50 81.60

34.20 038.30 3

890

. .

9 0.90 1.90 2.90 1490 4.90 .90

.70 3.0 1.80 10.70 11.70 1.80 12.70 12.80 13.70 13.80 14.70 14.80 16.70 15.80 18.70 16.80 17.70 17.80 18.70 18.80

2.40

3.10 310 8.10 38.10 39.10

S 00

38

7.80

8.60

.8 0.80 1.80 2.80 1480 .80

44 30 44.40 47.40 47.30 45.40 48. 0 4830 .48.1048.40 4.40 48.30

37.60 34,60 38.50 35.60 39.50

42.50

44,.50 44.80 47.60 45.80 48.50 47.60 48.60 48.60 48.60 48.60

44.70 4480 147.70 47.80 45.80 45.70 4.70 4 48.8 48.70

43

4 4. 0 144

stake islocated by the double entry method Inleft cound is nearly level the cut or . t side adjustment make stake. the slight isideit stake does not level estimate the dfference Innelevation between the It ground isnot necessary. to slope

Sstue

r

14CL i

iJ J /7oFOn c-, f

C-

j Llz

IMPROVED

TABLES

AND

INFORMAT ION

DIRECTIONS FOR USE OF TABLES TABLE No. XIV Distance of slope stake from side or shoulder stake for any width roadway, slope 1Y to 1.

If ground is nearly level, the cut or fill at side stake is located by the double entry method in left column and top row. The number in body of table in same row and column gives distance from side stake to slope stake. If ground is not level estimate the difference in elevation between the side stake and sloppe stake, lower target by this amount if cut, elevate if fill. 'Add this amount to cut or fill and find distance in table.

Set up rod at this point, and line of sight should cut target. If it does not make the slight adjustment necessary.

TABLE No. VIII To find Tangent and External for curve of any other degree, divide by degree of curve and add correction found in column of corrections.

Degree of curve with a given I may be found by dividing tangent, (or external). opposite I by given tangent, (or external).

The distance from a point on the tangent to the curve is very nearly the square of the tangent length divided by twice the radius.

TABLE II TRIGONOMETRIC FORMULE. LB =LPON = LOPL LA= LMOP R = OB= c= 1 sin A== a =a l C sosA= tan A=

=sosB= LP

b-- =-b=sin B=OL 1

b c

a

-

b

MQ

MQ

OM

1

MQ= cot B= MQ

= NT - tan B = NT cot A -= NT --1 ON

sec A

SOQ O .-.OQ Q = O OQ = csc B = OQ

OM

csc A = OT ON

1

=

OT = OT = sec B= OT I

= LM = covers B + OP OP-LP OP-LP = vers B covers A = OP exsec A = PQ = coexsec B coexsec A = PT= exsec B - / 1-Cos A sin vers A =

1/A=

/cos

A=

}1+Cos A 2

cos 2 A = cos' A-sin' A sin 2 A= 2 sin A cos A sin C' sin B sin A Law of Lines C a B cos C a'+b'-2 ab Law of Cosines c= a. b tan 1/s (A+ ) (A+ -tan Law of Tangents a-b tan /a (A-B)

TABLE II - Continued TRIGONOMETRIC FORMULJE (continued) in any triangle: Oiven a, b, C; to find c, B, A. Use Law of Lines.

Given A, B, c; to find a, b, C. Use Law of ULines.

Given' a, b, c; to find A, B, C.

a+b+c

(s-a) (s-b) (s-c)

2

-

A= o '1/s tan /IA

i

s-a

tan /

=bB

tan

=I

/TC

s (s-a)

sb

s-c Area of a triangle: Area = 1/2 ab Sin C

Area = s (s--a) (s-b) (s-c) PRISMOIDAL FORMULA.

INCHES AND FRACTIONS OF AN INCH IN DECIMAIS OF A FOOT 0

1

2

/s .0052 .0885 .1719 .0104 .0938 .1771 /s .0156 .0990 .123 1/4 .0208 .1042 .1875 /ss .0260 .1094 .1927 is .0313 .1146 .1979 7;a .0365 .1198 .2031 1/t .0417 .1250 .2083 */he .0469 .1302 .2135 s/s .0521 .1354 .2188 'I/is .0573 .1406 .2240 14 .0625 .1458 .2292 /s .0677 .1510 .2344 /s .0729 .1563 .2396 .0781 .1615 .2448 "/s 1 .06833 .1667 .2500 0

1

2

4

5

6

7

8

9

.2552 .204 .2656 .2708 .2760 .2813 .2865 .2917 .2969 .3021 .3073 .3125 .3177 .3229 .3281 .3333

.3385 .3438 .3490 .3542 .3594 .3646 .3698 .3750 .3803 .3854 .3906 .3958 .4010 .4063 .4115 .4167

.4219 .4271 .4323 .4375 .4427 .4479 .4531 .4583 .4635 .4688 .4740 .492 .4844 .4896 .4948 .5000

.5052 .5104 .5156 .5208 .5260 .5313 .5365 .5417 .5469 .5521 .5573 .5625 .5677 .5729 .5781 .5833

.5885 .5938 .5990 .6042 .6094 .6146 .6198 .6250 .6302 .6354 .6406 .6458 .6510 .6563 .6615 .6667

.6719 .677t .6823 .6875 .6927 .6979 .7031 .7083 .7135 .7188 *7240 .7292 .7344 .7396 .7448 .7500

.7552 .7604 .7656 .7708 .7760 .7813 .7865 .7917 .7969 .8021 .8073 .8125 .8177 .8229 .8281 .8333

3

4

5

6

7

8

9

10

11

.8385 .9219 .8438 .9271 .849 .9323 .8542 .9375 .8594 .9427 .8646 .9479 .8698 .9531 .8750 .9583 .8802 .9635 .8854 .9688 .8906 .9740 .8958 .9792 .9010 .9844 .9063 .9896 .9115 .9948 .90167 1.000 10

11

'p/*

1/s

3/s

14 6/s

*/s

7/1s

1/t

'/is

6/s 1'/is /*

1/s u, 1

TABLE IV

Dec.

10' 20 30 40 50

(p

-- RADII ORDINATES AND DrEFLcIONS

Mid. Ord.

Tan. Offset

34377.5 17188.8 11459.2 8594.42 6875.51

.036

.145 .291 .436 .582 .727

5729.61 4911.11 4207.28 3819.83 3437.87 3125.3

.218 .255 .291 .327

2864.93 2644.58 2455.70 2292.01 2148.79 2022.41

.436 .473 .545 .582 .618

1910.08 1F09.57 1719.12 1637.28

.655 .691 .727 764

Radius

.073

.109 .145 .182

Def. for 1 Foot

0.05'

0.10 0.15 0.20 0.25

.873 1.018 1.164 1.309 1.454 1.600

0.30 0.35 0.40 0.45 0.50 0.55

1.745 1.891 2.036 2.181 2.327 2.472

0.60 0.65 0.70 0.75 0.80 0.85

836

2.618 2.763 2.908 3.054 3.199 3.345

0.90 0.95 1.00 1.05 1.10 1.15

.873 .909 .945 1273. 51 .982 1228.11 1.018 1185.78 1.055

3.490 3.635 3.718 3.926 4.071 4.217

1.20 1.25 1.30 1.35 1.40 1.45

50

1146.28 1109.33 1074.88 1042.14 1011.51 982.64

1.091 1.127 1.164 1.200 1.237 1.273

4.362 4.507 4.653 4.798 4.943 5.088

1.50 1.55 1.60 1.65 1.70 1.75

10 20 30 40

955.37 1929.57 905.13 881.95 859.92

1.3009 1.346 1.382 1.418 1.455

5.234 5.379 5.524 5.669 5.814

1.80 1.85 1.90 1.95 2.00

10 20 30 40 50 3s 10 20 30 40 50 10 20 30 40 50 I5 10 20 30 40

.400

.509

.800

6

Note.

.364

De.

Chord Deflection ,2 times angent deflection.

Radius

Mid. Ord.

Tan. 0(et

Def. for 1 Foot

819.02 781.84 764.49 747.89

1.528 1.600 1.637 1.673

6.105 6.395 6.540 6.685

2.1 0' 2.20 2.25 2:30

716.78 688.16 674.69 661.74

1.746 1.819 1.855 1.892

6.976 7.266 7.411 7.556

2.40 2.50 2.55 2.60

637.28 614.56 603.80 593.42

1.965 2.037 2.074 2.110

7.846 8.136 8.281 8.426

2.70 2.80 2.85 2.90

573.69 546.44 521.67 499.06 478.34 459.28 441.68 425.40 410.28 396.20

2.183 2.292 2.402 2.511.

2.620 2.730 2.839 2.949 3.058 3.168

8.716 9.150 9.585 10.02 10.45 10.89 11.32 11.75 12.18 12.62

3.00

383.07 370.78 359.27 348.45 338.27 319.62 302.94

3.277 13.05 3.387 13.49 3.496 13.92 3.606 14.35 3.716 14.78 1 3.935 15.64 4.155 16.51

4.50 4.65 4.80 4.95 5.10 5.40 5.70

287.94 274.37 262.04 250.79 240.49

4.374 4.594 4.814 5.035 5.255

17.37 18.22 19.068 19.94 20.79

6 00 6.30 6.60 6.90 7.20

231.01 222.27 214.18 206.68 199.70 193.18

5.476 5.697 5.918 6.139 6.360 6.583

21.64 22.50 23.35 24.19 25.04 25.88

7.50 7.80 8.10 8.40 8.70 9.00

3.15

3.30 3.45 3.60 3.75 3.90 4.05 4.20 4.35

-----

,

TABLE V ,V CURVE FORMULE FOR SIMPLE CURVES COMPILED BY J. CALVIN LOCKE, C. E. (1)

c = 1/Ra

(2) c = fa'-+b

_

f2R

(3) c = (R - -/(R+b) (R-b) = 2R (R-R-b (4) c = 2 m (2R-m) (5) c = 2R sin /s 1 (6)c = 2T cos '/ (7) e=Rexsec 1/2 I (8) e = Rtan 1/tl tan '/ (9) e = Ttan / (10) b- }a (2R-a)

(11) b=

(12) b=Rsinl

a'+b'

(16) d =

s2ab R (2R-

(17) d =

2Rm

(14) R

(20) m - R +

-

)

(c+

Vc -

R)

(13) b= a cot s I c d' c'-4m' (15) R = 2a 2m 8m (2Rc) 2R-c)= R (2R- y4R'2I ) 18) d = 2R sin 1/4I R+

2

4 (R-

(19) m = 2R R

2

-

(21) m = R vers 'l1 (22) m = R sin '/1tan '/1 (23) m = 1/,c tan '1I (24) a =-~ (25) a = R - (Rb) (R--b) = R - R'-b' 26) a = 2 R (sin'i 11/1) (27) a = R vers I (28 a = R si Itan 1/i (29) a = b tan 1 /I (30) a = T sin (31) T = R tan t

(32) 1-

RX 57.295780

(33) R=

I X 57.295780

8d-c (34) L = IRX 0.01745329 (35) L = 3 LR-Rb LR-R' sin I (36Area S., 2 2

TABLE VI

SINES, COSINES, TANGENTS, COTANGENTS

wS

sin tan

sin

tan

sin

1

tan sin

tan sin

tan

000 0029 01750204 349 378 524 552 699 727 875 901 1051 1074 228 248 405 421 584 593

0029 005800580087 0204 0233 0233 0262 378 407 407 436 553 581 582610 729 756 758 785 904 929 934 958 1080 1103 1110 1132 257 279 287 305 435 449 465 478 614 622 644650

0087 0116 262 291 437 465 612 640 787 814 963 987 1139 1161 317 334 495 507 673 679

0116 0145 291 320 466 494 641 669 816 843 992 1016 1169 190 346 363 524 536 703 708

0145 89 320 88 495 87 67086 846 85 1022 84 198 83 376 82 554 81 733 80

10 736 11 908 12 2079 13250 14 419 15 588 16 756 17 924 18 3090 19256

763 765 944 937 2126 2108 309 278 493 447 679 616 867 784 3057 952 249 3118 443 283

793794 914 965 2156 2136 339 306 524 476 711644 899 812 3089 939 281 3145 476 311

823 822 2004 994 186 2164 370 334 555 504 742 672 931 840 3121 3007 314 173 508 338

853 851 2035 2022 217 193 401 363 586 532 773 700 962 868 3153 3035 346 201 541 365

883 880 2065 2051 247 221 432 391 617 560 805 728 994 896 3185 3062 378 228 574 393

914 79 2Q95 78 278 77 46276 648 75 83674 3026 73 217 72 411 71 607 70

20 420 21 584 22 746 23907 24 4067 25 226 26 384 27 540 28 695 29 848

640 448 839 611 4040 773 245934 4524094 663 253 877 410 5095 566 317 720 543 874

673 475 872 638 4074 800 279 961 4874120 699 279 913 436 5132 592 354 746 581899

706 502 906 665 4108 827 314 987 522 4147 734 305 950 4462 5169 617 392772 619924

739 529 939 692 4142 854 348 4014 557 173 770 331 986 488 5206 643 430797 658 950

772 557 973 719 4176 881 3834041 592 200 806 358 5022 514 243 669 467 823 96 975

805 4006 210 417 628 841 5059 280 505 735

30 5000 31 150 32 299 33446 34592 35 736 36 878 37 6018 38 157 39 293

774 5025 6009 175 29 494 471 745016 7002 760 265 901 536 6041 813 180 8098 316

5812 5050 6048 200 289 348 536 495 787640 7046 783 310 925 581 6065 860 202 8146 338

851 5975 6088 225 3305373 577 519 830 664 7089 807 355 948 627 6088 907 225 8195 361

890 5100 6128 250 371 398 619 544 873 688 7133 831 400972 673 6111 954 248 8243 383

930 5125 6168 275 412 422 661 568 916 712 7177 854 445 995 720 6134 8002 271 292 406

59 6208 58 453 70356 959 55 7221 54 490 53 766 52 8050 51 342 50

O'

0 0000 1 175 2 349.

3 523 4 698 5 872 6 1045 7 219 8 392 9 564

0'

10'

10'

20'

20'

30'

30'

40'

40'

sin

50'

tan

50'

69 08 67 66 65 64 63 62 61

60

40 428 391 450 441472 491494 54T517 5 91 539 64249 41 561 693 583 744 04 796 626 847 648 899 670 952 48 42 691 9004 713 9057 734 9110 756 9163 777 9217 799 9271 47 43 820 325 841 380 862 435 884 490 905 545 926 601 46 44 947 657 967 713 988 7707009 827 7030 8847050 94245 45 7071 1.0000 702 1.0058 7112 1.0117 133 1.0176 153 1.0235 173 1.0295 44

60'

.o cos

60'

cot

50'

cos

50' 40' cot

o os

40' 30' t

cos

3 30'

cot

2 20'

cos

20'

10'

cot cos

10'

cot

TABLE VI (continued)

SINES, COSINES. TANGENTS, COTANGENTS (continued) do S

sin

tan

'

0

sin

ten

10 10'

sin

tan

20'

20'

sin

30'

tan

'

sin

tan

sin

tan

40'

40'

50'

50'

'

46 7193 1.0355 7214 1.0416 7234 1.0477 7254 1.0533 7274 1.0599 7294 1.0661 43 412 .1041 42 392 .097 47 314 .024 333 .0786 353 .0850 373 .0913 48 431 .1106 451 .1171 470 .1237 490 .1303 .509 .1369 528 .1436 41 49 547 .1504 566 .1571 585 .1640 604 .1708 623 .1778 642 .1847 40 50 660 1.1918 7679 1.1988 7698 1.2059 7716 1.2131 7735 1.2203 7753 1.276 39 844 .2647 862 .2723 38 51 711 .2349 790 .2423 808 .2497 826 .2572 52 880 .2799 898 .2876 916 .2954 934 .3032 951 .3111 969 .3190 37 53 986 .3270 8004 .3351 8021 .3452 8039 .3514 8056 .3597 8073 .3680 36 175 .4193 35 158 .4106 141 .4019 .3848 124 .3934 .3764 107 54 8090 55 192 .4281 208 .4370 225 .4460 241 .4550 258 .4641 274 .4733 34 371 .5301 33 56 290 .4826 307 .4919 323 .5013 339 .5108 355 .5204 57 387 .5399 403 .5497 418 .5597 434 .5697 450 .5795 465 .5900 32 557 .6534 31 58 480 .6003 496 .6107 511 .6212 526 .6319 542 .6426 646 .7205 30 59 572 .6643 587 .6753 601 .6864 616 .6977 631 .7090

60

660 1.7321 8675 1.7437 8689 1.7556 8704 1.7675 8718 1.7797 61 746 .8040 760 .8165 774 .8291 788 .8418 802 .8546 62 829 .8807 843 .8940 857 .9074 870 .9210 884 .9347 63 910 .9626 923 .9768 936 .9912 949 2.007 962 2.0204 64 988 2.0503 9001 2.0655 9013 2.0809 9026 .0965 9038 .1123 65 9063 .1445 075 .1609 088 .1775 100 .1943 112 .2113 66 135 .2460 147 .2637 159 .2817 171 .2998 182 .3183 67 205 .3559 216 .3750 228 .3945 239 .4142 250 .4342 68 212 .4751 283 .4960 293 .5172 304 .5386 315 .5605 377 .6985 69 336 .6051 346 .6279 356 .6511 367 .6746

8732 1.7917 29 816 .8676 28 897 .9486 27 975 2.0353 26 9051 ,1283 25 124 .2286 24 194 .336 23 261 .4545 22 325 .5826 21 387 .7228 2

70 71 72 73 74 75 76 77 78 79

9446 502 555 605 652 696 737 775 811 843

2.8770 3.0475 .2371 .4495 .6891 .9617 4.2747 .6382 5.0658 .5164

19 18 17 16 15 14 13 12 11 10

5.9758 9868 6.0844 9872 6.6912 894 .8269 899 7.5958 918 7.7704 932 8.7769 939 9.0098 942 10.385 957 10.711 959 12.706 971 13.197 974 16.350 983 17.169 985 22.903 992 24.542 993 38.189 997 42.964 9998 114.58 1.000 171.88 1.000

6.1970 .9682 7.9530 9.2553 11.059 13.727 18.075 26.432 49.104 343.771

9 8 7 6 5 4 3 2 1 0

397 2.7475 9407 2.7725 9417 2.7980 9426 2.8239 9436 2.8502 455 .9042 465 .9319 474 .9600 483 .9887 492 3.0178 511 3.0777 520 3.1084 528 3.1397. 537 3.1716 546 .2041 596 .4124 563 .2709 572 .3052 580 .3402 588 .3759 613 .4874 621 .5261 628 .5656 636 .6059 644 .6470 659 .7321 667 .7760 674 .8208 681 .8657 689 .9136 703 4.0108 710 4.0611 717 4.1126 724 4.1653 730 4.2193 744 .3315 750 .3897 757 .4494 763 .5107 769 .5736 781 .7046 787 .7729 793 .8430 799 .9152 805 .9894 816 .1446 822 5.2257 827 5.3090 833 5.3955 838 5.4845

80 9848 5.6713 9853 5.7694 9858 5.8708 9863 81 877 6.3138 881 6.4348 886 6.5606 890 82 903 7.1154 907 7.2687 911 7.4287 914 83 925 8.1443 929 8.3450 932 8.5555 936 84 945 9.5144 948 9.7882 961 10.078 954 85 962 11.430 964 11.826 967 12.250 969 86 976 14.300 978 14.924 980 15.605 981 87 986 19.081 988 20.206 989 21.470 990 88 994 28.636 9995 31.242 9996 34.368 997 89 9998 57.290 9999 68.750 9999 85.940 9999

p

60

60'

50'

50'

40'

40' 30'

30'

20'

20'

10'

10'

cos

cot

cos

cot

cos

cot

cot

cos

cot

cos

cot

cos

.v

TABLE VII Deflections for Sub Chords for Short Radius Curves.

Y sub chord =

Degree

Radius

Curve

sin. def. ang

12.5 Ft.

30 320 34 36

193.18

1 5I' 10 9'

380 40

153.58 146.19

42440

of

_

460 480 50 520 54

56* 580 60~ --

T

=

T

181.39 171.01

161.8

20 06' 2 23 2 20'

_

°

def. angle .

2.Ft.

20 Ft.

2 58' 3o O' 3. 21' 3 33'

30431

58' 12' 4° 26'

101.15 101.33 101.48 Iol.66

4

101.85

17

3 4

40

27'

49 2° 57

3 44' 3551

139.52 133.47 127.97

2

4'

3 05

4 07'

' 454' 5 08'

2 41' 20 48

3 3

4 4

18' 29'

5 022' 5 36'

122.92

20 551

30 29

4

40

550'

118.31 I14.06

3002'

3

38'

4

3 3

46 54

11 110o.

Io6.50 o103.14 100.00 it ii

Rtan 5otan

Length of arc for 100 ft.

2 205' 2 0 33' 241

2

2

Sin. Sin.

sinof

__R

15 Ft.

2

°

13'

21'

I02.o6 102.29 102.53

102.76 103.00

°

°

30 o 3 16'

3022' 3 29 3 35'

4

"

2'

4

10' 4018'

51 5021 5 131

5 ' 5 34 50 44

'

103.24 103.54

604'

6o 631'

103.84 104.14 Io4.43

6 44' 60 ' 7

11'

ilI

104.72 l

I

R = T cot. ;I R Sin. I D

I

D

Chord def. = chord'

50

D =

E = R ex. sec

II

No. chords = I

D = 50 tan I E =T tan j I Tan. def.= I chord def.

T The square of any distance, divided by twice the radius, will equal the distance from tangent to curve, very nearly.

To find angle for a given distance and deflection.

Rule I. Multiply the given distance by .01745 (def. for ° for I ft. see Table II.), and divide given deflection by the product.

Rule 2. Multiply given deflection by 57.3, and divide the product by the given distance.

To find deflection for a given angle and distance. Multiply the angle by .01745, and the product by the distance.

GENERAL DATA RIGHT ANGLE TRIANGLES. Square the altitude, divide by twice, the base. Add quotient to base for hypotenuse.

Given Base -Ioo, Alt. bo.o+200oo=.5. oo0+.5=Ioo.5 hyp.

Sin.

Given Hyp. Too, Alt. 25.252

oo=3.125. 100-3.125=96.875= Base.

Error in first example, .oo2; in last, .045.

To find Tons of Rail in one mile of track: multiply weight per yard

by II,and divide by 7.

LEVELING. The correction for curvature and refraction, in feet and decimals of feet is equal to 0.574 d s , where d is the distance in miles.

The correction for curvature alone is closely, jd'. The combined correction is negative.

PROBABLE ERROR. If dl, d, ds, etc. are the discrepancies of various results from the mean, and if da2=the sum of the squares of these differences and n=the number of observations, then the probable error of the l

mean= -- 0.6745j n I

nn1)

TABLE VIII TANGENTS AND EXTERNALS 'TO A 1 CURVE. 1

T

.E I=

I

0

" 10 10' 20' 30' 40' 50' go0 10' 20' 30' 40' 50' 30

10' 20' 30' 40' 50' 40

10' 20' 30' 40' 50' So 10' 20' 30' 40' 50" 50' 50o

10' 20' 30' 40' 50' '70

10' 20'

go 40' 30' 50' 10' 50' 20' go 30' 40' 10' 20' 30' 40' 50'

50'

50.00 58.34 66.67 75.01 83.34 91.68 100.01 108.35 116.68 125.02 133.36 141.70 150.04 158.38 166.72 175.06 183.40 191.74 200.08 208.43 216.77 225.12 233.47 241.81 250.16 258.51 266.86 275.21 283.57 291.92 300.28 308.64 316.99 325.35 333.71 342.08 350.44 358.81 367.17 375.54 383.91 392.28 400.66 409.03 417.41 425.79 '434.17 442.55 450.93 459.32 467.71 476.10 484.49 492.88 501.28 509.68 518.08 526.48 534.89 543.29

E

T

n

551.70 560.11 568.53 + 576.95 T 585.36 .606 .03 593.79 .733 E 602.21 .873 .001 120 10' 610.64 1.024 20' 619.07 1.188 1.364 30' 627.50 1.552 40" 635.93 1.752 50' 644.37 o 652.81 1.964 100 C. 1i 2.188 10' 661.25 T 20' 669.70 2.425 .06 2.674 30' 678.15 E 2.934 40' 686.60 695.06 3.207 703.51 3.492 140 10' 711.97 3.790 4.099 20' 720.44 4.421 30' 728.90 4.755 40' 737.37 5.100 150 C 50' 745.85 T 5.459 754.32 762.80 5.829 .09 E 771.29 6.211 .004 6.606 30' 779.77 40' 788.26 7.013 7.432 50' 796.75 805.25 7.863 8.307 180 813.75 8.762 20' 822.25 830.76 9.230 200 C 9.710 40' 839.27 T 10.202 .13 50' 30' 847.78 E 856.30 10.707 .006 11.224 10' 864.82 11.753 470" 20' 873.35 12.294 30' 881.88 12.847 40' 890.41 13,413 50' 898.95 13.991 907.49 180 14.582 250 C. 10' 916.03 15.184 T 20' 924.58 15.799 .16 30' 933.13 16.426 40' 941.69 E 17.065 .0017 50' 950.25 958.81 17.717 190 18.381 10' 967.38 19.058 20' 975.96 19.746 30' 984.53 20.447 40' 993.12 21.161 300 C. 50' 1001.7 T 1010.3 21.887 200 22.624 .19 10' 1018.9 E 23.375 20' 1027.5 24.138 .008 30' 1036.1 1044.7 24.913 25.700 50' 1053.3 .218 .297 .388 .491

10' 120' 30' 40' 50'

so'

'----

T-

R tan /Q

T

i

I=20o

E

I=309

'"

o

E

26.500 27.313 28.137 28.974 29.824 30.686 31.561 32.447 33.347 34.259 35.183 36.120 37.070 38.031 39.006 39.993 40.992 42.004 43.029 44.066 45.116 46.178 47.253 48.341 49.441 50.554 51.679 52.818 53.969 55.132 56.309 57.498 58.699 59.914 61.141 62.381 63.634 64.900 66.178 67.470 68.774 70.091 71.421 72.764 74.119 75.488 76.869 78.264 79'.671 81.092 82.525 83.972 85.431 86.904 88.389 89.888 91.399 92.924 94.462 96.013

o40

10' 20' So C. 30' T 40' .06 50' x,.006 22o 10' 20' 30' 40' 50' 100 C. 230 10' 20' .13 30' E 40' .011 50' 240 10' 20' 30' 40' 150 C. 50' T 250 .19 10' E 20' .017 30' 40' 50' 260 10' 20' 200 C. 30' 40' T 50' .26 E .022 270 10' 20' 30' 40' 50' 280

10' 20' 30' 40' 50' 290 10' 20' 30' 40' 50' 300 C. T 30o .39 10' E 20' .034 30' 40' 50' 250o C.

T .32 E .028

"

1061.9 97.577 1070.6 99.155 1079.2 100.75 + 1087.8 102.35 So C. T 1096.4 103.97 1105.1 105.60 .10 E 1113.7 107.24 .013 1122.4 108.90 1131.0 110.57 1139.7 112.25 1148.4 113.95 1157.0 115.66 1165.7 117.38 100 C. 1174.4 119.12 T 1183.1 120.87 .19 1191.8122.63 E 1200.5 124.41 .025 1209.2. 126.20 1217.9 128.00 1226.6 129.82 1235.3 131.65 1244.0 133.50 1252.8 135.35 1261.5 137.23 150 C. T 1270.2 139.11 1279.0 141.01 .29 E 1287.7 142.93 .038 1296.5 144.85 1305.3 146.79 1314.0 148.75 1322.8 150.71 1331.6 *152.69 1340.4 154.69 1349.2 156.70 200 C. 1358.0 158.72 1366.8 160.76 .39 E 1375.6 162.81 1384.4 164.86 .051 1393.2 166.95 1402.0 169.04 1410.9 171.15 1419.7 173.27 1428.6 175.41 1437.4 177.55 250 C. T 1446.3 179.72 1455.1 181.89 .49 1464.0 184.08 E 1472.9 186.29 .065 1481.8 188.51 1490.7 190.74 1499.6 192.99 1508.5 195.25 1517.4 197.53 1526.3 199.82 300 C. T 1535.3 202.12 1544.2 204.44 .59 E 1553.1 206.77 1562.1 209.12 .078 1571.0 211.48 1580.0 213.86

R exsec'lj I

o TABLE VIII TANGENTS AND EXTERNALS TO A 10 CURVE

I

T

I=40

E

1589.0 216.3 1598.0 218.7 1606.9 221.1 1615.9 223.5 1624.9 226.0 1633.9 228.4 1643.0 230.9 1652.0 233.4 235.9 1661.0 1670.0 238.4 1679.1 241.0 243.5 1688.1 1697.21 246.1 248.7 1706.3 1715.3 251.3 1724.4 253.9 40' 1733.5 256.5 259.1 50' 1742.6 261.8 340o 1751.7 264.5 10' 1768.8 267.2 20' 1770.0 269.9 30' 1779.1 272.6 40' 1788.2 50 1797.4 275.3 350 1806.6 278.1 10' 1815.7 280.8 283.6 20' 1824.9 30' 1834.1 286.4 1843.3 289.2 40' 50' 1852.5 292.0 1861.7 294.9 360 10' 1870.9 297.7 1880.1 300.6 20' 30' 1889.4 303.5 40' 1898.6 306.4 305.3 50' 1907.9 1917.1 312.2 370 10' 1926.4 315.2 20' 1935.7 318.1 321.1 30' 1945.0 40' 1954.3 324.1 50' 1963.6 327.1' 1972.9 330.2 8 10' 1982.2 333.2 20' 1991.5 336.3 2000.9 339.3 30' 40' 2010.2 342.4 50' .2019.6 345.5 N00 2029.0 348.6 10' 2038.4 351.8 354.9 20' 2047.8 30' 2057.2 358.1 40' 2066.6 361.3 50' 2076.0 364.5 2085.4 367.7 400 10' 2094.9 371.0 20' 2104.3 374.2 377.5 30' 2113.8 40' 2123.3 380.8 384.1 50' 2132.7 o10

10' 20' 30' 40' 50' 320 10' 20' 30' 40' 50' 3830 10' 20' 30'

I

410 10' + 20' o C. 30' T .13 40' E 50' .023 42 o 20' 30' 40' 50' 100 C. 1T .26 E .046

15o C.

430

10' 20' 30' 40' 50' 440 10' 20' 30' 40' 50'

.40 E .070

10' 20' 30' 40' 50' 460 10' 20' 30' 200 C. 40' T 50' .53 E 470 .093 10' 20' 30' 40' 50' 480 10' 250 C. 20' T 30' .67 40' 50' .117 490 10' 20' 30' 40' 50' 0 30 C. 50o.

T 10' .80 20' E 30' *141 40' 50'

T= Rtan 's I

T

B I=50

2142.2 387.4 510o 2151.7 390.7 + 10' 2161.2 394.1 5° C. 20' 2170.8 397.4 30' T 2180.3 400.8 .17 40' 2189.9 404.2 E 50' 2199.4 407.6 .037 520 2209.0 411.1 10' 414.5 2218.6 20' 2228.1 418.0 30' 2237.7 421.4 40' 425.0 50' 2247.3 428.5 100 C. 53 o 2257.0 432.0 2266.6 10' T 2276.2 435.6 .34 20' 2285.9 439.2 30' E 442.8 .075 2295.6 40' 446.4 2305.2 50' 450.0 2314.9 540 2324.6 453.6 10' 457.3 2334.3 20' 2344.1 461.0 30' 464.6 2353.8 40' 2363.5 468.4 150 C. 50' T 2373.3 472.1 550 .51 2383.1 475.8 10' E 2392.8 479.6 20' 2402.6 483.4 .116 30' 2412.4 487.2 40' 2422.3 491.0 50' 494.8 2432.1 560 498.7 2441.9 10' 2451.8 502.5 20' 506.4 200 C. 30 2461.7 2471.5 510.3 40' 514.3 2481.4 50' .68 E 2491.3 518.2 570 2501.2 522.2 .151 10' 2511.2 526.1 20' 530.1 2561.1 30' 534.2 2531.1 40' 2541.0 538.2 50' 542.2 2551.0 58o 2561.0 546.3 250 C. 10' 2571.0 20' 550.4 T 2581.0 554.5 .85 2591.0 558.6 40' E 2601.1 562.8 .189 50' 566.9 2611.2 2621.2 571.1 10' 2631.3 575.3 20' 2641.4 579.5 30' 2651.5 583.8 40' 2661.6 588.0 50' 2671.8 592.3 300T C. 600 2681.9 10' 596.6 1.02 2692.1 600.9 20' E 2702.3 605.3 .227 30 2712.5 609.6 40' 2722.7 614.0 50'

.I

T

I

E

618.4 2732.9 2743.1 622.8 2753.4 627.2 2763.7 631.7 2773.9 636.2 2784,2 640.7 2794.5 645.2 2804.9 649.7 2815.2 654.3 2825.6 658.8 2835.9 663.4 2846.3 668.0 2856.7 672.7 2867.1 677.3 2877.5 682.0 2888.0 686.7 2898.4 691.4 2908.9 696.1 2919.4 700.9 2929.9 705.7 2940.4 710.5 2951.0 715.3 2961.5 720.1 2972.1 725.0 2982.7 729.9 2993.3 734.8 3003.9 739.7 3014.5 744.6 3025.2 749.6 3035.8 754.6 3046.5 759.6 3057.2 764.6 3067.9 769.7 3078.7 774.7 3089.4 779.8 3100.2 784.9 3110.9 790.1 795.2 3121.7 3132.6 800.4 3143.4 805.6 3154.2 810.9 3165.1 816.1 3176.0 821.4 3186.9 826.7 3197.8 832.0 3208.8 837.3 3219.7 842.7 3230.7 848.1 3241.7 853.5 3252.7 858.9 3263.7 864.3 869.8 3274.8 3285.8 875.3 3296.9 880.8 3308.0 886.4 3319.1 892.0 3330.3 897.5 3341.4 903.2 3352.6 908.8 3363.8 914.5

E=R exsec 112

I=to0

+C,

50

T .21 E .056

100 C. T .42 E .112

15 C. T .63 E .168

200 C. T .81 E .22

250 C.

T 1.05 E 2.83

30 C T 1.27 E .340

TABLE VIII TANGENTS AND EXTERNALS TO A 10 CURVE.

1 610 10' 20' 30' 40' 50' 10' 20' 30' 40' 50' 10' 20' 30' 40' 50' 11o

10' 20' 30' 40' 50' 10' 20' 30' 40' 50' U0

10' 20' 30' 40' 50' 870

10' 20' 30' 40' 50' 0

10' 20' 30' 40' 50' 10' 20' 30' 40' 50' 10' *20' 30' 40' 50'

T

E

I=700

3375.0 920.2 3386.3 925.9 3397.5 931.6 + 3408,8 937.3 5° C. T 3420.1 943.1 3431.4 948.9 .25 E 3442.7 954.8 .080 3454.1 960.6 3465.4 966.5 3476.8 972.4 3488.3 978.3 3499.7 984.3 3511.1 990.2 100 C. 3522.6 996.2 T 3534.1 1002.3 .51 3545.6 1008.3 E 3557.2 1014.4 159 3568.7 1020.5 3580.3 1026.6 3591,9 1032.8 3603.5 1039.0 3615.1 1045.2 3626.8 1051.4 3638.5 1057.7 150 C. T 3650.2 1063.9 3661.91070.2 .76 E 3673.7 1076.6 .240 3685.4 1082.9 3697.2 1089.3 3709.0 1095.7 3720.9 1102.2 3732.7 1108.6 3744.6 1115.1 3756.5 1121.7200 C. 3768.5 1128.2 T 3780.4 1134.8 1.02 3792.4 1141.4 E 3804.4 1148.0 .321 3816.4 1154.7 3828.4 1161.3 3840.5 1168.1 3852.6 1174.8 3864.7 1181.6 3876.8 1188.4 250 C. T 3889.0 1195.2 3901.2 1202.0 1.28 E 3913.4 1208.9 3925.6 1215.8 .403 3937.9 1222.7 3950.2 1229.7 3962.5 1236.7 3974.8 1243.7 3987.2 1250.8 3999.5 1257.9 300 C. T 4011.9 1265.0 4024.4 1272.1 1.54 E 4036.8 1279.3 4049.3 1286.5 .485 4061.8 1293.6 4074.4 1300.9

1

E 1I=80oI

T I

710o ' 'l 4086.9 1308.2 10' 4099.5 1315.6 20' 4112.1 1322.9 0 30' 4124.8 1330.3 5 C. T 40" 1137.4 1337.7 .30 50' 4150.1 1345.1 o E 72 0 4162.8 1352.6 .110 10' 4175.6 1300.1 20' 4188.5 1367.6 30' 4201.2 1375.2 40' 4214.0 1382.8 50' 4226.8 1390.4 7o0 4239.7 1398.0 100 C 10' 4252.6 1405.7 'F 20' 4265.6 1413.5 .61 30' 4278.5 1421.2 E 40' 4291.5 1429.0 .220 50' 4304.6 1436.8 4317.6 1444.6 740 T 10' 4330.7 1452.5 4343.8 1460.4 20' 30' 4356.9 1468.4 40' 4370.1 1476.4 50' 4383.3 1484.4 150 C T 4396.5 1492.4 .91 o50 1500.5 10' E 1508.6 .332 20' 1516.7 30' 1524.9 40' 1533.1 50' 1541.4 760 1549.7 10' 20' 4503.4 1558.0 45309.84 1566.3 200 C. 30' 4516.9 40' 4544.0 1574.7 T 1583.1 1.22 50' 4557463.1 E 1591.6 770 1600.1 .445 10' 4571.2 4626544.0 20' 4598.5 1608.6 4512.9 4553.6 1617.1 1625.7 40' 1634.4 50' 780 461295.21643.0 1651.7 250 C 10' T 20' 4681.3 1660.5 1669.2 1.53 30' E 40' 463709.21678.1 1686.9 .558 50' 4723.2 1695.8 790 10' 4737.2 1704.7 20' 4751.2 1713.7 30' 4765.3 1722.7 40' 4779.4 1731.7 50' 4793.6 1740.8 300 C. T 4807.7 1749.9 800 10' 4822.0 1759.0 1.84 E 20' 4836.2 1768.2 30' 4850.5 1777.4 .671 1786.7 4864.8 40' 50' 4879.2 1796.0

E=Rla~n1/

E

I=90

4893.6 1805.3 4908.,0) 1814.7 4922.5 1824.1 4937.0 1833,6 'T 1843.1 40'. 4951.5 .36 50' 4966.1 1852.6 E 4980.7 1862.2 .149 820 10' 4995.4 1871.8 20' 5010.0 1881.5 30' 5024.8 1891.2 40' 5039.5 1900.9 50' 5054.3 1910.7 5069.2 1920.5 10o C. 8340 10' 5084.0 1930.4 T 20' 5099.0 1940.3 .72 30' 5113.9 1950.3 40' 5128.9 1960.2 .299 5143.9 1970.3 50' 5159.0 1980.4 8540 10' 5174.1 1990.5 20' 5189.3 2000.6 30' 5204.4 2010.8 5219.7 2021.1 40' 50' 5234.9 2031.4 150 C. 5250.3 2041.7 85e 1.09 10' 5265.6 2052.1 20' 5281.0 2062.5 .450 30' 5296.4 2073.0 40' 5311.9 2083.5 50' 5327.4 2094.1 o 5343.0 2104.7 88 10' 5359.6 2115.3 20' 5374.2 2126.0 30' 5389.9 2136.7 20 0 .C. T 40' 5405.6 2147.5 50' 5421.4 2158.4 1.45 E 2169.2 5437.2 870 .603 10' 5453.1 2180.2 2191.1 5469.0 20' 30' 5484.9 2202.2 40' 5500.9 2213.2 50' 5517.0 2224.3 5533.1 2235.5 88 10' 5549.2 2246.7 250 C. T 20' 5565.4 2258.0 30' 5581.62269.3 1.83 E 40' 5597.8 2280.6 50' 5614.2 2292.0 .756 5630.5 2303.5 890 10' 5646.9 2315.0 20' 5663.4 2326.6 30' 5679.9 2338.2 5696.4 2349.8 5713.0 2361.5 30o C. 50' T 5729.7 2373.3 900 10' 5746.3 2385.1 2.20 E 20' 5763.1 2397.0 30' 5779.9 2408.9 .910 40' 5796.7 2420.9 50' 5813.6 2432.9 81o

10' 20' 30'

"

T

E=Resec1/a

TABLE VIII

I;EI=0 I 910 10' 20' 30' 40' 50' 920 10' 20' 30' 40' 50' S0 10' 20' 30' 40' 50' 940

10' 20' 30' 40' 50' 950

10' 20' 30' 40' *50' 96o 10' 20' 30' 40' 50' 970

10' 20' 30' 40' no~ 50' 980 10' 20' 30' 40' 50' 990

10' 20' 30' 40' 50' 1000 10' 20' 30' 40' 50'

TiT t=11l

TANGENTS AND EXTERNALS TO A 10 CURVE

5830.5 2444.9 5847.5 2457.1 -5864.6 2469.3 5 C. T 5881.72481.5 5898.8 2493.8 .43 5916.0 2506.1 E 5933.2 2518.5 .200 5950.5 2531.0 5967.9 2543.5 5985r3 2556.0 6002.7 2568.6 6020.2 2581.3 6037.8 2594.0 10 C. 6055.4 2606.8 T 6073.1 2619.7 .86 6090.8 2632.6 E 6108.6 2645.5 .401 6126.4 2658.5 6144.3 2671.6 6162.2 2684.7.

6180.2 2697.9 6198.3 2711.2 6216.4 2724.5 6234.6 2737.9 150 C. T 6252.8 2751.3 1.30 6271.1 2764.8 E 6289.4 2778.3 .604 6307.9 2792.0 6326.3 2805.6 6344.8 2819.4 6363.4 2833.2 6382.1 2847.0 6400.8 2861.0 6419.5 2875.0 20o C. 6438.4 2889.0 T 6457.3 2903.1 1.74 E 6476.2 2917.3 .809 .6495.2 2931.6 6514.3 2945.9 6533.4 2960.3 6552.6 2974.7 6571.9 2989.2 6591.2 3003.8 6610.6 3018.4 250 C. 6630,1 3033.1 T 6649.6 3047.9 2.18 6669.2 3062.8 E 6688.8 3077.7 1.02 6708.6 3092.7 6728.4 3107.7 6748.2 3122.9 6768.1 3138.1 6788.1 3153.3 6808.2 3168.7 0 30 C. 6828.3 3184.1 T 6848.5 3199.6 2.62 6868.8 3215.1 E 6889.2 3230.8 1.22 6909.6 3246.5 6930.1 3262.3

1610 10' 20' 30' 40' 50' 1020 10' 20' 30' 40' 50' 1030o 10' 20' 30' 40' 50'

6950.6 6971.3 6992.0 7012.7 7033.6 7054.5 7075.5 7096.6 7117.8 7139.0 7160.3 7181.7 7203.2 7224.7 7246.3 7268.0 7289.8 7311.7 104 7333.6 10' 7355.6 20' 7377.8 30' 7399.9 40' 7422.2 50' 7444.6 1050 7467.0 10' 7489.6 20' 7512.2 30' 7534.9 40' 7557.7 50' 7580.5 1060 7603.5 10' 7626.6 20' 7649.7 30' 7672.9 40' 7696.3 50' 7719.7 107o 7743.2 10' 7766.8 20' 7790.5 30' 7814.3 40' 7838.1 50' 7862.1 1080 78$6.2 10' 7910.4 20' 7934.6 30' 7959.0 40' 7983.5 50' 8008.0 1090 8032.7 10' 8057.4 20' 8082.3 30' 8107.3 40' 8132.3 50' 8157.5 1100 8182.8 10' 8208.2 20' 8233.7 30' 8259.3 40' 8285.0 50' 8310.8

--

T = R tan ' I

I T E1120

IEE1

3278.1 3294.1 3310.1 5o C. 3326.1 T 3342.3 .51 3358.5 E 3374.9 .268 3391.2 3407.7 3424.3 3440.9 3457.6 3474.4 10o C 3491.3 3508.2 .103 3525.2 E 3542.4 .536 3559.6 3576.8 3594.2 3611.7 3629.2 3646.8 3664.5 150 C T 3682.3 1.56 3700.2 E 3718.2 .806 3736.2 3754.4 3772.6 3791.0 3809.4 3827.9 3846.5 20o C 3865.2 T 3884.0 2.08 E 3902.9 3921.9 1.08 3940.9 3960.1 3979.4 3998.7 4018.2 4037.8 250 C, 4057.4 T 4077.2 2.61 4097.1 E 4117.0 1.36 4137.1 4157.3 4177.5 4197.9 4218.4 4239.0 300 C. 4259.7 T 4280.5 3.14 4301.4 E 4322.4 1.63 4343.6 4364.8

:;I

1110 10' 20' 30' 40' 50'

1120

10' 20' 30' 40' 50' 1130 itao 10' 20' 30' 40' 50'

0

114

10' 20' 30' 40' 50' 1150 10' 20' 30' 40' 50' 1160 10' 20' 30' 40' 50'

1179

10' 20' 30' 40' 50' 1180 10' 20' 30' 40' 50' 1190 10' 20' 30' 40' 50' 1200 10' 20' 30' 40' 50'

8336.7 4386.1 8362.7 4407.6 8388.9 4429.2 8415.1 4450.9 8441.54472.7 6468.0 4494.6 8494.6 4516.6 8521.3 4538.8 8548.1 4561.1 8575.0 4583.4 8602.1 4606.0 8629.3 4628.6 8656.6 4651.3 8684.0 4674.2 8711.5 4697.2 8739.2 4720.3 8767.0 4743.6 8794.9 4766.9 8822.9 4790.4 8851.0 4814.1 8879.3 4837.8 4861.7 4885.7 8965.0 4909 9 8993.8 4934.1 9022.7 4958.6 9051.7 4983 1 9080.9 5007.8 9110.3 5032.6 9139.8 5057.6 9169.4 5082.7 9199.1 5107.9 9229.0 5133.3 9259.0 5158.8 9289.2 5184.5 9319.5 5210.3 9349.9 5236.2 9380.5 5262.3 9411.3 5288.6 9442.2 5315.0 9473.2 5341.5 9504.4 5368.2 9535.7 5395.1 9567.2 5422.1 9598.9 5449.2 9630.7 5476.5 9662.6 5504.0 9694.7 5531.7 9727.0 5559.4 9759.4 5587.4 9792.0 5615.5 9824.8 5643.8 9857.7 5672.3 9890.8 5700.9 9924.0 5729.7 9957.5 5758.6 9991.0, 5787.7 10025.0 5817.0 10059.0 5846.5 10093.0 5876.1

I- I

E = R exsec l/2 I

o5C. T .62 E .360

T 1.25 *E

721

15 C. 1.93 E 1 09

T 2.52 E 1.46

250 C

T

3.16 1.83

300C. T 3.81

E 2.20

IX

TABLE

MIDDLE ORDINATES OF RAILS Length of Rail (feet)

c

O'

RI 30 128

Feet

0-20 17189 0-40 1-0 1-20 1-40 2 -0

4-20

2-40 3-0 3-20 3-40 4-0 4-20 4-40 5 6

7

8594 5730 4297 3438 2865 2456 2149 1910 1719 1563 1433 1323 1228 1146 955.3

26 I

Inch Inch Inch

.08

.07

24

.061 .05

.16 .14 .21 .24 .20 .18 .31 .27 .23 .39 .34 .29 .47 .41 .35 .55 .48 .41 .63 .55 .47 .71 .62 .53 .78 .68 .59 .86 .75 .65 .94 .82 .71 1.02 .89 .77 1.10 .96 .83 1.18 1.03 .89 1.41 1.23 1.06

22 20 C

nchl nchlch 0 .10 .15 . .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 75 .90

.041.03

.081.07 .13 .10 17 .13 .21 .17 .25 .20 .29 .23 .33 .27 .38 .31 .42 .35 .46 .38 .50 .42 .55 .45 .59 .48 .63 .52 .76 .62

819.0 1.65 1.44 1.24 1.05 .89 .73

R

30

Feet

8 716.8

9 10 11 12 13 14 15 16 17 18 19 20 22 24 26

28

26

24

Inch Inch Inch Inch

637.3 573.7 521.7 478.3 441.7 410.3 383.1 359.3 338.3 319.6 302.9 287.9 262.0 240.5 222.3

1.8811.64

2.12 1.84 2.36 2.05 2.59 2.26 3.83 2.47 3.05 2.66 3.30 2.87 3.54 3.08 3.76 3.28 3.7635 4.00 3.48 4.21 3.67 4.45 3.89 4.70 4.09 5.16 4.44 5.64 4.92 6.07 5.29

1.42

1.60 1.78 1.95 2.15 2.30 2.48 2.68 2.83 3.02 3.18 3.36 3.55 3.84 4.20 4.58

122

20 nch Inch

1.28 1.01

1.35 1.50 1.65 1.81 1.96 2.10 2.26 2.40 2.57 2.70 2.86 3.00 3.30 3.59 3.88

1.14 1.27 1.39 1.54 1.66 1.78 1.91 204 2.16 2.28 2.41 2.54 2:80 3.04 3.29

.84

.94 1.04 1.15 1.26 1.36 1.46 1.57 1.67 1.78 1.87 1.98 2.09 2.29 2.50 2.70

TABLE X

SHORT RADIUS CURVES Radius

Chord

Central

Deflection Angle

for 1 Foot

35 45 50 60 75 100 120 150 190 200 225 240 250 275 288 300 350 376 400 410

10 10 15 15 15 20 20 20 25 25 25 25 25 25 50 50 50 50 50 50

16-26 12-46 17-16 14-22 11-30 11-30 9-34 7-39 7-32 7-10 6-25 5 58 5 44 5-12 9-58 9 -32 8-12 '-40 7-1( 7-00

8-13 6-23 8-38 7-11 5-45 5-45 4-47 3-49 3-46 3-35 3-12 2-59 2-52 2-36 4-59 4-46 .4-06 3-50 3-35 3-30

49.3 38.3 34.5 28.8 23.0 17.3 14.3 11.5 9.15 8.6 7.7 7.2 6.9 6.2 6.0 5.7 4.9 4.6 4.3 4.2

Feet

Feet

Angle

Deflection

To find length of curve divide anele from P. C. to P. T. by central angle of chord, and multiply by length of chord.

TABLE XI INCLINED DISTANCE OF 100 FT. REDUCED TO HORIZONTAL Slope

Hori

nta Correction

Rise

Slope 800'

0.000

99.999 99.996 99.991 99.985 99.976 99.966 99.953 99.939 99.923 99.905 99.885 99.863 99.839 99.813 99.786

0.000

15' 30' 45' 1 00 15 30 45 2 00 15 30 45 3 00 15 30 45

0.001 0.004 0.009 0.015 0,024 0.034 0.047 0.061 0.077 0.095 0.115 0.137 0.161 0.187 0.214

0.00 0.009 0.013 0.017 0.022 0.026 0.031 0.035 0.039 0.044 0.048 0.052 0.057 0.061 0.065

15' 30' 45' 9 00 15 30 45 10 00 15 30 45 11 00 15 30 45

400

99.756

0.244

0.070

99.725 99.692 99.657 99.619 99.580 99.540 99.497 99.452 99.406 '99.357 99.307 99.255 99.200 99.144 99.087

0.275 0.308 0.343 0.381 0.420 0.460 0.503 0.548 0.594 0.643 0.693 0.745 0.800 0.856 0.913

0.074 0.078 0.083 0.087 0.092 0.096 0.100 0.105 0.109 0.113 0.118 0.122 0.126 0.131 0.135

0000'

15 30 45 5 00 15 30 45 6 00 15 30 45 7 00 15 30 45

100.000

Hoistan 99.027

Correction

Rise

0.913

0.139

98.965 98.902 98.836 98.769 98.700 98.629 98.556 98.481 98.404 98.325 .98.245 98.163 98.019 97.992 97.905

1.035 1.098 1.164 1.231 1.300 1.371 1.444 1.519 1.596 1.675 1.755 1.837 1.921 2.008 2.095

0.143 0.148 0.152 0.156 0.161 0.165 0.169 0.174 0.178 0.182 0.18 0.191 0.195 0.199 0.204

1200

97.815

2.185

0.208

15 30 45 13 00 15 30 45 14 00 15 30 45 15 00 15 30 45

97.723 97.630 97.534 97.437 97.338 97.237 97.134 97.030 96.923 96.815 96.705 96.593 96.479 96.363 96.246

2.277 2.370 2.466 2.563 2.662 2.763 2.866 2.970 3.077 3.185 3.295 3.407 3.521 3.637 3.754

0.212 0.216 0.221 0.225 0.229 0.233 0.238 0.242 0.246 0.250 0.255 0.259 0.263 0.267 0.271

For each foot take one one.hundredth of each reading.

TABLE XII MINUTES IN DECIMALS OF A DEGREE. '30" S00 30 200 30

.00833o0'30" .01667 11 00 .02500 3 .03333 1200 .04167 30

.17500 20'30" .18333 21 00 .191673 20000 22 00 .20833 30

.34167 30'30" .50833 40'30" .67500 50'30" .35000 31 00 .51667 41 00 ,68333 51 00 30 .69167 30 .35833 30.52500 .36667 3200 .5333 42 00 .70000 52 00 .37500 30.54167 30 .70833 30

.84167 .85000 .85833 .86667 .87500

4 00 30 5 0

.0667 14 00 .0750011 30 .08333 00 30 .016730 .100001600 .10833 30

.23332400 . .24167 0 .25000 2500 .25833 30 .2666726 00 .27500 30

.400003400 .56667 44 30 .57500 .40833 3500 .42500 30.59167 .433333600 .6000046 .44167 30 .60833

00 30

.45000 37 00

.73333 5400 30 00 75000 30 30.75833 00 .766675600 30 .77500 30

.61667 47 00

.78333 57 00

90000 .90833 9166 .92500 .93333 .94167

.12500

.29167

.45833

.62500

30

.79167

30

.95833

.80000 58 00 .80833 30 .81667 59 00 .82500 30

.96661 .97500 .98333 .99167

6 00 30

7 00

30

800 30 9 00 30

0 00

.11667 17 00 .28333 27 00 30

.13333 18 00 .14167 30 .15000 190 .15833 30

.166672000

30

.30000 2800 .30833 30 .3167 29 00 .32500 30

.333333000

30

.46667 38 00 .47500 30 .4833 3900 .49167 30

.50000 40 00

.63333 48 00 .64167 30 .65000 49 00 .65833 30

.6666750 00

.74167

.83333 6000

.95000

1.00000

TABLE

XII I-CORRECTIONS

FOR TANGENTS AND EXTERNALS

These.corrections are to be added to the approximate values, found by dividing the tangent, or external, for a 10 curve (Table V III) by the degree of curve, in order to obtain

the true tangents, or externals.

Intermediate values may be obtained by interpolation.

FOR TANGENTS ADD DEREE

Central

Angle

°

5

15*

10

°

20

250

OF

°

30

35

CURVE

450

40°

10 15 200 25'

03 04 06 .08

.06 10 .13 16

800 80 40 4P

10 11 13 15

19 22 .26 .30

.29 39 .34 47 .40 .53 44 60

800°

17 19 .21 23

.34 38 42 46

.51 .68 .85 1.02 1.19 1.36 1.54 .57 76 .95 1.14 1.32 1.52 1.72 .63 .84 1.05 1.27 1.49 1.71 1.94 .69 .93 1.161 401.41.882,13

55 600 65

T0

09 13 16 14 19 24 19 .26 .32 24 .33 40

°

25 .51

11t5

.59 .69 .80 91

.22 .34 .45 .58

50 .550

.25 .28 .39 45 .51 .58 67 .75

60

0

°

70P

65

.31 .34 38 42 .4 .51 .53 58 63 .68 .65 .72 79 .84 .90 .83 90 .99 1.06 1.14

.69 79 .89 .99 1.09 ,70 .81 .92 1.04 1.29 .93 1.06 1.20 1.34 1.49 1.06 1.21 1,37 1.52 1.70 1.72 1.91 1.92 2.14 2.17 2.38 2.382.63

1.20 1.42 1.64 1.87

1.29 1.54 1.79 2.04

1.39 1.66 1.94 2.21

2.102.29 2.48 2.35 2.56 2.77 2.60 2.83 3.07 2.883.133.3

761.02 1.28 1.54 1.80 2.06 2.33 2.60 2.88 3.16 3.44 3.

72

.83 1.12 1.401.691.982.27 2.57 2.87 3.16 3.47 3.78 4.09 .58 .61 .91 1.22 1.53 1.84 2.152.462 783.103.443.784.12 4.46 66 1.00 1.33 1.68 2.02 2.36 2 7 3.05 3.403.77 4.14 4.55 4 89

.27 30 33

60 80

49 .58 .67 76

19 29 .39 49

36 .72 1.09 1.45 1.83 2.20 2.57 2.94 3.32 3.70 4.10 4.50 4.91 5.32 39 79 r.l19 1.55 2.00 2.40 2.80 3.203.614.024.0 4.98 5.38 5.83 .43 .86 1.30 1.742.182.623.063,50 3.95 4.404.885.375.85 6.34 51 1.031.562.082.613.143.674.214 765.315 866.437 01 7 60

0 965 100 110 1S0

62 1.251 932.52 3.16 3.81 4 4655.11 5.77 6 44 7 12 7.80 8.50 9 22

FOR EXTERNALS ADD o DEGREE

Central Angle

°

5

°

10

°

15

°

20

250

°

30

°

35

F CUBVE °

°

85'

°

400

45

50

.011 .029 .045 .074

.012 .032 .051 .083

.014.015 .017 .018 .020 .035 . .043 .047 .051 .057 .063 .070 .076 .083 .093 .106 120 127 135

55°

600

70

100 15 10

85e

.001.003 .003 .007 .006.011 .009 .018

.004 .006 .007 .008 .009 .010 .014 .018 .023 .027 .017 .022 .028 .034 .038 .027 036 .046 .056 .065

80 805

.013 .025 .018 .035

400 45'

.023.046 .070 .093 .117 .141 .030 .060 .093 .119 .153 .184

.038 .051 .065 .078 .090 .103 .116 .129 .149 .170 .179 .188 .054 .072 .086 .109 .131 .153 .175 .197 .213 .230 .247 .264

.172 .203 .234 .265 .277 .290 .315 .341 .216 .254 .289 .325 .351 .378. .411 .445

600 65

.037.075 .046 .093 .056 .112 .067.135

.116 .142 .168 .204

.151 .188 .225 .273

.189 .236 .23 .343

.227 .283 .340 .412

.266 .332 .398 .483

70T 7S 80 85'

.080 .159 .095 .182 .110 .220 .128.259

.240 .286 .332 .391

.321 .383 .445 .524

.403 .480 .558 .657

.485 .578 .671 .790

.568 .652 .735 .819 .906 .994 .678 .777 .877 .977 1.07 1.18 .787 .903 1.02 1.13 1.25 1.38 .926 1.06 1.20 1.34 1.47 1.62

9 ° 95 100 1100

.149.299 .174 .350 .200.401 .268 .53

.450 .603 .756 .910 .522 .706 .985 1.06 .604 .8091.01 1.22 .806 1.08 1.35 1.63

110

.360.721 1.08

°

80 55

°

1.45

1.82 2.19

.305 .381 .457 .554

.345 .420 .516 .625

1.07 1.25 1.43 1.91

1.22 1.43 1.64 2.20

1.38 1.62 1.85 2.48

2.57

2.95 3.33

.34 .479 .575 .697

1.54 1.80 2.06 2.76

.425 .530 .636 .711

1.70 1.99 2.28 3.05

3.72 4.11

.467 .582 .697 .845

.50 .550 .641 .700 .774 .851 .922 1.01 1.08 1.29 1.50 1.76

1.17 1.39 1.62 1.91

1.87 2.18 2.50 3.35

2.03 2.38 2.73 3.66

2.20 2.58 2.96 3.96

4.50

4.91

5.32

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