VIII 



Table VI. Deflections for Sub Chords for Short Radius Curves. 



Degree 



of 

 Curve 



Radius 

 50 



sin. def. ang. 



3o u 



K 



36° 

 38 

 40 

 42 



44 

 46 

 48 

 50 

 52 

 54 

 56 

 58 

 6o° 



193.18 

 181.39 

 171 .01 

 161.80 



I53-58 

 146. 19 



139.52 



133.47 

 127.97 



122 .92 



118. 31 



114.06 



IIO. II 



106. 50 



103.14 



100.00 



14 sub chord 

 R 



= sin of def. angle 



12.5 Ft. 



i°59' 

 2° 06' 



2° I 3 ' 



2° 20' 

 2° 27' 



2 o 34' 

 2° 41' 



2° 48' 



2° 55' 

 3° 02' 

 3° 09' 

 3° 16' 



3° 22' 



3° 29' 

 3° 35' 



15 Ft. 



2° If 



2° 2 5 ' 



2° 33' 

 2° 41' 



2 0 49' 

 2 0 57' 

 3° 05' 

 3° 13' 

 3° 21' 

 3° 29' 

 3° 38' 

 3° 46' 



4° 02' 

 4° io' 

 4° 18' 



20 Ft. 



2 0 58' 

 3° io' 

 3° 21' 



3 o 44 



3° 55' 

 4° 07' 

 4° 18' 



4° 29' 

 4° 40' 

 4° 5i' 

 5° 02' 

 5° 13' 

 5° 23' 

 5° 34' 

 5° 44' 



25 Ft. 



3° 43' 

 3° 58' 

 4° 12' 

 4° 26' 

 4° 40' 



< 5 < 

 5° 08' 



5° 22' 



5° 36' 



5° 50' 



6° 04' 



6° 17' 



6° 31' 



6 l < 

 6° 57' 

 7 0 11' 



Length 

 of arc 

 for 100 ft. 



IOI.15 



101.33 

 IOI .48 

 101 .66 

 101.85 

 102 . 06 

 102 . 29 

 102 . 53 

 102 . 76 

 103 .00 

 103 . 24 



103 • 54 

 103.84 

 104. 14 



10443 

 104. 72 



T - R tan | I 



T - 



in. D 



Sin. D - 



— ^ _ 

 50 tan § I 



Sin. D 



50 

 R 



50 tan \ I 

 T 



CURVE FORMULAS 

 R = T cot. \ I 

 R = 50 

 Sin. D 

 E = R ex. sec \ I 

 E = T tan § I 



Chord def. *= chord 2 



_____ 



No. chords = | I 

 Tan. def. = | chord def. 



The square of any distance, divided by twice the radius, will equal the 

 distance from tangent to curve, very nearly. 



Table IV. contains Tangents and Externals to a i° curve. Tan. and Ext. 

 to any other radius may be found nearly enough, by dividing the Tan. or 

 Ext. opposite the given Central Angle by the given degree of curve. 



To find Deg. of Curve, having the Central Angle and Tangent: Divide 

 Tan. opposite the given Central Angle by the given Tangent. 



To find Deg. of Curve, having tfa* Central Angle and Tangent: Divide 

 Ext. opposite the given Central Angle by the given External. 



To find Nat. Tan. and Nat. Ex. Sec. for any angle by Table IV.: Tan. 

 or Ext. of twice the given angle divided by the radius of a 1° curve will be 

 the Nat. Tan. or Nat. Ex. Sec. 



To find angle for a given distance and deflection. 

 ^ Rule 1. Multiply the given distance by .01745 (def. for 1° for I ft.), 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. 



Right Angle Triangles. Square the altitude, divide by twice the base. 

 Add quotient to base for hypotenuse. 



Given Base 100, Alt. 10 io 2 -t-200 = .5. 100 +.5 = 100.5 hyp- 



Given Hyp. 100, Alt. 25 252-^200 = 3.125. 100 -3. 125 =96.875 = Base. 

 Error in first example, .002; in last, .045. 



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

 by ii, and divide by 7. 



IX 



Natural Sines 



Peg. 



0 



I 



2 



3 

 4 

 5 

 6 



7 



0' 



0000 



0175 

 0349 

 0523 

 0698 

 0872 

 1045 

 1219 

 1392 

 1564 



10' 



0029 

 0204 



0378 

 0552 



0727 

 0901 



1074 



1248 

 142 1 



1593 



20' 30' 



0058 

 0233 

 0407 

 0581 

 0756 

 0929 

 1 103 

 1279 

 1449 

 1622 



0087 

 0262 

 0436 

 0610 

 0785 

 0958 

 1 132 



1305 

 1478 

 1650 



40' 50' 



0116 

 0291 

 0465 

 0640 

 0814 

 0987 

 1161 



1334 



1507 

 1679 



0145 

 0320 



0494 



0669 



0843 



1016 



1 190 



1363 

 1536 

 1708 



dg. 

 89 40 



88 41 



84 



82 

 81 



87 

 86 



85 44 



42 

 43 



45 



83 46 



47 

 48 



49 



0' 



6428 

 6561 

 6691 

 6820 

 6947 

 7071 



7193 

 7314 

 7431 

 7547 



10' 



6450 



6583 

 6713 

 6841 

 6967 

 7092 



7214 

 7333 

 7451 



7566 



20' 



6472 

 6604 



6734 

 6862 

 6988 

 7112 



7234 



7353 

 7470 



7585 



30' 40' 50' peg. 

 649465176539 49 

 6626 6648 6670 48 

 6756 6777 6799 47 

 6884 6905 6926 46 

 7009 7030 7050 45 



7133 7153 7173 44 

 7254 7274 7294 43 



7373 7392 7412 42 

 7490 7509 7528 41 

 7604 7623 7642 40 



10 



11 



12 



13 



15 

 16 



17 

 18 



19 



1736 

 1908 

 2079 

 2250 

 2419 

 2588 



2756 

 2924 

 3090 

 3256 



1765 

 1937 

 2108 

 2278 

 2447 

 2616 

 2784 

 2952 

 3118 

 3283 



1794 

 1965 

 2136 

 2306 

 2476 

 2644 

 2812 

 2979 

 3145 

 33" 



1822 

 1994 

 2164 



2334 

 2504 



2672 



2840 



3007 



3173 



3338 



1851 

 2022 

 2193 

 2363 

 2532 

 2700 

 2868 



3035 

 3201 



3365 



1880 

 2051 

 2221 

 2391 

 2560 

 2728 

 2896 

 3062 

 3228 

 3393 



79150 



78 



77 

 76 



75 

 74 

 73 

 72 



7i 

 70 



5i 

 52 

 53 

 54 

 55 

 56 

 57 

 5 



59 



7660 

 7771 

 7880 

 7986 

 8090 

 8192 

 8290 



8387 

 8480 



8572 



7679 

 7790 



7898 

 8004 

 8107 

 8208 

 8307 

 8403 

 8496 

 8587 



7698 

 7808 

 7916 

 8021 

 8124 

 8225 



8323 

 8418 

 8511 

 8601 



7716 

 7826 



7934 

 8039 



8141 

 8241 



8339 

 8434 

 8526 

 8616 



7735 

 7844 



795i 

 8056 

 8158 

 8258 



8355 

 8450 



8542 



8631 



7753 

 7862 



7969 

 8073 

 8i75 

 8274 

 8371 

 8465 



8557 

 .8646 



39 



38 

 37 

 36 

 35 

 34 

 33 

 32 

 3i 

 30 



20 



21 



22 



23 

 24 

 25 

 26 

 27 

 28 



29 



3420 

 3584 

 3746 



3907 

 4067 

 4226 

 4384 

 4540 

 4695 

 4848 



3448 

 361 1 



3773 

 3934 

 4094 



4253 

 4410 



4566 



4720 



4874 



3475 

 3638 

 3800 

 3961 

 4120 

 4279 

 4436 

 4592 

 4746 

 4899 



3502 

 3665 

 3827 

 3987 

 4147 

 4305 

 4462 



4617 

 4772 

 4924 



3529 

 3692 



3854 

 4014 



4173 



4331 



4488 



4643 

 4797 

 4950 



3557 



3719 

 3881 



4041 



4200 



4358 



4514 

 4669 

 4823 

 4975 



69 60 



68 



64 



63 

 62 

 61 



61 



67 62 

 66 63 

 6564 



65 

 66 



67 

 68 



60 69 



8660 

 8746 

 8829 

 8910 

 8988 

 9063 



9135 

 9205 

 9272 



9336 



8675 

 8760 



8843 



8923 

 9001 



9075 

 9147 

 9216 

 9283 

 9346 



8689 

 8774 

 8857 

 8936 

 9013 

 9088 



9159 

 9228 



9293 

 9356 



8704 

 8788 

 8870 



8949 

 9026 

 9100 

 9171 



9239 

 9304 

 9367 



30 



3i 



32 

 33 

 34 

 35 

 36 

 37 

 38 



39. 

 Deg. 



5000 



5150 

 5299 

 5446 

 5592 

 5736 

 5878 

 6018 



6157 

 6293 



5025 

 5175 

 5324 



5471 

 5616 

 5760 

 59OI 

 604I 

 6l80 

 6316 



5050 

 5200 



5348 



5495 

 5640 



5783 

 5925 

 6065 

 6202 

 6338 



60' i 50' 



40' 



5075 

 5225 

 5373 



5519 

 5664 



5807 

 5948 

 6088 

 6225 

 6361 



30' 



5100 



5250 



5398 



5544 

 5688 



5831 

 5972 

 6111 

 6248 

 6383 



20' 



5125 



5275 

 5422 

 5568 

 5712 

 5854 

 5995 

 6134 

 6271 



59 

 58 

 57 

 56 

 55 

 54 

 53 

 52 

 5i 



6406 50 79 



10' Jdg 



70 



71 

 72 

 73 

 74 

 75 

 76 

 77 

 78 



9397 

 9455 

 95" 

 9563 

 9613 



9659 

 9703 

 9744 

 9781 

 9816 



60' 



9407 



9465 

 9520 



9572 

 9621 

 9667 

 9710 



975o 

 9787 

 9822 



50' 



9417 

 9474 

 9528 

 9580 

 9628 

 9674 

 9717 

 9757 



9793 

 9827 



9426 



9483 

 9537 

 9588 

 9636 

 9681 

 9724 



9763 

 9799 

 9833 



40' 



30' 



8718 

 8802 

 8884 

 8962 

 9038 

 9112 

 9182 

 9250 

 9315 

 9377 

 9436 

 9492 

 9546 

 9596 

 9644 

 9689 

 9730 

 9769 

 9805 

 9838 



20' 



8732 

 8816 

 8897 

 8975 

 9051 

 9124 

 9194 

 9261 

 9325 

 9387 

 9446 

 9502 



9555 

 9605 

 9652 

 9696 



9737 

 9775 

 981 1 



9843 

 10' 



29 



28 



27 

 26 



25 

 24 



23 

 22 



21 



20 



19 



18 



17 

 16 



15 

 14 

 13 

 12 

 11 

 10 

 Dig. 



Deg. 



0' 



10' [ 



20' 



30' 



40' 



50' 



Deg. 



80 



9848 



9853 9858 



9863 



9868 



9872 



9 



81 



9877 



9881 



9886 



9890 



9894 



9899 



8 



82 



9903 



9907 



991 1 



9914 



9918 



9922 



7 



83 



9925 



9929 



9932 



9936 



9939 



9942 



6 



84 



9945 



9948 



9951 



9954 



9957 



9959 



5 



85 



9962 



9964 



9967 



9969 



9971 



9974 



4 



86 



9976 



9978 



9980 



9981 



9983 



9985 



3 



87 



9986 



9988 



9989 



9990 



9992 



9993 



2 



88 



9994 



9995 



9996 



9997 



9997 



9998 



1 



89 



9998 



9999 



9999 



9999 



1 . 0000 



1 . 0000 



0 



Deg. 



60' 



50' 



40' 



30' 



20' 



10' 



] Deg. 



Natural Cosines 



