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11. To find the Radius of Outer Ruil of Turnout Curve. Subtract the throw of switch-rail from the gauge of track, both in decimals; call the remainder (l. Subtract the natural cosine of the inain frog angle from the natural cosine of the switch-rail angle; call the remainder 1. Divide a by b: the quotient will be radius.

12. To find the Main Frog Angle, the Radius of the Outer.

Rail being known. Call the natural cosine of the switch-rail angle a. Subtract the throw of switch-rail from the gauge of track, both in decimals. Divide the remainder by radius; call the quotient b. Subtract b from a: the remainder will be the natural cosine of the main frog angle.

13. To find the Angle of the Middle Frog, in the Case of

a Double Turnout. Call natural cosine of the main frog angle a. Divide half the gauge of track by the radius of outer rail of turnout curve; call the quotient b. Add a and b together. Their sum is the natural cosine of half the middle frog angle.

14. To find the Distance from Toe of Switch to Point of

Middle Frog. Subtract the throw of switch-rail from half the gauge of track, both in decimals; call the remainder a. Add together the switch-rail angle and half the middle frog angle. Find the natural sine of half this sum; call said natural sine b. Divide a by b: the quotient will be the distance sought.

15. The use of logarithms will be found convenient in working these rules.

Examples. 16. Switch-rail, 18 feet; throw, 5 inches 0.42 feet; frog angle, 5° 44'; gauge, 4.71 feet.

=

Sin. S= 0.42 • 18=.02334 sin. 1° 20'..

LO = (G – D) = sin. } (F + S) (4.71 0.42) = sin. 3° 32' = 4.29 = 0.0616 - 69.64 feet. R= (G — D) = (nat: cos. S nat. cos. F)= 4.29 = 0.00473 907 feet.

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907) 0.99759 = cos. 3° 587. Hence the angle of the middle frog 2 F 70 57'.

LI (1 G – D) • sin. I (F' + S) (2.354 - 0.42) = sin. # (3° 581' + 1° 20') 1.934 = 0.0463 41.8 feet.

17. In ordinary practice, frogs may be located with sufficient exactness by the following rules, deduced from the congruity of triangles. Great nicety in their location is not necessary. The important thing in practice is to l'ay the turnout curve so that the approach to the frog shall be fair and regular. How trackmen may do this without the use of instruinents, in a very simple way, will be shown hereafter. Not that frogs may be set hap-hazard, and the approaches forced to fit: they ought to be nearly where they mathematically belong, and they can be thus placed by means of the rules subjoined. 18. Let N stand for the number of the frog;

L the length of switch-rail in feet;
F the distance froin toe of outer switch-rail to point

of frog in feet. Then, for standard gauge, 4 feet 8 inches, straight switchrail, and 5 inches throw of switch.

8.6 L N
F
L + 0.42

N
"The above may be written roundly as a rule thus:-

Multiply the length of switch-rail in feet by the number of the frog, and set down the product. Multiply that product by 84, and call the result A. Next add together the lengih of switch-rail in feet and two-fifths of the frog nunber; call the sum B. Then divide A by B, and the quotient will be the distance in feet from toe of outer switch-rail to point of frog.

E.cumple. Switch-rail, 20 feet long; frog, No. 9. Length of switch-rail

20 Multiplied by frog number

9 Product

180 Multiplied by

81

1,530

3 A.
20
3.6

Length of switch-rail
Added to frog No. 9 .

.

A divided by B 1,530 divided by 23.6 = 64,8 feet, the frog distance; say, 65 feet.

19. If the switch-rail be curved, the forinula would stand thus:

8.6 L N F

L+ 0.84 N

Which may be made a written rule as follows:

Multiply the length of switch-rail in feet by the number of the frog, and their product by 8}; call the result A. Add together the length of switch-rail in feet and four-fifths of the frog number; call the sum B. Then divide A by B, and the quotient will be the distance from toe of outer switch-rai} to point of frog in feet.

20. The foregoing rules are applicable to turnouts from curves, as well as from straight lines.

21. To find the radius of outer rail of a turnout curve from straight track. Data same as in previous rules for frogs; R the required radius in feet.

8.6 L2 N2 If the switch-rail be straight, R=

L– 0.17 N2

8.6 L? N2 If the switch-rail be curved, R=

L2 – 0.68 N2

22. To find the radius of the outer rail of a turnout curve from curved track, proceed thus:

First find the radius as for a turnout from straight track by the preceding rule; call it, as before, R. Call the radius of the inain track R,, and the required radius of turnout curve r.

Then, if the turnout be towards the concave side of main track,

R2 XR
R2 + R

If the turnout be towards the convex side of main track,

R, XR r =

R.2 R

More explicitly, in the first case, r is equal to the product of the other radii divided by their sum; and, in the second case, r is equal to the product of the other radii divided by their

23. The angle of a frog is equal to 3,440' divided by the frog number.

24. To find the frog distances and radii for a three-foot gauge, find them by the preceding rules for standard gauge, and take five-eighths of the result, using a switch-rail reduced in like measure.

For a metre gauge, take seven-tenths of the result, using a switch-rail reduced in like measure.

Or these radii and distances may be found from the appended tables for standard gauge by pro-rating as above.

25. Three frog patterns are enough for general service. They should be so proportioned, that, taken in couples, the less may fit as middle frogs on double turnouts. Numbers 5}, 71, and 101 make an excellent suit; numbers 5, 7, and 9t also answer very well.

26. At the terminal stations, and about the shops of busy roads, patterns necessarily multiply. The better way in such cases is to plot the situation to a large scale, and to take the required distances and angles from the drawing.

TURNOUT TABLE. — SWITCH-RAILS, STRAIGHT; GAUGE OF TRACK, 4 FEET 84 INCHES; THROW OF SWITCH

RAIL, 5 INCHES.

MAIN FROG NUMBERS AND ANGLES.

4 5 53 6 63 7 7 8

8.1 9 9 10

10.

11 111 12 14° 20/ 12° 44/ 11° 28/ 10° 21' 9° 33' 8° 52/ 8° 11/ 7° 41/ 7° 10/ 6° 46/ 6° 22/ 6° 03/ 5°44/ 5° 28/ 5° 13' 5°00' 4° 47/

Main frog dist. 30.2 33.5 36.6 39.9 42.7 45.3 48.4 50.9 53.7 56.1 58.8 61.3 63.6 65.9 68.3 70.3 72.7 Switch-rail, 12 ft Rad. outer rail. | 140.6 | 178.9 221.8 274.9 325.0 378.4 450.2 515.0 595.8 675.0 772.0 870.0 976.7 1,089 1,224 1,342 1,506 Do.angle,2° 00' Mid. frog dist. 19.0 20.9 22.6 24.4 25.8

27.3 28.9 30.1 31.6 32.8 31.0 35.1 36.2 37.2 38.3 39.2 40.1 Mid. frog angle. 19° 28/ 17° 20' 15° 41' 14° 14' 13° 12/ 12° 16' 11° 24' 10° 46' 10° 04/ 9° 34' 9°03' 8° 40/ 8° 15' 7° 55' 7° 38/ 7° 21' 7°05/

Main frog dist. 31.2 34.6 38.0 41.6 44.6 47.5 50.9 53.7 56.8 59.5 62.6 65.3 68.0 70.0 73.4 75.7 78.5 Switch-rail, 16 ft Rad. outer rail. 139.4 176.9 218.8 270.3 318.6 369.7 438.0 499.2 575.0 648.0 737.0 826.0 921.0 1,020 1,137 1,263 1,377 Do.angle,1° 30' Mid. frog dist. 19.9 22.0 24.0 26.0 27.8 29.4

31.3 32.8 34.5 35.9 37.4 38.7 40.4 41.6 43.0 43.8 45.3 Mid. frog angle. 19° 22' 17° 14 15° 34' 14° 06' 13° 02' 12° 06' 11° 14' 10° 34' 9° 54' 9° 22 8° 51' 8° 27/ 8° 01' 7° 40' 7° 22' 7° 09/ 6° 49/

Main frog dist. 31.8 35.4 38.9 42.7 45.9 48.9 52.6 55.5 58.8 61.8 65.0 68.0 71.0 73.8 76.9 79.4 82.5 Switch-rail,20 ft Rad. outer rail. 138.8 176.0 217.4 268.2 315.7 365.8 432.6 492.1 565.0 636.0 721.0 807.0 897.0 992.0 1,102 1,196 1.325 Do.angle,1° 12 Mid. frog dist. 20.5 22.7 24.8 27.1 28.9 30.8 32.8 34.5

36.5 38.1 39.9 41.4 43.1 44.4 45.9 47.6 49.0 Mid. frog angle. 19° 19' 17° 10/15° 30' 14° 02' 12° 58' 12° 02' 11° 10' 10° 30' 9° 47/ 9°16 8° 44' 8° 21' 7° 54' 7° 34' 7° 15' 6° 57' 6° 41

Main frog dist. 32.2 35.9 39.5 43.5 46.8 49.9 53.7 56.8 60.3 63.4 66 8 70.0 73.176.1 79.4 82.0 85.3 Switch-rail,24 ft Rad.outer rail. 138.5 175.6 216.6 267.1 314.2 363.7 429.6 488.4 560.2 630.0 713.5 796.5 885.0 976.2 1,083 1,175 1,298 Do.angle,1°00' Mid. frog dist. 20.9 23.2 25.5

27.8 29.8 31.7 33.9 35.6

37.9 39.7 41.6 43.3' 45.2 46.8 48.3 49.9 51.4 Mid. frog angle. 19° 17' 17° 09/ 15° 28/ 14°00' 12° 56' 12° 00' 11° 06' 10° 28' 9° 44' 9° 14' 8°41' 8° 17' 7° 50' 7° 30' 7° 12' 6° 54' 6° 37/

Main frog dist. 32.6

36.4 40.1 44.2 47.7 50.9

54.9

58.1 61.8 65.0 68.7 72.0 75.3 78.5 82.0 84.8) 88.4 Switch-rail,30 ft Rad. outer rail. 138.3 175.2 215.6 206.2 312.9 362.0 427.3 485.3 556.2 624.8 707.0 788.5 875.0 964.2 1,068 1,157 1,276 Do.angle,0° 48' Mid. frog dist. 21.3 23.7 26.1 28.6 30.6 32.8 34.2 37.0 39.3 41.3 43.3 15.2 47.3 49.0 50.9 52.9) 54.2

Mid. frog angle. 19° 16/17° 08/ 15° 25' 13° 58/ 12° 54' 11°58' 11°04' 10° 25' 9° 42' 9° 10/ 8° 39/ 8° 14' 47' 7° 27/ 7° 09/6° 50' 6° 34

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