3. Cases of this kind are disposed of with great ease in the field by means of the curve-protractor.

TO FIND THE RADIUS OF A TURNOUT CURVE, THE FROG ANGLES, AND THE DISTANCES FROM THE TOE OF SWITCH TO THE FROG POINTS.

1. Draw the figure as in margin, C being the centre of the turnout curve, CK parallel to main track, and OK, IE, LM, perpendicular to it. Call the angle of the frogs at O, F; that of the intermediate frog at I, 2 F'; the throw of the switch-rail for single turnout, D; its angle with main track, S; the gauge of the track, G; and radius of outer rail, R.

2. Usually the length and throw of switch-rail and the angles of the frogs at O are given. In that case, to find R, F

3. The angle HN W, between the line of the switch-rail prolonged and a tangent to turnout curve at frog point O, NOP NHW FS. The angle NOL or NLO, behalf the intersection angle HNW

=

tween chord and tangent, =

=

=(FS). The angle NOB =NOL+LOB. But NOL is seen to be S), and NOB = F; then LOB=

=

= (F

NOB-NOL=F

· } (F — S) = (FS). The distance

LO, from toe of switch to point of main frog, = LB÷sin. LOB (GD) ÷ sin. (F + S).

=

4. Again: the angle LCY = NLO (F — S); LY =} LO=(G D) sin. (F+S). LY sin. LCY= LC; i.e., [ (G — D) ÷ sin. † (F + S)] ÷ sin. † (F — S)

=

= R.

5. R may be found otherwise, as follows:

=

=

OK OC cos. KOC = R cos. F; LM = LC cos. CLM =

R cos. S; LM — OK = LB; i.e., R (cos. S

cos. F) (G D). Hence R = (G— D) = (nat. cos. S―nat. cos. F).

6. If R be known, to find F. =nat. cos. S — [(G — D) ÷ R].

=

This equation gives nat. cos. F

7. To find the angle, 2 F', of the middle frog at I.

IE =

IPPE or OK; i.e., R cos. F'G+R cos. F. Hence nat. cos. F' = nat. cos. F+(† G ÷ R).

=

8. The angle LIV, by similar reasoning to that used in relation to LOB, is found to be (F′+S). The distance LI, from toe of switch to point of middle frog, = LV÷sin. LIV (GD) sin. † (F' + S).

RULES FOR FROGS AND SWITCHES.

9. To find the Angle of Switch-Rail with Main Track. Divide its throw, in decimals, by its length: the quotient will be the natural sine of the angle sought.

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

Main Frog.

Subtract the throw of switch-rail from the gauge of track, both in decimals; call the remainder a. Add together the angle of switch-rail with main track and the angle of the main frog; find the natural sine of half this sum, and call it b. Divide a by b: the quotient will be the distance

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

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

Subtract

Call the natural cosine of the switch-rail angle a. 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 the 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.

a by b: the quotient will be the distance sought.

Divide

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'.

R = (GD) (nat. cos. S - nat. cos. F)=4.29 0.00473

= 907 feet.

cos. 3° 58. Hence the angle of the middle

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 lay the turnout curve so that the approach to the frog shall be fair and regular. How trackmen may do this without the use of instruments, 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 from 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.

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 length of switch-rail in feet and two-fifths of the frog number; 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.

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 formula would stand thus:

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-rail 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.

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 main track R2, and the required radius of turnout curve r. Then, if the turnout be towards the concave side of main track,

r =

R2 X R
R2+R

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

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