to as the distance. As it is often difficult to determine the exact point a of the frog, a more accurate method of determining the frog number is to measure the entire length dl of the frog from mouth to heel, and divide this length by the sum of the mouth width b and the heel width h. The quotient will be the exact number of the frog.

For example, if in Fig. 2, the total length dl of the frog is 7 feet 4 inches, or 88 inches, and the width h is 15 inches, and the width b of the mouth is 7 inches, then the frog number is 88 + (157) 4. Frogs are known by their numbers. That in Fig.

2 is a No. 4 Frog.

By

The Frog Angle.-The frog angle is the angle formed by the gauge lines of the rails, which form its tongue. Thus, in Fig. 2, the frog angle is the angle la' k. The amount of the angle may be found as follows: The tongue and heel of the frog form an isosceles triangle (see Fig. 3). drawing a line from the point a of the frog to the middle point b of the heel c d, we form a rightangled triangle,

FIG. 3.

right-angled at b. The perpendicular line a b, bisects the angle

a, and, by Trigonometry, we have tan 1⁄2 a =

sions of the frog point given in Fig. 3 are not the same as those given in Fig. 2, but their relative proportions are the same; viz., the length is four times the breadth. The length a b = 4 and the width c d = 1; hence, b c = 2. Substituting these values,

we have tan 1⁄2α =

11⁄2
4

=

1% = 0.125.

Whence, a = 7° 71⁄2'

and a 14° 15'; that is, the angle of a No. 4 frog is 14° 15'.

Frog numbers run from 4 to 12, including half numbers, the spread of the frog increasing as the number decreases.

The Parts of a Turnout. -The several parts of a turnout are represented in Fig. 4. The distance pf from the P. C. of the turnout curve to the point of frog is called the frog dis. tance. The radius co of the turnout curve, the frog distance, the frog angle and the frog number bear certain relations to each other, which are expressed by the following formulas :

Tangent of half frog angle

Frog number =

Frog number

= gauge

frog distance.

radius co+ twice the gauge.

1+ half the tangent of half the frog angle.

Radius c o twice the gauge X square of the frog number.

Radius c o = (frog distance pf sine of frog angle) - 1⁄2 the

Frog distance pƒ = (radius c o half the gauge) X sine of frog angle.

Middle ordinate (approximate) = 1/4 the gauge.

Each side ordinate (approximate) = 34 the middle ordinate =

(or .188) of the gauge.

Switch length (approximate) =

throw in feet x 10.000

tan deflection for chords of 100 ft. for radius co of turnout curve

The tangent deflection may be obtained from the table on pages 286-288.

TURNOUTS FROM A STRAIGHT TRACK.

Gauge, 4 feet 81⁄2 inches. Throw of Switch, 5 inches.

The switch lengths in the above table merely denote the shortest length of stub switch that will at the same time form part of the turnout curve, and give 5 inches throw. Point or split switches require a throw of not more than 31⁄2 inches, though many have a throw of 5 inches, with an equal space between the gauge lines at the heel. The heels of a split switch which occupy the same position as the toes of a stub switch, should be placed at the point where the tangent deflection or offset is 5 inches. The point where the tangent deflection is but 41⁄2 inches will answer for many rail sections, but for those above 65 lb. per yard, 5 inches should be taken.

In the table on pages 286-288, tangent deflections for chords of 100 feet are given for all curves up to 20°, and for a curve of

higher degree, the tangent deflection may be found by applying

In complicated track work where space is limited, curves must be chosen to meet the existing conditions, and not with reference to particular frog angles, in which case the frogs are called special frogs, and are made to fit the particular curve used. The determination of the frog distance, switch length and frog angle may be understood by referring to Fig. 5.

Let the main track a b be a straight line; the gauge p q = 4 feet 8 inches (= 4.71 feet); the degree of the turnout curve = 13°; the chord q d= 100 feet; cd the tangent deflection of the chord qd, and pf the frog distance. From the table on page

-11.32

σ

FIG. 5.

288 we find the tangent deflection for a chord 100 feet long of a 13° curve is 11.32 feet.

Then, from Fig. 5, we have the proportion

c d e f qc : ge

Now, in curves of large radius q c and q d are assumed to be equal. Also, qe = p f, the frog distance, and substituting these equivalents we have the proportion

cd:ef: qd' pƒ2.

Substituting the above given quantities in the proportion, we have

If the space between the gauge lines at the heels of a split switch be taken at 5 inches = 0.42 of a foot, the distance from the P. C. of the turnout curve to the heel of the switch may be found as follows:

In Fig. 5, let h, the tangent off-set at the heel of the switch = 0.42 of a foot, we have the proportion

This locates the heel of a split switch and the toe of a stub switch.

The frog angle is the angle k fƒl (see Fig. 5) for.ned by the gauge line of the main rail ƒ k and the tangent to the outer rail qf of the turnout curve at the point where the two rails intersect. This angle is equal to the central angle qof. The arcs qf and rs are assumed to be of the same length. The turnout 13 X 60

curve being 13°, the central angle for a chord of 1 foot is

100

7.8', and the central angle for 64.5 feet, the frog distance, is 7.8'x 64.58° 23', the frog angle for a 13° curve. By this process the frog distance, switch length and frog angle may be calculated for curves of any radius.

To Lay Out a Turnout from a Curved Main Track.-There are two cases:

Case I. When the two curves deflect in opposite directions, illustrated in Fig. 6, and

Case II. When the two curves deflect in the same direction, illustrated in Fig. 7.

In Fig. 6, the curve a b is 3° 30', and it is proposed to use a No. 8 frog. By reference to the table on page 304 we find that