sions we take the mean of the dimensions of the given sections. These dimensions will be as follows:

With dimensions thus found, construct the section M shown in Fig. 3.

-20.2

-13.02

FIG. 3.

The area of section M is computed by the same method as that used with sections A and B in Figs. 1 and 2, and is as follows:

Total area of section M = 173.2 " 66

Denoting the distance between the sections by L, and the cubical contents of the prismoid by S, we have, by substituting in the prismoidal formula,

50

S =

6

- (223.3+4 x 173.2 + 128.3) = 8,703 cu. ft. =322.3 cu. yd.

TRACK WORK.

Curving Rails.-When laying track on curves, in order to have a smooth line, the rails themselves must conform to the curve of the center line. To accomplish this, the rails must be curved. The curving should be done with a rail bender or with a lever, preferably with the former.

To guide those in charge of this work, a table of middle and quarter ordinates for a 30-foot rail for all degrees of curve should be prepared.

The following table of middle ordinates for curving rails is calculated by using the formula

m =

C2

8 R

in which m is the middle ordinate; c, the chord, assumed to be of the same length as the rail, and R, the radius of the

curve.

The results obtained from this formula are not theoretically correct, yet the error is so small that it may be ignored in practical work.

In curving rails, the ordinate is measured by stretching a cord from end to end of the rail against the gauge side, as shown in Fig. 1. Suppose the rail A B is 30 feet in length, and the curve 8°.

FIG. 1.

Then, by the previous problem, the middle ordinate at a should be 1 inches. To insure a uniform curve to the rails, the ordinates at

the quarters and b' should be tested. In all cases the quarter ordinates should be three-quarters of the middle ordinate. In Fig. 1, if the rail has been properly curved, the quarter ordinates at b and b' will be x 17 in. = 1}, say 1 in.

In track work it is often necessary to ascertain the degree of a curve, though no transit is available for measuring it. The following table contains the middle ordinates of a one-degree curve for chords of various lengths:

The lengths of the chords are varied so that a longer or shorter chord may be used, according as the curve is regular or not.

The table is applied as follows: Suppose the middle ordinate of a 44-feet chord is 3 inches. We find in the table that the middle ordinate of a 44-feet chord of a 1° curve is 1⁄2 inch. Hence, the degree of the given curve is equal to the quotient of 3+ 1⁄2 = 6° curve.

Elevation of Curves.-To counteract the centrifugal force which is developed when a car passes around a curve, the outer rail is elevated. The amount of elevation will depend upon the radius of the curve and the speed at which trains are to be run. There is, however, a limit in track elevation as there is a limit in widening gauge, beyond which it is not safe to pass.

The best authorities on this subject place the maximum elevation at the gauge, or about 8 inches for standard gauge of 4 feet 8 inches. The gauge on a 10° curve elevated for a speed of 40 miles an hour, should be widened to 4 feet 914 inches.

All curves, when possible, should have an elevated approach on the straight main track, of such length that trains may pass on and off the curve without any sudden or disagreeable lurch. A good rule for curve approaches is the following: For each half inch or fraction thereof of curve elevation, add 30 feet or 1 rail length to the approach; that is, if a curve has an elevation of 2 inches, the approach will have as many rail lengths as contained in 2, which is 4 times. The approach will, therefore, have a length of 4 rails of 30 feet each, or 120 feet.

is

The following table for elevation of curves is a compromise between the extremes recommended by different engineers. It is a striking fact that experienced trackmen never elevate track above 6 inches, and many of them place the limit at 5 inches.

The Elevation of Turnout Curves.-The speed of all trains in passing over turnout curves and crossovers is greatly reduced, so that an elevation of 14 of an inch per degree is amply sufficient for all curves under 16 degrees. On curves exceeding 16 degrees, the elevation may be held at 4 inches until 20 degrees is reached, and on curves exceeding 20 degrees, of an inch of elevation per degree may be allowed until the total elevation amounts to 5 inches, which is sufficient for the shortest curves.

The Frog.-The frog is a device by means of which the rail at the turnout curve crosses the rail of the main track. The frog shown in Fig. 2 is made of rails having the same cross-section as those used in the track. Its parts are as follows: The wedge shaped part A is the tongue, of which the extreme end a is the point. The space b, between the ends c and d of the rails, is the

FIG. 2.

mouth, and the channel which they form at its narrowest point e is the throat. The curved ends ƒ and g are the wings.

That part of the frog between A and A' is called the heel. The width h of the frog is called its spread. Holes are drilled in the ends of the rails, c, d. k and to receive the bolts used in fastening the rail splices, so that the rails of which the frog is composed, form a part of the continuous track.

The Frog Number.-The number of a frog is the ratio of its length to its breadth; i. e., the quotient of its length divided by its breadth.

Thus, in Fig. 2, if the length a' l, from point to heel of frog is 5 feet, or 60 inches, and the breadth h of the heel is 15 inches, the number of the frog is the quotient of 60+15= 4. Theoretically, the length of the frog is the distance from a to the middle point of a line drawn from k to l; practically, we take from a