The distance of the P. C. or P. T. from the P. I. is called the tangent distance, and the chord connecting the P. C. and P. T. of a curve is commonly called its long chord. This term is also applied to chords more than one station long.

If I denotes the angle of intersection and R the radius of the curve, then the tangent distance

T=R tan } I

Laying Out a Curve With a Transit.-When the angle of intersection I has been measured and the degree of curve decided upon, the radius of the curve can be taken from the table of radii and deflections or it can be figured by the formula

R=

5,730
Dc

The tangent distance is then computed and measured back on each tangent from the P. I., thus determining the P. C. and P. T. Subtracting the tangent distance from the station number of the P. I. will give the station number of the P. C. Ordinarily, this will not be an even or full station. The length of the curve is then computed by dividing the angle I by the degree of curve, the quotient giving the length of the curve in stations of 100 ft. and decimals thereof. After having found the length of the curve, compute the deflection angles for the chords joining the P. C. with all the station points; set the transit at the P. C.; set the vernier at zero, sight to the intersection point, and turn off successively the deflection angles, at the same time measuring the chords and marking the stations. The station of the P. T. is found by adding the length of curve in chords of 100 ft. to the station of the P. C. If the entire curve cannot be run from the P. C. on account of obstructions to the view, run the curve as far as the stations are visible from the P. C. and run the remainder of the curve from the last station that can be seen. Suppose that in the 10° curve shown in Fig. 5 the station at H, 200 ft. from the P. C., which is at B, is the last point on the curve that can be set from the P. C. A plug is driven at H and centered carefully by a tack driven at the point. The transit is now moved forwards and set up at H. Since the deflection angle EBH is 10° to the right, an angle of 10° is turned to the left from zero and the vernier clamped. The instrument is then sighted to

a flag at B, the lower clamp set, and by means of the lower tangent screw the cross-hairs are made to bisect the flag exactly. The vernier clamp is then loosened, the vernier set at zero, and the telescope plunged. The line of sight will then be on the tangent IP, and the deflection angles to K and C can be turned off from this tangent, and the stations at K and C located in the same manner that the stations at G and H were

B

FIG. 5

located from B, because the angle at IHB between the tangent IH and the chord BH is equal to the angle EBH between the tangent EB and the same chord.

This method of setting the vernier for the backsight when the instrument is moved forwards to a new instrument point on the curve is sometimes called the method by zero tangent. The essential principle of the method is that the vernier always reads zero when the instrument is sighted on the tangent to the curve at the point where the instrument is set, and the deflection angles are made to read from the tangent to the

curve at this point in the same manner as if this point were the P. C. of the curve.

Tangent and Chord Deflections.-Let AB, Fig. 6, be a tangent joining the curve BCEH at B. If the tangent AB is prolonged to D, the perpendicular distance DC from the tangent to the curve is called a tangent deflection. If the chord BC is prolonged to the point G, so that CG=CE, the distance

GE is called a chord deflection. If the radius R of the curve and the length of the chord c are known, the tangent deflection ƒ can be determined by the formula

c2
f=
2 R

C2

If CE=BC, the chord deflection = 2ƒ== For this condition,

the table of radii and deflections gives the chord deflection and tangent deflection for 100-ft. chords and for degrees of curvature varying by intervals of 5' and 10' from 5' to 20°.

is the length of the first chord and c2 the length of the second chord preceding the station considered. When the tangent deflection f is known, the chord deflection

do= f (1+2)

Special Values of Chord and Tangent Deflection. For a chord of 100 ft. preceded by one of the same length the chord deflection for a 1° curve is 1.745; for a 2° curve, it is twice that amount, or 3.49; and so on. The tangent deflection, being half the chord deflection, will be .873 ft. for a 1° curve, 1.745 for a 2° curve, etc. The tangent deflection for a chord of any length equals the tangent deflection for a chord of 100 ft. multiplied by the square of the given chord expressed as the decimal part of a chord of 100 ft.

Application of Chord and Tangent Deflection.-Let it be required to restore center stakes on the 4° curve, Fig. 7, at each full station. The points A and B determine the direc

tion of the tangent, the point B being the P. C., which is at Station 8+25. For a 4° curve the regular chord deflection for 100 ft. is 4X 1.745-6.98 ft., and the tangent deflection is 3.49 ft. The distance from P. C. to the next full station is 75 ft.; hence, the tangent deflection CF = .752X3.491.96 ft. The point F is found by first measuring 75 ft. from B, thus locating the point C in the line AB prolonged, then from C measuring CF = 1.96 ft., at right angles to BC; the point F

This

thus determined will be Station 9. Next the chord BF is prolonged 100 ft. to D; BF is only 75 ft., DG is computed from the preceding formula; thus, do=3.49 (1+75%) = 6.11. distance is measured at right angles to BD; the point G thus determined will be Station 10. The point H, which is Station 11, and the P. T. of the curve, is determined in the same manner, except that, as the chords FG and GH are each 100 ft. long, the regular chord deflection of 6.98 ft. is used for EH. A stake is driven at each station thus located. Although a chord deflection is not at right angles to the chord theoretically, yet the deflection is so small, as compared with the length of the chord, that for curves of ordinary degree it is usually measured at right angles.

Middle Ordinate.-The middle ordinate of a chord is the ordinate to the curve at the middle point of the chord. The following formulas give the relation between the length of the chord c, the radius of the curve R, and the middle ordinate m.

To Determine Degree of Curve From Middle Ordinate.-It is sometimes necessary to determine the radius or the degree of a curve in an existing track when no transit is available. By measuring the middle ordinate of any convenient chord, the degree of the curve can be calculated from the relative values of the ordinate and chord. As the track is likely not to be in perfect alinement, it is well to measure the middle ordinate of different chords in different parts of the curve; as, also, the middle ordinate of a chord measured to the inner rail will somewhat exceed the middle ordinate of the same chord measured to the outer rail, the ordinate of each chord should be measured to both rails and the average of the two taken as the value of the ordinate. Having measured the middle ordinate of one or more chords, the degree of curve De can be found by the formula

45,840 m

n