For the terminal point of the curve (or P C C) since X = l, very nearly, and 7 200, we have l =

=

Tang. a = .005817 D.

a = 0° 20' X D = D°.

Since, when = 200, equation (10) gives ▲

3

= D, the deflection from the chord to the tangent at the PC C must be equal to D D D. From equation (20) it appears that tangent a varies as a2 and D, and since, for such values of a as would be likely to occur in practice, an angle varies very nearly as its tangent, we may say without appreciable error that the deflection angle from the tangent at the PTC to any point on the easement curve will vary directly as the degree of curvature of the central curve and as the square of the distance of the point from the P T C. By this principle we may very easily calculate in the field the deflections for intermediate points, and no table is essential, but a little time may sometimes be saved by the use of the following table of values of a:

When the curve is to be set in from the P C C toward the P T C the necessary deflections are given in the following table:

12

As these deflections are not easily remembered or calculated except those for the 100 and 200 feet points (51⁄2 D° and 3 D°) it will be most convenient to set only these two points from the PC C, and any other necessary points from the P T C.

It sometimes happens that the P T C is inaccessible. In this case set up at the P C, 100 feet (or 1) farther forward on the tangent, deflect D° from the tangent measure 100 feet and set the P C C, then set at the P C C and deflect D° from the chord, which will give the tangent. If the P C C cannot be set from the P T C, we may use the above method, or we may first set the middle point of the easement curve, and then set the P C C from this point. The deflections will be the following, as may readily be deduced from Tables III and IV:

Transit at P T C vernier reading 0° on tangent.

Deflect D°, vernier reading D°, gives chord to middle point.

Transit at middle point on chord produced.

Deflect D°, vernier reading D°, gives tangent at middle point.

Deflect D°, vernier reading D°, gives chord to P C C. Transit at P C C on chord produced.

Deflect D°, vernier reading D°, gives tangent at P C C.

General Case.

We now propose to show that the principle obtained above is general, and may be applied to any length of easement curve within ordinary limits of practice.

In equations (5), (3), (20), (21) and (22), substitute the

values of R, 1 and D from equations (6), (8) and (11) and we obtain

These equations being precisely analogous to (17), (18), (20), (21) and (22) it follows that the same principle governs in both cases, and Tables I, II, III and IV, may be used for any length of curve, except that in Table I the first, third and fifth lines should be used, and in Table II the first and sixth columns, while in Tables III and IV the only change necessary is to substitute for D. For convenience either in using these tables or in calculating them in the field, the curve should be divided into eight or four equal parts.

The difference between an easement curve of this form and a circular curve may now be very easily expressed. Supposing in each case the transit to be set at the origin of the curve, in a circular curve we deflect from the tangent the central angle to fix the other end of the curve, and at the other end deflect the remaining, and the deflections to intermediate points on the curve vary as their distances from the origin. In the easement curve the deflection from the tangent at the origin to the other end is the central angle, and at the other end the remaining is turned, and the deflections to intermediate points vary as the squares of their distances from the origin. The easement curve is also twice as long as a circular curve for the same central angle.

The field work of setting out the easement curve in all ordinary cases may now be summed up in the following very simple rules, which are designed for a length of 200 feet, but by using the expressions in brackets they become of general application:

1. Find the apex and the intersection angle (I) of the tangents and the apex distance (T) in the usual manner. Then T +100 will be the apex distance of the P T C. [T+ 50 L.]

2. Set transit at the PTC and deflect from the tangent 12 D° for the first chord of 100 feet, and D° for the second. The latter fixes the PC C. [2° for 3, and ° for l.]

3

3. Set at the PC C back sight to the P T C, and deflect D°. [°] This gives the tangent to the central curve. 4. Set out the central circular curve in the usual manner, making its angular length = I — 2 D°. [I − 2 ^°.] 5. At the end of the central curve (the second P C C) deflect from the tangent D° for the first chord of 100 feet (A for) and D° for the second (3° for 1). This gives the second P T C.

6. Set at the P T C, back sight to the P C C, and deflect } D°. [} ^°.] This gives the tangent.

Any intermediate points for grading or for track laying may be set out by the tables or by calculating the deflections.

It may sometimes occur that it is desirable to move the central curve inward a given distance, or, in other words, to assume a value for s. It then becomes necessary to determine the other elements of the curve. Transposing equations (25) and (26), we obtain

We

L = 3.708 S

D

A 1.854 VSD

e can then use tables III and IV as before.

(30)

(31)

By means of these formulas a very convenient and valuable application of the easement curve can be made in rectifying`a location. When a located line consisting of a series of tangents and circular curves has been laid out over difficult ground, as for instance along a steep hillside, it will often be found upon an inspection of the profile that some portions of the line are satisfactory and should not be disturbed, while other portions can be improved by slight changes, as by moving a tangent or a curve in or out the same distance at each end, or more at one end than at the other, or inward at one end and out at the other. To modify the entire location to fit these changes might be a serious task, and perhaps introduce undesirable changes elsewhere, but where a tangent is moved out or a curve moved in, the connection between the severed portions of the line can be made with great ease and convenience by an ease