the latter to curves of from 6° to 14° curvature; while for curves of from 5° to 8° either set may be employed.

Another objection which may be brought against it, and one which is often brought against transition curves generally, is that it is not worth the trouble taken in locating it. As regards this, the use of transition curves, not only theoretically but practically, is found to reduce the resistance of the curve very materially, to lessen the cost of maintenance of way, to reduce the chances of derailment, and considerably to ease the motion of the cars.

There is no need to set out the transition curves during the location, but the tangent in any instance should be run to c (Fig. 50) and the transit then offsetted to a, from which point the main curve can be located. The amount of the offset ac, and the distance oc, should be added to the notes of the curve, and also the distance ae, which represents C. The general plan of the location then shows the curves as in Fig. 16. Then when the engineer takes charge of the work for construction he has simply to "reference" the points o and e, and run in the curve by means of the above table, as easily as he would any simple

curve.

98. Method II.-Another form of transition curve is that

a

FIG. 51.

b

shown in Fig. 51. It is especially suitable in cases where it is more convenient to offset the curve than the tangent itself. It practically converts the original simple curve into a 3-centre one, but where the curvature of the main curve is light, it answers the purpose of easing off the curvature at its ends sufficiently in ordinary cases.

In Fig. 51, let r = radius of the original main curve ab.

Offset ab inwards by an amount afe; then if R = radius of the terminal curve cd, we have

from which we can find the position of d; and

ca= R(re) sin fod,

from which we can find the position of c.

The curve cd can

then be best located with a transit from the point c.

A convenient method of applying this principle in practice is to make e = 0.2 foot for every degree of curvature of ab, and to make R = 3(r — e); then if we make fd = 33.9 feet, d is the P.C.C., and

For ordinary curves ca then varies from 75 to 100 feet.

99. Method III.—Another method of substituting a 3-centre curve for a simple one, when we do not wish to change the original tangent-points, is as follows:

In Fig. 52 let o be the centre of the original simple curve afb, the radius of which = R; and let o, be the centre of the new main curve ced, whose radius R1. And let 02, 02 be the centre of the terminal curves ac and db, whose radii = R2.

e

Thus we obtain the position of the points c and d.

2. Given R1 and aoc = bod.

The curvature of the arc ced should never exceed that of ab by more than 1° (about 50′ excess is usually a suitable amount), and R2 should equal about 3R.

Suppose, however, in substituting the 3-centre curve for the simple one, it is advisable for the points e and f to coincide as in Fig. 53.

Then a must be put back on the tangent to u, and

100. We have already considered the dangers which arise from sudden changes of grade (see Sec. 29). Where these changes are considerable, amounting to, say, 0.5 p. c. in the difference of grade, it is advisable to round off the angle at the junction of the two grades by means of vertical curves. On bridge-work this should be more especially attended to. Theoretically, the curve which should be applied is a parabola, and this happens also to be the simplest form of curve to insert in practice.

In Fig. 54 let ac and cb be two grades between which it is required to insert a vertical curve.

=

Now of 2cd; therefore, if the letters a, b, and c stand respectively for the elevations at those points,

and the correction e at any other point is given by the equation

ac and co are usually made about 200 feet each.

Vertical curves are not usually inserted during location, or even shown on the location profile; but the corrections for them should be worked out before the cross-sectioning begins, and the grade as shown on the construction profile should be the corrected grade.

Note.-In dealing with deflection-angles and offsets of curves, the engineer-entirely ignorant of the Differential Calculus-may often save himself a considerable amount of labor by making use of the principle of Successive Differences, an application of which is given in Sec. 203, Part III. Thus, e.g., the deflection-angles given in Tables A and B, Sec. 97, may be calculated up to 300 feet merely by the application of the 2d differences, and may be extended considerably beyond that amount by using the 3d differences. More especially is this method applicable in calculating offsets to a curve which may be considered to vary as the Square of the tangential distance, for then their 2d differences will be SL constant. As an example of this, the values of (H — H')2

27 × 6'

given

in Sec. 130,-varying as the square of (H - H'),-have for their 2d difference 1.852, which does not change; therefore the differences of the differences of the values in the table increase regularly, the difference between any two values being greater than the preceding difference by this amount; thus the calculation of such a table as that is merely a matter of simple addition as soon as the 2d difference has been obtained. The engineer should be always on the lookout for this in the construction of tables, etc.