The sum of these two angles 18°

and 4° 30′ = 22° 30′, the half of

Fig. 5.

which is 11° 15', the required sub-deflexion angle abe.

Again, to find the sub-deflexion distance a e, of the sub-chord be; take from the table of sines, the natural sine of one-half the sub-deflexion angle, just found. Multiply this natural sine by 2, and multiply that product by the length of the sub-chord.

Example. The sub-deflexion angle is 11° 15'; one-half of it is 5° 37', the tabular natural sine of which is 0979, which multiplied by 2, gives 1958; and this multiplied by the sub-chord, 25 feet, gives 4-895 feet, the required sub-deflexion distance a e.

ARTICLE V.

Ordinates for Entire Chords.

It would be both tedious and liable to inaccuracy to attempt to fix all the necessary points in railroad curves by the foregoing means, which are employed only for entire chords, or for such sub-chords as may be required at the ends of curves.

Fig. 6.

The best method is to stretch a piece of twine a b, fig. 6, 100 feet long, between two adjacent chord stakes, and measure off as nearly as may be at right angles to it, with a graduated rod, the previously calculated ordinates c d, ef, gh,

yee

b

&c, placing pegs at d, f, h, &c. Our table of ordinates is calculated for distances apart bc, ce, eg, &c., of 5 feet; and for all curves likely to occur in practice. The 5 feet distances on the twine should be marked by knots or otherwise, and those at the centre, and half way between it and the ends, be further distinguished by tyeing on pieces of tape.

The 5 feet distances are only used (after the excavations and embankments are finished) for placing pegs to guide the laying of the rails, and then only for very sudden curves; for large ones, distances of 10 feet are quite sufficient, or even 25 feet for very easy curves. For guiding the curves of the cuttings and fillings, it is not necessary to place the stakes nearer than 50 feet apart; unless for those of less than about 1000 feet radius, when they may be placed 25 feet apart. Ordinates for radii intermediate of those in the table, may either be calculated by the rules

given further on, or they may be taken proportionally intermediate of the tabular ones, with sufficient accuracy for practice.

Ordinates for Sub-Chords.

These may readily be calculated approximately enough for railroad practice, for curves of over 300 feet radius, and for chords not exceeding 100 feet, thus: In a circle of given radius, not less than about 300 feet, the ordinates of an entire 100 feet chord may be assumed to be to those of a sub-chord as the square of the chord is to the square of the sub-chord. In all our tables the chord is supposed to be 100 feet, the square of which is 10,000; the rule therefore becomes, as 10,000 feet to square of sub-chord in feet :: Ord. of Chord: Ord. of Sub-chord approximately. Example. In a curve of 5730 feet radius, the middle ordinate of a 100 feet chord is 218 of a foot; what will be the length of the middle ordinate of a sub-chord of 50 feet? here,

and so of any other ordinate, always supposing the chord and sub-chord to be divided into the same number of parts.

ARTICLE VI.

Having given the angle a b d, fig. 7, it is required to find the point a or d, at which to commence a curve of given radius..

Fig. 7.

Rule. Subtract half the angle ab d, from 90°; the remainder will be the angle b c a, or bc d. From the table of tangents take the natural tangent of b c a, and multiply it by the given radius, the product will be ba or bd.

Example.-Let the angle abd be 120°, how far from b must we begin, at a or d, to lay out

a curve and of 2865 feet radius?

a=

Here, half of the angle a bd 60°, which taken from 90° leaves the angle bca 30°. The natural tangent of 30°5773, which multiplied by the radius of 2865 feet, gives 1653-96 feet for ba or bd. (See Art. XII.)

ARTICLE VII.

Having given the angle a b d, fig. 7, and the distance from b to a or d, at one of which we wish to commence a curve, it is required to find what radius a cor c d, the curve must have, in order to unite with b a and b d tangentially at a and d.

Rule.-Subtract the angle a b c, which is half the angle ab d, from 90°; the remainder will be the angle bca, or b cd. Then as nat. sine of b c a, or b c d, is to nat. sine of a bc, so is a b orb d, to a c or c d the radius required.

Example. Let the angle a bd be 120°, and the distance ba or b d, 1654 feet; what will be the radius a c or cd of a circle that shall touch a and d tangentially?

Here, the angle abc half the angle a b d, is 60°, which taken from 90°, leaves the angle b ca or bcd 30°. Then as the nat. sine of b c a

=

or b c d (30°) •5000, is to nat. sine of a b c or d bc, (60°)=8660, so is ba or b d (1654 feet) to a cor c d, 2865 feet, the radius required.

ARTICLE VIII.

Having given the radius ac, fig. 7, of a curve, and the angle a b d, it is required to find the number of chords of 100 feet that will constitute the

curve.

Rule.-Subtract the angle a bd from 180°, and divide the remainder by the angle of curvature, or deflexion of the curve. The quotient will be the required number of chords.

Example.-Let the angle ab d be 120°, and the radius a c, 2865 feet. Here the angle ab d, 120°, subtracted from 180°, leaves a remainder of 60°; which, divided by 20, the angle of deflexion for a curve of 2865 feet, gives a quotient of 30; which is the required number of chords of

100 feet.

N. B.-Had the quotient contained a fraction of a chord, it would have indicated that we should have had to employ a sub-chord at the end of the curve; for instance, had the number of chords been 301, a subchord of 50 feet would have been necessary.

ARTICLE IX.

Fig. 8.

d

How to proceed when the end of a curve does not correctly join the tangent. We sometimes find on running out a curve for the number of chords determined by the Rule in the preceding Article, that instead of uniting as it should with the previously deterinined tangent o m, fig. 8, at o, it ends tangentially to a line parallel to said tangent, either within it as at c; or beyond it as at b. Being first certain that no error has occurred

in tracing out the curve, ascertain with the compass the bearing of the line a d, and removing the compass to the end of the curve at c or b, (as the case may be,) run the line bo or c o, in the same course as a d, until it strikes the tangent do m; which may be ascertained by ranging two stakes placed on the tangent.

Then measure bo or co, (as the case may be,) and if the curve fall within the tangent om, as at c, measure forwards from t towards d, the distance t a, equal to co; or if the curve fall beyond the tangent, as at b, measure backwards from s, the distance sa equal to bo. Then the curve retraced from a, will terminate tangentially in dm at o.

N. B.-The direction of c o or bo may be ascertained without a compass, and better, thus: Multiply the tangential angle of the curve by twice the number of chords run, less one; subtract the product from 180°, and sighting back one chord to n, lay off the angle n cb, equal to the remainder. For example, if the tangential angle be 10°, and from t to c be 4 chords, then 7 times 10° taken from 180° leaves the angle ncb = 110°. When the product exceeds 180°, it must be subtracted from 360°, for the angle n c b.

This case occurs whenever an error has been made in measuring the distance from d to a. If d a be made too short, the curve s b is the result, and if too long, the curve t c.

If the error is small, it may be divided equally among the chords by measure, without retracing the curve with an instrument. This method may be employed with perfect security so long as the error does not exceed 1 foot to every chord of 100 feet; and it need never be greater if moderate care be taken.

Thus, if the curve be 20 chords long, and the error 20 feet, the last stake may be moved 20 feet, the next 19, the next 18, &c., as nearly at right angles to the curve as can be judged by eye.

The same ordinates that would have been used had the curve been correct, will answer for the one so adjusted, without perceptible differFor other cases, see Article X.

ence.

Again, it may happen that the error is not caused by a mismeasurement of the distance a e, figs. 9 and 10, as in the last case; but by a mistake in obtaining the angle aef.

If a e f, fig. 9, be measured in excess, as a eg, then the curve a b C, calculated for the incorrect angle a eg, will be found to fall beyond the true tangent ef, as at c; and the tangents e g and e f not being parallel, the curve cannot be adjusted by either of the methods given in the preceding Article, unless the error be within about 1 foot to each 100 feet length of the curve; in which case, (supposing no other error to exist,) either of those methods may be employed, with sufficient accuracy for practice.

Also, if a ef, fig. 10, be measured too small, as a e g, then the curve a bc, calculated for the incorrect angle aeg, will be found to fall within the true tangent e f, as at c; when so, the remarks contained in the preceding sentence are equally applicable here. If the error be within 1 foot to 100 feet length of curve, it may be equally divided among the chords. But if greater, we must either re-measure the angle a ef correctly, and go over the whole work again, or resort to some other mode of obviating the difficulty. The angle a ef may be difficult of access, or the curve may be so long that to retrace it would be a work of much labor. We may then adopt the method of compound curves, by which much trouble will be

avoided, and a considerable portion of the first part of the curve be allowed to remain as it is.

Thus, whether the curve a b c fall beyond the true tangent ef, as in fig. 9, or inside of it, as in fig. 10, place the instrument at b, figs. 9 and 10, (the point at which the change of radius is to take place,) and sighting back one chord to n, lay off the tangential angle nbm of the curve abc, and observe where the new tangent mb continued, strikes ef, as at o. Measure both bo and the angle bof. Half the angle bo ƒ taken from 90°, gives the angle b h o; then say, as the

Nat Sine of angle

so is

The given
side bo,

The required

to

side, or new radius bh.

Nat. Sine of angle bh o, opposite the is to boh opposite the given side bo, required side bh, Ascertain from the table or by calculation, the angle of deflexion, and the tangential angle corresponding to this new radius bh; and the new curve commenced a b will terminate tangentially to e fat i, as far from o as o is from b.

For the mode of uniting two curves at different radii, so as to form a compound curve, see Article XIII.

It will be observed, that when the first curve, a b c, fig. 10, falls inside the tangent ef, the new curve must be of greater radius; and when beyond, fig. 9, of a less one.

ARTICLE XI.

Having given the angles a b c fig. 11, and b c d, and the distance b c, it is required to find the greatest radius, gi or hi, that can be employed in a reverse curve, foinm, for uniting ab to c d.

Rule.-Half the angle a b c taken from 90°, leaves the angle bg i; and half the angle b c d taken from 90°, leaves the angle i h c.

From the table of tangents take the

natural tangent (bi) of angle b gi; and

އ

that (i c) of the angle ih c; and add them together.

Fig. 11.

h

Then as the sum of these two nat. tangents is to the nat. tang. of bg ï, so is bc to bi; and bi taken from bc, gives i c.

Again, in the triangle bgi, as the nat. sine of the angle bg i, opposite the given side bi, just found, is to the nat. sine of the angle g bi, opposite the required side gi, so is b i, the given side, to gi, the required side or radius.

Example.-Let the angle a b c be 71° 40', the angle b c d 129° 15', and the distance b c 950 feet. What is the length of radius hi or gi, of the easiest reverse curve that can be traced for uniting a b to c d?

Here, half the angle a b c (35° 50′) taken from 90°, leaves the angle bg i 54° 10′, and half the angle b c d (64° 37') taken from 90°, leaves the angle ihc 25° 221.

=

From the table of tangents, we have nat. tang. of bgi (54° 10′) = 1.3848; and nat. tang. of i hc (25° 22′) =·4743; their sum being 1.8591.