last offset; but the length of the offsets given in the table is

; therefore, if A p, 1 P2, &c., be taken as 2,

1

3, 4, &c., chains, the value of must be multiplied by 22 =

2 r

4, 32 = 9, 42 = 16, &c., respectively to find p q, and the result, in each case, multiplied by 2 for each of the offsets p2 92, P3 93, &c. In this manner the curve may be set out more speedily, and with less liability to error, on account of the less number and greater length of the lines required in the operation.

EXAMPLE.

Let A Or=120 chains, and 8 = 4 chains; then

=

× 16 = 3·3 × 16 = 52·8 inches 4 feet 4·8 inches = p 9; whence 4 feet 8 inches x 2 = 4.8 feet 9.6 inches P2 92, P3 93 = &c. NOTE 1. When the curve has been correctly set out, as in Case II., the intermediate stumps may be put in at the end of every chain, if required, by the method given in Case I. The distances of the intermediate stumps, thus put in, will not, in most cases, exceed a fraction of an inch; because the lengths of the offsets q, P2 2, &c., is so small, that the curvilinear lengths A q, qII 2) &c., can never greatly exceed those A p, q P2, &c.

NOTE 2. The method given in Case II., is sufficiently accurate when & does not exceed of the radius of the curve. Besides, at the closing point of the is most commonly less or greater than 8. Let (8 + d)d ̧ at the end of the curve is

curve, as at 94, the distance 43P 4

=

93 P4 d; then the offset P4 4'

when 8 = 1 chain, p1 q^

; and,

(1 + d)d.

2 r

2r ; or the tabular number for the given

radius must be multiplied by (8 + d)d, or by (1 + d)d, according as AP, IP21 &c., is taken chains or 1 chain, to give the last offset P4 4 of which P5 95, the offset to the tangent 94 95

is

=

EXAMPLE.

Let r = 120, and 8 = 4 chains, as in the last example, and

1

let

93 P1 = d = 2·68 chains; then på 94

= × (8 + d) d = 2 r

4

3.3 × (4 + 2·68) × 2·68 = 59.07 inches; of which, viz., 29.535 inches is = P5 95.

NOTE 3. When 8 exceeds of the radius r of the curve, the following formula ought to be used for finding the offsets.

See Baker's Land and Engineering Surveying, page 164.

NOTE 4. By this method the greater part of both British and foreign rail

way curves have been laid out. It was invented by the author about 30 years ago, when the Stockton and Darlington Railway was laid out, and eagerly adopted by engineers as it involves very little calculation, and does not require the use of a theodolite. It is, however, defective in practice, on account of its requiring so very many short lines connected together, as errors will unavoidably creep in and multiply, and more especially so where the ground is rough; thus the curve has frequently to be retraced several times before it can be got right; hence the author prepared the methods in the following Problem.

PROBLEM II.

To lay out a railway curve on the ground, by offsets from its tangents, no obstructions being supposed to prevent the use of the chain on the convex side of the curve.

CASE I.-When the length of the curve does not exceed of its radius.

Range the tangents A B, DC till they meet at T; and let the radius BO = 80 chains 1 mile; measure on BT the distance B q = 1 chain; and, at right angles to BT, lay off the offset qp = 4.95 inches, by Table No. 2, as in Problem I.; then p is the first point in the curve. Next measure ¶¶2 and lay off the offset -= P2 1 92 4.95 × 4 for the second point in the curve. The successive offsets, at the end of every chain, being 4, 9, 16, &c., or 22, 32, 42, &c., times the first offset pq, which may also be found opposite the given radius in the Table No. 2., as in Prob. I.

1

1 chain,

When the offsets have been thus laid out, till the last one 15 P5 falls little short of T; lay off the same offsets on TC as were laid off in BT, but in an inverted order, making the first distance on TC = Tq5; thus completing the curve to C.

NOTE. It can rarely happen in practice that the last offsets, from both tangents, will meet at the middle point p of the short curve, as shewn in the figure; but will either intersect one another or fall short of the middle point; but this is a matter of no consequence.

EXAMPLE.

Let the radius of the curve be 160 chains, required the offsets at the end of every chain, from the tangent to the curve.

CASE II. To lay out the curve when it is any required length.

In a long curve (of which there are some more than two miles in length) the tangents, if prolonged to their point of meeting, would necessarily fall at a great distance from the curve, thus giving an inconvenient length to the offsets, which in practice should never exceed two chains. To remedy this inconvenience the curve must be divided into two or more parts, by introducing one or more additional tangents, thus the offsets may be confined within their proper limits. Thus the tangent TC may, in this case, be extended, another tangent applied, and the offsets laid off, thus repeating the operation of Case I. a second time: if the curve be not yet completed, the operation may be repeated a third, fourth, &c., time, till it be completed.

NOTE. For a complete development of this important subject, see Baker's Land and Engineering Surveing, Part II., Chap. II., where two other methods of laying out railway curves are given; also methods of laying out compound, serpentine and deviation curves, with original formulæ ; all of which methods, as well as the two already given, were first drawn up by the author. See page 179 of the work above referred to, where a short history of the invention is given. See, also, Tate's Geometry, page 247.

CONTENTS OF RAILWAY CUTTINGS, &c.

TABLES.

The General Earthwork Tables, in conjunction with Two Auxiliary Tables, on the same sheet, in Baker's Railway Engineering, or the numbers for the slopes in Bidder's Table, are applicable to all varieties of ratio of slopes and widths of formation level in common use; and with the help of Barlow's table of square roots, these tables will apply to sectional areas, with all the mathematical accuracy that can be attained, with very little more calculation than adding the contents between every two cross-sections, as given by the General Table.-The contents in the General Table are calculated to the nearest unit, as are also

those in the Auxiliary Table, No. 2, which is for the decimals of feet in the depths. The Auxiliary Table, No. 1, shews the depths of the meeting of the side-slopes below the formationlevel, with the number of cubic yards to be subtracted from the contents of the General Table for each chain in length, for eight of the most common varieties of ratio of slope.

The following diagrams and explanations will further illustrate the method of taking the dimensions of railway cuttings, preparatory to using the above named tables.

Let ABDC cab d, be a railway cutting, of which A B D C, abdc are the cross sections, A B = a b = width of formation level, M M', m m' the middle depths of the two crosssections; the side-slopes A C, DB, a cbd, when prolonged two and two, will intersect at N and n, at which points the prolongations of

C

M

C

m

a

A

M

a

n

M M', m m' will also meet, thus constituting a prism A B N nab, the content of which is to be deducted from the whole content, given by the General Table, by means of the Table No. 1.; in which the depth M' N = mn is also given, as already stated, to several varieties of slope and bottom width.

с

To place this subject in a more practical point of view, let the annexed figure represent a longitudinal and vertical section of a cutting, passing through the middle A E of the formation level. H I, the line of the rails, and a h, the line on which the slopes, if prolonged, would meet. It will be seen that the cutting A b c d E commences and runs out on the formation level A E, and that the depth A a =

=

Be

b

d

H

I

Ai

B

Di

Cf=&c., is to be added to the several depths B b, Cc, D d of the cutting, the first and last depth at A and E being each=0; or, what amounts to the same thing, the several depths must be measured from the line ah: thus A a, be, cg, &c., are the depths to be used. And since the depth A a is given in Table No. 1, for all the most common cases, or it may be readily found by calculation for all cases, (see Railway Engineering), the line corresponding to a h must, therefore, be ruled on the railway section, at the proper distance below A E, from which the several depths must be measured; or the vertical

scale may be marked with Indian ink, (which may be readily rubbed off) at the same distance, and this mark may then be applied to the formation level A E, for the purpose of measuring the several depths. In the case of an embankment, the line for the several depths must be placed at a like distance above the formation level.

PROBLEM I.

The several depths of a railway cutting to the meeting of the side slopes, the width of formation level, and the ratio of the slopes being given, to find the content of the cutting in cubic yards, from the Tables referred to, the distances of the depths being one chain each.

RULE. Take the several quantities, corresponding to every two succeeding depths of a cutting, or embankment, measured to the meeting of the side slopes, at the distance of 1 chain each, from the General Table in Baker's Railway Engineering, and multiply their sum by the ratio of the slopes; from the product subtract the cubic yards, corresponding to the given bottom width and ratio of slopes from Table No. 1., multiplied by the whole length of the cutting, and the remainder will be the content of the cutting in cubic yards.

But, when the distances of the depths are greater or less than I chain, the quantities of the General Table must be multiplied by their respective distances.-And, when the distances are given in feet, the quantities must be multiplied by those distances, and the final result divided by 66 for the content in cubic yards, as in the following

EXAMPLES.

1. Let the depth of the railway cutting or embankment to the meeting of the side-slopes, at the end of every chain, be as in the following table, the bottom-width 30 feet, and the ratio of the slopes as 2 to 1; required the content in cubic yards.

NOTE. In the annexed table the quantity 1238, corresponds to the depths 10 and 33 feet, in the General Table; the quantity 3175 to the depths 33 and 39, and so on for the succeeding depths. By the Auxiliary Table No. 1, it will be seen, that the depth to be added below the formation level, for the given width and ratio of slopes, is 7.50 = 7 feet, therefore, the cutting begins and ends with a depth of 10-7 2 feet. The corresponding number of cubic yards, to be deducted for each chain

=