width M C (which consists partly of an embankment), multiply the distance M D (just found) by the difference of the bottom-width and the estimated half-width, and divide the product by the estimated half-width, and the quotient is the corrected half-width M C.*

Ex. Let the bottom-width A B = 30 ft., the depth M m = 3 ft., the ratio of the slopes 1 1, and the difference of level-reading 7 ft., at the distances of 25 and 24 ft. from the centre-stump on the surface-slope, and on the level respectively; required the corrected half-widths MD and M C.

If the same symbols be used for the given parts in this example, as in Prob. III., and w = bottom-width: then

NOTE. This operation for finding M C, it will be seen, is different from the preceding one.

See Demonstration.

Construction. The operation for this Problem is the same as that for Prob. III., excepting that A C is drawn parallel to B D.

By reversing the cross-section, it will be readily seen that the same

=

* Demonstration. Draw A c to C'D' prolonged, and let pq, which is to p C' D', be the difference of level-readings at M and 9: then, if a = Mm, Mc Am =w=bottom-width, the other symbols being the same as in the Demonstration to Prob. III., arc C' and C' M = Mcc C' w — ar; whence, by the Des (} w — ar). monstration above referred to, M C

=

1-rh

=

: but b

this value being substituted in that of MC, gives

=

w+ar, or arb−{w;

calculation and construction will apply in the case of A P C, being a cutting, and P B D an embankment, observing that the corrected half-width for the cutting is the shorter distance, and vice versâ.

PROBLEM V.

To find the Width of the Cuttings when the Surface of the Ground is laterally very uneven.

uneven surface of the ground, &c. The solution of this Problem will be best effected by giving an example in numbers.

Let A B 30 ft., M m = 30 ft., and the ratio of the sideslopes as 1:1; then M C' = M D' = 30 × 11 + 1 × 30 = 60 ft. Measure from M horizontally the distance M d= 60 feet, the point d is directly above D'. Place levelling-staves at M and d, and observe the difference of the readings at M and d, which, in this case, is 7 ft.; whence 7 x 1 = 10 ft. = approximate distance dyD, and M d + d D = MD = 60 + 10 = 70 ft. Now place a levelling-staff D, and observe the reading, which is found to be 7.8 ft. greater than that at M, or 0.8 ft. greater than that at d; whence 0.8 × 1 = 1.2 ft., and consequently 70+ 1.2 71.7 ft., which is a still nearer approximation to the true distance M D or M q, it being measured horizontally. The operation for finding M c, or MC measured horizontally, is the same as the preceding, excepting that the product of the stave-readings is subtracted from the estimated half-width, &c. In this manner the distance M c = horizontal distance M C is found to be 52.6 feet.

NOTE 1.

=

The widths R C, D S, of the side fences, must be added to the above results for the whole width.

NOTE 2. - If the difference of the stave-readings at M and d be very large, it will require three or four approximations similar to those given in the preceding example, to find the true corrected half-width.

Construction.

Take the levels of the several undulations of the surface C M D, making C ́ M D' the datum-line, and draw the crosssection A B D M C by the methods already given.

By reversing the cross-section A B D M C, its application to an embankment is obvious, observing also to reverse the distances M C, M D, as previously noticed.

Embankt. Cutting.

NOTE 3.

The truth of the method of approximation, used in this example, is

too obvious to require a demonstration.

NOTE. The depths of cuttings or embankments, in the 2d column of the preceding Level-Book, are found by calculation, or by carefully measuring them from the section by the vertical scale, but the latter method is not sufficiently correct. The computed half-widths in the 3d column, are found by Problems I. and II. The corrected half-widths, in columns 4th and 5th, by the five preceding Problems, according to the nature of the cuttings or embankments.

PROBLEM VI.

To find the Quantity of Land required for a projected Railway.

CASE I. In preparing the preliminary Estimates for a projected Railway, the Quantity of Land required for the Purpose is usually found without paying any Regard to the lateral Inclination of the Ground, by taking a considerable Length of the Section at once, especially if the Surface thereof have a regular Rise or Fall, and by measuring the Depth of the Ends of such Length with the vertical Scale.

RULE. ·Find the surface-widths, fences included, at each end of the given length, by Problems I. and II., add them together, multiply the sum by the length in chains, and divide the product by 1320 for the area in acres.

Ex. Let the length of the sectional surface be 18 chains, and the depths at the ends 22 and 38 feet, the bottom-width of the railway 33 feet, and the ratio of the slopes 1 to 1; required the area of the surface, the width of the side-fences being 9 feet each.

By Prob. II. w+2 ar+2ƒ=33 +3 × 22+2×9=117 w+2br+2f33+3 x 38+2×9=165

or, by putting = given length, and taking half the sum of the widths, there will result

(w+ra+b+2ƒ)l 660

=(33+13×22+38+2×9) = 3.84545 acres. 660

18

CASE II. When the exact Quantity of Land for the Railway is required.

-

RULE. Take the whole widths, at the end of every chain, from the 6th column of the Level-Book, for the several widths; add continually together the first and last widths, and twice the sum of all the intermediate widths, and divide the whole sum by 1320 for the area in acres.

Ex. Required the area corresponding to the several widths in the Level-Book at the end of Prob. V.

--

1408.92 = twice sum of intermediate widths,

142.30 first width,

144.62

12169.584

11 14.132

last width,

1.28473 = 1A. 1r. 51⁄2Р. = area required.

NOTE. It is very common in practice to find the areas of the quantities of land, required from the several proprietors, by actual measurement from the 2 chain maps, made for the use of the contractors, after the several widths have been laid down thereon: copies being taken, at the same time, from the maps, on tracing paper, showing the position and quantity of land required from each proprietor.

ON RAILWAY CUTTINGS IN GENERAL AND TABLES FOR FINDING

THEIR CONTENTS.

In preparing the preliminary estimates for a railway, the contents of the cuttings are usually found by tables for the purpose, the surface of the ground in the several cross-sections being assumed to be on a level with the centre of the line. But when power has been granted for constructing the line, the cross-sections are carefully taken at the end of every prominent variation of the surface of the ground, or, if consistent with accuracy, at the end of every one, two, or three chains in length; the several cross-sections are then plotted on a large scale (which may be done by the methods given in the preceding Problems), and their areas found by actual measurement; or reduced, where the surface of the ground is uneven, to horizontal sections, preparatory to finding the contents either by taking the mean of every two succeeding sections, which method is very erroneous where the areas of the sections differ greatly, or by finding the contents from the tables, by using the mean depths of the several sections, which method is correct; but the mean depth, used in this method, cannot be accurately found in many cases without considerable calculation. Some use a mean of the mean depths as the basis for a mean area, which method is also very inaccurate; especially where the areas of the extreme sections differ greatly. The magnitude of the errors in both cases, will be pointed out in the investigations at the end of these Problems.

On existing Earthwork Tables.

Tables for this purpose have been published by Sir John M'Neill, Mr. Bidder, Mr. Bashforth, Messrs. Sibley and Rutherford, and others; all of which are well adapted for finding the contents of cuttings, assuming the surface of the ground to be laterally level with respect to the direction of the cutting. But none of these tables are accompanied with directions for finding the contents from sectional areas, i. e. from the areas of working drawings, excepting Mr. Bashforth's tables; but his mathematical investigation of the rule for using them in finding the contents from working drawings, where the surface of the ground is laterally sloping or uneven, is founded on a false assumption, and, therefore, his results are erroneous, and especially so where the sectional areas differ considerably. Preface, and the investigations at the end of these Problems.)

(See

D D