Let A' = a, and B' = 36a, which is a case that may occur in practice, then

al

81

× 2.8007 = error in defect by Mr B.'s method.

See Ex. at the end of Problem X., where the error of Mr Bashforth's method is shown by giving actual areas.

NOTE. The errors of this method are not so prominent where the sectional areas approach near to equality, as in the case of the Burnley cross-sections, in the Ex. Art. 20, page 13, of Mr B's work: his method, however, is erroneous to a greater or lesser extent, in every case, except where the sectional areas are equal and similar; for they cannot be similar without being equal, while the bottom width remains the same.

Mr Bashforth says, in defence of his method of finding the contents of cuttings from sectional areas, in the "Mechanics' Magazine" for Sept. 11, 1847, p. 249, "In the case of contract estimates, numerous cross-sections ought to be taken; and it matters little whether we start from the intersection of the slopes or the formation-level." The above-noticed error (15) shows that it "matters" so much as 71 per cent. in defect. Besides, there is no need of taking "numerous cross-sections," as Mr B. recommends, especially where the surface of the ground is laterally sloping like a geometrical plane, or curved like a conical surface, one or other of which cases very frequently occurs, so that cross-sections, taken at a considerable distance from one another to the intersection of the slopes, may be considered as similar, or so very nearly so as not to induce any important mathematical error, which conditions may be easily determined by the eye. Moreover, the expense and trouble of taking "numerous cross-sections," plotting them, and finding their areas, are very considerable; and, therefore, ought to be avoided, together with all erroneous methods, such as Mr B.'s, of finding the contents of cuttings, as his

"numerous cross-sections" only tend to diminish the errors, without wholly getting rid of them.

The error of the method of finding contents by mean areas.

Let A and B be the areas of two cross-sections of a cutting to the intersection of the slopes, and its length: then the mean area is (A+B), and the content in cubic yards is x (A + B) = 1 × (A+B); from which subtract the true content, equation (11), p. 455,

Which error is very great when the areas A and B differ considerably. See (2), p. 451.

The error of the method of finding the content by mean depth.

Let a and b be the depths of two cross-sections to the intersection

of the slopes, the ratio of which is r to 1, and l ting; then (a + b) = mean depth, {r(a + b)2 the content in cubic yards

(a + b)2;

=

27 1 × fr (a +

length of the cut= mean area, and

b)2

=

=

32 rl (a - b)2

=

error in

which subtract from equation (1), p. 411, and there results rl {a2 + ab + b2 3 (a + b)2 } defect: Which error is very considerable, when the depths a and b differ greatly.

NOTE. The errors of these methods, as here shown, are the same as if the areas and depths had only extended to the formation-level, since a common quantity, i.e. the prism below the formation-level, is here included, and afterwards excluded, by taking the differences.

SECTION V.

TUNNELLING.

(1.) Previous to setting out the earthwork of a tunnel, the levelling operation must be repeated with great care, and should also be checked by the method given in Art. (22), Section I., especially if the tunnel pass under a very high summit: for, if the section be incorrect, the gradient or gradients, on which the tunnel is formed, will not meet at the points shown thereon, and thus embarrass the mining operation.

(2.) If the tunnel is formed on one gradient, as BD, Plate XIII.

the gradient must incline to one of the extremities of the tunnel, as at D, in order to discharge the water generated therein. Strong poles or masts must be firmly fixed on the surface, in the intended direction of the tunnel, of which one must be on the summit of the hill; at which place a temporary observatory is frequently erected, especially if the summit be a very high one, and the tunnel a very long one. Shafts must be sunk at the distance of four or five chains from one another, in the direction of the poles (and observatory, if there be one), in order to ventilate the tunnel, as well as to check the accuracy of the work as it proceeds. If the tunnel be a long one, it would be preferable, if convenient, to form it on two gradients, inclining to its opposite extremities to liberate the water, and thus to aid the mining operation, which is commonly commenced at both ends of the tunnel at the same time.

(3.) When it is necessary to have a curve in the direction of a part or of the whole of the tunnel, that direction must be carefully laid down on the surface, by the methods given in Section III., making allowance for acclivities and declivities, poles or masts being fixed. therein, as pointed out in Art. (2), that the shafts may be sunk so as to meet the mining operations of the tunnel, as well as to check their accuracy in point of direction, and this will be the more especially necessary in the curved part of the tunnel.

(4.) The mining operation of the tunnel should commence when the depth of the cuttings at each end is about 60 feet. The width and depth of the excavation of a tunnel, on the narrow gauge, should be about 30 feet each, and must be dug 5 or 6 feet below the intended line of the rails, to give space for the inverted arch and the ballasting, excepting where the excavation is made through rock sufficiently hard to form the side-walls of the tunnel, in which case 22 or 24 feet in width, and about 26 feet in height, will be sufficient, the excavation in this case being terminated below by the balanceline, or formation-level. The depth and width

of the excavation for a tunnel on the broad gauge must, in both cases, be proportionately larger.

The annexed figure is a cross-section of the masonry of a tunnel, which, of course, is such as is required where the tunnel is made through loose earth, only the arch above being required when made through hard rock.

NOTE. The remarks at Articles (25) and (27), Section II., refer only to the projection of tunnels on parliamentary maps.

A TABLE OF THE DIMENSIONS OF SEVERAL EXISTING TUNNELS.

Cromford and High Peak-Buxton

Great Western-The Box Tunnel

Manchester and } Littleborough

Leeds

Melford.

836

VIADUCTS, AQUEDUCTS, SKEW ARCHES, ETC.

What are here given on these subjects are necessarily only outlines, referring to some few existing structures, without entering into the details of specifications, working drawings, &c., which are foreign to the nature of this work, and properly belong to the department of the architect.

Railway Viaducts.-It would be impossible to enumerate the whole of the viaducts constructed since the introduction of railways. Structures in timber, brick, stone, and iron, of various designs, have been erected; and in some cases there is a novelty of principle accompanied with great boldness of execution.

:

The Brick Viaduct at Maidenhead, constructed by Mr Brunel for the Great Western Railway, is one of the best examples in that material it is composed of two elliptical arches spanning the Thames, each 128 feet, with a versed sine of 244 feet; the pier between the two arches is 30 feet in width. The arch in the middle is 5 feet in thickness, which gradually increases towards the abutments. Besides these two grand brick arches, on each bank of the

river are four others; those on the abutments span 21 feet each, and the six others 28 feet each.

The Viaduct over the Ouse, near York, consists of three arches, each 66 feet span; the piers are 10 feet in thickness, and the width of

the arches 28 feet on the soffite, and their thickness at the keystone 3 feet, the voussoirs gradually increasing towards the springing. This is a specimen of a stone viaduct combining great strength and elegance in its construction.

Aqueduct of Pont-y Cysyllte is 4 miles from Chirk, over the river Dee; its length is 1007 feet, and the river is 127 feet below the water-level of the canal carried over it.

To construct an aqueduct upon the usual principles, with piers and arches above 100 feet in height, and of a sufficient breadth and strength to afford room for a puddled water-way, would have been not only expensive but extremely hazardous. Telford, who had already carried the Shrewsbury canal by a cast iron trough 16 feet above the level of the ground, formed the idea of doing the same in the present instance, which was approved and finally adopted. The foundation on which the piers are erected is a hard sandstone; their height above low water in the river is 121 feet: at the botton they are 20 feet by 12, at the top 13 feet by 7. For a height of 70 feet above