OBSERVATIONS IMPORTANT TO THOSE CONCERNED IN THE CONSTRUCTION OF RAILWAYS, ON THE ERRONEOUS METHODS OF CALCULATING THE CONTENTS OF CUTTINGS, WHERE THE SURFACE OF THE GROUND IS UNEVEN; ALSO ON THE METHOD GIVEN IN THIS WORK. The magnitude of the errors of the methods (1), (2), and (3), in the last example, for calculating the contents of cuttings, where the surface of the ground is uneven, from sectional areas, is strikingly apparent; and since many tables are not accompanied by any directions for their practical application in this particular case, one or other of the last two of these defective methods is still frequently used by engineers and contractors; thus causing continual disputes concerning the contents in consequence of the different parties using irreconcilable methods, some preferring one and some the other, as giving, in their respective judgments, the true content. It is therefore the interest of those concerned in the construction of railways to adopt the method given in this work for finding the contents of cuttings, where the surface of the ground is laterally sloping, &c., as no other method combining all attainable mathematical accuracy has, to my knowledge, been yet published: those already published referring only to cuttings where the surface of the ground is either level, or the sections thereof assumed to be reduced to a level, which, in many cases, is a work of great labour to perform accurately. See the following investigations. INVESTIGATION OF THE CONSTRUCTION AND USE OF THE GENERAL AND AUXILIARY TABLES, AND OF THE ERRORS OF OTHER METHODS. Let ABDC, abdc, be vertical cross-sections of a railway cutting, the side slopes downwards and to the left till they meet in NV, and also meet the prolongation of the plane CDde in cv, dv. = Because AB NM, nm on CD, - ab, the cross-sections are to VN. Let fall the cd respectively, bisecting them in M and m, and also bisecting AB, ab in M' and m'. Put MN=a, mn=b, Nn = Aa = Bb = l, AB = ab=w, all in feet, and the ratio of the slopes, i.e., CM: MN: :r: 1. Then A CDN = a2r, A cdn = b2r. By similar figures ab: NV : NV -1, } }r Frustum CDNncd=} NV x a2r- (NV-1) b2r = fr (a2 - b2NV + b2l) = (by substituting the value of NV) = rl(a2 + ab + b2) cubic feet In the General Table r= 1, and 7 = 1 chain = 66 feet; and .. the solidity 22 27 S= (a2 + ab + b2) cubic yards wherein a and b have all integral values from 0 to 72. 4r From (3) and (5) the depths to be added, and the cubic yards to be deducted in Table No. 1, are calculated, by taking 1= 66 feet, and w and r all the values most commonly used in practice. w21 11w2 108r 18r 11w2L Let = 18r cubic yards to be deducted, Table No. 1. cubic yards to be deducted for the length L. (6) = sum of all the solidities s, s', &c. (per General Table), the Hw2 18r length of which is L, and z = from (6), then Er sum of soli . dities for slopes r to 1, from which deduct x × L, and there results Er-xL = cubic yards in the whole cutting (7) Whence the method of finding the contents of cutting in Problem VII. Cor. 1. The content of a cutting having only two given depths is (Sr-x)L cubic yards. (8) Cor. 2. If the ratio of the slopes of the two sides of a cutting be r to 1 and p to 1, and k, x the corresponding cubic yards to be deducted. (Table No. 1.) Let in (2) a and b, hitherto supposed to be integral numbers, have the increments or decimals a and B, so that a and b become respectively a + a and b + 3, which being substituted for a and b, neglecting the squares and product of a and B, as being comparatively small, there results: By subtracting (2) from (9), there results the sum of (9) which are the increments of the formula (2) arising from the addition of the decimal parts a and ẞ to the depths a and b, from which Table No. 2 has been calculated, the quantities 2a + b, and 2b + a being divided by 10, to prevent a too great extension of the Table; and the corresponding factors a and 6 multiplied by the same number, that the product might retain their original value, the decimal points being still affixed to the values of a and B in the horizontal line at the top of the Table. and 2b + a 2a + b In consulting Table No. 2 the nearest whole numbers to 10 are taken as the small errors, thus resulting, will usually balance one another in a long cutting, and can never in any case amount to much. Let the plane VCD revolve on the side vc, so that the lines CD, ed may incline from their hitherto assumed horizontal position, thus making the side slopes unequal; and let the area CDN = A, and the area can = B, the other symbols being the same as before; then the frustum CDNncd =S= VN × A - } (VN−1) × B = }} (A – B × VN + Bl), (A+B+ √Ax B), in cubic feet, and taking 166 feet 27 (A+B+ √ A× √ B) cubic yards (11) Therefore the Table must be consulted for depths √A and √B, the ratio of the slopes being included; for if a and b be the mean depths for the areas A and B respectively, the slopes being r to 1, then Therefore in reality the Table is consulted for the depths a √r= √A, and b r = √ B. If Σ' sum of all the solidities s, s', &c., for depths √ A and √ B, &c., and length L; then by subtracting L, as in (7), there results -xL cubic yards in the whole cutting Whence the Rule in Problem VIII. (12) Cor. 1. The content of a cutting, having only two given sectional areas, is (sx) L cubic yards (13) Cor. 2. Formula (11), (12), and (13) will evidently hold, if, instead of the straight lines CD, cd, being the surface edges of the cross-sections. CDN, cdn, of the cutting, these lines be the chords of similar curves forming the surface edges of the cross-sections, and the similarly situated points in the two curves be joined by right lines (which prolonged will all meet in the vertex V), thus forming the surface of the cutting, a necessary condition in taking crosssections; or the small inequalities of the earth's surface between the cross-sections must be balanced, as nearly as can be judged by the eye, so that the surface may fulfil this condition, in order that all attainable mathematical accuracy may be arrived at in finding the contents. THE ERRORS OF OTHER METHODS OF FINDING THE CONTENTS OF RAILWAY CUTTINGS. Mr Bashforth, in his investigations for finding the contents of railway cuttings, where the surface of the ground is level, or assumed to be so, includes the prism ABNnba, and afterwards deducts it which is mathematically accurate. But in finding the contents from sectional areas, where the surface of the ground is laterally sloping, or uneven, he altogether leaves out the above-named prism; for he says (Art. 15, p. 11 of his work), "The only way to proceed in such a case is find the areas of the cross-sections in square feet, take out of Barlow's Tables the square root of each, and treat these square roots precisely as if they had been measured heights, excepting that there will be nothing to be deducted for the prism, and no multiplication for the slopes, these having been already accounted for in finding the areas." Let A' and B' be the sectional areas CDBA, cdba respectively, as used by Mr B., and Nn = length of the cutting (fig. page 452), then according to his method, the content of the cutting is = which is erroneous in every case, except where the areas represented by A' and B' are similar and equal. For when the equal areas ABN, abn are omitted, the areas represented by A' and B' are dissimilar, and.. the proportion √A': √B′ :: NV: nv, on which Mr B.'s rule is founded, does not hold, it being true only in respect to the areas CDN, cdn, because the solids VCDN, vcdn are similar. Moreever, if the plane VCD revolve on the side CD so as to bring the lines cd, ab, to coincidence, or almost to coincidence, in the points b and d, the dissimilarity of the planes CDBA, cdba will become more strikingly evident, whereas with the addition of the AS ABN, abn, they still remain similar. = Let AABN abn a, then A'+a and B'+a are the sectional areas corresponding to A and B in formula (11) of my investigation, whence by that formula the true content, including the prisn ABNnba, is from which subtract the content according to Mr B. (14), and there |