Column 8 contains the product of column 1 multiplied by column 7. Column 9 states the ratio to the vertical, generally taken in banks as 1 to 1 and in cuttings as 1 to I, except rock; but the actual ratio depends upon the natural slope of the earths found upon the line of section. Column 10 gives the content of the slopes found by multiplying the figures in column 8 by the figures in column 9, and the last right hand column gives the total content in solid yards found by adding together the figures found in columns 6 and 10. Should the heights in column 2 exceed those in the table, the contents may be found by entering with one-half, one-third, or one-fourth, &c., of the measured heights. Thus, when they fall between 50 and 100 divide them by 2, and entering the table with the quotients, multiply the first content in the table by 4 and the second by 2. When they fall between 100 and 150 divide them by 3 before the table is entered, and multiply the first content in that case by 9 and the second by 3. The tables are calculated from the following formulæ :— Let x the number of feet which shows the greater height at the bottom of the table, And y the number of feet which shows the less height in the left hand vertical column of the table = = Then first the content (black figures) column 7 22 [(x + y) 2 xy], which is based upon the ordinary formula for the volume of a truncated pyramid with parallel bases. = 9 = Second content (red figures) column 3 11 (x + y) if x 8 and y 3 2 [(8 + 3) -24] 2297 = 79 (black figures) and (83) = 121 = 134 (red figures). 2 In column 5 a width of 30 feet has been assumed in the case of a cutting. In the case of a bank a width of 27 feet is generally assumed. When representing, upon the plan, the widths required for the formation of the railway, it is usual to include an addition of 10 feet to the transverse width at the toe of the bank (formation level) in the case of both banks and cuttings. This gives the boundary of the area required for the purpose of valuation = = Land Surveying and Levelling, pages 317, 318. CUTTINCS BLACK FICURES 7 79 156 105 123 221 244 221 84 3 for the estimate of expense. In some cases a recess of 7 feet upon either side of the centre line is considered necessary, giving in the case of a cutting half of 30 feet plus 7 feet recess, or 22 feet upon each side of the centre line constant width, plus width for slope derived from heights furnished by the section. In calculating earthwork quantities the following table of natural slopes of earths, giving a table of angles made with horizontal line, is useful for reference :— Gravel, average Vegetable Earth Shingle Rubble.... Clay, well drained ·40° ·38° 22° .28° 50° A form of level book sometimes employed in railway work is also shown under the diagram illustrating the section, in which book separate columns are provided for the heights of formation levels, embankments, and depths of excavations. 39° 45° .45° .. 16° 16 ft. 18 ft. ...20 ft. 28 ft. 30 ft. 33 ft. 38 ft. With the use of metrical earthwork tables, the lengths in column 1 are expressed in chains of 20 metres and the heights in column 2 are expressed in metres. The width of centre in column 5 is also to be stated in metres, and the total content will be given in cubic metres. It is useful to note that 30 centimetres equal very nearly one foot (305 millimetres 1200785 inches) and that one cubic metre contains 13069 cubic yards. Tables for finding the contents of earthworks, when the metre is the unit of measure employed, have been compiled by Messrs. = |