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The area of B is thus found to be 104.8 sq. feet, and of B',
Now, if a section of this prismoid be taken midway between the two ends, each of its several dimensions must be an arithmetical mean of the corresponding measurements of the end sections. Thus, the centre cutting is found to be 5 ft.; the dis
tance from the angle to slope stake, on the left, 9.9 ft., (10.8+9) the cutting on the extreme left is 6.6 ft., (7.2+6) &c.
65. Two other methods of computation are of frequent use among engineers. They are less laborious than the prismoidal rule, but they are also less accurate.
The first is known as the "Arithmetical Average" method. The rule is:
Multiply half the sum of the end areas by the length of the prismoid.
Applied to the example just solved, we should have,
vol. 100 X
= 11040 cu. ft.
66. The other method is called the "Mean Average" method. The rule is expressed thus:
Add together the two end areas and their geometrical mean proportional. Multiply this sum by one-third of the length of the prismoid.
This method, applied to the example, gives for the volume:
vol. = 100 (104.8 + 116 +√/104.8 × 116)
=11035 cubic feet.
When the difference between the end areas is considerable, the mean average method gives better results than the method of arithmetical average.
The last mentioned rule has been largely employed by the engineers of the public works of the State of New York.
Tables based upon the prismoidal formula, or modifications of it, are much used by engineers in earthwork computations.
67. In applying the prismoidal formula to an example in which one end section has more given dimensions than the other, the calculator is frequently in doubt how he shall average these dimensions to obtain the middle section. As a rule, each cutting of the most irregular section should be averaged with the cutting nearest opposite to it, in the other section.
We will illustrate this by a final example of earthwork: representing the sections by the field-notes only.
The half-width of the road, for which A is given in the notes, is to be considered as 8 feet. The length of the prismoid is expressed by the difference of the given distances, or 60 feet.
The dimensions of the middle section are found as follows:
The centre cut is half the sum of the given similar dimen15.8+10.4 sions, = 13.1 feet. 2
11.5 On the right, the average of the angle cuttings gives ; A for the next measurement, both cut and distance must be averaged; it is,
The last term in the upper section must be averaged with the last in the lower, thus:
× (8+ 8.4)= 8.2
On the left, the measurement of the upper section, must
be averaged with the centre cutting of the lower, being nearest opposite to that point. We have,
At the angle, in like manner, we have ; and finally at
The area of the upper section, after subtracting the tri
angles, as before, is 543.47 sq. feet.
The area of the lower end section is 325.44.
The middle section contains 429.8 sq. feet.
The volume of the prismoid is
× 60(543.47 +325.44 + 4 × 429.8)
Definitions and general Notions.
68. MINING SURVEYING comprises all the operations necessary to determine the relative positions of the parts of a mine with respect to each other, and also, with respect to the surface of the earth.
69. The general principles involved in this branch of surveying are the same as those used in surface surveying, but, from the nature of the case, certain modifications are required, both in the instruments employed and in the manner of using
stands instead of flag-rods; station points, if temporary, are marked by cross lines chipped in the rock, or sometimes by simple chalk lines, and, if permanent, by iron pegs driven into holes drilled for the purpose. Lines are measured along the slope, instead of on the horizontal, the chain men being guided by lamps instead of rods. Angles are measured by instruments specially devised for underground use; the compass, when used, is generally of a widely different form from the ordinary surveyor's compass; the theodolite, which is the principal angular instrument employed, differs from the ordinary theodolite in having a diagonal eye-piece, to permit observations to be made when the telescope is directed vertically upward, and also an arrangement for illuminating the cross hairs. These modifications will be more fully described in a subsequent article.
70. TRAVERSING is the operation of running a zig-zag line, from one point to another. The elements of the traverse are straight lines, determined by their lengths and by their inclinations to certain fixed planes. In mining surveying, three such planes are used; the first, is either a meridian plane through the origin of the traverse, or a vertical plane through the first course; the second, is a horizontal plane through the origin; and the third, is a vertical plane through the origin, and perpendicular to the other two.
71. WORKING, or REDUCING THE TRAVERSE, is the operation of finding the length and direction of a single line, equivalent to the zig-zag, that is, starting from, and terminating at, the same points. Such a line is called the resultant of the traverse.
The zig-zag line is run along the subterranean openings of a mine. For the sake of uniformity, such openings, when vertical, will be called shafts, and when not vertical, galleries.