A manual of the principles and practice of road-making: comprising the location, construction, and improvement of roads, (common, macadam, paved, plank, etc.) and rail-roads. By W. M. Gillespie ...

APPENDIX A. 381
same. The new end area is 567, and the true content becomes
42,300 cubic feet. But q = q'. Therefore, according to the principles established on page 378. the method of " Averaging areas"
should give the same result, and it does so. So too with the
method of "Middle areas." The method of "Equivalent mean
heights," however, still gives too little, because p x q' is not equal
to P' x q. On making the calculation (the equivalent heights being
9.97366 and 13.21475), we get a content = 41,600 cubic feet, or 700
cubic feet too little; and formula (7) gives the same result.
Example 4. In another warped surface solid, let one end area
have depths of 15 on the left and 5 on the right, and the other end
be 5 on the left and 15 on the right. Let the breadth of road bed
be 20 feet, and the side slopes 2 to 1. The true content will be
38,333 cubic feet. The " Averaging method" gives 35,000 cubic
feet; too little by formula (3'), because
5 x 5 + 15 x 15 > 15 x 5 + 15 x 5.
The " Middle area" method gives 40,000 cubic feet, an error in
excess of half the amount of the preceding deficiency. " Equivalent mean heights" give 35,000 cubic feet; not enough, because
P x q', or 20 x 20 (adding 20- 2 x 2 to the given depths) is not
equal to P' x q, or 20 x 20.
Example 5. Reverse one of these sections so that both may be 15
on the left, and both be 5 on the right. The surface is then a
plane, and the solid is a prism with a uniform section of 3500
square feet. For this solid all the methods give the same content;
and this is a final corroboration of our formulas. The " Averaging" method is now correct, because p = p', each being 15, or
because q = q', each being 5. The " Middle area" method is correct
for the same reason. The method of " Equivalent mean heights"
is now correct, because now P Q =- P' q.
The method of "Equivalent mean heights" which the preceding
investigation most particularly affects, seems to have been introduced by Telford, and has since been adopted without question
by most writers (the present one included), when perfect accuracy
was desired. The difficulty has been the want of any better
standard than itself with which to compare its results. But if the
positions which the writer endeavored to establish in the first part of
this paper be accepted as correct, this method should be at once and