1S40.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



335 



so that the solid figure, comprised between these two end areas is 

 composed of a middle part or core which is the frustrum of a wedge, 

 and of two side pieces, which together form the frustrum of a pyramid. 



It is evident that the content of the core is simply lb. — — and by 



file prismoidid formula the content of the side pieces is also readily 

 D' r + d-r + 4 (D- r 4-c?- r + 2 r D rf ) 



found = I. 



6 



which reduced becomes = l. „ (D^ -\- d'-[-V>d). 



This expression appears to be so simple as scarcely to require any 

 table by way of aid in the calculation. It is obvious, however, that 

 the only table which can at all be necessary in using this method of 

 computing sections is one of squares, such as may be found in the 

 Engineer's Pocket Book, and many other works of reference. 



The following example will show the manner in which the formulae 

 should be used. 



Fig. 4. 



Cutting 



Embankment. 



Let the above be a part of the section to be computed then the cal- 

 culation will be as under. 



Excavation, No. 1. 



Very little explanation will be required to render the preceding 

 calculation understood. It is evident that the multiplication by the 



22 11 

 fraction or — is necessiry (in consequence of the lengths being 



JX ^ 'J 



in chains, and the depths in feet,) to reduce the first results into cube 

 yards. 



Ar.d it will also be clear that as the numbers in the column headed 



7* 



"sides," are determined without multiplication by the fraction-, that 



o 



is for a slope of 3 to 1, the further division by 6 is necessary to reduce 

 them to aslope of i to 1. The quantities may be determined with 

 equal readiness for any slope, integral or fractional, by simply raulti- 



plying the numbers found as above, by the fraction -, where r is the 



rate of slope required. 



It will be found extremely convenient for engineers and others con- 

 sulting the sections of new lines of railways, or comparing together 

 two or more sections of the same line, to know the quantities for dif- 

 ferent slopes, and these may be readily exhibited by simple addition, 

 thus : 



(For a base of 30 ftet.) 



It may be useful now to glance at certain erroneous methods of cal- 

 culating earthwork which were at one time exceedingly prevalent. 

 These methods have often been the occasion of serious loss and disap- 

 pointment to contractors and others, by some of whom they are not 

 abandoned even at the present day. It will be shown that calculations 

 of earthwork made according to the common erroneous rules maybe 

 readily altered so as (o give a correct result. Hence the investigation 

 of these errors will furnish us with new and distinct rules for finding 

 the contents of earthwork sections, each rule being correct and giving 

 the same result as the formula already derived. 



I. Let it be required to determine the error occasioned by taking 

 the mean of the two enrf areas, and multiplying this mean by the length 

 for the solid contents of a prismoid. This method may be expressed 



thus -.—l 



D 6 -I- D= r + db + d'r _ 



= lb. ^f^+ I ^(D^ ■+ i-') from 



which it appears that the difference between this and the correct ex- 



T 



pression exists only in the side pieces, and is equal to - (D- + d^) — 



liiy^-^d^ + iid)=:lB^- + ^^d'-lB^-^d^ + l^d=y:B^ 



. irrf^_?ZD'+^''i^+|rDrf=^(D3 -{-^-2Drf). Excess 



tj i> O" D 



above the correct area. Now this excess is equal to one-sixth the 

 square of the difference of the depths multiplied by the ratio of the 

 slopes. 



H. The other erroneous method is more commonly in practice than 

 the preceding, and gives a result nearer to the correct one, but the 

 difference here is one of defect, not excess, that is on the wrong side 

 for the contractor. According to this method, an area is calculated 

 for the arithmetical mean of the depths, and this area is used as the 

 one which being multiplied by the length, is to give the content of 

 the figure. 



