APPLICATION AND DEVELOPMENT OF PRISMOIDAL FORMULA. 135 



area included within the heavy lines, 



A-HOD-Q (16). 



Thus the volume of a solid, in which the parallel terminal planes 

 are identical except as regards the position of the line FG, and in 

 which therefore the FG surface only is warped, is 



V=ilr fad +h(q + q 1 )C a C l +q 1 Cl-i?£-} (17)' 



For practical purposes q may be taken from tables of double 

 entry constructed with r and s as arguments. The formula for 

 the volume of m contiguous solids will be of the same type as 

 (13) § 6. If^be regarded as unity, and two longitudinal sections 

 be taken as forming one prismoid, (17) becomes simpler, and in 

 general sufficiently accurate for mere estimates of volume from 

 profile. 



8. Solids of pentagonal section with two warped surfaces. — 

 Reverting to (16) in the preceding section, it is immediately 

 evident that the volume of m longitudinally contiguous solids, 

 with two warped surfaces in each, viz., the FE and EG surfaces, 

 is expressed by 



r.=A! jC (2/) + A) + "- + C k (iV 1 + 4i> t + A t+1 )+... 



... + C m (D m ^ + 2D ia )-3m^\ (18) 



in which k has all values from 1 to m — 1. The quantity Imw 2 \\r 

 is the volume of the extended prism on ABD and lying under 

 the whole of the m sections. 2 The algorithm in practical compu- 

 tations is simple and convenient. 3 



1 In earthwork such solids present themselves in what is called ' two- 

 level ground/ that is ground where the natural surface on a 'cross-section* 

 may be regarded as of uniform slope on each side of the 'centre-line/ 



2 If the base of the solid ABBiAi is a parallelogram but not rectan- 

 gular, r is not the measure of the slope of the side planes with reference 

 to the base but only of their line of intersection with the parallel end 

 planes A B G E F. 



3 When FE and EG are not in the same straight line, the section is 

 known as a ' three-level section.' The data necessary for computation 

 are the horizontal distances of FG=D and the vertical heights DE= C. 



