152 
MISCELLANEOUS TABLES, ETC. 
The true content of the frustrum of 
the pyramid, and that of the ungula, 
may be found otherwise, thus : — 
Let ABCD and ABHI represent tlie end 
cross sections through the cutting. Let the 
angle DAB = angle CBA. Demit DG, IK, per- 
pendiculars upon EC, EH, respectively ; produce 
CB, DA, to intersect in E ; and draw EF perpen- 
dicular to AB, bisecting AB in F. Let the per- 
pendicular distance between the parallel planes 
DABC, IABH=L; and V" = volume of the whole 
frustrum of the pyramid limited by the same 
planes, viz. DEC, lEH ; then 
DG 
6 
\ IK 
EHrT^^ (7) 
The angle DAB = angle CBA being the same, as also the width of the road AB, the 
area of the triangular prism ABE is the same throughout the whole length of the cutting, 
and may therefore be deducted at one dimension ; namely, for the distance between the 
points in which D and I coincide with the eitremity of the road, as at A : moreover BE is 
constant, therefore by setting oif BE from zero on the scale, or by adding the same, 
the dimensions may be taken without intersecting CB, DA, in E. 
W W I ' 
If AB=W; then FE = --; BE =— n^ + l 
2n 2« ^^ 
Example. — Let the dimensions in feet be as follow, viz. : 
EH2 80x 80 'eC2 100x100 
64; DG = 60; — = — = 125; IK = 48; 
/ EH2 \ 
V" = (EC-fEH + — ) 
EC2 
^ X L. 
.(6) 
EC +EH + - 
EC = 100; EH = 
then we have 
EC 
From (6) 
100 
EH 
80 
60 
V"=(100 + 80+ 64)x — xL=2440xL. 
6 
48 
From (7) V" =(100 + 80 + 125) x — x L=2440 x L. 
the same in both cases, as it ought to be. 
If AB = 33 and n = l-5; then FE = 
= 11 ; and the area of the prism ABE 
2x1-5 
= 16-5 X 11 = 181-5. 
Therefore the true content of the solid, limited by the parallel planes DABC, lABH, is 
(2440 — 181-5) X L = 2258-5 x L. 
iVoie.— Equations 1, 2, 3, 4, 5 (p. 150 and 151) evideuily give the same result as tl> 
following more general equation : viz. — 
Let the middle area represented by cabd in Fig. 2 or DABC in Fig. 5=M; and Ir 
V = volume, as before described ; then 
n.zm ^ 
(8) 
V= M + - 
Printed by E. C. Osborne, 29, Bennett's Hill, Birmingham. 
