MISCELLANEOUS TABLES, ETC. 
151 
Fig. 5, let DABC reprc- 
From the values of the middle ordinate RQ, in col. 2, n = i-5, and rG=40; the middle 
widths EH in Fi^. 1 are, 55-0; 82-3; 103-9; 119-8 ; respectively ; vyhich being substituted 
in (2) and (3), the same result is obtained as by equation (1.) 
If the values of ZQ, be small or nearly equal, so that by a few trials a mean value 
of ZQ? may be found sufficiently near the true average value, then, the slope n to 1, the width 
of the road PG and the length L being measured off or given, the only dimension requi- 
site to be taken, in equation (1), is the middle ordinate RQ,, and in (2) and (3), the 
middle width EH ; the squares of which in (1) and (2), may be written down from a 
table of squares, as in the preceding example. 
If the railway pass through the side of a hill, then 
sent a vertical middle section through EH in 
Fig. 1. Let the angle DAB = angle CBA, and 
let DA and CB produced, meet in S ; draw DL 
parallel to AB, cutting BC in E ; also draw SH 
perpendicular to AB, DE, bisecting the same in 
Oand N; let ra.SN=ND=NE; demit CF per- 
pendicular to DL, intersecting DL in F ; upon DF 
as a diameter describe the semicircle DMF, cutting 
SH in M; make N6=NM; from G draw GI at 
right angles to DL, intersecting BC in I ; from I 
draw IK parallel to LD, intersecting SD produced 
in K, and cutting SH in P; then the 
area of the quadrilateral KDEIK = area of 
the triangle DEC. 
From C, draw CR parallel to LD, intersecting 
SH in R, then a/RCI^NB— OB NM— AO 
0P= = 
n n 
and the true content = V', of the solid, of which the middle section is DABC and the 
length L, may be found by either of the following equations, viz. : — 
, f RCxNB— 0B2 n.ZQ? ) 
1 — + — pi- W 
v 
= |«.0Nx0R + (0N + 0R) 0B + — xL 
.(5) 
in which ZQ,, as before, is one-half the difference of the two end ordinates NP, ST in 
Fig. 3 ; ON, OR, in (5) and Fig. 5, being perpendiculars let fall from the points D and C, upon 
AB produced both ways. Moreover, RQ in Fig. 3, = OP in Fig 5, is the common intersec- 
tion of the two planes SNPT, KABI, Figs. 3 and 5 : also the end sections of the solid are 
each parallel to the plane DABC ; and the length L is limited by the distauce between the 
points in which D coincides with the extreme width of the road as at A, near each end of the 
cutting. 
£xaOT;>/e.— Let RC = 74; NE = 56; OB = 20; ZQ (Fi^-. 3) =6 ; w = r5 : then ON = 
NE— OB 56—20 RC— OB 74—20 
■=24; 0R= =— TT— = 36; and we have 
1-5 
From (4) V 
1-5 
74 X 56— 20 X 20 1-5 x 36 
1-5 
From(5) V'= |l-5 x 24 x 36 + 60 x 20 + 
the true content of the whole solid in either case. 
3 
1-5 X 
2514 X L. 
X L = 2514 X L. 
