To Railway Earthworks. 83 
be generated as follows :—Let a line in the position c & slide 
along the lines C D, G H, in such a manner that it may pass 
over respective distances on each line continually propor- 
tional to C D, @ H; then, when this moving line arrives at D 
it arrives simultaneously at H. This moving line would 
generate a contorted surface that might be, for several rea- 
sons, assumed to be equivalent to that of the railway cutting 
DCGH. 
The volume of a solid founded on one side by a contorted 
surface generated in the manner just described can be mathe- 
matically determined. Thus, let aBCDEFGH(Fig. VI.) 
represent a solid whose base, A BC D, is a trapezoid, whose 
sides are planes perpendicular to the base, and which is 
bounded above by the contorted surface E F G H; now, if 
s denote area of triangle A B C 
Ss, ... area of triangle Ac D 
See 
Be acta OR 
pa saat Ee 
| CPE AS 5: 
It can be demonstrated that volume of solid 
3 
Reverting again to the prismoidal portion of cutting 
bounded aboye by the contorted surface assumed to be ge- 
nerated as described, 
Let denote the base a B 
d .. the length BF 
1 Hae ets DR 
U ae oe HO 
? pA ce ace 
re soe in Gee 
mtol ... the slopes 
V ... the required volume. 
V=RQPODCGH—ADREHO—BEQFGP 
of which 
RQPODCGH=}RQ. ar Cure See ree) $tor. Cr Gaeta ee) 
or, 
zaropcan—%} (Geese ical 9 naam Casas) 
ADBEHO—4G nt im+ilm) 
boISIP—1 i mirm teem) 
