To Railway Earthworks, 83 



be generated as follows : — ^Let a line in the position C G slide 

 along the lines c d, a n, in such a manner that it may pass 

 over respective distances on each line continually propor- 

 tional to C G h; then, when this moving line arrives at D 

 it arrives simultaneously at H. This moving Hne would 

 generate a contorted surface that might be, for several rea- 

 sons, assumed to be equivalent to that of the railway cutting 

 r> c a H. 



The volume of a sohd founded on one side by a contorted 

 surface generated in the manner just described can be mathe- 

 matically determined. Thus, let A B c r> E r g h (Fig. VL) 

 represent a solid whose base, A b c 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 ABC 

 5i ... area of triangle A C D 



... B F 



K C G 



A3 ... r> H 



K • • ■ A E 



It can be demonstrated that volume of solid 



ABCDEEGH = ^ V ' ^ 2 ^ 2^.-1 4^ "T 1 U 4 2 ^ j- J 



3 



Reverting again to the prisinoidal portion of cutting 

 bounded above by the contorted surface assumed to be ge- 

 nerated as described, 



Let b denote the base A B 

 d the length B P 



I ... ... D R 



T ... H O 



7 ... •.• C Q , 



/ ... ... G P 



m to I ... the slopes 



V ... the required volume. 



F^KQPODCGH — ADKEHO — BEQEGP 



of which 

 or, 



~\l m I m I I m J 



J c qfg p^d ( , ^ , , \ 



