136 



G. H. KNIBBS. 



9. Solids of heptagonal section with four or six warped surfaces. 



Another case of practical 

 importance is that illustra- 

 ted by Fig. 4, shewing a 

 section of the solid: AB, 

 and the angles GAB, ABH 

 are constant for each ter- 

 minal polygon : this con- 

 stancy of the angles how- 

 ever is unessential, and the 

 surfaces standing on GA, 

 and on HB, may be also 

 warped. It is essential, however, that the points E and F be 

 vertically over A and B. Let as before AB = w, CD = c, AE = e, 

 BF = e', and the sum of the projections of GE + ED = <i, and of 

 DF + FH = d\ so that d and d' will be the total horizontal distance 

 between the ' centre line ' and the points where the sides of the 

 road meet the natural surface. The area of a section is evidentry 



A = \ (cw + de + d'e) (19) 



consequently the volume of m longitudinally contiguous solids 

 whose parallel faces are the rectangular distance I apart is 1 : — 



Fig. 4. 



By writing these in columns I. and IV. as hereunder, column II. is 



obtained by multiplying the first and last D by 2, and the intermediate 



ones by 4. Diagonal addition gives the numerical values required in 



column III.: the products (7 2(D) can then be formed or taken out of 



suitable tables. For earthwork computations these may be constructed 



so as to give, not the products of C and D_. but those products not only 



multiplied by -i 2 - I, but also reduced to any required unit, as for example 



cubic yards in English practice. See for example, Table XL, Theory 



and Practice of Surveying. — J. B. Johnson, 1887, pp. 672 - 681. 



I. II. III. IV. 



Do 2D 2D + Di X Co 



Di^> 4D X D +4D 1 +D 2 X Ci 



D 2 4D 2 Di+4D 2 +D 3 X C 2 



D m 2D m An-1+ 2 D m X C m 



1 The case considered is for 'five-level sections/ where the 'intermediate 

 heights ' e and e' are taken over the sides of the road-bed : this simplifies 

 the computation of the area, see (19). 



