EARTHWORK 



209 



minus sign. One-half of the algebraic sum of these products 

 will be the desired area. This is evident, since, proceeding 

 according to the directions, the positive products are 



%by3, xsyz, xzyi, x\d, dxi', yi'xz, and yz'^b 

 and the negative products are 



ysxzf yzsfi, and xi'yz' 



One-half of the algebraic sum of these is identical with the 

 second member of the preceding formula. 



NOTE. The method just described for determining areas of 

 irregular sections is general and may also be used for three- 

 level sections. 



Following is an illustrative example showing the applica- 

 tion of the preceding method of determining the areas of 

 irregular sections. The field notes are given in the accompany- 

 ing table. The station numbers in column 1 run from the 



FIELD NOTES 



Roadbed 24 feet wide in cut. Slope 1.5 : 1. 



bottom of the page upwards, so that when one stands on the 

 line of the road looking forwards, the slope-stake fractions, 

 which give for each point the height and distance from the 

 center, will have on the notebook the same relative position 

 as they have on the ground. These figures for the left-hand 

 side are always given at the extreme left of the space in column 



