log (x z + z 2 ) at the points 1, 2, • • 8, 9 and from this sum subtracting the 

 values of log (x 2 + z 2 ) at the points 10, 11, ■ ■ 17, 18. The result multiplied by 

 kp(x 2 — x t ) would then be the value of A z at the point (x — 0, z = 0) . 



What is needed is a quick method of finding log (x 2 + z 2 ) at each of the 

 points 1, 2, • • 17, 18. One way of doing this is to draw a contour map of 

 log (x 2 + z 2 ) and superpose it on the cross section of the body. The principle 

 is the same as that used in showing elevations at points on a map by drawing 

 contour lines at suitable intervals through points of equal height above sea 

 level. The contour lines of \og(x 2 + z 2 ) are circles with centers at the origin. 

 Table 27-1 gives the radii of the circles corresponding to values of log r 2 = 

 log (x 2 + z 2 ) at intervals of 1/10 from log r 2 = to log r 2 = 9.20. 



The contours should be inked on cross section paper with certain contours 

 identified with the value of log r 2 , say every other one, similar to the way 

 contours are identified on topographic maps. The outline of the body may 

 then be drawn lightly with pencil on the contour map itself or drawn on a 

 piece of tracing paper and laid over the contour map. The lines dividing the 

 area into rectangles need not actually be drawn since the vertical lines ruled 

 on the cross section paper will serve the purpose just as well. There is an 

 advantage in drawing the outline of the figure on a separate sheet of paper 

 when there are several points to be computed along a profile across the body. 

 In this case it is much easier to move the outline of the figure with respect 

 to the origin on the contour map than it is to sketch the figure on the contour 

 map for each new point to be computed. 



To illustrate the method, we will use it to compute the value of A z at two 

 points above a horizontal cylinder of circular cross section. Suppose the 

 cylinder to have a radius of 100 meters, its center to be at a depth of 300 

 meters below the surface, and its density to be 1 gm. per cm. 3 higher than that 

 of the surrounding materials. Let one of the field points be vertically over the 

 center of the cylinder and the other 200 meters away from the projection of 

 the center of the cylinder on the surface. 



Figure 27-5 is a contour map of \og(x 2 + z 2 ) with the outline of the 

 cylinder drawn to proper scale and position. In this case the unit used in 

 drawing the contour map has been set equal to 100 meters. This choice will 

 place the center of the cylinder in the two positions 3 units below the x axis 

 or surface of the ground and will give the radius of the cylinder the value 

 1 unit. Any other choice for the value of the unit that would permit the out- 

 line of the cylinder to be drawn on the contour map would work just as well. 

 For example, if the unit had been set equal to 200 meters, the center of the 

 cylinder would have been li/9 units below the surface and its radius equal 1/2 

 unit. The width of the rectangles into which the body is divided has been 

 chosen to be 14 unit, which gives x 2 — x t the value 25 meters or 2500 cm. 



591 



