Chap. 7] GRAVITATIONAL METHODS 265 



Since Cmax. = ^o and xq ^^^ = To , the depth and the width, according 

 to Jung/" are 



and 



d = To cos 



$0 



vo = 2ro sin -^r- . 



> (7-94e) 



It is possible to approximate the outhne of two-dimensional masses of 

 irregular shape by using a polygon with straight sides and applying formula 

 (7-93c) repeatedly, as proposed by Matuyama and Higasinaka/^^ How- 

 ever, it is easier in such cases to use the graphical methods described in 

 the following paragraphs. 



3. Graphical interpretation methods make use of diagrams containing 

 mass elements in section or plan view in such an arrangement that their 

 effect on a station (0-point) is identical irrespective of distance or azimuth 

 (see also pages 153 and 227). For three-dimensional subsurface features of 

 moderate relief the diagrams calculated by Numerov, discussed on page 

 228 and illustrated in Figs. 7-78a and 7-786, are appHed in connection 

 with subsurface contour maps, and strips bounded by successive contours 

 are evaluated. Then the height of the instrument above ground f cor- 

 responds to the depth, D, of the effective geologic feature beneath the in- 

 strument. The mean "elevation" of a contour strip with respect to this 

 point is h = D — d where c? = (di + c?2)/2, or the mean of the depth values 

 of two contours. 8 is the density contrast. If iic is the number of elements 

 comprised by a contour strip in the curvature diagram and n„ the cor- 

 responding number in the gradient diagram, the effect of one strip, (sub- 

 script st), is 



(U.;)st = ^8.n„.(D' - d') 1 



(7-95a) 

 {-U^)st = S.nc-{D - d). J 



For determinations of Uyz and 2Uxy the diagrams are rotated 90° or 45°, 

 respectively, as in the terrain applications. For steep slopes of subsurface 

 features the accuracy of horizontal (contour line) diagrams is insufficient, 

 and vertical diagrams such as shown in Figs. 7-82 and 7-83 must be 

 applied with geologic sections through the geologic body in a number of 

 azimuths. Although the diagrams under discussion have been calculated 



1" Ibid. 



1" Japan. J. Astron. and Geophys., 7, 47-81 (1930). 



