236 GRAVITATIONAL METHODS [Chap. 7 



of the diagrams in different azimuths are combined by vectorial addition 

 to a vector of the azimuth a for gradients and another vector of the azi- 

 muth 2a! for the curvatures. The justification for this procedure is seen 

 by reference to formula (7-86), since the azimuth terms may be written 



in the form 2 sin -r- , . v for gradients and 2 sin Aa , . „ ^ for curvatures. 

 2 (sma) (sm2Q:) 



The gradient and curvature vectors of the terrain may then be resolved 



into their x- and y-components to obtain the corrections for all four 



gradient and curvature components. 



(c) Planimeter and integraph methods. If planimeters or integraphs are 

 used, terrain effects are determined by outlining the terrain profile or 

 terrain contours with the stylus of one of these instruments rather than 

 by counting elements as in the graphical methods. Two procedures are 

 applicable: (1) use of ordinary planimeters with contour lines or terrain 

 profiles distorted in respect to horizontal azimuth or vertical angle and 

 distance scale in such a manner that the effect of the mass area is inde- 

 pendent of azimuth and distance ; (2) use of special integraphs with contour 

 lines or terrain profiles drawn to undistorted scale. 



The first procedure was proposed by Below and has been described 

 before in connection with the interpretation of gravity anomalies (page 155). 

 Its applications have been worked out: (a) for the entire surrounding 

 topography; (6) for small elevations at greater distance.^" The relations 

 that apply in the first case have been given in eq. 7-85. They contain 



integrals of the form / — for both gradients and curvature values which 

 P 



express the effect of distance, and of the form / , x for gradients 



and / , „ X for curvatures which express the effect of azimuth. 

 J (cos 2a) 



These integrals may be reduced to surface integrals of the form 



// 



R dR d^ by the substitution R = \/2 loge p and $ = sin a, cos a, 



sin 2a:/2 and cos 2q;/2. These substitutions result in a diagram which is 

 distorted in respect to both scale and azimuth (see Figs. 7-85 and 7-86). 

 On such diagrams lines of equal elevation angle ^ are plotted and the area 

 S between successive lines is determined with a planimeter so that for the 

 gradients 



(sm a) p 36 J J 36 



"»K. Jung, Zeit. Geophys., 6(2), 114 (1930). 



