144 GRAVITATIONAL METHODS [Chap. 7 



(1) analytical methods, (2) graphical methods, and (3) integraphs. The 

 first method is equivalent to a calculation, by the formulas given on pages 

 146-153, of the attraction of a body of given dimensions and depth. In 

 the second method use is made of diagrams (graticules) described before 

 in connection with the calculation of terrain anomalies. The geo- 

 logic section is superimposed on these diagrams, and the anomaly is calcu- 

 lated by counting the number of elements included within the outline of the 

 body. In the third method the area of the section of a geologic body and 

 its effect is obtained mechanically by tracing the outline with integraphs 

 of special design. In all of these methods considerable simplification in 

 the calculation is possible by assuming that the geologic bodies are two 

 dimensional instead of three dimensional, that is, that they have virtually 

 an infinite extent in the strike direction. Regardless of whether analytical, 

 graphical, or mechanical-integration methods are applied, the fundamental 

 mathematics are the same in each method and differ only in such details 

 of solution as lend themselves best to the application of each method. 

 These relations will be discussed first before the analytical, graphical, and 

 mechanical-integration methods are described in detail. 



Fundamentally, the calculation of the gravitational effect of subsurface 

 masses rests on Newton's law. Since the force of attraction, F, between 

 two masses, m and m', is F — kmm' /r , the gravity anomaly or vertical 

 component of the force may be obtained by letting m' = 1 and Z = 

 Ag = F cos 7. Since cos 7 = z/r, the effect of a point mass on gravity is 



Ag = km--. (7-39a) 



For the calculation of the anomalies of all other types of masses it is more 

 convenient to follow a different line of approach. It is based on the gravity 

 potential of a mass element which is given by the expression U = k- dm/r. 

 Since dm = 8-dv, where 8 is density and dv is element of volume, the 



potential is U = k8 i I I (1/r) dv or 



V 



U = k8 I I I - dxdydz 



V 



in rectilinear coordinates and f (7-3%) 



U = k8 j I I rdr cos <p d(p da 



V 



in polar spherical coordinates with cp as latitude and a ss longitude, as- 

 suming the density to be constant throughout the volume. 



