254 



GRAVITATIONAL METHODS 



[Chap. 7 



[7-39c] and [7-39/]) and horizontal gravity components. Then the gradi- 

 ents and curvatures are 

 (a) for three-dimensional bodies (v = volume): 



U,^ = 3A;5 fjf ^ dy = 3B jfj ^ dxdydz 



V V 



Uy^ = 3fc5 fjf ^dv = Sk8 fff ^ dxdydz 



V V 



U^ = 3k8 fff ^-^ dv = 3fc5 fff ^ ~ ^ dxdydz 



} (7-91a) 



2U^ = Sk8 fff ^dv = SkS fff ^ dxdydz 



V V 



(h) for two-dimensional bodies (S = surface) : 



U,', = 4k8 ff-^dS = ik8 ff^dxdz 



U.'^ = 



' y'z 



2U,',' = 



■ x'y 



(7-916) 



-U'^ = 2k8 ] j ^-—^ dS = 2U 



II 



x — z 



r* 



dxdz 



or in j)olar coordinates: 



dS = 2k8 



// 



sin 2(p 



drd<p 



-K = 2kff'^dS = 2k8ff'^drd^. 



\ (7-91c) 



2. In analytical methods of interpretation, the above equations are used 

 for calculating the anomalies of bodies of simple geometric shape. To 

 illustrate the development of these formulas, it is useful to begin with the 

 simpler forms (point element, line element, spherical body, cylindrical 

 body), although it is, of course, realized that geologic bodies occurring in 

 nature never have such shapes and rarely approach them. 



The, gradients and curvatures of a point element are given by eqs. (7-66) 



