146 



GRAVITATIONAL METHODS 



[Chap. 7 



in polar coordinates. Therefore, the anomaly in gravity is 



= 2k8 J f \dxdz 

 = 2kb I I sin <pdrd<p 



or 



in rectilinear and 



> (7-39/) 



in polar coordinates. 



Calculations of torsion balance anomalies are likewise based upon these 

 equations by forming the second partial derivatives with respect to the 

 horizontal directions (in the case of the curvature values) and with respect 

 to horizontal and vertical directions (in the case of the gradients) (see 

 page 254). 



P 2. "Analytical" methods involve calcula- 



tions of the anomalies produced by sub- 

 surface bodies and a substitution of their 

 assumed dimensions and depths in the 

 formulas derived below. For three-di- 

 mensional bodies, the gravity anomalies 

 may frequently be estimated with suffi- 

 cient accuracy by assuming them to be 

 spherical, although it is, of course, under- 

 stood that geologic bodies seldom have 

 such simple geometric shapes. The effect 

 of a sphere follows directly from eq. (7- 

 39c), thus: 



kMz 4 



___L_J._.. 



= — - = - irSkz 



©•■ 



(7-40a) 



Fig. 7-41. Cylinder. 



with M as the mass and R the radius of 

 the sphere. 



A number of useful relations may be 

 obtained by calculating the gravity anom- 

 aly, due to a vertical cylinder, for a point 

 in its axis (see Fig. 7-41).^ Let the cylinder have the length I and the 

 radius p and let d be the distance between its upper surface and the point 

 P. Consider a mass element in a thin disk of the thickness Al at a dis- 



•' F. R. Helmert, Theorieen der hoeheren Geodaesie, Vol. 2, p. 141 (Leipzig, 1884). 



