Chap. 7J 



GRAVITATIONAL METHODS 



145 



Therefore, the anomaly in gravity is 



(7-39c) 



= k8 I I I —dx dy dz 



V 



in rectilinear coordinates, \vith r = \/x^ + 2/^ + z^, and 

 Ag = kd I I ~ dr cos <p dcp da 



in polar coordinates as above. 



The gravity attraction of three-dimensional bodies derives from the 

 "Newtonian" potential. This is not true for two-dimensional bodies, i.e. 

 bodies which are much longer in one direction than in the other two. This 

 direction of greatest extension is usually the strike; therefore, the attrac- 

 tion of such bodies is proportional to their section in a plane perpendicular 

 to the strike. The attraction between masses in a plane derives from the 

 "logarithmic" instead of the Newtonian potential. This follows also from 

 eq. (7-396) by carrying out the integration in the strike (^)-direction be- 

 tween the limits of + and — infinity. Therefore, while the Newtonian 

 potential of a point mass is proportional 1/r, the logarithmic potential 

 of a line mass is proportional 2 log« 1/r. The attraction in Nev/ton's law 

 is proportional 1/r , but the attraction of two-dimensional bodies is pro- 

 portional 1/r: 



F = 



kmm' 



(7-39d) 



In the potential and gravity expressions a surface integral takes the place 

 of the volume integral : 



U = 2k8 f flog,^ 



- dS or 



2k8fJ\og' 



dxdz 



In rectilinear, and 



U = 2k8 f f \oge--rdrd<p 



(7-39e) 



** This is also true for electrical and magnetic attractions which follow Coulomb's 

 law. The attraction of a magnetic "line" is proportional 1/r while the attraction 

 of a magnetic pole is proportional 1/r*. 



