354 EXPLORATION GEOPHYSICS 



between two points at which the gradient is one-half the maximum gra- 

 dient is equal to twice the depth to the top of the body; (2) for anomalies 

 of the type B and C, the horizontal distance between the two points hav- 

 ing the greatest absolute values of the gradient is equal to twice the depth 

 to the top of the geologic body. For anomalies of the type B, rule (1) is 

 used either with the right hand portion or the left hand portion of the 

 anomaly. 



Formulas for Computing Effects of Simple Forms. — The deriva- 

 tions of the analytical expression for anomalies associated with geologic 

 bodies of simple geometric form utilize the same potential theory used in 

 the solution of magnetostatic and electrostatic problems. The charac- 

 teristic feature of the potential theory as employed in gravitational ex- 

 ploration work is that three types of quantities are evaluated : ( 1 ) the 

 first derivative of the potential, i.e., the component of gravity in a par- 

 ticular direction; (2) second derivatives of the form Uxx, i.e., the rate of 

 change of the x component in the x direction; (3) second derivatives of 

 the form Uyx, i.e., the rate of change of the x component in the y direction. 



The total gravitational effect of any mass is the algebraic sum, or in- 

 tegral, of the individual effects of its constituent elements. The elements 

 usually employed are particles of mass, i.e., point elements, or thin cylin- 

 drical masses, i.e., line elements. f 



The gravitational potential due to a particle of mass m (point element) 

 at a point P located at a distance r from m follows directly from the defini- 

 tion of the potential. That is, 



U=CG'-^.dr=^ (111) 



;r r- r 



where G is the gravitational constant (6.68* 10~^ c.g.s. units.) The effect 

 of this potential is to produce a differential curvature and gravity gradient 

 at the torsion balance which may be computed as follows : 



and 



Similarly 



and 



-dU ^ X 



— - = — Gin -A 

 ^x r^ 



r-T= U^j; = 6Gm -ir — G -r 

 ^x^ r"" r"^ 



Uyy = 3Gni ^ - G ^ 



Uyy - Uxx — 3Gm (y- — x^ ) 



-1-2 



(112) 



t See footnote (***) page 333. 



