GRAVITATIONAL METHODS 



355 



Also 



and 



U.,= 



U,.= 



3Gnivi 



ZGmzx 



(113) 



A line element is equivalent to a thin cylinder of uniform cross section 

 and density (Figure 206). The cross section of the line element PN will 

 be designated by 8 and its density by a. is the center of the balance. 



The equations for the Eotvos effects in this case are :t 



3v^ 9-v 



f 



/ 







= 3G"c 



/c 



/o 



-3(;(78aj -^ = GaaS l^^^jq^-^^ - ^ J 

 / 



!> (114) 



'dx'ds 

 dyds 



— 3G'ctS& I — ip — Gct&S ,,o , — ^TTi 5" 



1 ,.0 L(^'' + a)" r J 



/o 



Utilizing the above formulas for point and line elements, it is possible 

 to derive the Eotvos gravity effects: (a) for structures bounded by plane 

 surfaces, e.g., infinite layers of finite 

 rectangular cross section, semi-infinite 

 layer with sloping edge, etc., and (b) 

 for various irregular structures which 

 can be represented, approximately, as 

 the sum of several regular bodies. 



The more usable formulas for the 

 gradient and the differential curvature 

 anomalies produced by different types 

 of bodies in different orientations are 

 given below.* In these formulas, G is the gravitational constant and a 



t E. Lancaster Jones, "Computation of Eotvos Gravity Effects," A.I.M.E. Geophysical Pros- 

 pecting, 1929, pp. 506-509. 



* The summary of formulas given here is taken from D. C. Barton, "Calculations 

 in the Interpretation of Observations with the Eotvos Torsion Balance," A.I.M.E. 

 Geophysical Prospecting, 1929, pp. 481-486. The derivations of the formulas are given 

 by Lancaster Jones in the article cited above, pp. 517-529. 



Fig. 206. — Coordinates of attracting line ele- 

 ment NP referred to center of balance 0. 



