} (7-73) 



218 GRAVITATIONAL METHODS [Chap. 7 



Coefficients for each ring resulting from the evaluation of these integrals 

 and the expansion of the coefficients c', b', e', and d' in terms of elevations 

 in eight and sixteen azimuths are given in the article referred to. 



As stated before, this procedure is permissible only for greater distances 

 (cartographic correction). In the application of terrain methods to 

 smaller distances, some variation of elevation between successive azimuths 

 and radii must be assumed in order to arrive at a mean elevation of a 

 sector. The simplest assumption has been mac^e by Eotvos. If the 

 sector is bounded by angles ai and a^ and by radii p' and p" ; if the elevations 

 hi and /i2 have been determined on the radius p' in the azimuths ai and 02 ; 

 and if the elevations hi and /i2 have been determined on the radius p" in 

 the same azimuths, the mean elevation is assumed to be given by 



h = Ci -{- da -\- Capa + Cip, 

 so that the coefficients 



Ci(a2 — oci)(p" — p) = p"(a2hi — aihz) — p'ia^hi — ajli) 



C2(«2 - «i)(p" - P) = P"{h', - h[) - p'{h'; - h'l) 

 Zz{a2 - ai){p" - p') = hi - h'l - (hi — h'l) 



Ci{ai — a\){p" — p') = o-iQii — hi) — aiihi — hi). 



The effect of each ring is calculated from integrals similar to those 

 contained in eq. (7-72). However, Eotvos does not use the squares of the 

 elevations as in eqs. (7-71) to represent the azimuthal variations but breaks 

 off the power series of (7-686) after the first term, which makes it possible 

 to use the first powers of elevations. By considering only the first term in 

 eq. (7-686), eq. (7-69) therefore takes the following form for the gradients: 



where Q indicates a quadratic term Q = h'^/2. Because of this simplifica- 

 tion, the formulas are applicable to small elevations and gentle terrain only. 

 In the Eotvos formulas given below, the innermost ring of 1.5-m radius is 

 considered a plane with the inclination of ^ in the north and t in the east 

 directions. A quadratic term is included for the 5-m radius, which is 

 generally negligible for- the gentle terrain to which Eotvos' formula is 

 applied. The radii are 



Pi P2 P3 P4 Pb 



1.5 m 5 ra 20 m 50 m 100 m 



Density is assumed to be 1.8 and ^ is 100 cm. 



» Ibid. 



»• Op. cit., 358. 



