220 



GRAVITATIONAL METHODS 



[Chap. 7 



and linear elevation term in the power series of eq. (7-686); the second uses 

 the squares of the elevations as in eq. (7-69). In both methods the varia- 

 tion of elevation with azimuth is represented by Fourier series. This 

 makes the method more flexible and it is possible to apply the formulas 

 to any desired number of azimuths. In any given azimuth, the variation 

 of elevation is assumed to be linear between rings in the first method and 

 quadratic in the second. In both methods the variation is assumed to be 

 linear in the curvatures. 



Considering the linear variation between rings first, the following propor- 

 tion exists: 



hr 



Pn 



hp — hg 

 p — Pn 



. Pn+i 



Therefore the elevation is 



hp(pn+i ~ Pn) = ^nPn+1 ~ ^n+iPn "" p(^n ~ ^n+i). 



Substituting this in eq. (7-68a), and considering in the expansion of eq. 

 (7-686) only the linear terms, the terrain effect of one ring between the 

 radii Pn+i and pn is 



3A;5f >" >"' 



t/„ = 



Pn-M 



'nL •'0 



COS ada(haPn+i — ^n+iPn) 



+ /i / COS adaihg — /in+i) 



^k8t r r^^ 



Uyt = 1 3 / sin ada{haPa+i — /in+ipJ 



Pn+i — Pn L •'0 



+ 7i j sin a da(ha - ha+i) 



Ui, = 7i / COS 2ada{haPa+i — /in+iPn) 



Pn+1 — PnL •'0 



— hi COS 2a da(ha — hg+i) 



1 1 I sin 2adaihaPa+i — /in+iPn) 



+ /z / sin ada{hn - K+i) . 



} (7-75a) 



2C/x„ = 



Sk5 



Pc+i ~ Pni 



The integral Iz is the same as the integral in the gradients of formula , 

 (7-72), and integral h is the same as the integral in the curvature expres- 



