216 



GRAVITATIONAL METHODS 



[Chap. 7 



Extending the limits of integration over the entire surrounding topog- 

 raphy, 



U^z = 3H 



{/.. = 3A;5 



Jo ♦'O •'0 



/77 



Jq ->o jo 

 Jo Jo Jo \t 



i Jo i f-3 -4- r^ - hV]w 



[p2 + (^ _ ,,)2]5/2 



'^ p" sin adadpdhii; — h) 

 ^0 ^1p^ + (f - hyf'^~~' 



" -'' ^ cos 2a da dp dh 



I r— 







> (7-68a) 



2f/x„ = 3k8 



"0 .0 [p^ + (r - /i)^]^/^ 



" "'' "" p' sin 2a da dp dh 







h [p« + (r-/i)^r 



In practicfe the integration of the effects of the concentric circles is not 

 carried to infinity. The circles are calculated individually; the calculation 

 is stopped when a circle exerts an influence of less than 0.1 E.U.; and all 

 circles are added. Within each circle the effect of the elevation /i is a 

 function of azimuth, as shown further below. For the last integration 

 the denominator in eq. (7-68a) is more conveniently written 



\p' + (f - h)-V = 





-5/2 



P^ + f^ 



For fairly gentle slopes not exceeding 8°-10°, the last term is generally 

 small and the above expression may be expanded in the form of a power 



series, so that after multiplication by / dh {^ — h) = ^ {2^h — 1i) the 



•'0 



series is 



\ [(P^ + fr"] \_2-^h -h' + \ ^^J^^^' - • • • ] (7-686) 



Only the first two terms of the last factor are considered. This approxima- 

 tion is made in all analytical terrain-correction methods. Carrying out 



the integrations / dh(^ — h) and / dh, the four derivatives are: 

 Jo Jo 



rr r,i 5 r°° /* "^ P COS a dadp ,,% ^..s 



^'" = -^^ 2 i I V+n^ ^' - '^" 



TT ^1 ^ f^ f ' P sin adadp yr2 o>.i,\ 



,« rir ^3 ^Qg 2q, (]^c( dp , 



-U^= -\-6k 



2U^ = -\-6k 



2 Jo •'0 



8 r f 



2 Jo Jo (p2 + r^)^/2 



(p2 + ^2)5/2 



p^ sin 2a da dp 



(7-69) 



