Chap. 7] 



GRAVITATIONAL METHODS 



217 



From this point on, various analytical terrain methods make different 

 assumptions regarding the variation of h with azimuth and radius. For 

 greater distances it is permissible to make no assumption whatever regard- 

 ing a variation and to assume merely that the elevation measured in the 

 field is representative of the mean elevation of a sector which is so bounded 

 that the station at which the elevation is measured is at -its center^* (and 

 not at its corners as assumed in other analytical methods). 



The effect of each sector is then obtained by integration between the 

 limits Pa and Pa+i and am and am+i. Considering that 



r' 2 r^' 2 r^^ r^' . 



/ f COS a da — I ^ sm a da = f cos 2a da = I f sin 2ada = 



Jo Jo Jo Jo 



and that, therefore, the substitutions 



h' - 2^h = h' -2^h- f = H' and h-^ = H 



(7-70) 



are permissible, the azimuthal effects of all sectors in a concentric ring are 

 given by 



»2r 



I 



cos adaH^ = — Zl H^ cos a = c' 

 mm 



/ sin a da H^ = —^ H^ sin « = b' 

 •'0 mm 



(7-71) 



/ cos 2a daH = — Yl H COS 2a = e' 

 Jo mm 



f sm2adaH = — J2H sin 2a = d' 

 Jo mm 



where m is the number of azimuths (eight or sixteen) in which the eleva- 

 tions are measured. 



Then the effect of one ring bounded by the radii pn and Pn+i is 



8 r^n+i 



U. 



2 la (p^ + rT^" 



5 /-"n+l 



8 fn+l 



Jo- 



p^dp 



(p2 + ^2)6/2 



• b' 



p^dp 



2Uxy = 6fc 



2 J._ (p2 + r)6/2 



8 r''n+i ^ 



/ 



P dp 



(p2 + ^2)5/2 



d' 



(7-72) 



»* C. A. Heiland, A.I.M.E. Geophysical Prospecting, 554 (1929). 



