228 



GRAVITATIONAL METHODS 



[Chap. 7 



C/« = Sk8(^h - hh') / cos a da ^ 



J«™ •'pn P 



•Pn+i 



Uy^ = Sk8(^h 



hh' 



am+i 



/•am 



)/ 



•'am 



sin a da 





dp 



/•"m+i /"Pn + i J 



— Ua = Sk8'h I cos 2a da I -^ 



•'oTTi •'on P 



dp 



2U^y = 3k8.h 



/: 



sin 2a da 





dp 



(7-82) 



Carrying out the integration for a sector of constant elevation h, 



= ^k8(^h - ^h )(sin a^a+i - sin 



(7-83) 



«m)(-2 - ^-) 



^v» = - o ^^^^^ ~ hh^)(cos am+i - cos an»)(-2 - -2- ) 



^ \pn Pn+i/ 



- ^A = 7^ A;5/i(sin 2an,+i - sin 2an,)( ) 



^ \Pn Pn+y 



2t/iy = — ^ A;5/i(cos 2ann.i — cos 2ani)(— ) 



^ Vpn Pn+i/ 



Assuming now that f /i — ^h^ = constant = 1 ; that 6 = 1; that the unit 



efifect is 10 for gradients and 10 for curvatures; and further that 



(sin tti 



sin am) = const. 



and -2 — 



Pn 



-J— = const.; 



Pn+i 



(7-84) 



it is possible to divide the entire surrounding topography into elements so 

 dimensioned that regardless of distance and azimuth all produce the same 

 effect. It is not necessary to construct separate diagrams for the east 

 gradient and the northeast curvature. The north gradient and north 

 curvature diagrams may be used for the purpose by rotation through 90° 

 and 45°, respectively. 



Fig. 7-78a shows a diagram for the calculation of terrain gradients. 

 With the arrow north, the diagram is used for Ux^; with the arrow east, for 

 Uyz. The unit effect of the five interior rings is 10~^ , that of the others 

 10~^^ C.G.S. The diagrams are applicable from 50 to 500 m or 5 to 50 m; 

 however, f must be expressed in the same units as h. If the elevation be- 

 tween two successive contour lines is h, the effect of the contour "strip" is 



n5(fA - P'), 



