Chap. 7] 



are 



dm 



GRAVITATIONAL METHODS 



Sin (p cos (p cos a 



233 



Uyz = Sk I dm 

 Ut, = -Sk f d 



2U, 



= Skf 



dm 





Since the mass element dm = 

 B-dr- rdip • r map • da, the gradients and 

 curvatures of one element are 



U„ = dk8 

 Uyz = Sk8 



/•am+i r<Pv+i f 



•'am *'«'? •''■ 



rava+l /"fiV+1 r 



Jam J an •'r 



Fig. 7-81. Spherical shell mass- 

 element. 



COS a da sin^ <p cos <p d<p dr 



'am "fv 'r-n. T 



•om+i /-vp+i fn+i ~:_ J -:_2 



"m+i /•iPp+i /•'"n+i 



/•«m+i (•vp+i i« 



Ui, = Sk8 / / 



•'am •'*'P •'''1 



sin a da sin (p cos v? d(p dr 



cos 2a da sin^ <p d<p dr 



•'am •'*'? *^»"n ^ 



These four equations may be reduced to two because the azimuthal 

 differences (sin, cos) in the components can be taken care of by rotation 

 of the diagrams. Then the effect of one mass element is 



Uxz = kd (sin am+i — sin am) (sin^ <pp+i — sin^ <pp) loge 

 Ua = — ^k8 (sin 2ani+i — sin 2am) (cos^ v^p+i — cos' <p. 



rn 



^ (7-88) 



— 3 cos ipp+i + 3 cos v?p) logs 



rn 



Evaluation of gradients and curvatures in eight sections, that is, in 

 sixteen azimuths, does not require thirty-two but only six diagrams. Of 

 these, the GI and the CI diagrams are reproduced in Figs. 7-82 and 7-83. 

 Unit effect is 1 • 10~^°; scale is arbitrary; and assumed density is 1.0. Their 

 application above and below the horizon is indicated in the scheme in 

 Fig. 7-84. 



