am 



-3 ,^ dp 



Chap. 7] GRAVITATIONAL METHODS 



/•Pn+i /•«m+i J 



U^^ = -k8 I (1 - cos" ^) - cos a da 



/pn+i /• 

 Mil •'a 



/•pn+1 /""m+i J 



Ua = k8 I (3 sin i^ - sin" ^) -^ cos 2a da 



*'pn •'«m P 



rpn+i ram+i j 



2t/^ = A;5 / / (3 sin ^ - sin" ^) -^ sin 2a da. 



231 



(1 — cos yp) — sin a da 



'am P 



) (7-85) 



am P 



3 , n/ ,\ JO _• I _• 3 



In these expressions the terms 1 — cos yp = G(^) and 3 sin ^ — sin ^ = 

 X(^) may be obtained for any terrain profile from Fig. 7-39. Carrying 

 out the integrations in eq. (7-85), the following equations are obtained 



sin am) 



} (7-86) 



C/„ = -UG{-^) loge ^' (sin am- 



Pn 



Uyz = -kdGiip) loge — ' (cos am+1 " COS a^) 

 Pn 



Ua = hkSKirP) loge ^' (sin 2am+i - sin 2am) 



Pn 



2C/xv = -hk8K(rP) loge ^' (cos 2am+i - cos 2am) 



Pn 



Again only two diagrams are necessary, since the N gradient diagram may 

 be used by rotation through 90° for the E gradient, and the curvature 

 diagram by rotation through 45°. The graticules shown in Figs. 7-79a 

 and 7-796 were calculated by making 



loge — ^ = const. = 0.1, sin am+i — sin am = const. = 0.05 



Pn 



and Ksin 2am+i — sin 2am) = const. = 0.05, so that for both diagrams the 

 product of distance and azimuth factors is 0.005. Hence, for the curvature 

 diagram C the unit effect is 0.333- 10~'-5X(^), and for the gradient 

 diagram D the unit effect is 0.333 10~^5G(^). For calculating Ua, 

 the C diagram is oriented with the M axis toward N, E, S, and W. The 

 effects of the N and S quarters are negative and those of the E and W 

 quarters positive when elevations are positive, and vice versa. For 

 calculating 2U^, the M lines are oriented NE, SE, SW, and NW. The 

 NE and SW quarters are positive and the SE and NW quarters negative 

 when elevations are positive. Diagram D, for calculating t/«, is oriented 



