Chap. 7] GRAVITATIONAL METHODS 263 



In the calculation of symmetrical anticlines, the curvatures due to two 

 opposite slopes given by eq. (7-93c) are added and the gradients sub- 

 tracted so that 



Ux'z = 2k8 sin i sin i log« - -f cos t(4> 



L ^2 



— Ua' = 2k8 sin i sin t($i + $2) — cos i loge ^ 



(7-93/) 



For a symmetrical syncline, 



Ux'z = 2k8 sin i sin i log« - — cos i(^i — $2) 

 — Ua' = 2k8 sin z sin i(*i + ^2) + cos i loge ""2" ■ 



(7-93^) 



It should be noted that in Figs. 7-98c and 7-98d 4> stands for difference 

 in angle. K. Jung"^ and H. Shaw^'^ have calculated a number of diagrams 

 to assist in direct determinations of the characteristics of anticlines and 

 synclines from gradients and curvature values. 



To obtain the gradients and curvatures for an inclined dike as in Fig. 

 7-98e, two slopes (eq. [7-93c]) are deducted from each other, so that with 

 the notation indicated in the figure. 



{7-9Sh) 



Ux'z = 2A;5 sin i sin i log* -~ + cos z(<p2 — fi — <Pi -\- <Pd) 

 L nvi J 



— ?7a' = 2k8 sin i sin t(</92 — v>i — v'4 + <P3) — cos i log* — - . I 

 L ^1^4 J J 



Usually the depth extent of the dike is considerable, so that 

 Ux'z — 2k8 sin i sin i loge - + cos z(<^3 — <^i) 



— U A' = 2k8 sin i\ sin {(v?.-? — ^\) — cos i log* - . 



If the gradient at the origin (x = above the center of the upper face) 

 (see Fig. 7-99) is Go = Gmax. -|- Gmin. and if the curvature Co = 

 Cmax. + Cmin. , thcn cotan i = Co/Gq , and Go = § sin 2t$o (*o = $ for 

 X = 0); d = (w/2) cotan $/2. For the last calculation, special diagrams 



"8 Ibid., 3, 257-280 (1927); 5, 238-252 (1929). 



"9 H. Shaw, A.I.M.E. Geophysical Prospecting, 336-366 (1932). 



