Chap. 7] 

 Uxz = k8 log, 

 Uyz = k8 log J 



GRAVITATIONAL METHODS 



n+j/i ^-H/2 tt -\- y2 n -\- yi 

 _n + ?/2 r2 + 2/1 rs + ?/i rg + yi 



n + Xi r4 + a;2 ry + 0:2 re + a;i 



257 



U. 



kB log 



L^2 



n + X2 r2 + xi n + xi rs + X2 



+ 2i r6 + 22 r4 + Z2 r-i + 21 

 + 22 rs + Zi rs + Zi rs + Z2_ 



= /c5 1 



TT 7 fix -lyi^i X -12:222 , , -1 yi2i , -1X221 



— C/a = ^5 I tan ^ — — tan — + tan - — — tan 



+ tan" 



X2r% 

 -1 Z/221 



XxTi 



ytn 



X2rz 



, -1 a;i2i , , -1 2/1 22 , 

 tan — + tan - — — tan 



2/2^6 



X -1 yiZi , ^ -1 a:i2i 

 — tan -— + tan — tan 



XiTi 



yin 



X\r2 

 -1 ^222 



XxTf, 



+ tan 



, -1 2/122 , , -1 2:222 , -1 2/221 



— tan - — + tan — — tan - — 



2:2 ^4 2/1 r4 X2ri 



tan 



2/1^3 



-1 a;iZ2 

 2/1^2 



-1 X\Z2 



y%H 

 -1 a:2 2i 



) (7-92c) 



2/2 ^7 J 



When the slab is symmetrically disposed in respect to the profile plane, 

 y ^ y^ = —y2,ri = ri,,r2 = r6,r3 = r7, and u = rs ] the Uxy and Uyz 

 components vanish and the gradients and curvatures become 



u 



.n 



y 7-2 + 2/ ^3 + 2/ u - y_ 



U^ = 2k8\t2^n-' ^ - tan-^ ^^ - tan'^ ^ + tan 

 L 2:2 r4 yn X2r3 



-1 2:2 2i 

 yrz 



I + -1 2/2l 

 + tan -^^ — 



XiTi 



_i a;i2i , -1 yzi 



tan — tan - — + tan 



yri XiTi 



-1 a:i22 "j 

 yr-i\' ^ 



(7-92d) 



It is convenient to consider the gradients and curvatures due to a hori- 

 zontal line of limited strike extent since this gives the possibility of deriving 

 an approximate formula which will indicate when it is permissible to con- 

 sider a three-dimensional feature as two-dimensional. If p is the radius 

 vector from the station to the line of the section dS in the profile plane 

 and if ± 6 is its extension at right angles thereto, then Uy'z and 2Ux'y' 

 are zero, and the gradient and curvature component follow from integra- 

 tion of eq. (7-91a) and series expansion of the expressions involving 

 r = 6(1 + p/6)*: 



