486 EXPLORATION GEOPHYSICS 



The condition ^— = r — dXz — d requires that 



pi OS p% OS 



iOO 00 ) 



^Sos 4- Y M2U2kd-z) _ Y iJVa, (2^J + g) ( 

 " [r^ + s'V^ Z [r'+ (2^d-2)']''^ Z [r^+ {2kd + 2yM 

 k = l h—l J 3 = d 



pa] {r' + s']'^ Zu[r^+(2kd + 2yM 



3 = d 



oo 



iSod V iM2,.(2k — l)d , V iN2u(2k + l)d 





pi Hr' + d']^ Ai[r'+ i2k — iy-d']'y^ ^ [r + (2k + ly d"-]^'' 

 k=i k=i 



__ 2Sod + Y 2No^(2k + l)d 



Equating corresponding terms of the summation yields 



10 1M2 200 . , ^ 



— — for k = 1. 



pi pi P2 



and 



— {- (2k — l)rf iM« + (2k - l)d lAT^c*-!)} = — (2^ - l)d 2Ar2(».i, 



Pi P2 



or 



lAfgfc , iA^2(iii-i) _ 2A^2(fc-i) , 7> = ? ■? 4 . . . . 



pi pi pa 



Thus, the boundary conditions have yielded the following relations between the con- 

 stants So, Mk and AT* : 



(a) oSo = iSo (d) 15*0 + 1M2 = 25*0 



(b) oM2k = iM2k+iN2k for ^ = 1,2,3 • • • (e) 1^2*+ iAr2(*-i) =2iV2(s-i) 



foryfe = 2,3,4 



(c) 



oM2fc _ lM21> liVzfc 



PO pi pi , f\ lOO 1M2 _ 2-JO 



for & = 1, 2, 3 • • • • pi pi P2 



(g) _ ij^ ^ iAr2(,-i) ^ 2iV2(.-i) ^^^ ^ = 2, 3, 4 • . . 



pi pi pa 



On combining Equations (b) and (c), we obtain 



0^2*= (^^) iM2, and iiV2fc = r^^) iM2H 



\po + piJ Vpo + pi/ 



From Equations (e) and (g), we obtain 



2A^2(*-i) = ( r I iM2(.k-i} and 1^/2* = ( ^' . ^ I iN2(*-i) 



\P1 + P2/ \P2 + Pl/ 



From Equations (d) and (f), we obtain 



25'o=C-^)i5'; and iM2=f^^)i5-o 



\P2 + Pl/ \P2 + Pl/ 



Let _ 



po + pi P2 + P1 



Then 



-^=l + Oi. ^^=1 + 02 



PoTpi P2"l"Pl 



