STRUCTTJRAL RESULTS OF FAULTING. 169 



in which S has the positive sign if the surface sloped in the same sense 

 as the fissure plane, and the negative sign if the dip and the slope were in 

 opposite directions.^ This formula therefore makes it possible to recon- 

 struct the original surface, in so far as it is unmodified by other causes. 



Reduction of equation to simplest form. — Equatlou (3) is the most conveuieut form 

 for the calculation of the constants involved, because the direction of the 

 y-axis, and commonly also the position of origin, can be directly observed,' 

 but for plotting and for some purposes of discussion the equation can be 

 advantageously reduced to another form. The equation of the asymptote is 



y-\-Az=.x tan 5. 



If, therefore, we refer equation (3) to the intersection of the asymptote and 

 the //-axis and adopt the asymptote as a new a;-axis, (3) will reduce to the 

 form 



' I have preferred to chnracterize these angles in this way rather than to adopt the ordinary but 

 not nuiversal convention as to positive and negative angles, because this is a discussion of structural 

 geology. The niatbeniatieal question involved is simply whether /iand li lie in tliesanie quadrant or in 

 adjoining ones. 



For similar reasons common logarithms instead of natural logarithms have been used iu all for- 

 mulas, the direct applicability of which to natural occurrencesreudersit possible that computations may 

 be based upon them. 



^In computing the logarithmic curve which most nearly apjjlies to a given surveyed .section line 

 it is necessary to know the dip of the fissure and the position of three points on the surface relatively 

 to the re.ctan;ular coordinates the origin of which is the cropping and the i/-axis the dip-line. The 

 computation is greatly simplified by so selecting the arbitrary values of x (.r,, Xg, X3) that ar, =\ n.—l ./j. 

 The three equations then become 



^i = J (ift-^' — 1 ) — x, tan 9 ; 

 !h = A (m-'^' — l) — 2xi tan 9; 

 1/3 = ^ (m--*^'— 1) — 4J-1 tan 9. 

 Solving these equations for the three constants, it will be found tliat 



log ; 



(„t-x._i)2' 



