172 



GEOLOGY OF THE COMSTOOK LODE. 



^ y the constant vertical difference between contour planes. But the equa- 

 tion for the case of an oblique fault is so complicated as to be of no value. 

 The ideal map would be one in which the contour planes were so close that 



— would be sensibly equal to ,- ; and, indeed, where the slope is consid- 

 ^y -^ ^ dy ^ 



erable this is often the case, but when the .surface-line becomes nearly 



horizontal the difference between the two ratios is large. 



Angle of tangent to the horizonai. — The angle wliicli a taugeut to the curve 



referred to inclined coordinates makes with the horizontal ma}^ be found as 



follows, without going through a 

 troublesome transformation of coor- 

 dinates. Let dx and dy be the differ- 

 entials at the point of tangency 

 obtained from the above equation, 

 and c?rC] and dy^ the differentials for 

 the same point if the //-axis were 

 Fig. -.—Explanation of ;i lanin-.i snrt-ice. vertical and the ,T-axis horizontal. 

 Consider /? as a positive acute angle and d also as a positive acute angle 

 when it falls in the same quadrant with /?, but as negative when it falls in 

 an adjacent quadrant. Let a be the angle which the tangent makes with 

 the horizontal. Then, as appears from the figure, 



tan a 



dyi 

 dxy 



and the equation of the curve referred to inclined coordinates gives 



ylnlO:^-^^ 

 dx 



and b}' a simple projection 



or by reduction 



— dy sm fi-\- dx sin 6 

 tan a := — -^ — ; 



— dy cos fi-\-dx cos 8 



win 10 sin /? 4- sin d 



tan a rr ~<~~^^ ~- ^. 



«/ In 1 cos p -f cos o 



