Figure 4. — Coordinate system (hypothetical faiZvcre surface). 



It is convenient to express the equation for the failure surface in terms of 

 rectangular coordinates. Considering the hypothetical failure surface shown in 

 figure 4 and choosing the coordinates x = y = to denote the intersection of the 

 failure surface with the ground surface, the equation for the failixre surface can be 

 written 



y = z + X tan 6 



The slope of the failure surface is given by the expression 



dx 



(21) 



(22) 



Taking the derivative of the equation for the failure surface, equation 21, the ex- 

 pression for x is obtained as follows: 



dy dz ^ . 

 = J = J— + tan G, 

 dx dx 



(23) 



and 



X z 

 X = / dx = / ( 



m - tan 



-) dz 



(24) 



9 



