136 STATICS. [222. 



This equation determines the normal reaction N of the 

 surface. 



To obtain an expression .for pN, multiply the second of the 

 equations (6) by (f> z , the third by <f> y and subtract ; owing to 

 the relations (4) this gives 



Similarly, we find 



-2 K- $ = 



The left-hand members as well as the parentheses on the right 

 are determinants of the second order; hence, squaring and 

 adding, we find 



(8) 



If TVbe now eliminated between (7) and (8), we find the final 

 condition of equilibrium that must be fulfilled by the given forces 

 independently of the reaction of the surface : 



Putting this equation into the form 

 i 



R R' R R' R 



where ^ 2 =(2^) 2 -f(2F) 2 +(2Z) 2 and ^' 2 = x 2 + y 2 + ^ 2 , it is 

 seen to express the fact that the resultant R of the given forces 

 makes the friction angle with the normal, each member of the 

 equation being an expression for the cosine of this angle. 



