206 1 ARTICLE ON SURFACE. 



fore cos 1 6 < (1 - cos' ff) ; 



therefore cos f 0< 



. , 

 fore 



easy to shew that this result includes that of the former 

 Article by putting n = 0, m = , /= ' .' . 



17G. A particle t* constrained to remain on a rough sur- 

 face: determine the condition o 



Let P be the resultant force on the particle exclusive of 

 the action of the surface; </> the angle between the direction <>i 

 Pand the normal to the surface at the point where the particle 

 is placed ; u = the equation to the surface ; x, y, z the co- 

 ordinates of the particle. The resolved part of P alon^ r the 

 normal is Pcos </>, and that at right angles to the normal is 

 Psin <. Hence, for equilibrium we must have 



Psin <!>< pP cos </>; 

 therefore sin 1 ^ < ^ cos 1 $ ; 



therefore cos 1 <j> > - - t ; 



( X 4- F 4- Z^ U \ 



\ dx du dz ; 1 



, - 

 therefore 



177. In tlie following Articles of this Chapter we shall 

 investigate certain equations which hold when the equilibrium 

 of different machines is on the point of being disturbed. The 

 equations in such cases will involve the forces acting on the 

 machine and /A the coefficient of friction. When we have 

 found one of these limitinj us, we can draw the follow- 



ing r -s: 



(1) If in order to satisfy the equation for a given set of 

 forces it is necessary to ascribe to p a value greater than i;n 



