222.] CONSTRAINTS. 



The magnitude of the reaction N is found from (3) : 



222. If the surface be rough, the total reaction of the surface 

 lies within the cone of friction (Art. 187), and the resultant R of 

 all other forces acting on the particle must therefore also fall 

 within this cone. 



The boundaries of the regions on the surface within which 

 equilibrium is possible are .found by considering the total reac- 

 tion in its limiting position, i.e. when it makes the friction angle 

 tan" 1 //, with the normal to the surface. 



Let N represent the normal component of the total reaction ; 

 N x) N y , N g its components ; the force of friction pN lies in the 

 tangent plane, and has, therefore, the components pNdx/ds, 

 l*,Ndy/ds, pNdz/ds, for motion along any curve s on the sur- 

 face (2). These components of the friction must be given the 

 double sign T, because the force of friction may act in either 

 sense along the curve s. Thus, the conditions of equilibrium 

 are 



=o,, 

 as 



o, (6) 



as 



. 

 as 



To eliminate the reactions, multiply these equations by <f> x , 

 <f>, and add ; this gives 



since the differentiation of the equation of the surface (2) gives 

 $ x dx+$ y dy + $ z dz=v. Substituting for N x , N v , N, their values 

 from (4), the equation becomes 



(7) 



