I 3 4 STATICS. [221. 



The condition that the particle should remain on this surface 

 can be replaced by introducing the reaction of the sujface, i.e. a 

 force that is always so directed as not to allow the particle to 

 leave the surface. Combining this force with the given forces 

 acting on the particle, this particle can be regarded as free, and 

 the general conditions of equilibrium must hold. 



221. If the surface be smooth, i.e. if the particle move along 

 it without friction, the reaction of the surface must be directed 

 along the normal to the surface (2). Let N denote this normal 

 reaction ; N x , N v , N g its components ; then the conditions of 

 equilibrium are 



2X+N, = o, 2F+^ = o, 2Z+^. = o. (3) 



The condition that N has the direction of the normal is 

 expressed by the relations 



where <,,= -, < y =-2, <^ = _ are obtained from (2). 

 ox oj/ dz 



Eliminating the reactions by means of (3), we find the two con- 

 ditions of equilibrium, 



<#> </>y </>* 



The meaning of these equations is obvious ; they express that 

 the resultant of the given forces must have the direction of the 

 normal to the surface. 



The problem generally consists in finding the positions of 

 equilibrium of the particle on the surface. The two equations 

 (5) represent a curve whose intersections with the surface (2) 

 give the required positions. 



