224-] 



CONSTRAINTS. 



137 



The regions of the surface (2) on which the particle is in 

 equilibrium are cut out of this surface by the surface (9). 



223. Particle constrained to a Curve. If the particle be sub- 

 ject to two conditions, 



$ (x, y, *) =o, i/r (x, 7, z) = 0, 



(10) 



so that it has two constraints and but one degree of freedom, its 

 motion is restricted to the curve of intersection of the two sur- 

 faces (10). The particle may be imagined as a small sphere 

 moving within a tube, or as a small ring or bead sliding along a 

 thin wire. 



224. Let .the curve be smooth, so that its total reaction is 

 along the normal to the curve. Denoting this normal reaction 

 again by N, its components by N t1 N y , JV Z) the conditions of 

 equilibrium are 



: 0) (II) 



or 



(12) 



The condition that N has the direction of the normal can be 

 expressed in the form 



N x dx + N y dy + N t dz = o, 

 which, by (12), reduces to fc 



Differentiating the equations of the curve (10), we find 



and eliminating the differentials between the last three equa- 

 tions, the single condition of equilibrium, independent of the 

 reactions, is found in the form 



