220.] CONSTRAINTS. 133 



conditions, or constraints, may be of various kinds ; the body 

 may have a fixed point, or a fixed axis, or one of its points may 

 be constrained to move along a given curve or to remain on a 

 given surface, etc. 



As explained in Kinematics, Art. 37, a free rigid body is said to 

 have six degrees of freedom. The most general form of motion 

 that it can have is a screw-motion, or twist, consisting of a rota- 

 tion about a certain axis, and a translation along this axis ; each 

 of these resolves itself analytically into three rectangular com- 

 ponents, and these six components may be regarded as consti- 

 tuting the six possible motions of the body, on account of which 

 it is said to have six degrees of freedom. 



Equilibrium will exist only when these six possible motions are 

 prevented ; hence there must be six conditions of equilibrium. 



219. We proceed to consider some forms of constraint and 

 the corresponding changes in the equations of equilibrium. 



It is generally convenient in dynamics to replace such restrain- 

 ing conditions by forces, usually called reactions. Whenever'it 

 is possible to introduce such forces having the same effect as 

 the given conditions, the body may be regarded as free, and the 

 general equations of equilibrium can be applied. 



Before considering the constraints of a rigid body, those of a 

 single particle, or point, must be briefly discussed. 



220. Particle constrained to a Surface. A free particle has 

 three degrees of freedom ; and accordingly its equilibrium is 

 determined by three conditions (Art. 101) : 



o. (i) 



If the co-ordinates determining the position of the particle be 

 subject to one condition, expressed by an equation between these 

 co-ordinates, the particle is said to have two degrees of freedom 

 and one constraint. Its motion is restricted to the surface repre- 

 sented by the equation between its co-ordinates, say 



4>(x,y,z)=o. (2) 



