86 KINETICS OF A PARTICLE. [162. 



Thus, let X', Y f , Z f be the components of the resultant of all 

 the constraints ; X, Y, Z those of the resultant of all the other 

 impressed forces. Then the equations of motion are : 



162. It must, however, be noticed that the reactions repre- 

 senting the constraints, such as the tension of the string in 

 the example referred to, are generally not given beforehand. 

 Moreover, the constraints are often expressed more conveniently 

 by conditional equations. Thus, if the motion of a particle be 

 restricted to a surface, the equation of this surface, say 



<!>{?, y t z)=o, (2) 



may be given as a constraining condition to be fulfilled by the 

 co-ordinates of the moving particle. 



163. As a particle has but three degrees of freedom, it can 

 be subjected to only one or two conditions of the form (2). 

 One such condition confines it to a surface ; two to the curve 

 of intersection of the two surfaces represented by the two 

 conditional equations ; three conditions would evidently prevent 

 it entirely from moving. 



164. The curve or surface to which a particle is constrained 

 may vary its position and even its shape in the course of time. 

 In this case the conditional equations, referred to fixed axes, 

 will contain not only the co-ordinates, but also the time. That 

 is, they will be of the more general form 



4>(x,y,z,t)=o. (3) 



165. To constrain a particle completely to a surface, we may 

 imagine it confined between two infinitely near impenetrable 

 surfaces. The complete constraint to a curve might be realized 

 by confining the particle to an infinitely narrow tube having 

 the shape of the curve, or by regarding it as a ring sliding along 

 a wire. 



