104 KINETICS OF A PARTICLE. [192. 



4. MOTION ON A MOVING OR VARIABLE CURVE OR SURFACE. 



192, If the constraining curve or surface be not fixed and 

 invariable, the conditional equations will contain the time t 

 explicitly, besides the co-ordinates x t y, z of the moving particle 

 (Art. 164). For the sake of simplicity we here assume the 

 curve or surface to be smooth, so as to offer only a normal 

 resistance N\ if there be friction, the components of the 

 frictional resistance may be regarded as included in the com- 

 ponents Jf, Y, Z of the resultant force acting on the particle. 



The treatment of this general problem of constrained motion 

 of a particle is here presented not so much on account of its 

 application to the solution of particular problems, as for the 

 reason that it offers an opportunity of explaining the meaning 

 of d'Alembert's principle and illustrating its application in a 

 comparatively simple case. 



193. Two Constraints. Let the equations of the curve to 

 which the particle is constrained be 



< (x, y t z, i) =o, ^ (x, y, z, t) =0. (i) 



To apply d'Alembert's principle (Arts. 97-102), let the particle 

 be subjected, at any given time /, to an infinitesimal displace- 

 ment Bs. If this displacement be selected along the curve (i), 

 the reaction N of the curve, being at right angles to Bs, will 

 do no work during the displacement ; hence the equation of 

 motion will be the same as that for a free particle (see 

 Art. 101), viz. 



(-mx+X)%x+(-my + Y)fy + ( mz+Z)z=o. (2) 



In this equation, then, the forces X, Y, Z do not involve the 

 normal reaction of the curve ; but the components Bx, By, 

 of the displacement Bs must be selected so that Bs should lie on 

 the curve (i) at the time /; this is usually expressed by saying 

 that the displacement should be compatible with the conditions (\\ 



