204-] VARIABLE CONSTRAINTS. 



202. If the equations (10) be combined according to the 

 principle of kinetic energy, we find 



where again the coefficient of X vanishes only when the sur 

 face is fixed, in which case 



(12) 



while in the general case of a moving or variable surface we 

 have 



(13) 



203. Plane Motion. If a particle be constrained to move in a 

 plane curve under the action of forces lying in the plane of the 

 curve, d'Alembert's principle gives the equation of motion 



and the equation of the curve 



<f>(x, y, /)=o ( l $) 



gives by differentiation for a virtual displacement Ss on the 

 curve at a given time /, 



Hence, proceeding as in Art. 200, the equations of motion can 

 be written in the form 



mx = X + \<> x , my-= Y-\-\<f) v , (17) 



while the normal reaction of the curve is 



(18) 



204. The process of solution is now as follows for the case of 

 plane motion. Differentiate the equation of condition (15), 

 which holds at any time, with respect to the time, remembering 

 that x and y are functions of the time ; this gives : 



o. (19) 



