I 9 9-] VARIABLE CONSTRAINTS. 107 



which show that the coefficients of \ and /JL in the equation 

 of kinetic energy vanish. 



Hence, for motion on a fixed curve we have 



d(\ mv 2 ) = Xdx + Ydy + Zdz, (6) 



which agrees with the equation (7) of Art. 173, considering 

 that the frictional resistance is supposed to be included among 

 the forces X, Y, Z. 



198. In the general case, the complete differentiation of the 

 equations (i) gives 



and the equation of kinetic energy -for motion on a moving or 

 variable curve becomes 



(7) 



The distinction between the virtual displacement Ss along 

 the curve in its position at the time t and the actual displace- 

 ment ds of the particle along the moving curve should be 

 clearly understood. The virtual displacement &s = PP' joins 

 the position P (x, y, z) of the particle at the time / to a point 

 P' (x+x, y + ty, z + z), which is on the curve, and infinitely 

 near to P at the time /, while the actual displacement ds = PP" 

 joins P (x, y, z) to the position P" (x+dx, y + dy, z-\-dz) of the 

 particle at the time t + dt\ P" lies, therefore, on the position 

 that the curve has, not at the time t, but at the time t + dt. 

 The reaction N of the curve at the time t is normal to Ss, 

 but not to ds. 



199. One Constraint. Let 



$(x,y, z, *)=o (8) 



be the equation of the surface on which the particle is assumed 

 to remain throughout its motion. The reaction N of this sur- 

 face will do no work if the displacement Ss be taken along the 



