173-] MOTION ON A FIXED CURVE. 91 



the resultant F of the given forces, and 7V Z , N y , N g those of the 

 normal reaction TV of the curve. If there be friction, the fric- 

 tional resistance pN, being directed along the tangent to the 

 path opposite to the sense of the motion, has the direction 

 cosines dx/ds, dy/ds, dz/ds, so that the components of the 

 force of friction are ^Ndx/ds, ^Ndy/ds^ pNdz/ds. The 

 general equations of motion are, therefore, 



If the acceleration of the particle be zero, the left-hand mem- 

 bers are all =o, and the equations reduce to the conditions of 

 equilibrium of a particle on a curve, as given in Statics (Part 



ii., P . 138, (i4. 



In addition to the equations (3) we have of course the equa- 

 tions of the curve, say 



< (*, y, *) =o, ^r (*, y, *)=o, (4) 



and the relations 



(5) 



the latter expressing that TV is perpendicular to the element ds 

 of the path. 



173. Multiplying the equations (3) by dx, dy, dz, and adding, 

 we find the equation of kinetic energy 



d(%mi?) = Xdx+ Ydy+Zdz-^Nds. (7) 



This relation might have been obtained directly from the con- 

 sideration that for a displacement ds along the fixed curve the 

 normal reaction N does no work, while the work of friction is 



