200-203] TORTUOUS CURVE. 189 



and B for the component along the binormal. Also we take R 1 

 for the component of the pressure along the principal normal 

 towards the centre of curvature, and R 2 for the component of the 

 pressure along the binormal in the same sense as B. Further if 

 the curve is rough we take F for the friction. 



We take s to be the arc of the curve from some point to the 

 position of the particle at time t, p to be the radius of curvature, 

 and v to be the velocity, and we suppose the sense in which s 

 increases to be that of v. Then the equations of motion are 



dv _ ~ p 

 mV els ~ ~ 



When the curve is smooth F is zero, and we can integrate the 

 first equation, in the same way as in Article 195, in the form 



* = I Sds + const., 



and this result can be expressed in the form 



change of kinetic energy = work done, 



so that the velocity is determined in terms of the position. The 

 other two equations then determine the pressure. As in Article 

 195 the integral equation is expressed more simply when the 

 system of forces whose components are S, N, B is conservative. 



When the curve is rough we have to eliminate F, RI, R 2 by 

 means of the equation 



w r hich expresses that the friction is proportional to the resultant 

 pressure. There results a differential equation for v&quot;, and, if we 

 can integrate this equation, we shall obtain an equation giving the 

 velocity in terms of the position. As in Article 200 the velocity in 

 any position depends partly on the way in which that position has 

 been reached. 



203. Motion on a smooth surface of revolution under 

 gravity. When a particle moves under gravity on a smooth 

 surface of revolution whose axis is vertical we can always obtain 



