192-195] SMOOTH CURVE. 183 



the particle at time t. Let S be the tangential component of the 

 forces in the direction in which s increases, and N the component 

 along the normal inwards. Let v be the velocity of the particle 

 in the direction in which s increases, and R the pressure of the 

 curve on the particle. We shall write down the equations for the 

 case where the particle is on the inside of the curve, and R 

 accordingly acts inwards. The equations for the case in which 

 R acts outwards can be obtained by changing the sign of R. 



By resolving along the tangent and normal we obtain 



dv 

 mv -jssSf 



as 



m-=N+R 

 P 



These are the equations of motion. The former can be inte 

 grated so as to express the velocity in terms of the position. In 

 fact, since the forces $ and N depend only on position, and since 

 the coordinates of any point of the curve can be regarded as 

 functions of a single parameter, it is clear that, on the curve, S is 

 a function of the parameter, and ds is the product of a function of 

 the parameter and its differential. Hence we have 



mv 2 = I Sds + const., 



re 

 where the integral is of the form I f(B) &amp;lt;/&amp;gt; (6) dO in which is the 



parameter, 8 is/(#), and ds is (f&amp;gt; (0) dO. 



This is a case in which the work done by the forces between 

 two positions can be calculated whether the system of forces is 

 conservative or not, and the equation can be interpreted in the 

 form 



change of kinetic energy = work done, 



where the &quot;change of kinetic energy&quot; on the left means the excess 

 of the kinetic energy in the position at time t above that in some 

 definite standard position, and the &quot;work done&quot; on the right 

 means the work done by the forces in the displacement of the 

 particle along the curve from the definite standard position to the 

 position at time t. 



In the case of conservative positional forces I Sds is the excess 



