112.] 



CENTRAL FORCES. 



59 



111. The second of the polar equations (3) gives immediately, 

 if c denote the constant of integration : 



dt 



(4) 



and the meaning of this equation is that the sectorial velocity is 

 constant and equal to \ c. The same result can be obtained from 

 the equations (2) by applying the principle of areas (see 

 Arts. 87, 88). 



To express the constant c in terms of the initial conditions, 

 let v denote the velocity, / the perpendicular to it from the 

 centre of force, and ty the angle between the radius vector r 

 and the velocity v, all at the time / (Fig. 17) ; and let the initial 

 values of these quantities, at the time *=o, be distinguished by 



Fig. 17. 



zero-subscripts. Then it follows from the equation (4) that we 

 have (see Art. 89 and Part I., Art. 230) 



= z sn 



(5) 



i.e. the velocity is inversely proportional to its perpendicular dis- 

 tance from the centre, or, as it is sometimes expressed, the 

 moment of the velocity about the centre of force is constant. 



112. Another first integral of the equations of motion is 

 obtained by combining the equations (i) according to the prin- 

 ciple of kinetic energy (Art. 71). This gives 



mv 2 ) = - Fdr, or d(% v 2 ) = -f(r)dr y (6) 



