30 KINETICS OF A PARTICLE. [57. 



a force whose direction constantly passes through a fixed centre 

 (9, while its magnitude is a function of the distance s from the 

 centre alone. If the initial velocity be zero, or if its direction 

 pass through the centre O, the motion of the particle will be 

 rectilinear, the line of motion passing through the centre of 

 force, O. The most important special cases of this kind have 

 been treated in kinematics (Part I., Arts. 117-124, 176). 



57. Let the force be due to a mass m' concentrated at the 

 centre O, and attracting according to Newton's law of the 

 inverse square of the distance (Part II., Art. 257). 

 Counting the distances s from the centre O as origin 

 (Fig. u), we have, for the force acting on the par- 

 ticle m y 



F N 



where K is a constant whose value may be deter- 

 mined, as indicated in Part I., Art. 119, and Part II., 

 Arts. 262, 263. 



The principle of kinetic energy, equation (12), Art. 

 lg ' ' 47, gives at once 



The quantity m' /s, or in absolute measure tcm'/s, is the 

 potential at P due to m' (Part II., Art. 278), and /cm'/s Q is the 

 potential at P due to the same mass m'. The increase in 

 kinetic energy is, therefore, proportional to the decrease in 

 potential. 



The quantity /cmm'/s is sometimes called the mutual potential 

 of the masses m and m' ; hence, the increase of the kinetic 

 energy can be said to be equal to the difference of the mutual 

 potentials in the final and initial positions. 



The negative of the mutual potential is designated as the 



