56 KINETICS OF A PARTICLE. [104. 



2. CENTRAL FORCES. 



104. We proceed to apply the general principles developed 

 in the preceding articles to the motion of a particle under the 

 action of central forces. 



The term central force is generally understood to imply two- 

 conditions, viz. (a) that the direction of the force passes constantly 

 through a fixed point, usually called the centre of force ; and (b) 

 that the magnitude of the force is a function of the distance from 

 the centre alone (comp. Art. 56). 



Let O be the centre of force, P the position of the moving 

 particle at any time t, m the mass of the particle, and OP=r its 

 distance from the centre ; then the general expression for a 

 central force Fis 



where the function F(r) represents the law of force, and the 

 function f(r) the law of the acceleration produced by this force 

 in the particle m. 



105. The most important special case is that of a force pro- 

 portional to some power of the distance r, say 



where //, and n are constants. The constant /*, which represents 

 the value of the force at unit distance from the centre, is often 

 called the intensity of the force, or of the centre. 



In the case of Newton's law of universal gravitation (Part II., 

 Art. 257) we have n= 2, p^icmm', where K is a constant, viz. 

 the acceleration produced by a unit of mass acting' on a unit of 

 mass at unit distance, while m is the mass of the attracted par- 

 ticle, and m 1 that of the attracting centre ; that is, Newton's 

 law is expressed by the formula 



106. From the physical point of view, attractions following Newton's 

 law, and indeed, central forces generally, are usually regarded as due to 



