io8.] CENTRAL FORCES. 57 



the presence of mass, not only in the moving particle, but also at the 

 centre of force ; and the action between these two masses is then a 

 mutual action, being of the nature of a stress, i.e. consisting of two 

 equal and opposite forces. It follows that what we have called the 

 centre of force is not a fixed point. 



It will, however, be shown later (Arts. 150-157) that a simple modi- 

 fication allows us to apply to this case the results deduced on the assump- 

 tion that the centre is fixed. 



Again, the attracting or repelling masses will here be regarded as con- 

 centrated at points. It should be remembered that a homogeneous 

 sphere, according to Newton's law, attracts a particle outside of its 

 mass as if the whole mass of the sphere were concentrated at the 

 centre of the sphere (Part II., Arts. 272-276). The attraction of any 

 other mass on a particle can, of course, always be reduced to a single 

 force ; but as the particle moves, the direction of this force will not 

 in general pass through a fixed point ; such a force is, therefore, not 

 central. 



107. If a particle P of mass m be acted upon by a single 

 central force 



F=mf(r), 



its acceleration j=F/m=f(r) will pass through the centre of 

 force and be a function of r alone. The problem reduces, 

 therefore, at once to the kinematical problem of central motion 

 (Part I., Art. 223). Although the leading ideas of the solution 

 of this problem have been indicated in kinematics (Part L, Arts. 

 225-234, 237-238), the importance of the subject of central 

 forces demands a restatement in this place of the principal 

 methods in the language of kinetics, and a more complete expo- 

 sition of some special cases. 



108. A particle of mass m acted upon by a single central 

 force F=mf(r) will describe a curvilinear path whenever the 

 initial velocity is different from zero and does not pass through 

 the centre of force (see Art. 56). As shown in kinematics 

 (Part L, Art. 225), the path of the particle, here usually called 

 the orbit, is always a plane curve. 



