5 8 KINETICS OF A PARTICLE. [109. 



Taking the plane of motion as ;rj/-plane and the centre O 

 as origin (Fig. 16), the direction cosines 

 of the force F are ^x/r, ~3-y/r, the 

 upper sign corresponding to an attrac- 

 tive force, the lower to a repulsion. 

 L_ Hence, the dynamical equations of 



Fig. 16. motion are 



If mf(r) be substituted for F, the factor m disappears, and 

 the equations become purely kinematical. 



109. To avoid the use of the double sign, we shall give the 

 equations in the form corresponding to the more important case 

 of attraction ; for a repulsive force it will only be necessary to 

 change throughout the sign of F or f(r). Thus the funda- 

 mental equations of motion are (comp. Part I., Art. 226): 



If polar co-ordinates r, 6 (Fig. 16), with the centre as pole, be 

 used, the equations of motion are, since the total acceleration 

 is along the radius vector : 



, I d 



110. Two principal problems present themselves : (a) the 

 problem of finding the orbit for a given law of force, and (b) 

 the converse problem of determining the law of force, i.e. the 

 function /(r), when the orbit is given. The solution of the former 

 problem is effected by obtaining first integrals of the equations 

 of motion from the principle of areas and from the principle 

 of kinetic energy, and by combining these integrals so as to 

 effect a second integration. Formulae for the solution of the 

 latter problem will be found incidentally. 



