1 5 o.] PROBLEM OF TWO BODIES. 79, 



If the law of force is given as a single-valued function f(u), 

 there can exist only two different apsidal distances (although 

 there may be any number of apses). The angle between 

 these two different apsidal distances is called the apsidal angle. 



149. Exercises, 



(1) Find the law of force when the equation of the orbit is 

 r n = a cos nO + b, and investigate the particular cases n i, n = 2, 



n = i, n = 2. 



(2) A particle moves in a circle under the action of a single force of 

 constant direction ; determine the law of force and discuss the motion. 



(3) Find the law of the central force directed to the origin under 

 whose action a particle will describe the following curves : (a) the 

 spiral of Archimedes raO; (b} the hyperbolic spiral 6r= a - } (c} the 

 logarithmic or equiangular spiral r=ae aQ ; (d) the curve r=acosn&. 



(4) A particle moves in a circle under the action of a central force 

 directed towards a point on the circumference ; find the law of force. 



(5) A particle is acted upon by a force perpendicular to a given 

 plane and inversely proportional to the cube of the distance from the 

 plane. Determine its motion. 



(6) A particle moves in a semi-ellipse under the action of a force 

 perpendicular to the axis joining the ends of the semi-ellipse. Deter- 

 mine the law of force and the velocity at the ends. 



3. THE PROBLEM OF TWO BODIES. 



150. In the preceding discussion of the motion of a particle 

 under the action of, a central force, it has been assumed that 

 the centre of force is fixed. In the applications of the theory 

 of central forces this assumption is in general not satisfied. 

 Thus, in considering the motion of a planet around the sun, 

 the force of attraction is, according to Newton's law of universal 

 gravitation (Art. 129), regarded as due to the presence of a 

 mass M at the centre (sun), and of a mass m at the attracted 

 point (planet) ; and the action between these two masses is a 



