154 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



Now the curve can be described under all the forces if there exists a 

 velocity V satisfying the two equations 



and it is clear that these are satisfied by 



Thus the condition is that the kinetic energy when all the forces act must 

 be the sum of the kinetic energies when they act separately. 



4. Prove that a lemniscate r/ = c 2 , where 2c is the distance between the 

 points from which r and r are measured, can be described under the action of 

 forces mn/r and m/z/r directed to those points. 



5. A particle describes a plane orbit under the action of two central 

 forces each varying inversely as the square of the distance, directed towards 

 two points symmetrically situated in a line perpendicular to the plane of the 

 orbit. Show that the general (p, r) equation of the orbit, referred to the 

 point where the line joining the centres of force meets the plane as origin, is 

 of the form 



where c is the distance of either centre of force from the plane, and a and 6 

 are constants. 



6. A point describes a semi-ellipse, bounded by the axis minor, and its 

 velocity, at a distance r from the nearer focus, is a ^{/ (a r)/r (2a r)}, 

 2a being the axis major, and / a constant. Prove that its acceleration is 

 compounded of two, each varying inversely as the square of the distance, one 

 tending to the nearer focus, and the other from the farther focus. 



175. The Problem of Two Bodies*. Two particles which 

 attract each other according to the law of gravitation are projected 

 in any manner. It is required to show that the relative motion is 

 parallel to a fixed plane, and that the relative orbits are conies, and 

 to determine the periodic time when the orbits are elliptic. 



The principle of the conservation of linear momentum shows 

 that the centre of inertia of the two particles moves uniformly in 

 a straight line. The accelerations of the particles, and the 

 velocity of either relative to the other, are unaltered, if we refer 

 them to a frame whose axes are parallel to those of the original 

 frame of reference, and whose origin is at the centre of inertia. 

 We shall suppose this to be done. 



* The Problem of Two Bodies was solved by Newton, Principia, Lib. i. Sect. xi. 

 Props. 57 63. 



