174, 175] 



PROBLEM OF TWO BODIES. 



155 



Then the acceleration of each particle is in the line joining it 

 to the origin, and the velocities of the particles are localised in 

 lines which lie in a plane containing the origin, the motion of each 

 particle therefore takes place in this plane. 



Fig. 44. 



Now let G be the centre of inertia, m lt ra 2 the masses of the 

 particles, r lt r 2 their distances from G at time t, 6 the angle the 

 line joining them makes with any fixed line in the plane of motion, 

 also let r, = r x + r 2 , be the distance between the particles at time 

 t, and let the force between them be 7ra 1 ra 2 /r* 2 . 



Then the equations of motion of m l are 



Since r x = m 2 r/(m l -f m 2 ), these equations become 

 r r6 2 = y (m l + m 2 )/r 2 , 



dt 



and it is clear that the equations of motion of m 2 would lead us to 

 the same two equations. 



The equations last written show that the acceleration of m^ 

 relative to m 2 , or of m 2 relative to m l , is r y( / m l + m 2 )/r 2 , and that 

 there is no transverse acceleration. Thus either particle describes 

 a central orbit about the other with acceleration varying inversely 

 as the square of the distance, and by Example 1, p. 63, this orbit 

 is a conic described about a focus. 



