158.] PROBLEM OF TWO BODIES. 83 



Then the sun will be reduced to equilibrium, while the resulting accel- 

 eration of the planet, which is its relative acceleration with respect to 

 the sun, will evidently be the sum of the acceleration exerted on it by 

 the sun, and the acceleration exerted on the sun by the planet. This is 

 just the result expressed by the equations (3). 



156. It can here only be mentioned in passing that, while 

 the problem of two bodies thus leads to equations that can 

 easily be integrated, the problem of three bodies is one of exceed- 

 ing difficulty, and has been solved only in a few very special 

 cases. Much less has it been possible to integrate the 3/2 

 equations of the problem of n bodies. 



157. According to the equations (3), the first and second laws 

 of Kepler can be said to hold for the relative motion of a planet 

 about the sun (or of a satellite about its primary). The third 

 law of Kepler requires some modification, since the intensity of 

 the centre p should not be =/cM, but =fc(M+m). Thus we 

 have, by (26), Art. 139, 



in other words, the quotient a?/T 2 is not independent of the 

 mass m of the planet. 



Thus, if m lt m 2 be the masses of two planets, a v a 2 the major 

 semi-axes of their orbits, and 7\, T 2 their periodic times, we 

 have 



This quotient is approximately equal to one if M is very large 

 in comparison with both m^ and m^ ; hence, for the orbits of the 

 planets about the sun, Kepler's law is very nearly true. 



158. Exercises. 



( i ) Two particles of masses m lf m 2 attract each other with a force 

 which is any function of the distance r between them, say ^=m 1 m 2 /(r). 

 Show that their common centroid moves uniformly in a straight line, 

 and find the equations of this line. 



