.So KINETICS OF A PARTICLE. [151. 



mutual action, being of the nature of a stress , i.e. consisting 

 of two equal and opposite forces, each equal to 



Hence, the mass m of the planet attracts the mass M of the 

 sun with precisely the same force with which the mass M 

 of the sun attracts the mass m of the planet. The attraction 

 affects, therefore, the motions of both bodies. 



151. The accelerations produced by the two forces are, of 

 course, not equal. Indeed, the acceleration F/m = rcM/r 2 , 

 produced in the planet by the sun, is very much greater than 

 the acceleration F/M=Km/r 2 y produced by the planet in the 

 sun ; for the mass of even the largest planet (Jupiter) is less 

 than one thousandth of that of the sun. The assumption 

 of a fixed centre can therefore be regarded as a first approxima- 

 tion in the problem of the motion of a planet about the sun. 



In the case of the earth and moon, the difference of the 

 masses is not so great, the mass of the moon being nearly 

 one eightieth of that of the earth. 



It can, however, be shown that the results deduced on the 

 assumption of a fixed centre can, by a simple modification 

 be made available for the solution of the general problem of tJu 

 "motions of two particles of masses, m, M, subject to no forces 

 besides their mutual attraction. In astronomy, this is callec 

 the problem of two bodies. In the solution below we assume the 

 attraction to follow Newton's law of the inverse square o: 

 the distance. It will be convenient to speak of the two 

 particles, or bodies, as planet (m) and sun (M). 



152. With regard to any fixed system of rectangular axes 

 let x, y, z be the co-ordinates of the planet (m), at the time t 

 x', y', z 1 those of the sun (M), at the same time ; so that for 

 their distance r we have 



