153.] PROBLEM OF TWO BODIES. 8l 



Then the equations of motion of the planet are 



Mm x'-x 



Mm v' 





t r r 



while the equations of motion of the sun are 



mM xx* 



n 



M=- =K 

 dt* 



_ mM z- 

 " ~ 



153. By adding the corresponding equations of the two sets, 

 we find 



If it be remembered that the centroid of the two masses m, M 

 has the co-ordinates 



_ _ 



' 



m+M m + M m + M 



it appears that these equations can be written in the form 

 d*x 



in words : the acceleration of the common centroid of planet and 

 sun is zero ; i.e. this centroid moves with constant velocity in 

 a straight line. 



It may be noticed that this result is merely a special case 

 of a more general proposition to be proved hereafter, viz. 

 that the centroid of any system acted upon by no forces 

 external to the system moves uniformly in a straight line 

 (Art. 381). 



PART III 6 





