16, 17] GRAVITATION CONSTANT. THREE BODIES. 31 



and for 2 are 



W 2 dt z ~~f r 12 2 r 12 



m ^ = ?Y 



2 r 



12 '12 



The integration of these six equations is easily carried out (see 

 102), and gives us for the case of the sun and a planet a slight 

 modification of Kepler's laws, for the sun does not remain absolutely 

 at rest. If there are three bodies their equations of motion are 

 similarly, 



v 



__. / I /vi/i *^ ^ ^ * I n/i/i K 



dt* ~? 



- = 7 (^2 r^ir + m z~-ir 



\ r i9. r t* 



~d^ = r 



The problem of integrating these equations is known as ''the 

 problem of three bodies" and has not been completely solved. The 

 problem of the solar system is still more complicated, but by means 

 of approximations, the perturbations of the different planets upon 

 each other, causing slight variations from Kepler's laws, have been 

 calculated. It is in this manner that the observations of astronomers 

 from the time of Newton until the present have furnished the most 

 brilliant verification of Newton's great discovery. 



17. Absolute Systems. The above system of units, which 

 has for its fundamental units the centimeter, gram, and second, is 

 called the C. Gr. S. system, and was recommended by a committee of 

 the British Association for the Advancement of Science in 1861. It 

 is sometimes incorrectly spoken of as the absolute system of units. 



