$ 228] Newton's Problem : the Problem of Three Bodies 289 



when required, as in the cases of precession and of 

 other motions of a planet or satellite about its centre, and 

 of the corresponding action of a non -spherical planet on its 

 satellites ; to this group of problems belongs also that of the 

 tides and other cases of the motion of parts of a body of 

 any form relative to the rest. 



Again, the solar system happens to be so constituted that 

 each body's motion can be treated as determined primarily 

 by one other body only. A planet, for example, moves 

 nearly as if no other body but the sun existed, and the 

 moon's motion relative to the earth is roughly the same as 

 if the other bodies of the solar system were non-existent. 



The problem of the motion of two mutually gravitating 

 spheres was completely solved by Newton, and was 

 shewn to lead to Kepler's first two laws. Hence each 

 body of the solar system could be regarded as moving 

 nearly in an ellipse round some one body, but as slightly 

 disturbed by the action of others. Moreover, by a general 

 mathematical principle applicable in problems of motion, 

 the effect of a number of small disturbing causes acting 

 conjointly is nearly the same as that which results from 

 adding together their separate effects. Hence each body 

 could, without great error, be regarded as disturbed by one 

 body at a time; the several disturbing effects could then 

 be added together, and a fresh calculation could be made 

 to further diminish the error. The kernel of Newton's 

 problem is thus seen to be a special case of the so-called 

 problem of three bodies, viz. : 



Given at any time the positions and motions of three 

 1, initially gravitating bodies, to determine their positions and 

 motions at any other time, 



Even this apparently simple problem in its general form 

 entirely transcends the powers, not only of the mathe- 

 matical methods of the early i8th century, but also of 

 those that have been devised since. Certain special cases 

 have been solved, so that it has been shewn to be possible 

 to suppose three bodies initially moving in such a way that 

 their future motion can be completely determined. But 

 these cases do not occur in nature. 



In the case of the solar system the problem is simplified, 

 not only by the consideration already mentioned that one 



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