INEQUALITY OF JUPITER AND SATURN. 



331 



We have just explained how Laplace demonstrated 

 that the solar system can experience only small periodic 



the arrows. Assuming that the mean motion of Jupiter is to that of 

 Saturn exactly in the proportion of five to two, it follows that when 

 Jupiter has completed one revolution, Saturn will have advanced 

 tnrough two fifths of a revolution. Similarly, when Jupiter has corn- 



R 



pleted a revolution and a half, Saturn will have effected three fifths of 

 a revolution. Hence when Jupiter arrives at T, Saturn will be a little 

 in advance of T'. Let us suppose that the two planets come again 

 into conjunction at Q, Q'. It is plain that while Jupiter has completed 

 one revolution, and, advanced through the angle p s Q (measured in 

 the direction of the arrow), Saturn has simply described around s the 

 angle p' s' Q'. Hence the excess of the angle described around s, by 

 Jupiter, over the angle similarly described by Saturn, will amount to 

 one complete revolution, or, 360 a . But since the mean motions of the 

 two planets are in the proportion of five to two, the angles described 

 by them around s in any given time will be in the same proportion, 

 and therefore the excess of the angle described by Jupiter over that 

 described by Saturn will be to the angle described by Saturn in the 

 proportion of three to two. But we have just found that the excess of 

 thes'e two angles in the present case amounts to 360, and the angle de- 

 scribed by Saturn is represented by p' s' Q' ; consequently 360 is to the 

 angle P' s' Q' in the proportion of three to two, in other words P' s' Q' is 

 equal to two thirds of the circumference or 240. In the same way it 

 may be shown that the two planets will come into conjunction again 

 at R, when Saturn has described another arc of 240. Finally, when 

 Saturn has advanced through a third arc of 240, the two planets will 



