170 SECULAR VARIATIONS OF THE ELEMENTS OF 



5. It thus appears that the mean motion of the nodes of Jupiter and Saturn, on 

 the invariable plane, are exactly the same. This indicates a relation of a perma- 

 nent character between the positions of the nodes of these two planets, the nature 

 of which we shall now examine. 



If we divide equation (547) by (548) we shall obtain an equation similar to (147) 

 and (148). And if in this equation we substitute the numbers corresponding to 

 Jupiter and Saturn, we find that the mean places of the nodes, on the invariable 

 plane, will be the same if N 7 17 and iV 7 r have the same sign, and that they will dif- 

 fer by 180° if N/ r and N 7 T have different signs. The computed numbers show that 

 the signs of these two quantities are different: it therefore follows that the mean 

 longitudes of the nodes of Jupiter and Saturn, on the invariable plane of the 

 planetary system, always differ by 180°. We shall find, by the analysis of Chapter 

 I, § 21, that the actual place of Jupiter's node may differ from its mean place to 

 the extent of 19° 38', while that of Saturn can deviate from its mean place only to 

 the extent of 7° 7'. It therefore follows that the longitudes of the nodes of these 

 two planets can differ from 180° to the extent of 26° 45'. Their nearest possible 

 approach is therefore 153° 15, while their present distance apart is 166° 27'. 



We shall also find that the actual place of Mercury's node can differ from its 

 mean place to the extent of 18° 31'; while the nodes of Uranus and Neptune can 

 respectively deviate from their mean places to the extent of 6° 0' and 9° 40'. 



