106 HISTORY OF MECHANICS. 



place, in 1787, showed that this inequality arose 

 from the circumstance that two of Saturn's years 

 are very nearly equal to five of Jupiter's. 



The problems relating to Jupiter's Satellites, 

 were found to be even more complex than those 

 which refer to the planets : for it was necessary to 

 consider each satellite as disturbed by the other 

 three at once; and thus there occurred the Problem 

 of Five Bodies. This problem was resolved by 

 Lagrange 3 . 



Again, the newly-discovered small Planets, Juno, 

 Ceres, Vesta, Pallas, whose orbits almost coincide 

 with each other, and are more inclined and more 

 eccentric than those of the ancient planets, give 

 rise, by their perturbations, to new forms of the 

 problem, and require new artifices. 



In the course of these researches respecting 

 Jupiter, Lagrange and Laplace were led to consider 

 particularly the secular Inequalities of the solar 

 system; that is, those inequalities in which the 

 duration of the cycle of change embraces very 

 many revolutions of the bodies themselves. Euler 

 in 1749, and 1755, and Lagrange 4 , in 1766, had 

 introduced the method of the Variation of the 

 Elements of the orbit; which consists in tracing 

 the effect of the perturbing forces, not as directly 

 altering the place of the planet, but as producing a 

 change from one instant to another, in the dimen- 



3 Bailly, Ast. Mod. iii. 178. 



4 Giiuticr, Prob. de Trois Corps, p. 155. 



