370 HISTORY OF MECHANICS. 



tions which succeed in the Lunar Theory fail here. Artifices somewhat 

 modified are required to overcome the difficulties of this case. 



Euler had investigated, in particular, the motions of Jupiter and 

 Saturn, in which there was a secular acceleration and retardation, 

 known hy observation, but not easily explicable by theory. Euler's 

 memoirs, which gained the prize of the French Academy, in 1748 and 

 1*752, contained much beautiful analysis ; and Lagrange published also 

 a theory of Jupiter and Saturn, in which he obtained results different 

 from those of Euler. Laplace, 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 neces- 

 sary 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 in- 

 clined 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 dimensions and position of the El- 

 liptical orbit which the planet describes. 5 Taking this view, he deter- 



5 Bailly, Ast. Mod. iii. 178. 4 Gautier, Prob. de Trois Corps, p. 155. 



5 In the first edition of this History, I had ascribed to Lagrange the invention of 

 the Method of Variation of Elements in the theory of Perturbations. But justice to 

 Euler requires that we should assign this distinction to him ; at least, next to New- 

 ton, whose mode of representing the paths of bodies by means of a Revolving Orlit, 

 in the Ninth Section of the Principia, may be considered as an anticipation of the 

 method of variation of elements. In the fifth volume of the Mecanique Celeste, livre 

 xv. p. 305, is an abstract of Euler's paper of 1749 ; where Laplace adds, " C'est le 

 premier essai de la mcthode de la variation des constantes arbitrages." And in 

 page 310 is an abstract of the paper of 1756 : and speaking of the method, Laplace 



