SEQUEL TO THE GENERALIZATION. 107 



sions and position of the Elliptical orbit which the 

 planet describes (D). Taking this view, he deter- 

 mines the secular changes of each of the elements, 

 or determining quantities of the orbit. In 1773, 

 Laplace also attacked this subject of secular 

 changes, and obtained expressions for them. On 

 this occasion, he proved the celebrated proposition 

 that, " the mean motions of the planets are invari- 

 able :" that is, that there is, in the revolutions of the 

 system, no progressive change which is not finally 

 stopped and reversed ; no increase, which is not, 

 after some period, changed into decrease; no re- 

 tardation which is not at last succeeded by accele- 

 ration ; although, in some cases, millions of years 

 may elapse before the system reaches the turning 

 point. Thomas Simpson noticed the same conse- 

 quence of the laws of universal attraction. In 1774 

 and 1776 Lagrange 5 still laboured at the secular 

 equations; extending his researches to the nodes 

 and inclinations ; and showed that the invariability 

 of the mean motions of the planets, which Laplace 

 had proved, neglecting the fourth powers of the 

 eccentricities and inclinations of the orbits 6 , was 

 true, however far the approximation was carried, 

 so long as the squares of the disturbing masses 

 were neglected. He afterwards improved his me- 

 thods 7 ; and, in 1783, he endeavoured to extend the 

 calculation of the changes of the elements to the 

 periodical equations, as well as the secular. 



5 Gauticr, p. 104. * Ib. p. 184. 7 Ib. p. 196. 



