D'ALEMBERT. 417 



that having originally taken the radius vector r, (the re- 

 ciprocal of u in our former equation.) = , 



1 cos. m v 



k 

 he now takes fully that reciprocal u or = 1 e 



cos. m v + Q cos. 7 cos. ( m ] v 4- cos. 



n \n / 



(2 \ / 2 \ 

 h m ) v % cos. ( 2m) v, terms obtained 

 n / \n / 



by the first or trial integration, which he had fully ex- 

 plained in his first Memoir to be the more correct 

 mode of proceeding, (' Mem.,' 1745, p. 352 ;) and the 

 consequence of this is to give the multiplier, on which 

 depenids the progression of the apogee, a different value 

 from what it was found to have in the former process. 

 It is never to be forgotten that the original investiga- 

 tion was accurate as far as it went ; but by further 

 extending the approximation a more correct value of 

 m was obtained, in consequence of which the expres- 

 sion for the motion of the apogee became double that 

 which had been calculated before. 



It should be observed, in closing the subject of the 

 Problem of Three Bodies, that Euler no sooner heard 

 of Clairaut's final discovery, than he confirmed it by 

 his own investigation of the subject, as did D'Alembert. 

 But in the meantime Matthew Stewart, (Life of Sim- 

 son, p. 137,) had undertaken to assail this question by 

 the mere help of the ancient geometry, and had mar- 

 vellously succeeded in reconciling the Newtonian 

 theory with observation. Father Walmisley, a young 

 English priest of the Benedictine order, also gave an 

 analytical solution of the difficulty in 1749. 



The other great problem, the investigation of which 

 occupied D'Alembert, was the Precession of the equi- 

 noxes and the Nutation of the earth's axis, according 

 to the theory of gravitation. Sir Isaac Newton, in the 

 xxxix. prop, of the third book, had given an indirect 

 2 E 



