CALCULUS OF PARTIAL DIFFERENCES. 55 



/ 2 \ 



2 m } v, terras obtained by the first or trial integration, 



V / 



which he had fully explained 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 depends 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 investigation 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 expression 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, Mat- 

 thew Stewart had undertaken to assail this question by the 

 mere help of the ancient geometry, and had marvellously suc- 

 ceeded in reconciling the Newtonian theory with observation. 

 Father \Yalmisley, 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 occu- 

 pied D'Alembert, was the Precession of the equinoxes and the 

 Nutation of the earth's axis, according to the theory of gravi- 

 tation. Sir Isaac Newton, in the xxxix. prop, of the third 

 book, had given an indirect solution of the Problem concern- 

 ing the Precession ; the Nutation had only been by his un- 

 rivalled sagacity conjectured a priori, and was proved by the 

 observations of Bradley. The solution of the Precession had 

 not proved satisfactory; and objections were taken to the 

 hypotheses on which it rested, that the accumulation of matter 

 at the equator might be regarded as a belt of moons, that its 

 movement might be reckoned in the proportion of its mass to 

 that of the earth, and that the proportion of the terrestrial 

 axes is that of 229 to 230 ; that the earth is homogeneous, and 

 that the action of the sun and moon ad mare movendum, are as 



