TO SEVERAL CENTRES. 193 



SxSE S 



5^pii' It was in consequence of this mvestiga- 



o M 3 o -ili 



tion that Clairaut for some time announced, as did also 

 Eulcr and D'Alembert, that there was a material error in 

 the Newtonian theory of the Moon^s motion. The error, 

 which afterwards was found to arise from their having 

 omitted the consideration of certain quantities, was acknow- 

 ledged by Clairaut three years later (Mem. Acad. 1748, 

 pp. 421, 434), but no one can read that paper without 

 feeling that the acknowledgment was too coldly made, after 

 he had gone so far as to suppose that the whole Newtonian 

 doctrine was overthrown, and to propose a new law of 



-| , the whole of this arising from his own error. 



r r* 



It is to be remarked, however, that the investigation of 1745 

 was in all respects most accurately conducted, and must have 

 led to the same result as in 1748 but for the supposition that 

 certain quantities might safely be neglected. Eve'n in 1745, 

 Clairaut, upon Newton's assumption of the excentricity of M 

 being nothing, comes to his conclusion that the proportion of 

 the axes is as 69 to 70. 



4. Legendre treats the subject very fully, as far as regards 

 two centres, and also confining himself to the forces being 

 inversely as the square of the distance (Exercises de Calcul. 

 Integral, part iv. sect. 2). He deduces from his analysis 

 several theorems, two of which he regards as very remark- 

 able. The first apparently strikes him in this light, because 

 it shows the same orbit to be produced by the combined 

 action of the two forces directed towards two foci, as either 

 force would produce acting on the body, and directed to one 

 of the foci. If V is the velocity at the vertex of the ellipse 

 which would make the body describe that curve when acted 

 upon by the force directed to one focus, v tie velocity at the 

 same point which would make the body desciibe the ellipse 

 when acted upon by the other force directed to the other 

 focus ; then if the two forces act together upon the body, and 



o 



