DYNAMICAL PRINCIPLE. 



tion, the value of the third term above given in that equa- 

 tion 



, .-. 5 (as above given.) 



u a (h* + &c. N 



This omission he now supplied, and he found that the 

 result, when applied to the case, made the progression of the 

 moon's apogee twice as quick as the former operation had 

 given it, or nine years, agreeing with the actual observation. 

 He deposited, in July, 1746, with the secretary of the 

 Academy, as well as with Sir Martin Folkes, president of 

 the Royal Society, a sealed paper containing the heads of his 

 analysis, but delayed the publication of it until he should 

 complete the whole to his satisfaction : a most praiseworthy 

 caution, after the error that had been committed in the first 

 instance. He announced, however, the result, and its con- 

 firming the Newtonian theory, in May of the same year ; and 

 added, that his reasoning was purely geometrical, and had no 

 reference to vague topics, giving, at the same time, a con- 

 clusive exposition of Buffon's ignorance in his hot attack, 

 which showed him to be wholly incapable of appreciating any 

 part of the argument. In May, 1752, the Memoir itself was 

 given to the Academy, and it appears in the volume for 1748. 

 It is entitled, " De 1'Orbite de la Lune, en ne negligeant pas 

 les quarres des quantites de meme ordre avec les forces per- 

 turbatrices ;" which has misled many in their conception of 

 the cause to which the error must be ascribed. But in the 

 volume for 1748, p. 433, he leaves no doubt on that cause ; for 

 he states that having originally taken the radius vector r, 



k 



(the reciprocal of u in our former equation,) = , 



1 cos. m v 



k 



he now takes fully that reciprocal M or = 1 e cos. m v 



r 



2v /2 \ /2 \ 



+ 8 cos. y cos. m }v + d cos. f- m }v 4 cos. 



n \ n J \ n J 



