57 



" de faire, en doit voir que, puisque les valeiirs de o et de 

 " Qo — Ro"' — So^ — &c. sont exprimees en s6ries qui pro- 

 " cedent suivant les puissances de d, il n'est pas permis de 

 *' pousser rapproximation au-del^ de cette nifime puissance 

 " dans I'equation resultant de relimination de o: d'ou il suit 

 " que le terme que contient ^' dans cette equation, dont 

 " necessairement ^tre incomplet; et puisque c'est de ce 

 " m6me terme que depend le rapport claerche de ^, on en 

 " doit conclure que le valeur trouvee de ce rapport est 

 " inexacte." 



Notwithstanding the remarks of the ingenious author, it 

 is not very clear, that the error of the result in this solution 

 must necessarily be the same as in that of Newton, if the 

 error of the Newtonian solution have been rightly pointed 

 out. These solutions have nothing in common ; and, there- 

 fore, as they give the same result, the error in each must 

 flow from a common source. In this solution of Lagrange, 

 he computes the increments of the ordinate and abscissa in 

 the time d, by supposing the resistance to act, during that 

 time, in the direction of the tangent; and thus the deviation 

 from the tangent in the time 6 is expressed by ^, depending 

 only on the time and force of gravity. In the Newtonian 

 solution, fg and FG, the deviations from tlie tangent in 

 equal times, are taken accurately equal, and therefore made 

 to depend on the force of gravity only. Hence, a common 

 source of error; and these solutions, so entirely different in 

 their progress, might be expected to produce the same result. 



M.- Lagrange concludes his observations on this problem, 



VOL. XI. I by 



