51 



ultim6 s/Jg—s/ld : ,/m—s/YL : : b—b" : b'-b". Now 

 there is nothing in the Newtonian solution by which it can 

 be shewn that this ratio is a ratio of equality. It may or 

 may not be so; and, therefore, from this step, the Newto- 

 nian investigation ceases to be supported by demonstration. 



Let us suppose that ultimc!> 



C/— CF : CF : : m {^JG—^JI) : ^Tl 



If m be found to be unity, then Newton's proportion is 

 accurate, otherwise not. 



Proceeding with this corollary, in the manner of Newton, 



we have 



C/— CF : CF : : m (FG—M) : CG +hL 



Also for Cor, 2. 



2HF : CF : : vi (FG— H) : 2FG 



CF : 2FG : : CF : 2FG 



Therefore 



2HF : 2FG : : m (FG—M ) CF : 4FG"- 



And hence Resist. : Grav. : ; mS^l-^Q' : 2R2 



But according to the corrected Newtonian solution, as 



given in the second and third editions of the Principia, as 



given also by Lagrange, and as is likewise shewn at the end 



of this paper. 



Resist. : Grav. : : SSyn-Q- : 4R' 



Hence m = i, and therefore ultimo 



\/Jg—\/ld : V'FG— v/I/ : : 3 : 2, 



instead of the ratio of equality assumed by Newton. 



The ultimate ratio of ^/fg — \/kl : v/FG — \/kl, or the 



ratio m : 1 may also be fluxionally investigated as follows: 



H 2 Let 



