50 



" BC ; (ut mos est) et valor ordinatim applicatae resolvatur 

 " in seriem convergentem : Problema per prinios serei termi- 

 " nes expedite solveretur: * * * 



a * « » * « « * 



" * * * Si designetur series universaliter 



" his terminis tQo — Ro- — So^ &c. erit CF sequalis v/o-+Qo" 

 « * * YG—kl ajqualis 2So\ * * Deducendo 

 " igitur Problema unumquodque ad seriem convergentem, 

 " et hie pro Q, R, S scribendo terminos serei ipsis respon- 

 " dentes; deinde etiam ponendo resistentiam medii in loco 

 " quovis C esse ad gravitatem ut S\/H-Q- ad 2R" * 

 " solvetur problema. » # * 



Now with respect to this solution it may be remarked, 

 that in Cor. 1. it is stated that 



fC : kC :: \/fg seu v^PU : ^B et _ 



divisim JTc : kC id est C/— CF : CF : : ^J^—Jkl : Jkl 

 But although ultinic!> ^fg : ,JVG is a ratio of equality, it 

 does not follow that ultimo v^FCj — ^kl : ^fg — s/kl is a 

 ratio of equality. It is easy to see, that ultim6 fg : kl is a 

 ratio of equality; and, therefore, it by no means necessarily 

 follows, from the method of prime and ultimate ratios, that 

 if ultim^ ^fg : ^t'G is a ratio of equality, that ultimo 

 ^FG — s/kl : ,/fg — ^/kl is also a ratio of equality. Had 

 ultimo J'g : kl not been a ratio of equality, then 

 v^FG — i^kl : ^fg — ^/kl must necessarily have been a ratio 

 of equality. Because ultim6 >Jfg, v/FG, ^kl,. are all 

 equal, we may represent them by ao-\-ho"--'r &c. ao-\-'bo--\- «S:c, 

 fio-\-"bo"-\- kc. where o may be diminished indefinitely. Then 



ultimO> 



