■470 mmVMrvL. EULERl OPERA POSTHUMA. Analysis. 



5. Froblema 1. Dalo arcu elllptico Be in vertice B terminato, abscindere a quovis puncto 

 dato /* alium arcum fg^ ut eorum differentia fg — Be geometrice assig;nari queat. 



ISolutio. Sint abscissae datae CE = e, CF=f et quaesita Cg = g, erit Arc.Be = IT.e et 

 Arc, fg = 11. g — U.f; ut igitur arcuum fg et Be differentia fiat geometrica, necesse est, ut sit 

 n.e — (n.g — /7./*) = quantitati algebraicae. Hoc autem, ut vidimus, evenit si 



eV(l-fr){l-nff)-*-fV{l-ee)(i-nee) ^ 



,n- '^ .t„' ,<-..- ♦> ^ \-neefr 



. ' Quodsi ergo abscissae CG=g hic tribuatur valor, erit Arc.Be — Arc.fg = nefg, posito scilicet 

 CA=i et CB = k, atque n=i—kk. Q. E. L 



6. Coroll. 1. Poterit etiam a puncto dato f versus B accedendo ejusmodi arcus fy abscindi, 

 ut difFerentia Be — fy fiat algebraica. Posita enim abscissa CT=y capiatur 



fV(\ — ee) (i — nee) —eV({— ff) (1 — nff) 



i—neeff , 



eritque Arc . Be — Arc . fy = nefy. 



T.^^^CopoW. !l. Erit ergo quoque 'k'fciaum fy et fg differentia geometrice assignabilis; habe- 

 bitur enim Arc. fy — Arcfg = nef{g — y). Est autem 



■ . ^ ^eV(^{-fr)(i-nff) 



^ ^ 1 neeff ' 



• sive cum sit 2fgy(i — ee) (i — nee) = ff-\~gg — ee — neeffgg et 



-4-2/Vy(l — ee){i — nee) = ff-^yy — ee — neeffyy, erit 



• '' = X~^g «* 9-r = ^V{^-ff){^-'nff){ff-yg)(i-nffyg) 

 atque Arc.fy - Avc.fg=2nf(ff-^ yg)V(i -ff) (i --nff). 



Tjifji 8. CopoH. 3. Cum sit 



eV(\—fr)(i— nff) -*-fV({-ee)(i— nee) 

 n _ _ _ — . 1 . — , 



«^ i — neeff 



erit ' V(i s^ V(i-ee)(i-fr)-efV{i-nee)(i-nfn ^ 



^ ^ "">' 1 —neeff 



^ ""^ i—neefr 



hincque , ^ .y(|-«)(,-„f).HMl-f)(1-««) 



(, .V{i—gg) i—ee—fr-t-neefr 



-('i^- V{A — ngg) _ V(i - ee) (1 - rfee) (i - ff) {i - nff) -t- (j - n) ef 



>^(i— flfl) i—ee — ff-t-neeff 



gV{i—n gg) e (1 — 'inff-h- nf*) V{i — ee) (1 — nee) -4- /'(l — SJnge-nng*) -/(1 — ^ (1 — n^ 



V{i—gg) ~ (1 - e6 — ff-t- neeff) (1 — neeff) 



y/| nn)(i j^ \_ ^f{^n^ee-i-ff)-[n-t-\){i-^neeff))-*-(i-^neeff)V{i-ee){i-nee)(i-ff)(i-nff) 



1 



