De comparatione arcnum nirrarnm irrecfiflcahthnm, 469 



(Fig-. 57) y4CB quadrans ellipticus, cujus altcr scmiaxis C^=iy altcr CB = k, Eritque posita 

 abscissa quacunque CP = Zy arcus ei rcspondens Bp = fdzV—^^ — —• Sit brevitatis gratia 

 1 — kk = nj ita ut V/i denotet distantiam foci a centro C, hincque fiet Arc . Bp = f--~y7rzijz)~' 

 2. Reddatur formulae hujus numerator rationalis, ut prodeat ^ 



. „ r dz(i — nzz) U 'V. 



Arc ^Up — ] y(i _(„_^ j)^^_^„,4)' 



ad quam formam ut formulae superiores reducantur, poni oportet ^=1, C = — n — 1, E = nf 

 51=1, 6 = — rt, (§, = 0; quo facto habebimus pro differentia duorum arcuum ellipticorum 



fdx V^-^ - fdy yi^ = Const. -i- ncxy 



siquidem abscissa y ex abscissa x ita determinetur, ut sit 



c V(l — xx) (1 — nxx) — X V{i — cc) (I —ncc) 



y = i 5 



•^ ' 1 — nccxx 



sive = — cc -\- XX -\- yy -i~ 2xy "/(1 — cc) (1 — ncc) — nccxxyy. 



3. Denotet 11. z arcum eUipsis abscissae z respondentem, ac nostra aequatio inventa hanc 

 induet formam 



n.x — /7. j = Const. -H/icaT*, 



posito autem a5 = 0, fit ;y = c, unde Const. = — II. c. Ergo 



n .c-^n .X — n.y = ncxy. 

 Sin autem sumto y(l — cc) (^ — ^^^) negativo, ut sit 



cV (l^acx)(l—nxx)-t-xV(i — ce)(i—nec) 



i — nccxx 



fiet n.y — n.c — n.x = — ncxy^ sive H.c — {Il.y — II.x) = ncxy, ut ante. 



k. Temae autem quantitates c, cc, ;y ita a se invicem pendent, ut habita signorum ratione inter 

 se permutari possint; si enim ad abbreviandum ponatur 



y(l— cc)(l —ncc) = Cy -V{i—xx){i—nxx) = Xy y(l — >y) (1 -— /i>r)= 5^, 



eX-*-xC yC — cY yX—xY 



erit r = -. > X = :: ) ^* — 



1 — nccxx 1 — nccyy i — nxxyy 



ex quibus per combinationem eliciuntur sequentes formulae 



yy — xx = c {yX-*-xY) xX-^yY={ncca!y-+-C){yX~*'xY)y 



yy — cc = £c {yC -i- cY) cC — xX= {ncxyy — Y) {xC — cX), 



XX — cc = y{xC — cX) cC-»-jrF= (ncasocy-HX) (jC-i-cF) 



ac denique 



^ocyC^xx-i-yy — cc — necxxyy 



2cyX=cc -i-yy — xx~~ nccxxyy 



— 2cajF=cc H-£caj — yy — nccxxyy. 



