De comparattone arcuum curvarum trrecttficabiltum. 477 



Sit e = — » ut habeatur a?* — 2x^~i~2nx — /i = 0, ac ponatur ad secundum terminum tollendum 

 X = j -♦- I , prodibit 



r * — -^yr -*-(2/i— i)r — ^ = o, 



cujus factores fingantur yy-^cey-^/S et yy — ccy-*-yj eritque 



/?-*-y = aa — — , y_^ = __ et ^y=—- 

 unde elicimus 



(/5 -4- y)* — (y — /?)=» = a* — 3 «^ H- - — '— — ^ = * ^y = — - , 



ideoque a®— 3«*-»- 3a*= (2/j — 1)^; 

 subtrahatur utrinque 1, ut cubus fiat completus . r : . :i ,. 



3 3 3 



(aa — 1)'= 4-/1/1 — 4/1, ergo aa=i-t-ykn(n — 1) = 1 — Y^nkk et « = l/(l— y* 



Invento ergo cc erit onoq •luJcD^a 



^ 1 3 (2n-l) ^ 1 3 (2n-l) 



1 _. ^//3 1 . (2n— l)v — aadby^Saa — a4±2(2n — 1)«) , r,.. 



mdeque y = -^-cc±y{j^-jaa±-^^) = ^ -' 



unde obtinetur e=^ r» Porro debet esse ^fg=pq-^qr-^rs, seu 



3/j = (l-4-e)T/i^. ideoque /j = ± (H- e) l^if^. 

 ex quo obtinemus 



ff -H ^r^f = ee ■+. y /icc ( 1 -H e) * . ^^ -i- 1 ( i -^ e) ( 1 — ce). 



Cognitis igitur valoribus fg et ff-^-ggt seorsim abscissae CF = f et CG = g reperientur, quae 

 arcum determinabunt fg praecise subtriplum totius quadrantis j4B. Q. E. I. 



Conipapatio apcaum Hyperbolae. 



32. (Fig. 59). Sit C centrum hyperbolae, cujus semiaxis transversus Cj4 = k, et semiaxis con- 



jugatus =1. Hinc sumta super axe conjugato a centro C abscissa quacunque CZ = Zy erit appli- 



cata Zz = k V(i •+- zz) , unde 



arrus Jz — fdz i/ i-^a-^*^)« _ f dz H -*- {i ^ kk) z^) 



arcus yiz — jaz Y ^ ^ ^^ — y^^j _f- (2 -*- W) «« ^ (1 -#- kk) x^) Uk^-io j» .u i. 



33. Ponatur brevitatis gratia 1h-M = /i, ita ut /i sit numerus affirmativus unitate major, 

 eritque arcus hyperbolae quicunque 



JV(l-*-{n-*-i)zz-+-nz*) 



Poni igitur in § XI oportet ^=1, C = /i-h1, E=n, 01=1, S = /i et @=0. Unde si fucrit 



c V(l -»- xx) (1 -4- nxa;) — xV{i -t- cc) (1 -h ncc) 



y =: 



1 — nccxx 



ibebimus fdx V Jl!^ — fdy y ~*~**^ — Const. — ncay. 



J i-*-xx J "^ l-*-W 



w Ki> I0p 



