m V L. EULERI OPERA POSTHUMA. Anaiym. 



tumque sumta abscissa 



,, _j ',~I*T"T7yr[ \2(66-»-ftJfc) '^ ^66-*-JWi a(bb-t-kk) ^ Aibb-t-kk)*^ J' 



O^T) 



ii8 



ni ivimich mmii ai moi- ^j..^ arcus .^/p = ^^^ "*~ ^^ ^^ 



43. Coroll. a, Si hyperbola fuerit aequilatera , seu /c = 6 = 1 , poni debet n = 2 , Getque 

 a = 2l/3 et arcus rectificabilis Je abscissa prodit 

 9noHfjTf>'»<^ !>rff . r^y f^iiUh hmUm^^ y. ^, -i /t/3 -h "/(3 -*- 2 "/3) 





et ipsa hujus arcus longitudo reperitur 



j _ -/3 -f- -/(3 -4- 2/3) i /y3 -I- VjS -4- 2-/3) 



^e— . 3 V 2 



1 -f- "/(«'-H 1) 



kk. Copoll. 3. Si ponatur kn{n — 1)=5^, ut sit n = — — ^» signa radicalia cubica 



cx c^lculo evanescent; prodit enim 



a=y{2-t-ss-*-2V{s^-i-i)) = y{i --5-4-w)-i-y(i -h*), 



unde fit ( "*" 2 "*^ }^^~ c . . 



|y(l^_5)-H4y(l— 5-»-5*)±y(l — y55H-y{l-l-53)H-(l — :|5)y(l-H5)-4-(l-H|5)y(l— ^ 



y(l-»-«)H-y(l — *-4-m) dbl/(4 -s» -f- 41/(1 -♦-53) -t- 2 (2-s) /(l-t-») -»-2 (2-H») -/(i -»-hm)) 

 sive ee = ^^ y-. =r 



45. Coroll. 4. Pro hyperbola aequilatera, ubi /i = 2, si radicalia per fractiones decimales 

 evolvantur, reperitur CE= e= 1,4619354 et ^e= 1, 4248368«, seu Arc.^e=2,0830191, semiaxe 

 transverso existente CA=i, quos numeros ideo adjeci, quo veritas hujus rectificationis facilius 

 perspici queat. 



46. CoiroII. 5. Casus etiam satis simplex prodit si 5=1 et /i = — - — = 1 -h-kk, ita ut sit 

 k = V — r — > hinc enim fit 



Ergo sumta abscissa CE = y(l -i-l/3), erit arcus Je = -^ — — -• In fractionibus 



decimalibus fit /c = 0,45509, e=^l,65289 et Arc.^e= 1,81701. 



47. Copoll. 6. Si sit 5=0, quo casu fit /i=l et A: = 0, hyperbola autem abit in lineam 

 rectam CE, erit ee=3 et e = V3 = CE, arcusque Ae evadit =V^=CEy uti natura rei postulat. 



48. Problema 8. Invenire alios arcus hyperbolicos rectificabilcs. 



Solntlo. Sumta abscissa CE=e, capiantur aliae duae abscissae CP=p et CQ = q, ut sit 



eVH -f- pp) (\ -*- npp) -t- p V{1 -*- ee) (1 ■+• nee) 



q z= f 



l — neepp 



