112 L.svEULERI OPERA POSTHUMA. ArMmettca. 



cujus radix ponatur ftn — ^y; hujus quadratum 



rAiwnol ?udiiip ,« OYq ioibq bsffltia Biajis 9ma'' 9a{bb — n) 



b^iVlH^htt^: "" • .u-jol ^;;;:/;!*^ iiui ^*^^ 



- 3a 9an a 9a« - W' • ■ ,. . , Zmaa . ■ 



unde y= — ^vr, ergo a" = — — -- .. Sin autem radix ponatnr oh — ^ y-^pyy-> ent 



, _,, 9mma* 3mpaaj/' . 



3maijy -l- my^ = 2bpyy h- -^^^ V!/ H ^ >- PPy • 



1 £ A » ai. 9mma* ^ 9am(66— n) 



Jam fiat ima =2bp-i — — = 2bp-i -^^^ 9 ergo 



lesdob A -,.q n.«^ boup ,^-. 26/, = 3m« - ?^=^^ = ^ -t- ^ , ergo 



466 4 466 ° 



nnm^l .inlooiCqiihJin -^ tdq ' 3^^ 



r~ »=»4 • 3ma 9amn 3ma / , 3n\ 



^tamii^^ . . »b 8*' 86 V bb J 



n i • 3paa ppy 

 Superest haec aequatio 1 = — 1 , ergo 



•*U4 HtU loJiribttUir »i"' ppy _ ^ 3paa _ _ 9ma' _ Wmn _ _ }_ 9n _ 27n 27nn 

 "»ir "^6 866 86* ¥"^866 m ~^ ^b^ 



1 9n 27nn m / , 18n 27nn\ 



= -8 -4^-*--86f' "^^« y^sppKr^-^^^^h^T 



(Lexell.) 

 Annofatio ad superiorem formulam mx^-*-n = n? wW ex daio casu ma^-f-n = bh ope tramformationis 

 elicuimus novum casum. 



Omni attentione dignum videtur, quod si n fuerit numerus quadratus =kk, ex casu cognito ma^-\-kk = hb 

 immediate duo alii elici queant hoc modo: Ponatur x=^az, ut habeatur ma^ z^-\-kk ^= O ■> hoc est 



{bb — kk)z^-V-kk = U==-yy-, ita ut sit {bb — kk)z^=yy — kk = (y-i-k){y — k). 

 Resolvetur ergo formula {bb — kk)z^ etiam in duos factores, quod duplici modo fieri potest: 



I. Sit unus factor {b-t-k)zz = y-\-k, eritque alter {b — k)z = y — k, haec aequatio ab illa subtracta relin- 

 quil (b-\-k)zz — {b — k)z=:2k, cujus una radix manifesto est ,2=1, unde pro altera fit 



— 2fc — 2aJfc 



5 ergo X 



b~t-k " 6-f-A; ' 



II. Sit jam prior factor {b — k)zz = y — k, et alter dabit {b-\-k)z = y-t-k, unde fit differentia 



'^"-'' {b — k)zz — {b-\-k)z = —2k, 



^'^' ,. 2A 2aft 



CMjus una radix est z=i, et allera z = et x 



b—k b—k 

 Unde conficimus istud egregium TheOrema: 



Si formula mx^-\-kk fuerit quadratum casu x = a, ita ut sit ma^-\-kk = bb, tum etiam quadratum 

 erit his duobus casibus: 



-2a* ^ - -V-2a* 



. primo: x=- — -5 et altero: x = -r — r-« 



b-t-k b — k 



Exempli gratia, cum formula 90ic^-f-1 fiat quadratum casu x=-- ; fiet enim 90'--H-i =-jj ubi est 



* — -g"' * — ^ ^*^*~T' hini casus derivati erunt ar= — — el a; = -i- — ; ex priore enim fit ' -f-1 = — ^ 



2 ' 8 4 



>10.8 

 81 • "^81 



et ex posteriore ^-f- 1 = l^ .+. 1 — 1^. 



5^ 25 25 A. m. T. I. p. 120. 121. 



