A^mMnK .fragmenlaexAdver^sdrtisdepromla. 205 



III. Si « = 23, «umatur ar=156« et y;=133*, erit a;y (a?*— yy)=23. 156*. 133*. 289. 42025. 



IV. Ut a = *l, capialur ae = 21» et y = 20*, erit ary (apo? — yy) =*1. 21». 20* 29», 



V. Ut fiat a = 31 , sumatur a; = 40* et y = 9*, fit enim 



^ - a?y(a;a; — yy) = 31. 9*. 40*. 7*. 41». ^*^ 



VI. In genere si capiatur x = \pnqq et y=ivp — qq]^, tiel , • ., . 



" ^^ ■ ^^ ^^' iiiiiiiini»» fi)i V )'j «) fa. Iuifty». iiotjp 



Unde si sit 2pq-i-pp — qq = aa, fit a = 2p7 — pp-^qq, al illa formula 2pq-\-pp — qq fit quadratum «umendo 



' Vn. Deinde vero si «umatur -r = (^pp ■+■ ^^)* et y =^ {2pp - qq)\ fiet ''*'** '**" ,AA^ = Wt^-i~)V> »i» 



iFy(a?a; — yy)=8jip.9^(8p*-t-29*) n=(4p*-H7*)D 



Utlde tit *- an = kp*-\-q*={2pp-^2pq-\-qq){2pp — 2pq-\-qq). ' ' 



iT j . /. •» rt rt . • ^ f^ . .1. . .- !■> 5'> »^^» — rt -- ". i'l»ii'jitiii'i 



Unde si luent 2pp -4- 2p9 -4- 97 = C » lunc ent a = 2pp—2pq-\-qq. At lUud evenit 



fii j) = 2r« et q = rr — ss — 2rs, unde fil a=pp~t-{p — qf. 



VIII. Ex casu VII, fii p = 5 et q = T, capialur a; = 99* et y=l, erit a = 29 et a?y(a-ar— yy)=2dn; 



vel s\ capiatur a; = 29.l3* et y = 70» '«n — 



uiHit .JirmiiiiM.» .iiji».» »mi'.''».^ UiWMt: ulidii! ■ j^^ p,^ j^ j^ p^ <^\\ 22^ 



39. 



{Lexell.) 

 Problema. Invenire nnmeros a?, y, z, ut fiat axx-^3yy = yzz, siquidem cognitus fuerit casus 



(•*r»i| •♦•|jyit>is>4li lo.im'»» tni( <^lf •+• ^99 =yhh. i«03tfT 



SoLUTio. Statuatur axx-t- ^yy = {aff-t- ^gg) {app-i- ^qq)^, tum enim erit . _ d = ^j^*^ -*- xT.it it» Ju 

 axx-^§yy = yhh{app-\-^qq)^, sicque erit z = h {app -¥- (iqq); oiTkarnAovia 

 illud autem hoc modo per factores praestetur. Sit xVa-t-yV — /? = (^Va-i-yV — /?)(pVa-4-7V — /?)*, tum enim 

 sponte fit a;Va — yV— /? = (/■ Va—^V--/?)(p Va — jV— /?)*, quarum formularum productum ipsa est aequatio 

 supposita. Prior autem evoluta dat ' 



a: Va -I- y V— /? = a/p;> Va — pfqq Va ^r 2/?yp^ Va 



.+,agppy—§ — PgqqV—§-\-2afpqV—p 

 X = f{app—^qq)^2(igpq 

 y = g{app — ^qq)-t-2afpq 

 ac tum erit z=h{app-¥--Pqq). 



Verum haec solutio nondiim est generalis, eodem modo enim ponere potuissemus 



axx -\-pyy = {aff-h- ^gg) {pp-+- apqq)^, 

 unde fit z = h{pp-i-a(Sqq). Pro hoc ergo casu statuatur 



xVa-t- y V-^ — {fVa -*- y V- /?) (p -♦- 9 V-a^)», 

 cujus evolutio praebet x = f{pp — a^qq)—2g^pq 



y = 9{PP — a^qq)-\-2fapq. 

 Verum ne hi ambo quidem casus solntionem praebent generalem, cum sine dubio ejusmodi casus dentur, qui- 

 bus z non per h fit divisibile, quare pro solutione generali statuatur 



axx -\- /?yy = {aff h- ^gg) {anpp -\- /?n qq)^ , unde fit z = hn {app -\- fiqq) , 

 ubi forte n potest esse fraclio denominatoris h. Statuatur igitur 



a;Va.+ yV-/3 = (/-Va-i-j7V-/?) (pVan -t- ?V- |ffn)», MimUnP. 



cujus evolutio praebet 



