a52 L. EULERl OPERA POSTHUMA. Arithmettca. 



Ita fiumto s=l et < = T jpro *Gasu > == 2 eritj[)==7t, g = 85, <k = 71, y = 20. Hic loco p et gr eorum 

 semisumma et semidifferentia sumi possunt fietque p = 2.3.13 et g = 7. Hinc enim fiet 

 ^^, .^ ,^ pqipp — qg)=z2. 3.5.7. n.\7. 11 et a;y(^a;—i/t/) = 3.5.7.13.17.71. 



Sin autem sumatur w= — 2 manente 8 = 1=1, prodit p = 9, 0=5.9, sive » = i et o = 5, sive etiam 



,,■■;■:: ;:- •:-!';./,: .:,>■•")';;:«<.> • ._ ^> , - ■ ,■*.,;.,, :.r^- . . ;; -■,• \rf ';,aiVii!r nuuuiii .:'ii; / 



»j=3 et q = 2, tum vero eril x = d, t/ = 4.9, sive a?=l et w = 4, tum enim erit ... ,, . , . 



pgf (pp — g^g) =2.3.5 et xy{xx — i/i/) = 3.5. 



A. m. T. III. p. 121— 123. 



6) Quaestiones ad resolutionem plurium aequationum ducentes. 



^ (TF. £. Krafft.) 



' " 1 



Problema. Efficere, ut fiat x-t-y-i-z = a et xyz=.i, ideoque z = — . 



Sequentes solutignes particulares prodierunt 



, 1 9 4 



I. x = ^, y=^, z = - 



„ 2 50 81 



II. x = -^, y = -, z = ~ 



Isv ^i^qi l9V jitoHq liigflM&r i!/ ii^j^ __ 13 t _^147- . -. > 392* ..wioD 



.•.JsoU ^r^-ijinoa filinkj -Uid y, i;: ?92 , ; f .,.;[;. 39^;, ,.,..,, ^ -, 9'13 .;fqiif> minoo 



IV. x = -~. 



2 18 25 



z 



5 ' ^ 5 ' ^.36 



V. aj=~2, v = 



2^ _ 9. 



9' ^— 4 



VI. ar=i, ^ = A, r.-^ = 72. 



r 



o 



1 



Si ratio a; ad v sumatur a.b, et ponatur a) = ma et y = mb, erit .s= r, unde fieri debet 



m(a~t-b)^ — = n, sive m'(aHht)H — r = Q» 



seu *n*aa6fc (a-»-fc)-f- a6 = n. 



A.m. T.I. p. 121. 



' " ' (Lexell.) 



Ut formulae aa~\-Mbh et aa-\-Nbb quadrata reddi queant, posito M=mn caipiSLUir N= {app-t-m){aqqHr-fi), 

 «icque pro dato numero M infiniti valores idonei pro iV reperiuntur, veluti si M=l, ideoque m = fi = dzl, 

 erit N={appzizl){aqqz*zl). Si p = l, erit N={a±z l){aqq±il), cujusmodi formulae sunt 



pro a=l, N=2{qq-h-l) 



-^-^'^ M^—^in^^f^ii^ ' a=:2, N^^i^qq-^l), ^qq — l :* Midit" i 



iqio Mi mfiwp ^filiaoq rn^jtt_gip., jy_. jj. (3^^^^ ^j^ 2(3yj — 1) '^'* *^^* 



.ivU««»n J«»aI)Oiq i»ra6il9 JiH)wtwbiafej(V/ij:^_^ 3(^j-l). 



