254 L. EULERI OPERA POSTHUMA. Anthmettca. 



quae commode per x — y divisa fit {dd-^i){x — y)^2c{x~^y) = 0, unde reperilur . . .. 



X _dd-+-2c— 1 r oHfi? r 



~ir dd — 2c — l' 



Sumatur ergo x = dd-t-2c — l et y = dd—2c—l, fietque 



AH-i? = 2(<W— 1) et >l — ^ = W, ergo A^dd-^^cd—l et B==dd — 2cd—i. 



Haec autem solutio tantum est particularis. '>fctej|«(jff < 



Problema. Resolvere has duas aequalitates -. 

 .iii; . 



^ XX ■+- 2axy ~h- nccyy = .4^^ et xx ^ 2bxy -+- nddyy = B"^. 



, SoLUTio. Eliminetur littera n, quaerendo A^dd—B^^cc^xx^dd—cc^-t-^xy^add—bcc). Ponatur nunc 



' i'^"0" J^ \^M)&^ ^^ ^Bc = x{d~t-c), eritque >lrf — 5c = a? (rf— c) -+-2^ (^^^^^) ? 



fJHmwl evifU mei; dd — h' dd—b 



unde erit 2i4<? = 2rfa? -i- 2t/ . — ^, unde fit A = x-i-y .—— — r, quo valore substituto erit 



^ d-t-c ^ d(d-i-c) ^ 



add — bcc /add — bcc\ * 



_, ^ add — 6cc /add — bcc\^ 



2ax -\- nccy = 2x . -— r -i- y /.^ - ) j 



^ d(d-i-c) ^ \d(d-t-c)y 



a? (add — 6cc) * — nccdd (d -\ 



y 2cd(d-+-c) (ad-i-6c) 



. . . , />, a? (add — 6cc)* — nccdd (d -*- c)^ . 



ijil jaiii/j iuu unde fit — = ^^ 7: ^-^ — -. uuji>4ia. 



A. m. T. II. p. 157. 



8i-t .q 



74. 



Resolutio aequationum 

 i :; , : XX -+- 2axy ~\- cyy = A^ et xx-i- 2bxy -4- dyy = B^. 



Cum sit A^—B^=2{a — b)xy-v-{c — d)yy, sumatur A— J5 = (a— 6) t/, erit 



A-\- B = 2x-i y, hinc additis quadratis erit 



2 (^.4 -I- BB) = kxx H- 4 . ^^ xy -h T^^) W 



-H(a — 5)*t/j/. 



Cum igitur 2{AA -\- BB) = h-xx -+ k- {a -\- b) xy -^ 2 {c -\- d) yy , inde sequitur 



4 .^^ic — 4 (a-+-6)a;-i- IC-^) -H(cf — ft)'^ — 2(c-l-d) j/ = 0, hincque fit 



X (c - d)' -«- (g - 6)^- 2 (c -^- d) (g — b)^ ^ 



y 4(a-6)(aa-66-c-i-d) ^j-j a,.)/i„^., 



Sumi ergo poterit «- = (6 — ^^''-♦-(a — ftj*— 2(c-f-rf) (a — ft)* et y = 4(a — 6) (aa— 66— c^rf). Haec solutio differt 

 ab ea, quae supra est tradita, ubi loco c et rf habuimus ncc et ndd, quod mirum non est, cum utraque solutio 

 tanlum sit particularis. 



Eodem etiam modo hae aequalitates resolvi possunt 



(2-t-a)«a;-i- (nn — 4) a;t/-l- (2 — a)j/j/ = 4* et (2-f-6)a;a7 -♦- (nn — 4)a7t/-l-(2 — 6)yt/ = ^*. 

 Facta enim simili operatione, dividi poterit per x — t/, ac reperietur inde 



a: = n*— Snn-f- (a— 6)*-H2nn(a-»-6) et y = n*— 8nn-i-(a— 6)*— 2nn(a-*-6). 



In his autem formulis continetur casus supra tractatus, quando n = 2. 



A. m. T. II. p. 158. 



