4.52 L. EULERl OPERA POSTHUMA. Anaiysis. 



XXII. 

 De eoiiiparatioue areiiiiiii eiirTariiiii irreetifieabiliiiiii. 



Sectio prima 



continens evolutionem hu^us aequationis: 



I. 



Si ex hac aequatione sigillatim utriusque variabilis x e.t y valor extrahatur, reperietur 



—p — 8x-i-V((i^ — ay-+-'i^{8—y]a;-i-i88 — yy)xx) 



y _ . , 



— p — 8y-V(^^ — ay-i-^^[8 — y)y-t-[88 — yy)yy) 



X 



y 

 Ponatur brevitatis gratia /9/9 — ay = Jp,/3{d — y) = Bp et 88 — yy = Cp^ eritque 



^ ■^y'y-^8x = -\-y[A-^2Bx'-*-Cxx)pj 

 /3-\-yx-^8y = — y{A-\-2By-\-Cyy)p. 



11. 



Litteris jam A, B, C pro lubitu assumtis, ex iis litterae «, ^, y, 8 eX p sequenti modo de- 



finientur: Primo ex acqualitate secunda fit 8 — ;/ = — , qui valor in lcrtia 8-\-y=^-~ substitutus 

 dat 8-\- y = -^i ita ut sit 



"^—25^2/3 ^* ^ — 25 2/J 

 Flinc autem aequalitas prima abit in hanc 



8—y 



^fi-^^-^^^^^P 



j P • * §^(^B^-Ca) . , 



ex qua dcnmetur p = ^, __ -? mdeque porro 



^ ^(AC^-i-BB^ — BCa) (i^(AC — BB) 



BCiA^ — Ba) ^ ^ B(^A^-Ba) 



Sic ergo litterae a et /? arbitrio nostro relinquuntur, quarum altera quidem unitate exprimi poterit, 

 altcra vero constantcm arbitrariam, a coefficientibus A^ B, C non pendentem, exhibebit. 



