284 L. EULERI OPERA POSTHUMA. Anaiysis. 



Aequatio haec octo dimensionum resolvatur in binas biquadraticas, quarum alterius radices sint -+-/), 

 -f-g, -f-r, -+-5, alterius autem — /), — g, — r, — 5, quae sint 



Z*-i- «Z^-i- ^Z^-H 7ZH- 5 = 0, 



in quibus erit per naturam aequationum 



a =p-^-q-t-r-t-s 



^ = pq -t- pr -t- ps -*- qr -^ qs -t- rs 



y = pqr -\- pqs -\- prs -\- qrs 



8 = pqrs. 



Quoniam igitur productum ex his duabus aequationibus biquadralicis illi aequationi octo dimensionum 

 aequale esse debet, erit 



Q = j3^—2ay-t-2d 

 R = f^2^d 

 ^'^ S = d^ . . 



Et cum sit ci=p-i-q-+-r~t-Sy erit a=2t, ideoque a^=htty unde fit 



a^—2/3 = ktt^2/3 = P=ktt-—J, 



ergo /? = -g-' Secunda aequatio Q = ^^ — ^ay-t-^o dabit 



^f^^Att-t-B^^ — kyt-^^by 



sive B=^t' — ^Att-^2yt — ^-^^- 



Tertia vero aequatio ii = y^ — 2^h praebebit 



kt^—dAt'-t-2Btt^C=y^ — Ad, 

 sive Ad^ — kt^^-t-^At^—^Btt-t-C-t-y^ 

 hinc cum superiori fit 



kt^^^A^tt-t-^Ayt — — ] 



1 



Extracta radice quadrata obtinebitur 



y=At±:y(^kt^-t-{2B — ^A^)tt^^A^-¥-^AB—C^ 

 hincque 



d = 3t'-i-^Att — ^A^-i-^Bzt:2tV(kt^-i-(2B-^jA^)tt'^^A^-t-~AB^Cy 

 cujus awadratum erit 



