488 L. EULERI OPEIU POSTHUMA. Analysis diaphantea. 



auae reducitur ad hanc formam — — — r— = n , ita ut habeatur haec conditio 



^ (za — 0) {60 — 4a) 



X (2a — h) (36 — ha) {aa -+- bb) = d . 

 NoTA. Si N*^ III posuissemus d = — a, habuissemus cc — bb -^ ac — cf6==0, quae per c — b divia 

 praebet c -i- 6 -+- a = 0, sive c = — a — 6, unde porro fit 



p =: (a H- hf- -H aa = cc -I- dd, q =. — (3a h- 6) (a -i- 6) , r = s = — {aa — ab — bb) 

 p H- s = (26 -*- a) (6 H- a) , p — s = a (3a -t- 6) , q-v-r = — a (4a -t- 6} , q — r = — (2a h- 6) (26-i-a) 



Unde quadratum o.sse. debet haec forma 



X {aa -*- 66) 



— (3a -H 6) (a -+- 6) (26 ■+- a) (6 -+- a) a (3a -i- 6) a (4a -+- 36) (2a h- 6) (26 -h o) 

 sicque quaestio reducitur ad hanc formam — a (aa -h 66) (2a h- 6) (4a -f- 36) = n . Hic imprimis notalu dignum 

 occurrit, quod per positionem tertiam, qua fecimus p = cc -^h dd, praeter expectationem, quatuor paria simpli- 

 cium factorum ex calciilo discesserunt. 



Conditioni tertiae p = cc -i- dd sequenti modo generaliter satisfieri potest : Quum sit 



cc -I- dd =: aa — ac -+- bb — bd, erit cc -h ac — aa = bb — bd — dd, sive 



(2c H- a)* — 5aa = (26 — d)^ — 5dd, sive (2c -i- a)^ — (26 — rf)« = 5 {aa - dd) et 

 (2c H- a — 26 -*- rf) (2c -»- a -4- 26 — rf) = 5 (a -+- d) (a — d) = 5mntu ; 



unde colligitur 2c -i- a -- 26 -t- rf = wtt, 2c-*-a-i-26 — d=5mt et a~t-d=mu et a—d=nt. Ex his concludilur 



mu -*- nt , mu — nt , 



a = -^ et d= — ^, 



inde vero 4-c -i- 2a = 5mt ■+- nu et 46 — 2d = 5mt — nu, unde fit 



(5«i — n) t -t-(n — m) u , , (om — n)t — (n — m)u 



c = — 7—^^ — et b = 



4 4 



Hinc p = [5 (5mm — 2mn -+- nn) tt — 2 (5mm — 2mn -+- nn) tn -+- (5mm — 2mn -+- nn) uu] k 16 



q = [ — 5 (5m#fl — 2tnn -+ nn) tt +- & {5mm — 2mn -t- nn) tu — (5mm — 2mn -f- nn) uw] : 16 



5mm — 2mn -+- «n ,„ „ 



sive p = j^ (»« — dtu -t- uu) 



(5mm — 2»nn -H nn) (5mm — 2mn -♦- nn) , , ,^ 



q= — ■ ^ (5« — &tu -t-uu) = — ^ — -' {t — u) {5t — u) 



5mm — 2mn -t- nn , ^ , 



r = s =0= — ( — a// -I- uu). 



lo 



„. , .^ .. . (5mm — 2mn -I- nn) _ 



Sit brevitalis gratia ^^ — = C, ut sit 



lo 



p = C (5« — 2/M -*- uu) , q = — C {t ^ u) {5t — u), r = s = C {— 5tt -t- uu) 

 eritque p -\- s = — 2Cu {t — u), p — s = 2Cl {5t — u) 



7 -H r = — 2Cw (3/ — w) , q^r = 2Ct {5t — 3m) 

 aa -\- bb = C {5tt -\- 2tu -+• uu) ; 



quare formula quadratum reddenda est 



X (5« -+- 2;m -t- uu) 

 {t — u) {5t - m) . — 2« (< - m) 2f {5t ~ u) . — "2m (3< — «) 2« (5^ - 3m) ' 



quae reducitur ad hanc conditionem : A (5« -i-2/m -+- mm) (3/ — - w) (5/ — 3w) = n . Statuatur u = t- — t, fielque 

 X (4ff H- w^ (4.« — t^-) {St — 3i^) = n ; seu posHo 2< = w erit X {ww -+- i^i') {2w — u) (4-iv — 3^^) = n : quo facto 

 habebitur p = ivw -h (iv — ^^)*, q = {w — i^) (3w — v^) et r = s = t^p- — p-w — ww. 



Quae soluMo cum praecedente prorsus congruit, ex quo patet lllam solutionem multo esse generaliorem. quam 

 initio videbatur. 



