Of y I ocrvi J

IN SOLUTIONEM AHQUATIONUM ALGEBRAICARUMDISQ UISITIONES.

QUARUM PARTEM DECIMAM QUINT AM

CONSENSU AMPL. FAC. PH1L0S. UPSAL.

PR JESIDE

mag. JÖNS SVANBERG,EQUITK REG, ORD. DE STELLA POLART,

MATH. PROFESSORE REG. ET ORD.,

REG. SCIENT. ACAD. HOLM. ATQUE SOG, SCIENT. UPS.

REGI-3EQUE SCIENT. MILIT. ACAD. HOLM. ET SOC. SCIENT.PHILADELPH. MEMBRO, NEC NON REG. ACAD. AGRTC. HOLM»

REG. ACAD. PARIS. ET REG. SCIENT. ACAD.AMSTELiED. CORRESP.

PRO GRADU PHILOSOPHICQ

p. p.

PETRUS BENJAMIN SKÖLDBERG5 UDERMANN O - NERI Cl US.

IN AUDIT. GUST. DIE XVI MAJI MDCCCXXVII.

H. A. M. S.

UPSALIAE

EXCUDEBANT REGIJE ACADEMIJE TYPOGRAPHT,

IN

. SACRAM REGIAM MAJESTATEM

MAXIMJE FIDEI VIRO

MARESCHALLO AULICO,

CHILIARCHJE tOCUM TENENTI

REGII ORDINIS ENSIFERORÜM EQUITI

GENEROSlSSlMO DOMINO

AUGUSTO von FRANCKEN

Msecenati Summ o

v. d.

PETRUS BENJ. SKÖLDBERG.

) MW (

igitur 4ec* - 4Pa -+- 4Q = a2 - zPcc + P2 - u,

atque fic quidem3a2 - zPcc - P2 -h 4Q u = o;

fed ex ipfa hypothefiec* - Pcc2 4- Q# - ü? rz o,

igitur, fi per folitam exterminätioni.s metbodum exhisce novisfimis aequationibus ipfam eljiinnemus cc, pro-dibit utique tandem

11* - z(P2 - sQJii1 -+- CP2 -+ 4P'R - p*q? + 4q-- + 27

p! - 4pq + pR - p«tum vero a —P2 - 3« - 3»

atque « = P* - 4PQ. + fiR - (P* - .?Q)« .P-3*

§. XXXIV.

Quod II jam ifta ad aequationem accommodemusC120), cujus equidem radices funt ac 4. bd, ab 4. cd,et bc + ady atque ex qua

P = q, Q = pr - 4sy et R = p2f 4. r2 - 4^fiet adeo u zzr (nb —J- cd — bc — ad)2

= {a - c)2, {b - d)2Q

) *22 C

(,122%

z{P5 - 4PQ + pR - P»)206 = V =S ZZ

P2, - jQ. - 5«

2(4 * - 4pqr - 20^ -t-_pr4 -pp2s - qtijq2 - jpr -t- 12S - 311

t = pz - 2q - v

!p*q* - 4q* - 3p3 r + iqpqr - z^2V- 6p*s 4- rtqs - (JP2 ~ Sq)u \ < > > /-?2Z),qz - 3pr H- i2s - 311

tpiq* _ 4£3 - 3p?r -t- iqpqv - 7£r2lv - 6p*s -f- 16qs - (q2 - jpr + 72j)£ )

-- ♦ '

3V% - Sq - 3*

atque ipfa tandem u aequatione definietur«■* - 2(qz - 3pr 4-121)11* 4 (q* - 3pr+ 12s)*. n

[p2q2r2 - 8opq*rs - i28qzs2 - 6pzrzs - 27z*4,14« i^qr*s -f- 144p* qsz Hh i8pqr3 4 igp3qrsr \ 220.(123),

- 4p-3 r3 - igzprs2 4- 236s3 - 4

5 iså C

rnanerenrms, qusenam ei conveniret alterius radix,multoque lane maxime a vero aberraremus fcopo, fiunamquamlibet unius unicuilibet alterius adjungendamesfe ftatueremus radicem. Scilicet alteram lemper cumper alteram definire Jiceat, patebit adeo non nifi unamunius alteri alterius convenire radicem, ideoque vel neliciendam esfe ex formula (/22), fi t ex aequationedeterminata fuerifc (t/j?)* vel t ex formula (121), fi u exsequatione determinata fuerit (/2j). Determinatis au-tern rite n et t, quorum adeo per demonftrata tria da-buntur valorum fyftemata, ex quibus utrumlibet omni»no ad arbitrium adhibere licebit, prodibit ex aequa¬tione (z/7)

ex qua formula, prout in eadem fignum adhibeatur 4*vel obtinebuntur, pro diverfis ipforum t et u valoiribus, vel valöres functionum fyftematibus (A), {b) et(c) comprehenfarum, vel denique valöres functionumfyftematibus (fl), (B) et (C) comprehenfarum. Ideo¬que fi denique ponamus

pofito videlicet brevitatis caufa a = V - 7, cum fita —fl -f» c -J- d = p»

a -f- a b - c — ad = Asa - b -f- ctc - ad s= B, eta -{■ ab - ac - d = C,

) "4 (

fiet adeo (oc 4- z) b - (oc 4- z) c = A - B ,(« - z) f - (a - i) d zz A-C,

atque fic - (oc2 - i) b 4- (a2 - z) r = (oc - i) B - (oc - z)y^,- (ct2 - z) c -f- vos2 - i) d =: (os + z) C- (ot 4" z)

hoc eft 2[b - c) zz (oc - i) B - (oc - i) A,2(c - d) z= (cz-h i) C - (oc + z) yd? ;

atque per harum denique additionem2\b t- d) •+■ 0 C -f- (a — z) 5 - 2ccA;

fed et ulterius

2(6 -f- d zz p -f- A - B - C,

igitur 4*} ■— Z7 (' "" 2c^) -A *4* (os - 2) 2? 4" ccCy4C ^b 4- 2{CC - l) A - 2(C6 - l) B

zz p r- A -* ocB «+• ccC4d zz 4c 4- 2(00 4" z) A - 2(00 4- z) C

P Hr C! 4" 20c) A - «5 - (oc 4~2)4(6 4- c 4- d) = 3p 4- A - (a + 2) 5 4 (a - ,2) C

4a zz 4p - 4(b 4- c -J- d)= P - A 4" {oc 4* z) B .Hh (2 - a) C.

§. XXXV.Quod fi 33m /, t" ejt t'" ipfas exbihere ponaotur

faequationis (iip) rådiges, atque u" et u" liisce corre-latas aequationts (z2j) radices, ficebit utique pro A,

) "*5 ('

ipfis t' et u conveniente, quamcunque adeo Iibuerit ad-hibere radicem, five ex formula

Vt* - $u 4- - t2 ur five exKt2 - 8u - 4^411* - t2uy

prodeuntem; tum vero, utramlibet (modo eandem) ad-hibeamus formulam, patebic infuper quamcunque iti-dem quae ipfis t" et u" conveniat radicem pro B ad-hiberi posfe, atque fimiiiter etiam pro C quamcunqueadhiberi posfe radicem, quse ipfis conveniat t"' et n".Igitur pro dato quolibet ipfius A valöre quatuor et-iamnum diverfos pro B adhibere licebit valöres, fcili-cet vel ipfum adeo -4- B, vel -4- «5, vel - J5, vet- uBt atque fimiliter etiam pro C, vel ipfum -4- C,vel -f- ocC, vel - C, vel - ctC; atque fic denique hosomnes fi pro lubitu adhiberemus, fequentium omninoprodirent diverfa fedecim valorum a, by c et d indeorientium fyftemata:4a p — A -4»* (2 «4- otj B -f- (2 — ecj C ,4b zz p 4* (t - 20c) A 4* (ot> ~ 2) B '4- ocC, m /t\

^ * • »♦ v.1;:4C = p - A - uB 4- ccCy4d z= p 4|- (1 4- 2cc) A - ctB - (2 + cc)C>4a z= p ~ A - (1 2») B 4- (2 -

) »6 C4a == p - A - (et 2) B (2 - et)C 14b m p "4™ (f — 20o) A (et —1 2^} ß , 06 C I4.c zz p - A aB -{- ccC I4d =3/7-4- (1 zct) A coB - (2 -f- et) C4a == p - A - (20c - 1) B (2 - et) C4b =3/7-1- (1 - zet) A -4- (20t "4- 1) B + etC4c 3= p - A - B -4- ccC4d 3= p -f- (1 -+" 200) A - B - (2 -4- et) C4a =3/7 - A (2 et) B -fr» (2ct -4- O C4h 33 p -f- (1 - 20c) A -\r (et - 2) B - C4c 3= p - A - et B - C4cl — p -fr- (1 —fr- 200) A - coB - (200 — 1) C4a zz p - A - (1 - 200) B -f- (200 -fr- 1) C4b 33 p -f- (l - 2ct) Å - (1 + 20t) B - C l ^ (Vi),4c 33 p - A -fr- B - C 14d zz p -h 0 ~h 200) A -+■ B - (206 - 1) C ]4a zz p - A - (et + 2) B + (206 ■+■ O C \46 = p -+■ (1 - A - (

) I2S (

40 sr p - A - (ct 4 2) B - (zoc 4 i) C4b zzz p 4- (1 - 20c) A - {cc - 2) B 4- C 1

^ (XV)4c = p - A 4 ccB 4 C4)5-C2a4-|>) C

(XVI).~

p 4- ^4 ^« 4* 'J B 4r C4c 22Z. p — A — 7? 4" ^ -«i-—.... il [""'44dz2: p 4 (1 4 20c) A - B + (zoc - 1) C

Ex allatis adeo fyftematibus, unicum folummodoveras exhibet, quse datae conveniant aequationi -px3 4 qx* - rx 4- s z= o, radices; atque hoc ipfumfi ftatuamus esfe C1)» rebqua oinnia fpurias tantumfuppeditabunt radices, quartim femper in aperto ver-fabimur discrimine pro veris venditandarum, quamdiupro eisdem determinandis ad ipfas B et C delcendereopus erit; quippe pluribus omnino aequationibus cumeaedem conveniant reductae, atque fi modo ftatuatur

3P2 - SQ. = SP2 - Sq3P* - i6P*Q 4 20Q2 4 4PR - joS)Z2Z jy4" - i6p2q 4" zöq* 4- 4prP6 - 8P*Q 4 2oP2Q* - 16Q} - 8R2- 24p*S -4 fyQß - 8PQ.R 4 4p'RS= p6 - 8p*q 4 20p*qz - i6q% - ßr2

- 24p2s 4 Hqt - 8pqr 4 4p"reaedem certe reductae aequationibus conveniant

x* - px2 4 qx2 - w + / = o, atquex* - Px5 4 Qx2 - Rx 4- S = o.

i? - 7