Arkit: 1 arkintunnukseton lehti, A4 B1.Painovuosi nimekkeestä

Hällström, Gustavus Gabriel

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DEINVENIENDO NAXIMO VEL MINI.MOVALORE FUNCTIONISy^Aax+Bbx+ Ccx +—- + Mm* + Ntf* +P,DISSERTATIOiQUAMCONSENTrENTE AMPL. FAC. PHIL. REG. ACAD. ABOENSIS,PRyESIDEmag. GUST. GABR. HALLSTR6M,PHr.ICES PROFES.. REC. ET ORDIN. ATQUE REG. SOCIET.OECON. FENN. MEMBRO.rr;o gradu philosophicoPUBLICO EXAMINI OFFERTERICUS GABRIEL MELARTIN,STITEND. ARCKENHOLTZ. WIBUBG E N SI S.In Auditorio Majori die ff/ Maji MDCCCII.Horis 4-F m. confuetis.ABOAE, TYPIS Frenckellianis,In legibus natune eruendis id praeclpue Phyfico «s--fe agendum judicamus, vt primum experimenta,mutatis rebus illa fpeftantifous, plura & accurata in-ftituat. Deinde autem haec mter fe conferendo ipfatninveniet legem phaenomenorum omnrum intra limitescosdem, gui fa&a experimenta -conrprehendebant,contentorum. Vulgo turn sequationem aliquam autalgebraicnm aut etiam tranfcendentem fibi fingunt,cujus quantitates incognitas conferendo cum valori-bus ope experimentorum inventis determinant.Inter plures tales interpoland-o infervientes se-quationes faepe magno calculi compendio adhiberipoteft haec: y = Aax * BY * Cc* "*" * Mm* *Nnx * P, in qua pro cognitis quibusdam valoribuso, xJ, 2x', fsx>, jfx', &c, quantitatis X per experimen-ta dati habebuntur valores refpondentes funftionis y,&, ii modo pofitivas fint quantitates «*, b*, c x,m , n, non difficulter determinantur conftantes A, a t8,b,.C,£~-- r.M,m,N,n,&P.A Deter-i2Determinatis omnibus quantitatibus confi:antibcH_vfcepenumero utile eft fcire, num pro aliquo certo va-lore quantitatis x maxima vel minima fiat y, nee nonquantae in hoc cafu fint x&y. Methodo vulgari utrius-que membri aequationis fluxio quaeritur, & evanefcensponitur *), vt, faclro in praefenti cafu Log. hyp. a —ot, Log. hyp. B =■= /3, Log. hyp. c ___=. y,. - Log.hyp. m = f/, & Log. hyp. n ==. v, habeatur dy _____"Jx~xdx + Bfib*dx'+ Cycxdx + * M\j.mxdx ~-Nwdx, adeoque *1 = Axa*'* B~bx * Cyc* *dx* M\imx* Nvn* =o„Ex hae ultima aequationedeterminanda eft quarc-titas ;*., quas in apquatione ■// ~ Aax ~- Bb'x * Cc* +* 7f//«'v * Nnx *_P fubftituta dat maximumvel minunum valorem quantitatis y.. Cum autern,,etiamfi reales fint omnes quantitates conftantes, nonfemper inveniri queat aliquis valor quantitatis x, quaaequationi o — Aza* + Bs,6* * * M\imx +Nvt* fatisfit, nee femper ifte vaJor fun&ionis y de-terminari poteft..Ut*) Vide: Infiiitutiones calculi dijfercntialis, Auttore Lk-ONH. EuJ-_.RO,. i*. 11. Cap. X. pag. ~7%,jrjrj.3Ut vero, quando fieri poteft, ex liac a-quationeeruatur x, omnes ejus termini in feriem funt evol-vendi. Hoc modo invenitur:jfjta'**- J*<! +_f + _!f" + !___!*-^ *&c);i7/3_ * =i?/s(i*^*£f:*___d *____!.+ &C0;i 1.2. 1.2.3. 12.3 4.Cyc* =___ Cy (i *_5 * ____!* Z!^! * _!!___ * &c;i 1.2. i 23. 1.2.3 4-Mjtm" = M\i (i *t* + (L%fl + *!£, * Ji!?_L * &c.) ■I 1.2. 1.2.3. 1.2.3.4.Nv n* =_ M(i*HL * v!x!*!! * ___!^!_ *&c.):I 1.2. 1.2.3. 1.2.3.4.unde colligitur esfeo _= + Ax * /fo2 j** ril!»*- * d!\x* * _^_!>*+&c.I 12. ' 1.2.3 ' 1.2.3.4.'-■Bfi+BQ' * ——. +—— +——■12. 1.2.3 1-2.3.4.+Cy* L y 2 + ——| + ——| +——1.2. 1.2.3. 1.2.34._. » »-f__-' - « - _-_-+ 2l *_^i! *^/i! *^!_12. 1.23. 1.2.34--*Nv*Nv~ *^1 *£-_ *J*LJ i( 2. J 1.3.3.J M.3.4JA 3 Fa_>Sf, ) 4 C <&Fa&is itaqueJ° = Aa* B'& * Cy * ----_-* * M,J =*" Ax2 * 8~~ . Cy- * * M\x~ + 2Vv\___= *Ml * £*1 * +_€ + hHr1.2. 1.2. 1.2. 1--. 1.2.j" == ttK! +Ml+ £*1 * + &L4*J£_!l,].2.3. J'2.3. 1.2.3 1.23. 1'.2.3s..2.3.4. 1-2.3.4. 1.2.3.4. 1.2.3.4. »-__3-4_t'&-C. ***- &c.erit o —A° + A'x + Afx* + A'"xr + A'-'x' + &c.Ut haric feriem revertendo inveniamus alram,qune determinet quantitatem x, & cujus termini con-ftantibus & datis quaotitatibus, progrediantur, iinga-tur esfex = 717-.° * -Vv./02 + P'AaJ * QA°* * &__,vt habeanturJx=M'AA° + N'A'A°' + PJ'A°-*Q:AAo**&j~.0 **&j~.J"x'=M'~A''A°' *2M'N'JA°~: + 2M'P'A /_,°*+&c,+ N' 7 A"\J"'x* ■**= M'-J'"AoJ * sM'~N'A'"J°* * &c.//ivx*_ M'*AivA°~ * &c;quibus omnibus colie&s & in ferie revertenda fub-ftitutis, proditO __=_-3fc jr $ ( «gfor+r \A~^~N'A'\Af- 7>\>?'A' ]A^±QA' ]A°^Sccr±M'A'\ ±M"A"\ *iM'N~A") ±zM'P'A"\*M'-A'"\ *hN'~A"---M'--N'A'"\±M''A-'r)In determinandis quantitatibus adhue incognitisM', _V, P'i Q', &C, obfervandurn eft, quum aquatio-nis hujus membrum unum fit____.o, quascunque etiauicoeflicientes quantitatis J° in altero membro evane-fcentes esie ponendas. Hac autern ratione obtiner_-tur fequentes asquationes:i ■> M'A' =__ o }NA + M'2A" __- o?PA + 2M'NA' -h M,rJ" ___= o,-Q'A' + 2M'P'A" + il^yf' ~~*3M'2N'J'" -. M'"J'v = o?72'^' * 2M'Q'J" * 2N'P'A" * 3M'ZP'J" *3M'N'~J'" * 4M'~N'J-~- * _¥'f^v ___= oj&c. __=. &c.unde fequentes inveniuntur valoresM'■=.-!'#-_ - _t.A"-P =-!*!*'!!-A'~ AA $ £,«=--fe, ) 6 t &g, -A"' -A"A^ _ Aiv.~A7~~ ~~A~~~~ ~~~ji, _4_-_>" 4 *iA"~-A'" 6A"Aw 3A"'* A_\— ~~j7~~~ * A7l ~v~ ~A~~ * 2'"1'&c. s= &c.Si hi jam hoc"modo determinati valores fubfii-tuuntur in a>quatione antea asfunita, provenit aequa-tio, cujus ope computari poteft ille valor quantitatisx, qui funetionem y niinimam vel maximam reddit.Habetür enira turnA A' 1 A" i A' 1A" <-A"A'"* A". A' sAwSie quidem generaliter ifte valor quantitatis xdeterminatur, qui funclionem i/ maximam vel mini-mam, fi fieri poslit, reddit. In cafibus autern qui-busdam fpecialioribus prolixo hoc calculo uti opusnon elr. Quando r nimirum in aequatione o —A<%a* B $bx* * M\iinx* Nvn* omnia produftaAotax, Bsb*- &c, funt vel pofitiva vel negativa, vi-dendurn eft, an omnes quantitates a,b,c,- - - - m & nfintfe- > % $ cgfmt vel majores vel minores unitate', adeoqne enrnmLogarithmi 01,'% y, yc & v vel pofitivi vel neg-r--tivi. Si hi omnes funt pofitivi, fumma produttorum* (A<ta* + B\V)X* Mtvn * IfaT) evarrefcerenon potc-ft, ni(i fumnrur x-~ — 00, qui valor reddicminimum u— _L +' P = P. Si autern omnes Loqa--00rithmi x, /3, 7, >y- & » funt negativi, fumma pro-duftorum _ (Axax ~h B$bx * -- - * Mftm* * iVvnxjnon evanefcit nifi pofita a,*= 00, qni vaforreddit fun-clionem r/ -= oo , unde mox apparet, maximam veiminimam r/ fecundum receptam loquendi rationern inhoc cafu non darh.Hufus rer exemphmi prsebet sequatfo y =. Aa*'~~ P, übt primo fumatur a> r. Pro cafii maximaevel minimae y habetür aequatio JL — Ax~x=o, qua?fdxfi reaiis eft, dat x—— &~, adeoque, fa&a fubftitutio-ne, y __= Aa~~~~°° ~h P= P. Si itaque eft A quantitaspofitiva , invemtur minima y =P. Sed pro J nega-tiva evadit maxima y— P. Si autern eft a < i, ~e-quatio Axa* =■= o dat x "= co, quae reddit i/ == Aa°°■h P== ± 00. Idem qnoque ope feriei allata? inveni-tur. Eft enim in hoc exemplo J° =--Ax; A'~~-Jx*rA~'=-fe ) * i A*J" ~ *?.*!", A" —H - jiv = !!—- Jv = datl *1.2. 1.23. -.2.3.4, 1.2.3.4.5.'&C, atque _¥' =_= — JL; N=— _J ;P' = —Ax2 2A-ct-_^_; Q' == - _i_; K=- _L__, &c; undesA^cC- 4AA.-x- sA^u6pro cafu mil.im.a2 // iuvenitur x == — L(l~*L * l^.«231. + -L -h C£C.). flujus vero feriei fumma eft .____= co °),4 5adeoque fi x eft pofitivus, pro maxima vel minima y00erit x=— ~, &y= P. Si vero eft 0. negativus,faafaetur x = —,;& y— ± 00, vt antea.Praeterea quoque interdum, quando funt C ■= o,i^ =o, Ar-o, utfito-yfoa* * B&bx, fine adhibitaferie inveniri poteft maxima vel minima y. E(. enim/^\* =_ S!, adeoque fi quantitas —.r eft pofiti-\b/ Ax Axva, invenitnr x (Log. /r. — Log. b~) = Log. (— 2?/3)~— Log. _4.*, feu == Log. Bfi — Log. (— Jx), &X =*) Cfr. Introdutt. in Analyfin infinitortim,, Autt, Legnh.Eulero, Laufann-t 1748, Tora. I, pag, 229.Sfe ) 9 C &X= Log'( L°g-^« feu y =Lo^o-LorJ- Xf«)Log. <7 — Log. £ Log, a — Log. BHujus cafus exemplum prasbet aequatio y == 2* ~- 0,3*.* i. Efi: enimJ==i; B=i7a = 2; h = 0,3f* = 0.693T47"r /3 = — 1,2039728.Invenkur itaque & —.#/3=1,2039728',,adeoque * =L°g- '-g°39728 - Log. 0,693 »478 __Log, 2 — Log, 0,30,2397912. qUare er jt Log. x-- 0,4639542 —ir0,8239087& x = 0,2910410. Hoc autern valore fubftituto e-ruitur minima y = 2,9279253.Si ulterius omnia preduSa Axa* ,. BfiY, &crita funt vel pofitiva vel negativa, vt unus vel plures,,non autern oranes, Logarithmi &> /3, >', &c.,. fint poiiri-vi, aequatio o — ± (jitLO* + Bjlb* * "** MvmK* lVvnx) eft imaginaria. FacYis ex. Gr. && ,3 nega-tivis at(]ue |u & v p-ofitivis, pro aliquo finito valorequantitatis # non evaneicit membrum pofterius a?_qu.'tionis* pro x-= oo autern fit o — :: ( M\x ■*J\'v) co , nee non pro # — — oo , o _ ± (Ad * B$+ _ . i_) 00, quarum a*quationum utraque- eft irapos-fibilis. In taiibus itaque eaiibus nullus valor quantitatib. X d~>10x dari poteft, qui funcAionem y maximam vel mini-.nam redderet.Generaliter quidem, quando inventus eft aliquismaximus vel minimus valor funclionis //, judicare li-cet, an maximus vel minimus ille iit, ii majores &minores vajores quantitatis x fuccesfive fubftituuntur.Directeautemilludefignisquantitatuminaequationeda-tarum faciliuscognofcitur,fi facto^v=o valorfuncHionis""?_ eruitur. Nam pro 1~ v invenitur minima ff,dT1 <lx~fed pro Y-l = — v maxima, fi v eft quantitas quae-dx2dam aiTirniativa ; ). Habeturautem exrrquationenoftragenera!i,£_! = Axax * Bsb* * Cycx + ---- +JxMvmx + Nvn. — o , unde erit AV == - Axax—Bfrb*— Cyc* — Mum . Hac fafta fubftitutione in sequa-tione(U1 = Aota* + B&b* + Cy*c" +- - - + Mu*mKJx** /Vv■■'■//", invenitur i_il == A(*—v) a" + ff/3 (/3 —</■*■*"») bx +Cy(y— v) cx + +Mu(u— v) mx, übiquantitates conftantes determinant an pofitivus velnegativus iit valor V.2, adeoque an vel minima veldx-maxima fiat funolio //.~") Ck. L. EuLERI Inftitut. Ctilculi differcntitilis, pag. 583,

De inveniendo maximo vel minimo valore functionis: y = Aa [... kaava], dissertatio; quam consentiente ampl. fac. phil. Reg. Acad. Aboënsis, praeside mag. Gust. Gabr. Hällström, physices profess. reg. et ordin. atque Reg. Societ. Oecon. Fenn. membro. Pro gradu philosophico publico examini offert Ericus Gabriel Melartin, stipend. Arckenholtz. Wiburgensis. In auditorio majori die [ ] Maji MDCCCII. Horis p. m. consuetis

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