Bernoulli, Daniel, Hydrodynamica, sive De viribus et motibus fluidorum commentarii

Table of figures

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          <pb o="263" file="0277" n="277" rhead="SECTIO DUODECIMA."/>
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        <div xml:id="echoid-div269" type="section" level="1" n="204">
          <head xml:id="echoid-head261" xml:space="preserve">Solutio.</head>
          <p>
            <s xml:id="echoid-s7708" xml:space="preserve">Sit canalis A C D (Fig. </s>
            <s xml:id="echoid-s7709" xml:space="preserve">74.) </s>
            <s xml:id="echoid-s7710" xml:space="preserve">per cujus foramen o transfluere ponantur
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              <note position="right" xlink:label="note-0277-01" xlink:href="note-0277-01a" xml:space="preserve">Fig. 74.</note>
            aquæ velocitate uniformi & </s>
            <s xml:id="echoid-s7711" xml:space="preserve">tali quæ debeatur altitudini verticali o S: </s>
            <s xml:id="echoid-s7712" xml:space="preserve">ducatur
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            S N & </s>
            <s xml:id="echoid-s7713" xml:space="preserve">fingatur vas infinite amplum N M Q Paquis plenum usque in N P, ex
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            quo canalis aquas ſuas perpetuo & </s>
            <s xml:id="echoid-s7714" xml:space="preserve">æquabiliter hauriat: </s>
            <s xml:id="echoid-s7715" xml:space="preserve">hæc ideo ſic fingo, ut
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            cauſa adſit ſeu vis propellens uniformis, quæ aquas data velocitate propellat
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            ſeu fluxum aquarum conſervet æquabilem: </s>
            <s xml:id="echoid-s7716" xml:space="preserve">Et ſine hac hypothef
              <unsure/>
            i problema
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            noſtrum foret indeterminatum, quia velocitas eadem in eodem canali infini-
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            tis modis ad temporis punctum generari poteſt & </s>
            <s xml:id="echoid-s7717" xml:space="preserve">propterea, ut habeatur
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            menſura cauſæ aquas propellentis, fingenda eſt uniformitas in motu aquarum.</s>
            <s xml:id="echoid-s7718" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7719" xml:space="preserve">Fuerit nunc aquarum preſſio definienda in C F (aut c f): </s>
            <s xml:id="echoid-s7720" xml:space="preserve">huncque in
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            finem putabimus rurſus abrumpi canalem in C E (aut c e) ſectione ad cana-
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            lem perpendiculari examinaturi, quamnam accelerationem retardationemve
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            guttula C E G F (vel c e g f) poſt primum rupturæ momentum receptura ſit:
              <lb/>
            </s>
            <s xml:id="echoid-s7721" xml:space="preserve">quâ de cauſa generaliter motum momentaneum per vas decurtatum N M E C A
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            Q P (vel N M c e A Q P) definiendum habemus. </s>
            <s xml:id="echoid-s7722" xml:space="preserve">Sitigitur velocitas guttulæ in-
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            finite parvæ CEGF (ſeu c e g f) ipſo decurtationis puncto = v: </s>
            <s xml:id="echoid-s7723" xml:space="preserve">maſſa ejus
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            = dx: </s>
            <s xml:id="echoid-s7724" xml:space="preserve">erit vis viva aquæ in vaſe decurtato motæ proportionalis quantitati
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            v v, eamque proinde faciemus = α v v, intelligendo per litteram a quantita-
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            tem quamcunque conſtantem, quæ pendet ab amplitudinibus canalis abrupti; </s>
            <s xml:id="echoid-s7725" xml:space="preserve">
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            præciſa autem ejus determinatio hic non requiritur. </s>
            <s xml:id="echoid-s7726" xml:space="preserve">Notetur vim vivam aquæ
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            in vaſe ficto N M QP negligi ob infinitam ejus amplitudinem: </s>
            <s xml:id="echoid-s7727" xml:space="preserve">nulla tamen ſi
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            vel infinitæ non eſſet amplitudinis inde in calculo oritura fuiſſet variatio. </s>
            <s xml:id="echoid-s7728" xml:space="preserve">Ha. </s>
            <s xml:id="echoid-s7729" xml:space="preserve">
              <lb/>
            bemus jam incrementum vis vivæ aquæ in vaſe decurtato motæ = 2avdv, cui
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            ſi addatur vis viva ſimul genita in guttula ejecta, oritur 2avdv + vvdx, quod
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            eſt incrementum vis vivæ totale, debitum deſcenſui actuali guttulæ dx per alti-
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            tudinem verticalem aquæ ſupra punctum C (vel c,) quam deſignabimus per a: </s>
            <s xml:id="echoid-s7730" xml:space="preserve">
              <lb/>
            hinc igitur iſtud incrementum vis vivæ totale faciendum eſt æquale adx, ſic
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            ut ſit
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            2avdv + vvdx = adx vel
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            {vdv/dx} = {a - vv/2a}.</s>
            <s xml:id="echoid-s7731" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s7732" xml:space="preserve">Reliqua ſi fiant, ut in paragrapho quinto & </s>
            <s xml:id="echoid-s7733" xml:space="preserve">ponatur velocitas v talis
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            quæ debeatur altitudini b, invenietur preſſionem aquæ in C F (aut cf) </s>
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