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

Table of contents

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[81.] Solutio.
[82.] Scholium.
[83.] Problema.
[84.] Solutio.
[85.] Corollarium 1.
[86.] Corollarium 2.
[87.] Scholium.
[88.] Experimenta quæ ad Sectionem V. pertinent. Ad §. 5.
[89.] HYDRODYNAMICÆ SECTIO SEXTA. De fluidis non effluentibus ſeu intra latera vaſorum motis. §. 1.
[90.] De motu aquarum per canales indefinite longos. Caſus 1.
[91.] Exemplum 1.
[92.] Exemplum 2.
[93.] De oſcillationibus fluidorum in tubisrecurvis. Caſus II.
[94.] Lemma.
[95.] Solutio.
[96.] Problema.
[97.] Solutio.
[98.] Corollarium 1.
[99.] Corollarium 2.
[100.] Corollarium 3.
[101.] Corollarium 4.
[102.] Theorema.
[103.] Demonſtratio.
[104.] Problema.
[105.] Solutio.
[106.] Corollarium. 1.
[107.] Corollarium 2.
[108.] Scholion.
[109.] Theorema.
[110.] Demonſtratio.
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              <pb o="73" file="0087" n="87" rhead="SECTIO QUARTA."/>
            Geometrico infinitum, non ſolum non fit tempore infinite parvo, prouti in
              <lb/>
            caſu foraminis ſimplicis, ſed tempore infinitè magno, intereaque etiam quan-
              <lb/>
            titas aquæ infinita effluit, cum per foramen quantitas cæteris paribus infinite
              <lb/>
            parva effluat. </s>
            <s xml:id="echoid-s2001" xml:space="preserve">Hæc autem ut eruerem, opus habui aliam elicere æquationem
              <lb/>
            ex æquatione generali §. </s>
            <s xml:id="echoid-s2002" xml:space="preserve">23. </s>
            <s xml:id="echoid-s2003" xml:space="preserve">ſect. </s>
            <s xml:id="echoid-s2004" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2005" xml:space="preserve">quam ſimpliciſſimam hanc s = x, poſita
              <lb/>
            s pro altitudine, quæ velocitati aquæ effluentis reſpondeat & </s>
            <s xml:id="echoid-s2006" xml:space="preserve">x pro altitudi-
              <lb/>
            ne aquæ ſupra orificium effluxus; </s>
            <s xml:id="echoid-s2007" xml:space="preserve">intelliget autem quisque rem pro inſtitu-
              <lb/>
            to noſtro ita eſſe efficiendam, ut habeatur ratio incrementorum velocitatis,
              <lb/>
            quod antea non requirebatur.</s>
            <s xml:id="echoid-s2008" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2009" xml:space="preserve">§. </s>
            <s xml:id="echoid-s2010" xml:space="preserve">16. </s>
            <s xml:id="echoid-s2011" xml:space="preserve">Fuerit igitur ut in paragrapho 22. </s>
            <s xml:id="echoid-s2012" xml:space="preserve">ſect. </s>
            <s xml:id="echoid-s2013" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2014" xml:space="preserve">cylindrus A E H B
              <lb/>
            (Fig. </s>
            <s xml:id="echoid-s2015" xml:space="preserve">18.) </s>
            <s xml:id="echoid-s2016" xml:space="preserve">is que cenſeatur infinite amplus & </s>
            <s xml:id="echoid-s2017" xml:space="preserve">aqua plenus, habeatque tubum
              <lb/>
            annexum F M N G finitæ amplitudinis formæ coni truncati, ſive creſcentis
              <lb/>
            amplitudine ſive decreſcentis verſus orificium M N, per quod aquæ effluunt:
              <lb/>
            </s>
            <s xml:id="echoid-s2018" xml:space="preserve">ſit ut ibi altitudo initialis aquæ ſupra foramen M N, nempe N G + H B = a; </s>
            <s xml:id="echoid-s2019" xml:space="preserve">
              <lb/>
            altitudo ſuperficiei aqueæ in ſitu C D ſupra M N, id eſt, N G + H D = x; </s>
            <s xml:id="echoid-s2020" xml:space="preserve">
              <lb/>
            longitudo tubi annexi ſeu N G = b, amplitudo orificii M N = n, amplitudo
              <lb/>
            orificii F G = g, amplitudo cylindri, quæ eſt infinita, = m; </s>
            <s xml:id="echoid-s2021" xml:space="preserve">ſitque tandem
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            velocitas ſuperficiei aquæ in ſitu C D talis quæ conveniat altitudini v, quæ
              <lb/>
            altitudo utique infinite parva erit. </s>
            <s xml:id="echoid-s2022" xml:space="preserve">His poſitis vidimus loco citato obtinere
              <lb/>
            generaliter hanc æquationem: </s>
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              <lb/>
            m(x - b)dv + {bmm/√gn}dv - {m
              <emph style="super">3</emph>
            /nn}vdx + mvdx = - mxdx
              <lb/>
            in quâ patet, poſſe nunc negligi terminum primum m(x - b)dv præ ſe-
              <lb/>
            cundo {bmm/√gn}dv, ut & </s>
            <s xml:id="echoid-s2024" xml:space="preserve">quartum mvdx præ tertio - {m
              <emph style="super">3</emph>
            /nn}vdx, atque ſic aſſumi
              <lb/>
            {bmm/√gn}dv - {m
              <emph style="super">3</emph>
            v/nn}dx = - mxdx. </s>
            <s xml:id="echoid-s2025" xml:space="preserve">
              <lb/>
            in qua æquatione ſi rurſus negligatur primus terminus, quod fieri poteſt,
              <lb/>
            niſi mutationes etiam deſiderentur, quæ durante primo deſcenſu, etſi infi-
              <lb/>
            nite parvo fiunt, orietur regula vulgaris aſcenſus potentialis aquæ effluentis ad
              <lb/>
            altitudinem integram aquæ: </s>
            <s xml:id="echoid-s2026" xml:space="preserve">nunc vero pro noſtro negotio, quo mutatio-
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            nes illas primas deſideramus, terminus iſte retinendus erit, atque ſic æqua-
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            tio ultima in tota ſua extenſione pertractanda.</s>
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