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

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        <div xml:id="echoid-div110" type="section" level="1" n="83">
          <pb o="103" file="0117" n="117" rhead="SECTIO QUINTA."/>
        </div>
        <div xml:id="echoid-div111" type="section" level="1" n="84">
          <head xml:id="echoid-head109" xml:space="preserve">Solutio.</head>
          <p>
            <s xml:id="echoid-s2907" xml:space="preserve">Adhibitis rurſus poſitionibus & </s>
            <s xml:id="echoid-s2908" xml:space="preserve">denominationibus paragraphi tertii & </s>
            <s xml:id="echoid-s2909" xml:space="preserve">
              <lb/>
            duodecimi, invenienda nunc erit æquatio inter x & </s>
            <s xml:id="echoid-s2910" xml:space="preserve">t: </s>
            <s xml:id="echoid-s2911" xml:space="preserve">quia vero, ut vidi-
              <lb/>
            mus §. </s>
            <s xml:id="echoid-s2912" xml:space="preserve">12. </s>
            <s xml:id="echoid-s2913" xml:space="preserve">eſt d t = {γdx/√v}, erit √ v = {γdx/dt}, hicque valor ſubſtituendus
              <lb/>
            erit in æquationibus, quas dedimus §. </s>
            <s xml:id="echoid-s2914" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2915" xml:space="preserve">integratis; </s>
            <s xml:id="echoid-s2916" xml:space="preserve">prior harum æquationum
              <lb/>
            hæc fuit: </s>
            <s xml:id="echoid-s2917" xml:space="preserve">v = {mma/mm - nn} X (1 - c{n
              <emph style="super">3</emph>
            - nmm/mmN} x)
              <lb/>
            quæ pro præſecuti inſtituto mutatur in hanc
              <lb/>
            (I) {γγdx
              <emph style="super">2</emph>
            /dt
              <emph style="super">2</emph>
            } = {mma/mm - nn} X (1 - c{n
              <emph style="super">3</emph>
            - nmm/mmN} x)
              <lb/>
            altera ex §. </s>
            <s xml:id="echoid-s2918" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2919" xml:space="preserve">allegatarum æquationum talis fuit
              <lb/>
            v = a X (1 - c
              <emph style="super">{- n/N} x</emph>
            )
              <lb/>
            quæ adeoque ſubminiſtrat in præſenti caſu ſequentem
              <lb/>
            (II) {γγdx
              <emph style="super">2</emph>
            /dt
              <emph style="super">2</emph>
            } = a X (1 - c
              <emph style="super">{- n/N} x</emph>
            )</s>
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          <p>
            <s xml:id="echoid-s2920" xml:space="preserve">Erunt nunc æquationes (I) & </s>
            <s xml:id="echoid-s2921" xml:space="preserve">(II) integrandæ, quod quidem facile
              <lb/>
            eſt & </s>
            <s xml:id="echoid-s2922" xml:space="preserve">quia prior alteram continet (utraque enim eadem eſt ſi m = ∞)
              <lb/>
            hanc ſolam pertractabimus, eamque nunc ſub hâc forma conſiderabimus.
              <lb/>
            </s>
            <s xml:id="echoid-s2923" xml:space="preserve">dt = {γ√(mm - nn)/m√a}dx:</s>
            <s xml:id="echoid-s2924" xml:space="preserve">√(1 - c{n
              <emph style="super">3</emph>
            - nmm/mmN}x)</s>
          </p>
          <p>
            <s xml:id="echoid-s2925" xml:space="preserve">Ponatur autem ut integrationis modus eo magis pateſcat
              <lb/>
            c{n
              <emph style="super">3</emph>
            - nmm/mmN}x = z, atque proin dx = {mmNdz/(n
              <emph style="super">3</emph>
            - nmm)z},
              <lb/>
            dein brevitatis ergo indice
              <unsure/>
            tur quantitas conſtans
              <lb/>
            {γ√(mm - nn)/m√a} X {mmN/n
              <emph style="super">3</emph>
            - nmm}, ſeu {- γmN/n√(mm - nn) a} per α,
              <lb/>
            & </s>
            <s xml:id="echoid-s2926" xml:space="preserve">habebitur dt = {αdz/z√(1 - </s>
          </p>
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