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

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        <div xml:id="echoid-div107" type="section" level="1" n="80">
          <pb o="100" file="0114" n="114" rhead="HYDRODYNAMICÆ"/>
        </div>
        <div xml:id="echoid-div108" type="section" level="1" n="81">
          <head xml:id="echoid-head106" xml:space="preserve">Solutio.</head>
          <p>
            <s xml:id="echoid-s2818" xml:space="preserve">Retentis hypotheſibus & </s>
            <s xml:id="echoid-s2819" xml:space="preserve">denominationibus omnibus, quas in §. </s>
            <s xml:id="echoid-s2820" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2821" xml:space="preserve">adhi-
              <lb/>
            buimus, poſitoque inſuper tempore à fluxus initio præterito = t, mutan-
              <lb/>
            das habebimus æquationes in dicto paragrapho datas in alias, quæ relatio-
              <lb/>
            nem exprimant inter t & </s>
            <s xml:id="echoid-s2822" xml:space="preserve">v, eliminatis quantitatibus x & </s>
            <s xml:id="echoid-s2823" xml:space="preserve">d x. </s>
            <s xml:id="echoid-s2824" xml:space="preserve">Eſt vero elemen-
              <lb/>
            tum tempuſculi d t proportionale minimo ſpatiolo d x, quod percurritur, di-
              <lb/>
            viſo per velocitatem √v: </s>
            <s xml:id="echoid-s2825" xml:space="preserve">ponemus igitur d t = {γdx/√v}, & </s>
            <s xml:id="echoid-s2826" xml:space="preserve">ſic mutabitur æquatio
              <lb/>
            dx = Ndv: </s>
            <s xml:id="echoid-s2827" xml:space="preserve">(na - nv + {n
              <emph style="super">3</emph>
            /mm} v)
              <lb/>
            quæ data fuit pro affuſione verticali debita velocitate inſtituenda in hanc
              <lb/>
            (I) dt = N γdv:</s>
            <s xml:id="echoid-s2828" xml:space="preserve">(na√v - nv√v + {n
              <emph style="super">3</emph>
            /mm} v√v)
              <lb/>
            altera vero affuſioni inſerviens laterali, nempe dx = Ndv: </s>
            <s xml:id="echoid-s2829" xml:space="preserve">(na - nv)
              <lb/>
            abit in hanc poſt eandem ſubſtitutionem
              <lb/>
            (II) dt = N γdv:</s>
            <s xml:id="echoid-s2830" xml:space="preserve">(na√v - nv√v)
              <lb/>
            Hæ vero æquationes debito modo integratæ dant pro prima
              <lb/>
            (α) t = {mNγ/n√(mma - nna)} X log. </s>
            <s xml:id="echoid-s2831" xml:space="preserve">{m√a + √(mmv - nnv)/m√a - √(mmv - nnv)}
              <lb/>
            & </s>
            <s xml:id="echoid-s2832" xml:space="preserve">pro altera, quæ ex priori deducitur, poſito m = ∞
              <lb/>
            (β) t = {Nγ/n√a} X log. </s>
            <s xml:id="echoid-s2833" xml:space="preserve">{√a + √v/√a - √v}. </s>
            <s xml:id="echoid-s2834" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s2835" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2836" xml:space="preserve">I.</s>
            <s xml:id="echoid-s2837" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div109" type="section" level="1" n="82">
          <head xml:id="echoid-head107" xml:space="preserve">Scholium.</head>
          <p>
            <s xml:id="echoid-s2838" xml:space="preserve">§. </s>
            <s xml:id="echoid-s2839" xml:space="preserve">13. </s>
            <s xml:id="echoid-s2840" xml:space="preserve">Si vas de quo ſermo eſt ſit cylindricum utcunque intortum & </s>
            <s xml:id="echoid-s2841" xml:space="preserve">
              <lb/>
            inclinatum, cujus longitudo ponatur = b, manente altitudine ſuperficiei
              <lb/>
            aqueæ ſupra foramen = a, erit rurſus, ut §. </s>
            <s xml:id="echoid-s2842" xml:space="preserve">11. </s>
            <s xml:id="echoid-s2843" xml:space="preserve">N = {nn/m}b.</s>
            <s xml:id="echoid-s2844" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2845" xml:space="preserve">Quoniam autem, ut conſtat, 2γ√A exprimit tempus, quod corpus
              <lb/>
            inſumit cadendo libere & </s>
            <s xml:id="echoid-s2846" xml:space="preserve">à quiete per altitudinem A, patet quantitatem
              <lb/>
            {2mNγ/nn√a} (= 2γ√{bb/a}) exprimere tempus quo corpus moveri incipiens à
              <lb/>
            quiete liberè deſcendit per altitudinem {bb/a}: </s>
            <s xml:id="echoid-s2847" xml:space="preserve">accipiemus iſtud tempus </s>
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