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

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        <div xml:id="echoid-div54" type="section" level="1" n="39">
          <pb o="37" file="0051" n="51" rhead="SECTIO TERTIA."/>
        </div>
        <div xml:id="echoid-div55" type="section" level="1" n="40">
          <head xml:id="echoid-head49" style="it" xml:space="preserve">De his quæ pertinent ad effluxum aquarum ex Cy-
            <lb/>
          lindris verticaliter poſitis, per Lumen quod-
            <lb/>
          cunque, quod eſt in fundo horizontali.</head>
          <head xml:id="echoid-head50" xml:space="preserve">§. 13.</head>
          <p>
            <s xml:id="echoid-s1003" xml:space="preserve">GEometræ, quibus de aquis ex vaſe erumpentibus ſermo fuit, con-
              <lb/>
            ſiderare potiſſimum ſolent cylindros verticaliter poſitos: </s>
            <s xml:id="echoid-s1004" xml:space="preserve">Igitur haud
              <lb/>
            abs re erit ex theoria noſtra generali conſectaria illa, quæ huc per-
              <lb/>
            tinent, deducere. </s>
            <s xml:id="echoid-s1005" xml:space="preserve">Sit amplitudo cylindri ad amplitudinem foraminis ut m
              <lb/>
            ad n; </s>
            <s xml:id="echoid-s1006" xml:space="preserve">altitudo aquæ ſupra foramen, cum fluxus incipit = a; </s>
            <s xml:id="echoid-s1007" xml:space="preserve">altitudo aquæ
              <lb/>
            reſiduæ = x, altitudo velocitati aquæ internæ debita = v; </s>
            <s xml:id="echoid-s1008" xml:space="preserve">erit in æquatio-
              <lb/>
            ne canonica paragraphi octavi y = m, N = mx (per §. </s>
            <s xml:id="echoid-s1009" xml:space="preserve">6.) </s>
            <s xml:id="echoid-s1010" xml:space="preserve">quæ adeoque
              <lb/>
            abit in hanc æquationem.
              <lb/>
            </s>
            <s xml:id="echoid-s1011" xml:space="preserve">mxdv - {m
              <emph style="super">3</emph>
            /nn}vdx + mvdx = - mxdx, vel
              <lb/>
            (1 - {mm/nn})vdx + xdv = - xdx
              <lb/>
            multiplicetur hæc poſterior æquatio per x
              <emph style="super">{- mm/nn}</emph>
            , ut habeatur
              <lb/>
            (1 - {mm/nn})x
              <emph style="super">- {mm/nn}</emph>
            vdx + x
              <emph style="super">1 - {mm/nn}</emph>
            dv = - x
              <emph style="super">1 - {mm/nn}</emph>
            dx.</s>
            <s xml:id="echoid-s1012" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1013" xml:space="preserve">Poteſt jam hæc æquatio integrari: </s>
            <s xml:id="echoid-s1014" xml:space="preserve">obſervanda autem eſt in Integratio-
              <lb/>
            ne conſtantis additio, talis nempe, ut a fluxus initio, id eſt, cum x = a,
              <lb/>
            ſit velocitas fluidi nulla, ipſaque proin v pariter = o: </s>
            <s xml:id="echoid-s1015" xml:space="preserve">ita vero oritur:
              <lb/>
            </s>
            <s xml:id="echoid-s1016" xml:space="preserve">x
              <emph style="super">1 - {mm/nn}</emph>
            v = {nn/2nn - mm}(a
              <emph style="super">2 - {mm/nn}</emph>
            - x
              <emph style="super">2 - {mm/nn}</emph>
            ) vel
              <lb/>
            v = {nna/2nn - mm}(({a/x})
              <emph style="super">1 - {mm/nn}</emph>
            - {x/a})</s>
          </p>
          <p>
            <s xml:id="echoid-s1017" xml:space="preserve">§. </s>
            <s xml:id="echoid-s1018" xml:space="preserve">14. </s>
            <s xml:id="echoid-s1019" xml:space="preserve">Ex hâc igitur æquatione cognoſcitur altitudo generans velocita-
              <lb/>
            tem aquæ internæ; </s>
            <s xml:id="echoid-s1020" xml:space="preserve">ubi notari meretur, ſi vas ſit ampliſſimum, mox poſſe
              <lb/>
            cenſeri v = {nn/mm}x, poſtquam ſcilicet vel tantillum deſcendit aqua, id </s>
          </p>
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