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

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          <pb o="66" file="0080" n="80" rhead="HYDRODYNAMICÆ"/>
          <p>
            <s xml:id="echoid-s1777" xml:space="preserve">§. </s>
            <s xml:id="echoid-s1778" xml:space="preserve">9. </s>
            <s xml:id="echoid-s1779" xml:space="preserve">Fuerit igitur vas cylindricum verticaliter poſitum aqua ple-
              <lb/>
            num, ſitque altitudo aquæ ab initio fluxus = a, amplitudo cylindri = m, am-
              <lb/>
            plitudo foraminis = n, Sectio venæ ſolidæ = {n/α} effluxerit jam aqua per tempus
              <lb/>
            t; </s>
            <s xml:id="echoid-s1780" xml:space="preserve">ſitque tunc altitudo aquæ reſidua ſupra foramen = x, eodemque temporis
              <lb/>
            puncto habeat ſuperficies aquæ internæ velocitatem, quæ reſpondeat altitudini
              <lb/>
            v: </s>
            <s xml:id="echoid-s1781" xml:space="preserve">erit velocitas ipſa = √ v, eſt autem elementum temporis d t proportio-
              <lb/>
            nale elemento ſpatii - d x diviſo per velocitatem √v, unde dt = {- dx/√v}.</s>
            <s xml:id="echoid-s1782" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1783" xml:space="preserve">Determinatus @equidem fuit valor ipſius v in ſect. </s>
            <s xml:id="echoid-s1784" xml:space="preserve">3. </s>
            <s xml:id="echoid-s1785" xml:space="preserve">ubi iisdem denomi-
              <lb/>
            nationibus uſi ſumus, quibus nunc utimur. </s>
            <s xml:id="echoid-s1786" xml:space="preserve">At quoniam pro recta aquarum
              <lb/>
            erogatarum menſura requiritur, ut foramini n ſubſtituatur ſectio venæ con-
              <lb/>
            tractæ {n/α}, ſequitur, ut in valore ipſius v eadem fiat ſubſtitutio atque ſic ſta-
              <lb/>
            tuatur v = {nna/2nn - mmαα}(({a/x})
              <emph style="super">{1 - mmαα/nn}</emph>
            - {x/a})</s>
          </p>
          <p>
            <s xml:id="echoid-s1787" xml:space="preserve">Hic vero valor ſi ſubſtituatur in æquatione
              <lb/>
            dt = {- dx/√v}, oritur
              <lb/>
            dt = - dx: </s>
            <s xml:id="echoid-s1788" xml:space="preserve">√[{nna/2nn - mmαα} (({a/x})
              <emph style="super">{1 - mmαα/nn}</emph>
            - {x/a})]
              <lb/>
            ope cujus æquationis omnia tempora deſiderata definiri poſſunt per approxi-
              <lb/>
            mationes, ſeu ſeries, ſi modo in ſingulis punctis valor ipſius α innoteſcat:</s>
            <s xml:id="echoid-s1789" xml:space="preserve">
              <unsure/>
              <lb/>
            Aſſumemus autem eſſe illum conſtantis valoris, quandoquidem in præſenti
              <lb/>
            caſu nihil ſit, à quo mutari poſſit præter diverſas altitudines & </s>
            <s xml:id="echoid-s1790" xml:space="preserve">velocitates
              <lb/>
            fluidi, quæ parum vel nihil quantum ſenſibus percipi poteſt ad id negotii
              <lb/>
            conferunt.</s>
            <s xml:id="echoid-s1791" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1792" xml:space="preserve">§. </s>
            <s xml:id="echoid-s1793" xml:space="preserve">10. </s>
            <s xml:id="echoid-s1794" xml:space="preserve">Jam ut æquatio deſiderata per ſeries exhiberi poſſit, conſiderabi-
              <lb/>
            mus quantitatem.</s>
            <s xml:id="echoid-s1795" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1796" xml:space="preserve">1:</s>
            <s xml:id="echoid-s1797" xml:space="preserve">√[{nna/2nn - mmαα} (({a/x})
              <emph style="super">{1 - mmαα/nn}</emph>
            - {x/a})] fub hâc forma
              <lb/>
            ({nnx/mmαα - 2nn})
              <emph style="super">- {1/2}</emph>
            X (1 - ({x/a})
              <emph style="super">{mmαα/nn} - 2</emph>
            ) -
              <emph style="super">{1/2}</emph>
            </s>
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