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

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            <s xml:id="echoid-s1797" xml:space="preserve">
              <pb o="67" file="0081" n="81" rhead="SECTIO QUARTA."/>
            poſteriorem per regulas ſolitas reſolvemus in hanc ſeriem
              <lb/>
            1 + {1/2} ({x/a})
              <emph style="super">{mmαα/nn} - 2</emph>
            + {1. </s>
            <s xml:id="echoid-s1798" xml:space="preserve">3/1. </s>
            <s xml:id="echoid-s1799" xml:space="preserve">2. </s>
            <s xml:id="echoid-s1800" xml:space="preserve">4} - ({x/a})
              <emph style="super">{2mmαα/nn}</emph>
            - 4 + {1.</s>
            <s xml:id="echoid-s1801" xml:space="preserve">3.</s>
            <s xml:id="echoid-s1802" xml:space="preserve">5/1.</s>
            <s xml:id="echoid-s1803" xml:space="preserve">2.</s>
            <s xml:id="echoid-s1804" xml:space="preserve">3.</s>
            <s xml:id="echoid-s1805" xml:space="preserve">8}({x/a})
              <emph style="super">{3mmαα/nn} - 6</emph>
              <lb/>
            + &</s>
            <s xml:id="echoid-s1806" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1807" xml:space="preserve">unde nunc habetur mutata paullulum æquationis forma:
              <lb/>
            </s>
            <s xml:id="echoid-s1808" xml:space="preserve">dt = - {dx√mmαα - 2nn}/n√a} X [({x/a})
              <emph style="super">- {1/2}</emph>
            + {1/2} ({x/a})
              <emph style="super">{mmαα/nn} - {@/z}</emph>
              <lb/>
            + {1.</s>
            <s xml:id="echoid-s1809" xml:space="preserve">3/1.</s>
            <s xml:id="echoid-s1810" xml:space="preserve">2.</s>
            <s xml:id="echoid-s1811" xml:space="preserve">4} ({x/a})
              <emph style="super">{2mmαα/nn} -{9/2}</emph>
            + {1.</s>
            <s xml:id="echoid-s1812" xml:space="preserve">3, 5/1.</s>
            <s xml:id="echoid-s1813" xml:space="preserve">2.</s>
            <s xml:id="echoid-s1814" xml:space="preserve">3.</s>
            <s xml:id="echoid-s1815" xml:space="preserve">8} ({x/a})
              <emph style="super">{3mmαα/nn} - {13/2}</emph>
            + &</s>
            <s xml:id="echoid-s1816" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1817" xml:space="preserve">]
              <lb/>
            Hæc æquatio ita eſt integranda, ut poſita x = a fiat t = 0; </s>
            <s xml:id="echoid-s1818" xml:space="preserve">ſic autem oritur
              <lb/>
            t = [2 + {nn/2mmαα - 3nn} + {3nn/16mmαα - 28nn} + &</s>
            <s xml:id="echoid-s1819" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1820" xml:space="preserve">] X {√(mmαα - 2nn).</s>
            <s xml:id="echoid-s1821" xml:space="preserve">a/n}
              <lb/>
            - [2{(x/a)}
              <emph style="super">{1/2}</emph>
            + {nn/2mmαα - 3nn} ({x/a})
              <emph style="super">{mmαα/nn}-{3/2}</emph>
              <lb/>
            + {3nn/16mmαα - 28nn} ({x/a})
              <emph style="super">{2mmαα/nn} - {7/2}</emph>
            + &</s>
            <s xml:id="echoid-s1822" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1823" xml:space="preserve">] X
              <lb/>
            X {√(mmαα - 2nn).</s>
            <s xml:id="echoid-s1824" xml:space="preserve">a/n},
              <lb/>
            ubi 2 √ a exprimit tempus quod corpus impendit dum libere delabitur per
              <lb/>
            altitudinem a. </s>
            <s xml:id="echoid-s1825" xml:space="preserve">Si vero in iſta æquatione ponatur
              <lb/>
            x = a:</s>
            <s xml:id="echoid-s1826" xml:space="preserve">({mmαα - nn/nn})
              <emph style="super">nn: ({mmαα - 2nn})</emph>
              <lb/>
            quæ eſt altitudo aquæ cum velocitas maxima eſt (per §. 16. </s>
            <s xml:id="echoid-s1827" xml:space="preserve">ſect. </s>
            <s xml:id="echoid-s1828" xml:space="preserve">3. </s>
            <s xml:id="echoid-s1829" xml:space="preserve">& </s>
            <s xml:id="echoid-s1830" xml:space="preserve">§. </s>
            <s xml:id="echoid-s1831" xml:space="preserve">8. </s>
            <s xml:id="echoid-s1832" xml:space="preserve">ſect
              <lb/>
            4.)</s>
            <s xml:id="echoid-s1833" xml:space="preserve">, tum obtinetur tempus quod à fluxus principio ad punctum maximæ ve-
              <lb/>
            locitatis usque præterit; </s>
            <s xml:id="echoid-s1834" xml:space="preserve">& </s>
            <s xml:id="echoid-s1835" xml:space="preserve">cum ponitur x = o, oritur tempus, quo vas to-
              <lb/>
            tum depletur, ac denique ſi ponatur x = cuicunque quantitati c, exprimet t
              <lb/>
            tempus quod ſuperficies inſumit in deſcenſum per altitudinem a - c; </s>
            <s xml:id="echoid-s1836" xml:space="preserve">Videbi-
              <lb/>
            mus autem pro his caſibus, quid fieri debeat, cum vas eſt valde amplum,
              <lb/>
            numerusque m alterum n ſic pluries continet.</s>
            <s xml:id="echoid-s1837" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1838" xml:space="preserve">§. </s>
            <s xml:id="echoid-s1839" xml:space="preserve">11. </s>
            <s xml:id="echoid-s1840" xml:space="preserve">Fuerit primo {m/n} numerus infinitus
              <unsure/>
            , erit altitudo aquæ puncto
              <lb/>
            maximæ velocitatis reſpondens ſeu</s>
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