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

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          <p>
            <s xml:id="echoid-s7969" xml:space="preserve">
              <pb o="272" file="0286" n="286" rhead="HYDRODYNAMICÆ"/>
            gpv:</s>
            <s xml:id="echoid-s7970" xml:space="preserve">[pv + m√(a - {ppvv/nn})].</s>
            <s xml:id="echoid-s7971" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7972" xml:space="preserve">Si hæ partes multiplicentur reſpective per quadrata ſuarum velocitatum,
              <lb/>
            habebuntur earundem vires vivæ, quarum aggregatum æquandum eſt
              <lb/>
            cum g X a, id eſt, cum deſcenſu actuali guttulæ g per altitudinem a. </s>
            <s xml:id="echoid-s7973" xml:space="preserve">Sic ob-
              <lb/>
            tinetur talis æquatio, ſi reducatur
              <lb/>
            n
              <emph style="super">3</emph>
            vv - n
              <emph style="super">3</emph>
            a = mpv√(nna - ppvv) ſive
              <lb/>
            vv = {2n
              <emph style="super">6</emph>
            + mmnnpp + nnmp√4n
              <emph style="super">4</emph>
            + mmpp - 4nnpp)/2n
              <emph style="super">6</emph>
            + 2mmp
              <emph style="super">4</emph>
            .</s>
            <s xml:id="echoid-s7974" xml:space="preserve">}a,
              <lb/>
            hæcque quantitas exprimit altitudinem pro velocitate aquæ in o effluentis, qua
              <lb/>
            cognita habetur quoque altitudo ſimilis pro altero foramine ac, quæ nempe
              <lb/>
            eſt = a - {ppvv/nn}.</s>
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          <p>
            <s xml:id="echoid-s7976" xml:space="preserve">§. </s>
            <s xml:id="echoid-s7977" xml:space="preserve">22. </s>
            <s xml:id="echoid-s7978" xml:space="preserve">Si p = n, fit vv = a; </s>
            <s xml:id="echoid-s7979" xml:space="preserve">ergo tunc aquæ tota velocitate exiliunt
              <lb/>
            ſolita per foramen o, & </s>
            <s xml:id="echoid-s7980" xml:space="preserve">per alterum foramen a c nihil effluit. </s>
            <s xml:id="echoid-s7981" xml:space="preserve">In utroque
              <lb/>
            porro foramine velocitas reſpondet integræ altitudini a, ſi p eſt veluti infini-
              <lb/>
            te parva: </s>
            <s xml:id="echoid-s7982" xml:space="preserve">Si vero m eſt infinite parva, fit quidem v v = a, ſed altitudo ve-
              <lb/>
            locitatis pro foraminulo ac eſt = a - {pp/nn}a, ut §. </s>
            <s xml:id="echoid-s7983" xml:space="preserve">7. </s>
            <s xml:id="echoid-s7984" xml:space="preserve">jam indicatum fuit:
              <lb/>
            </s>
            <s xml:id="echoid-s7985" xml:space="preserve">Si m = p, fit vv = {n
              <emph style="super">4</emph>
            a/n
              <emph style="super">4</emph>
            - nnpp + p
              <emph style="super">4</emph>
            }; </s>
            <s xml:id="echoid-s7986" xml:space="preserve">& </s>
            <s xml:id="echoid-s7987" xml:space="preserve">a - {ppvv/nn} = {(nn - pp)
              <emph style="super">2</emph>
            a/n
              <emph style="super">4</emph>
            - nnpp + p
              <emph style="super">4</emph>
            }.</s>
            <s xml:id="echoid-s7988" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7989" xml:space="preserve">Denique obſervari poteſt, aquas per foramen o ſemper majori velo-
              <lb/>
            citate ejici, quam quæ altitudini a reſpondet, quod utique fit, quia aquæ
              <lb/>
            in E d veluti impetum faciunt in aquas d F.</s>
            <s xml:id="echoid-s7990" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7991" xml:space="preserve">Interim quamvis omnia hæc Corollaria egregie cum indole argumenti
              <lb/>
            conſentiunt, non poteſt tamen ſolutio iſtius problematis aliter quam proxi-
              <lb/>
            me vera cenſeri.</s>
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