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

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        <div xml:id="echoid-div111" type="section" level="1" n="84">
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            <s xml:id="echoid-s2926" xml:space="preserve">
              <pb o="104" file="0118" n="118" rhead="HYDRODYNAMICÆ."/>
            in quâ ſi præterea fiat 1 - z = qq, ſeu z = 1 - qq, dz = - 2qdq,
              <lb/>
            oritur
              <lb/>
            dt = {- 2αdq/1 - qq} = {- αdq/1 + q} {- αdq/1 - q}
              <lb/>
            cujus integralis eſt
              <lb/>
            t = - α log. </s>
            <s xml:id="echoid-s2927" xml:space="preserve">(1 + q) + α log. </s>
            <s xml:id="echoid-s2928" xml:space="preserve">(1 - q) = α log. </s>
            <s xml:id="echoid-s2929" xml:space="preserve">{1 - q/1 + q}.</s>
            <s xml:id="echoid-s2930" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2931" xml:space="preserve">Nec opus eſt conſtante, quandoquidem ex natura rei t & </s>
            <s xml:id="echoid-s2932" xml:space="preserve">x, ſimul
              <lb/>
            evaneſcere debent, poſito autem x = o, fit z = 1, & </s>
            <s xml:id="echoid-s2933" xml:space="preserve">q = o, igitur pa-
              <lb/>
            riter t & </s>
            <s xml:id="echoid-s2934" xml:space="preserve">q ſimul à nihilo incipere debent, cui conditioni ſatisfacit æquatio
              <lb/>
            inventa t = α log. </s>
            <s xml:id="echoid-s2935" xml:space="preserve">{1 - q/1 + q}: </s>
            <s xml:id="echoid-s2936" xml:space="preserve">Supereſt ut retrogrado ordine valores priſtinos
              <lb/>
            reaſſumamus, ita vero fit
              <lb/>
            t = α log. </s>
            <s xml:id="echoid-s2937" xml:space="preserve">{1 - √(1 - z)/1 + √(1 - z)} vel
              <lb/>
            t = {γmN/n√(mm - nn)a} X log. </s>
            <s xml:id="echoid-s2938" xml:space="preserve">{1 + √(1 - z)/1 - √(1 - z)} vel denique
              <lb/>
            (I) t = {γmN/n√(mm - nn) a} X [log. </s>
            <s xml:id="echoid-s2939" xml:space="preserve">[1 + √(1 - c{n
              <emph style="super">3</emph>
            - nmm/mmN} x)]
              <lb/>
            - log. </s>
            <s xml:id="echoid-s2940" xml:space="preserve">[1 - √(1 - c{n
              <emph style="super">3</emph>
            - nmm/mmN} x)]]
              <lb/>
            Iſtaque æquatio poſito m = ∞ dat alteram æquationem quæſitam
              <lb/>
            (II) t = {γN/n√a} X [log. </s>
            <s xml:id="echoid-s2941" xml:space="preserve">[1 + √(1 - c
              <emph style="super">{- n/N} x</emph>
            )]
              <lb/>
            - log. </s>
            <s xml:id="echoid-s2942" xml:space="preserve">[1 - √(1 - c
              <emph style="super">{-n/N} x</emph>
            )]] Q. </s>
            <s xml:id="echoid-s2943" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2944" xml:space="preserve">I.</s>
            <s xml:id="echoid-s2945" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div112" type="section" level="1" n="85">
          <head xml:id="echoid-head110" xml:space="preserve">Corollarium 1.</head>
          <p>
            <s xml:id="echoid-s2946" xml:space="preserve">§. </s>
            <s xml:id="echoid-s2947" xml:space="preserve">15. </s>
            <s xml:id="echoid-s2948" xml:space="preserve">Si ponatur x = ∞, ut appareat natura rei, cum infinita jam
              <lb/>
            transfluxit aquæ quantitas aſſumaturque m major quam n, prouti plerumque
              <lb/>
            eſſe ſolet, evaneſcere cenſenda eſt, in utroque logarithmo affirmative ſum-
              <lb/>
            to, quantitas exponentialis & </s>
            <s xml:id="echoid-s2949" xml:space="preserve">habebitur utrobique log. </s>
            <s xml:id="echoid-s2950" xml:space="preserve">2. </s>
            <s xml:id="echoid-s2951" xml:space="preserve">At vero in logarith-
              <lb/>
            mo negative ſumto ſtatuenda eſt
              <lb/>
            √(1 - c{n
              <emph style="super">3</emph>
            - nmm/mmN} x) = 1 - {1/2} c{n
              <emph style="super">3</emph>
            - nmm/mmN} x & </s>
            <s xml:id="echoid-s2952" xml:space="preserve"/>
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