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

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          <p>
            <s xml:id="echoid-s2952" xml:space="preserve">
              <pb o="105" file="0119" n="119" rhead="SECTIO QUINTA."/>
            log. </s>
            <s xml:id="echoid-s2953" xml:space="preserve">[1 - √(1 - c{n
              <emph style="super">3</emph>
            - nmm/mmN} x)] = log.</s>
            <s xml:id="echoid-s2954" xml:space="preserve">{1/2}c{n
              <emph style="super">3</emph>
            - nmm/mmN} x = {n
              <emph style="super">3</emph>
            - nmm/mmN} x - log. </s>
            <s xml:id="echoid-s2955" xml:space="preserve">2@</s>
          </p>
          <p>
            <s xml:id="echoid-s2956" xml:space="preserve">Hæ ſubſtitutiones ſi recte fiant, erit pro primo quem finximus affuſio-
              <lb/>
            nis modo
              <lb/>
            (I) t = {γmN/n√(mm - nn) a} X (2 log. </s>
            <s xml:id="echoid-s2957" xml:space="preserve">2 + {mmn - n
              <emph style="super">3</emph>
            /mmN} x)
              <lb/>
            quæ poſito rurſus m = ∞ dat pro altero caſu
              <lb/>
            (II) t = {γN/n√a} X (2. </s>
            <s xml:id="echoid-s2958" xml:space="preserve">log. </s>
            <s xml:id="echoid-s2959" xml:space="preserve">2 + {n/N} x).</s>
            <s xml:id="echoid-s2960" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2961" xml:space="preserve">Sequitur ex iſtis formulis, minori quidem quantitate transfluere aquas,
              <lb/>
            ac ſi ſtatim ab initio omni velocitate, quam in utroque caſu poſt tempus
              <lb/>
            infinitum acquirunt, effluerent: </s>
            <s xml:id="echoid-s2962" xml:space="preserve">differentiam tamen nunquam certum trans-
              <lb/>
            gredi terminum & </s>
            <s xml:id="echoid-s2963" xml:space="preserve">poſt tempus infinitum finitis comprehendi terminis.</s>
            <s xml:id="echoid-s2964" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div113" type="section" level="1" n="86">
          <head xml:id="echoid-head111" xml:space="preserve">Corollarium 2.</head>
          <p>
            <s xml:id="echoid-s2965" xml:space="preserve">§. </s>
            <s xml:id="echoid-s2966" xml:space="preserve">16. </s>
            <s xml:id="echoid-s2967" xml:space="preserve">Quum convertimus æquationes inventas, obtinemus
              <lb/>
            (I) x = {2mmN/mmn - n
              <emph style="super">3</emph>
            } - [log. </s>
            <s xml:id="echoid-s2968" xml:space="preserve">(1 + c
              <emph style="super">{-t/α</emph>
            }) - log. </s>
            <s xml:id="echoid-s2969" xml:space="preserve">2 + {t/2α}], & </s>
            <s xml:id="echoid-s2970" xml:space="preserve">
              <lb/>
            (II) x = {2N/n} X [log. </s>
            <s xml:id="echoid-s2971" xml:space="preserve">(1 + c
              <emph style="super">{-t/β}</emph>
            ) - log. </s>
            <s xml:id="echoid-s2972" xml:space="preserve">2 + {t/2β}]
              <lb/>
            ubi α, ut ſupra, = {-γmN/n√(mm - nn)a} & </s>
            <s xml:id="echoid-s2973" xml:space="preserve">β = {-γN/n√a}.</s>
            <s xml:id="echoid-s2974" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2975" xml:space="preserve">Si præterea, ut in proximo Corollario, ponatur t = ∞, evaneſcit
              <lb/>
            unitas præ quantitatibus, exponentialibus, quæ ſupra omnem ordinem infinitæ
              <lb/>
            ſunt, & </s>
            <s xml:id="echoid-s2976" xml:space="preserve">fit log. </s>
            <s xml:id="echoid-s2977" xml:space="preserve">(1 + c
              <emph style="super">{-t/α}</emph>
            ) = -{t/α} atque log. </s>
            <s xml:id="echoid-s2978" xml:space="preserve">(1 + c
              <emph style="super">{-t/β}</emph>
            ) = -{t/β}:
              <lb/>
            </s>
            <s xml:id="echoid-s2979" xml:space="preserve">unde tunc erit reſumtis valoribus litterarum α & </s>
            <s xml:id="echoid-s2980" xml:space="preserve">β. </s>
            <s xml:id="echoid-s2981" xml:space="preserve">
              <lb/>
            (I) x = {mt√a/γ√(mm - nn)} - {2mmN/mmn - n
              <emph style="super">3</emph>
            } log. </s>
            <s xml:id="echoid-s2982" xml:space="preserve">2. </s>
            <s xml:id="echoid-s2983" xml:space="preserve">& </s>
            <s xml:id="echoid-s2984" xml:space="preserve">
              <lb/>
            (II) x = {t√a/γ} - {2N/n} log. </s>
            <s xml:id="echoid-s2985" xml:space="preserve">2.</s>
            <s xml:id="echoid-s2986" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2987" xml:space="preserve">Igitur ſi ſtatim à fluxus initio utrobique aquæ omni, quam </s>
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