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

Table of handwritten notes

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            <s xml:id="echoid-s2028" xml:space="preserve">Ponatur autem pro ſeparandis ab invicem indeterminatis {mm/nn}v - x = s, ſive
              <lb/>
            v = {nn/mm}(s + x), atque dv = {nn/mm} (ds + dx) ſicque fiet
              <lb/>
            dx = {- nnbds/nnb - ms√gn},
              <lb/>
            quæ ita eſt integranda, ut facta x = a, prodeat v = o, hincque s = - a,
              <lb/>
            ita vero fit
              <lb/>
            x - a = {nnb/m√gn}log.</s>
            <s xml:id="echoid-s2029" xml:space="preserve">{nnb - ms√gn/nnb + ma√gn}
              <lb/>
            & </s>
            <s xml:id="echoid-s2030" xml:space="preserve">poſito pro s valore ejus aſſumto {mm/nn}v - x, prodit
              <lb/>
            x - a = {nnb/m√gn}log.</s>
            <s xml:id="echoid-s2031" xml:space="preserve">{n
              <emph style="super">4</emph>
            b - m
              <emph style="super">3</emph>
            v√gn + mnnx√gn/n
              <emph style="super">4</emph>
            b + mnna√gn}</s>
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          <p>
            <s xml:id="echoid-s2032" xml:space="preserve">Hic rurſus in quantitate ſigno logarithmicali involuta poteſt ex nume-
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            ratore eliminari terminus n
              <emph style="super">4</emph>
            b, infinities nempe minor termino mnnx√gn
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            nec non ex denominatore terminus n
              <emph style="super">4</emph>
            b infinities pariter minor altero
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            mnna√gn. </s>
            <s xml:id="echoid-s2033" xml:space="preserve">Et ſic fit
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            x - a = {nnb/m√gn}log.</s>
            <s xml:id="echoid-s2034" xml:space="preserve">{nnx - mma/nna}</s>
          </p>
          <p>
            <s xml:id="echoid-s2035" xml:space="preserve">Inde habetur, poſito c pro numero cujus logarithmus eſt unitas:
              <lb/>
            </s>
            <s xml:id="echoid-s2036" xml:space="preserve">v = {nnx/mm} - {nna/mm} X c {m.</s>
            <s xml:id="echoid-s2037" xml:space="preserve">(x - a)√gn/nnb}
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            aut poſita a - x = z, ſic ut z denotet ſpatium, per quod ſuperficies aquæ
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            jam deſcendit, poterit æquationi hæc conciliari forma: </s>
            <s xml:id="echoid-s2038" xml:space="preserve">
              <lb/>
            v = {nn.</s>
            <s xml:id="echoid-s2039" xml:space="preserve">(a - z)/mm} - {nna/mm}:</s>
            <s xml:id="echoid-s2040" xml:space="preserve">c
              <emph style="super">{mz/nb}</emph>
            √{g/n}
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            de qua iterum liquet quod cum z vel minimam habuerit rationem ad b, fiat
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            denominator alterius termini infinitus & </s>
            <s xml:id="echoid-s2041" xml:space="preserve">v = {nn.</s>
            <s xml:id="echoid-s2042" xml:space="preserve">(a - z)/mm} = {nnx/mm}: </s>
            <s xml:id="echoid-s2043" xml:space="preserve">at vero ali-
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            ter ſe res habet, quamdiu deſcenſus z infinite parvus eſt, quem caſum nunc
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            conſideramus.</s>
            <s xml:id="echoid-s2044" xml:space="preserve"/>
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            <s xml:id="echoid-s2045" xml:space="preserve">§. </s>
            <s xml:id="echoid-s2046" xml:space="preserve">17. </s>
            <s xml:id="echoid-s2047" xml:space="preserve">Hiſce præmiſſis facile nunc eſt definire per quantulum ſpatium
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            deſcendat fluidum, dum maximam velocitatem acquirit, faciendo </s>
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