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

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            <s xml:id="echoid-s1171" xml:space="preserve">
              <pb o="45" file="0059" n="59" rhead="SECTIO TERTIA."/>
            (1 - {mm/nn})vdz + zdv = - zdz - bdz + {mbdz/√gn}
              <lb/>
            quæ multiplicata per z
              <emph style="super">{-mm/nn}</emph>
            facit
              <lb/>
            (1 - {mm/nn})z
              <emph style="super">- {mm/nn}</emph>
            vdz + z
              <emph style="super">1 - {mm/nn}</emph>
            dv = - z
              <emph style="super">1 - {mm/nn}</emph>
            dz - bz
              <emph style="super">- {mm/nn}</emph>
            dz +
              <lb/>
            {mbz
              <emph style="super">- {mm/nn}</emph>
            dz/√gn}
              <lb/>
            poſt cujus integrationem addita conſtante Coritur
              <lb/>
            z
              <emph style="super">{nn - mm/nn}</emph>
            v = C - {nn/2nn - mm} z
              <emph style="super">{2nn - mm/nn}</emph>
            - {nnb/nn - mm} z
              <emph style="super">{nn - mm/nn}</emph>
              <lb/>
            + {mnnb/(nn - mm)√gn} z
              <emph style="super">{nn - mm/nn}</emph>
              <lb/>
            in quo valor quantitatis conſtantis C ex eo definitur quod ab initio fluxus
              <lb/>
            (cum nempe x = a ſive z = a - b + {mb/√gn}) ſit v = o quia non poteſt motus
              <lb/>
            oriri in inſtanti temporis puncto; </s>
            <s xml:id="echoid-s1172" xml:space="preserve">hinc igitur fit C =
              <lb/>
            [(a - b + {mb/√gn}) X {nn/2nn - mm} + {nnb√gn - mnnb/(nn - mm)√gn}] X (a - b + {mb/√gn})
              <emph style="super">{nn - mm/nn}</emph>
              <lb/>
            Ex his quidem æquationibus definiuntur omnia; </s>
            <s xml:id="echoid-s1173" xml:space="preserve">quia verò calculus fit paullo
              <lb/>
            prolixior, niſi amplitudo vaſis ſuperioris indicata per m tanta ſit, ut poſſit ra-
              <lb/>
            tione amplitudinum g & </s>
            <s xml:id="echoid-s1174" xml:space="preserve">n infinita cenſeri, hunc ſolum conſiderabimus caſum,
              <lb/>
            idque eo magis quod error notabilis inde non oriatur, etſi mediocris ſit ma-
              <lb/>
            gnitudinis numerus {m/n} aut {m/g}</s>
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          <p>
            <s xml:id="echoid-s1175" xml:space="preserve">§. </s>
            <s xml:id="echoid-s1176" xml:space="preserve">23. </s>
            <s xml:id="echoid-s1177" xml:space="preserve">Quod ſi proinde ponamus m = ∞, ſimulque utamur pri-
              <lb/>
            mâ æquatione differentiali proximi paragraphi, atque in hâc ponatur
              <lb/>
            v = {nn/mm}s, ut ſic inveniatur ex valore litteræ s altitudo ad quam aqua per ori-
              <lb/>
            ficium M N effluens ſuâ velocitate aſcendere poſſit, erit primo
              <lb/>
            {nn/m} (x - b)ds + {bnn/√gn}ds - msdx + {nn/m}sdx = - mxdx
              <lb/>
            & </s>
            <s xml:id="echoid-s1178" xml:space="preserve">quia m = ∞ atque facile prævidetur rationem ſore finitam inter s & </s>
            <s xml:id="echoid-s1179" xml:space="preserve">x, at-
              <lb/>
            que inter ds & </s>
            <s xml:id="echoid-s1180" xml:space="preserve">dx, hæc eadem æquatio mutabitur rejectis terminis rejiciendis
              <lb/>
            rurſus in hanc - msdx = - mxdx vel s = x, quod pariter paragr. </s>
            <s xml:id="echoid-s1181" xml:space="preserve">10.</s>
            <s xml:id="echoid-s1182" xml:space="preserve"/>
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