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

Table of figures

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            <s xml:id="echoid-s3763" xml:space="preserve">
              <pb o="131" file="0145" n="145" rhead="SECTIO SEPTIMA."/>
            cujus ſecundus terminus z d v rurſus præ primo negligi poteſt, ita vero
              <lb/>
            habetur
              <lb/>
            adv + nnvdz = (c - z)dz.</s>
            <s xml:id="echoid-s3764" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3765" xml:space="preserve">Ponatur hic (ſumto α pro numero, cujus logarithmus hyperbolicus eſt
              <lb/>
            unitas) v = {1/nn}α
              <emph style="super">{-nnz/a}</emph>
            q; </s>
            <s xml:id="echoid-s3766" xml:space="preserve">hoc modo mutabitur poſtrema æquatio in hanc
              <lb/>
            α{-nnz/a}adq = nn (c - z)dz, vel
              <lb/>
            adq = nnα
              <emph style="super">{nnz/a}</emph>
            X (c - z)dz:</s>
            <s xml:id="echoid-s3767" xml:space="preserve"/>
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            <s xml:id="echoid-s3768" xml:space="preserve">Hæc vero ita eſt integranda, ut z & </s>
            <s xml:id="echoid-s3769" xml:space="preserve">v vel etiam z & </s>
            <s xml:id="echoid-s3770" xml:space="preserve">q ſimul evane-
              <lb/>
            ſcant; </s>
            <s xml:id="echoid-s3771" xml:space="preserve">habebitur igitur
              <lb/>
            q = (c + {a/nn} - z)α
              <emph style="super">{nnz/a}</emph>
            - c - {a/nn}, vel denique
              <lb/>
            v = {1/nn} (c + {a/nn} - z) - {1/nn} (c + {a/nn})α
              <emph style="super">{-nnz/a}</emph>
            ;</s>
            <s xml:id="echoid-s3772" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3773" xml:space="preserve">Ex iſta vero æquatione deducitur:</s>
            <s xml:id="echoid-s3774" xml:space="preserve"/>
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            <s xml:id="echoid-s3775" xml:space="preserve">I. </s>
            <s xml:id="echoid-s3776" xml:space="preserve">Oriri rurſus, ut paragrapho decimo alia mathodo inventum fuit,
              <lb/>
            v = {2cz - zz/2a}, ſi nempe rurſus ponatur {nnz/a} numerus valde parvus, Id ve-
              <lb/>
            ro ut pateat, reſolvenda eſt quantitas exponentialis α
              <emph style="super">{-nnz/a}</emph>
            in ſeriem, quæ
              <lb/>
            eſt ipſi æqualis, 1 - {nnz/a} + {n
              <emph style="super">4</emph>
            zz/2aa} - {n
              <emph style="super">6</emph>
            z
              <emph style="super">3</emph>
            /2. </s>
            <s xml:id="echoid-s3777" xml:space="preserve">3a
              <emph style="super">3</emph>
            } + &</s>
            <s xml:id="echoid-s3778" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3779" xml:space="preserve">ex quâ pro noſtro
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            ſcopo tres priores termini ſufficiunt; </s>
            <s xml:id="echoid-s3780" xml:space="preserve">eo autem ſubſtituto valore rejectoque
              <lb/>
            termino rejiciendo, reperitur ut dixi
              <lb/>
            v = {2cz - zz/2a}</s>
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            <s xml:id="echoid-s3781" xml:space="preserve">II. </s>
            <s xml:id="echoid-s3782" xml:space="preserve">At ſi viciſſim {nn/1} infinites major ponatur quam {a/z} aut {a/c}, quia tunc
              <lb/>
            α{-nnz/a} = o, ut & </s>
            <s xml:id="echoid-s3783" xml:space="preserve">{a/nn} = o, fieri intelligitur v = c - z, ſive v = x - b,
              <lb/>
            ut §. </s>
            <s xml:id="echoid-s3784" xml:space="preserve">4.</s>
            <s xml:id="echoid-s3785" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s3786" xml:space="preserve">III. </s>
            <s xml:id="echoid-s3787" xml:space="preserve">Neutram vero præmiſſarum formularum ſine notabili errore lo-
              <lb/>
            cum habere patet, cum {nnc/a}, numerus eſt mediocris, nempe nec infinitus,
              <lb/>
            nec infinite parvus, & </s>
            <s xml:id="echoid-s3788" xml:space="preserve">tamen utraque quantitas {nn/1} & </s>
            <s xml:id="echoid-s3789" xml:space="preserve">{a/c} infinita.</s>
            <s xml:id="echoid-s3790" xml:space="preserve"/>
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