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

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            <s xml:id="echoid-s3957" xml:space="preserve">
              <pb o="137" file="0151" n="151" rhead="SECTIO SEPTIMA."/>
            Sic eſt ξ = α + z & </s>
            <s xml:id="echoid-s3958" xml:space="preserve">b = α + c, ubi quantitates z & </s>
            <s xml:id="echoid-s3959" xml:space="preserve">c ſunt ceu infinite par-
              <lb/>
            væ conſiderandæ ratione quantitatis α. </s>
            <s xml:id="echoid-s3960" xml:space="preserve">Habetur hinc
              <lb/>
            ({α/ξ})
              <emph style="super">nn</emph>
            = ({α/α + z})
              <emph style="super">nn</emph>
            = (1 + {z/α})
              <emph style="super">-nn</emph>
            = adhibendo ſeriem notam
              <lb/>
            & </s>
            <s xml:id="echoid-s3961" xml:space="preserve">ex illa ſumendo tres primos terminos 1 - {nnz/α} + {nn.</s>
            <s xml:id="echoid-s3962" xml:space="preserve">
              <emph style="ol">nn + 1</emph>
            zz/2αα}. </s>
            <s xml:id="echoid-s3963" xml:space="preserve">Subſtitu-
              <lb/>
            tis iſtis valoribus pro b, ξ & </s>
            <s xml:id="echoid-s3964" xml:space="preserve">({α/ξ})
              <emph style="super">nn</emph>
            mutatur æquatio ultima paragraphi de-
              <lb/>
            cimi quarti in hanc, v = {α + c/nn} X ({nnz/α} - {nn x
              <emph style="ol">nn + 1</emph>
            zz/2αα}) -
              <lb/>
            {1/nn + 1} X (α + z - α + nnz - {nn.</s>
            <s xml:id="echoid-s3965" xml:space="preserve">
              <emph style="ol">nn + 1</emph>
            zz/2α}) =
              <lb/>
            (α + c) X ({z/α} - {
              <emph style="ol">nn + 1</emph>
            zz/2αα}) - (z - {nnzz/2α}) =
              <lb/>
            {cz/α} - {zz/2α} - {
              <emph style="ol">nn + 1</emph>
            czz/2αα}: </s>
            <s xml:id="echoid-s3966" xml:space="preserve">Poteſt autem negligi iſte ultimus terminus & </s>
            <s xml:id="echoid-s3967" xml:space="preserve">ſic
              <lb/>
            fit ſimpliciter
              <lb/>
            v = {2cz - zz/2α},
              <lb/>
            quam æquationem n non amplius ingreditur: </s>
            <s xml:id="echoid-s3968" xml:space="preserve">Neque illa differt ab æquatio-
              <lb/>
            ne pro deſcenſu §. </s>
            <s xml:id="echoid-s3969" xml:space="preserve">10. </s>
            <s xml:id="echoid-s3970" xml:space="preserve">data, nempe v = {2cz - zz/2a}, quandoquidem quan-
              <lb/>
            titas a & </s>
            <s xml:id="echoid-s3971" xml:space="preserve">α non differunt niſi quantitate minima 2 c.</s>
            <s xml:id="echoid-s3972" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3973" xml:space="preserve">Cæterum hic omnia etiam ſunt ſubintelligenda, quæ eodem §. </s>
            <s xml:id="echoid-s3974" xml:space="preserve">10. </s>
            <s xml:id="echoid-s3975" xml:space="preserve">de
              <lb/>
            tubo non nimis obſtruendo dicta ſunt.</s>
            <s xml:id="echoid-s3976" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3977" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3978" xml:space="preserve">19. </s>
            <s xml:id="echoid-s3979" xml:space="preserve">Sunt igitur deſcenſus & </s>
            <s xml:id="echoid-s3980" xml:space="preserve">aſcenſus ſibi æquales; </s>
            <s xml:id="echoid-s3981" xml:space="preserve">nam ex æquatio-
              <lb/>
            nibus noſtris patet, liquorem æqualiter librari ultra ſuperficiem aquæ externæ.
              <lb/>
            </s>
            <s xml:id="echoid-s3982" xml:space="preserve">Deinde vero potiſſimum ſequitur ex iſtis formulis, eſſe vel oſcillationes inæqua-
              <lb/>
            les inter ſe iſochronas, modo omnes poſſint infinite parvæ cenſeri ratione ſub-
              <lb/>
            merſionis: </s>
            <s xml:id="echoid-s3983" xml:space="preserve">Pendulum autem ſimplex tautochronum eſſe ejuſdem longitudinis
              <lb/>
            cum parte tubi ſubmerſa.</s>
            <s xml:id="echoid-s3984" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3985" xml:space="preserve">Differt iſtud theorema ab illo, quod §. </s>
            <s xml:id="echoid-s3986" xml:space="preserve">4. </s>
            <s xml:id="echoid-s3987" xml:space="preserve">ſect. </s>
            <s xml:id="echoid-s3988" xml:space="preserve">6. </s>
            <s xml:id="echoid-s3989" xml:space="preserve">de oſcillationibus in
              <lb/>
            tubo cylindrico ex duobus cruribus verticalibus compoſito citatum fuit, in eo,
              <lb/>
            quod ibi oſcillationes omnes non excluſis oſcillationibus finitæ magnitudinis
              <lb/>
            ſint tautochronæ, cum@in præſenti caſu oſcillationes finitæ ſint inæqualis </s>
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