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

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        <div xml:id="echoid-div243" type="section" level="1" n="187">
          <pb o="226" file="0240" n="240" rhead="HYDRODYNAMICÆ"/>
        </div>
        <div xml:id="echoid-div244" type="section" level="1" n="188">
          <head xml:id="echoid-head239" xml:space="preserve">Problema.</head>
          <p>
            <s xml:id="echoid-s6639" xml:space="preserve">§. </s>
            <s xml:id="echoid-s6640" xml:space="preserve">35. </s>
            <s xml:id="echoid-s6641" xml:space="preserve">Quæritur motus aëris denſioris in aërem externum rariorem
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            infinitum ex vaſe per foramen valde parvum erumpentis, poſito in utroque
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            aëre eodem caloris gradu.</s>
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        <div xml:id="echoid-div245" type="section" level="1" n="189">
          <head xml:id="echoid-head240" xml:space="preserve">Solutio.</head>
          <p>
            <s xml:id="echoid-s6643" xml:space="preserve">Sit denſitas aëris interni initialis = D; </s>
            <s xml:id="echoid-s6644" xml:space="preserve">denfitas aëris externi = δ:
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            </s>
            <s xml:id="echoid-s6645" xml:space="preserve">denſitas aëris interni poſt datum tempus t reſidui = x, altitudo aëris homo-
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            genei, (ſive ratione aëris interni ſive externi, nec enim diverſa eſſe poteſt,
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            ſi uterque aër eodem calore præditus ſit, ſicque denſitates & </s>
            <s xml:id="echoid-s6646" xml:space="preserve">elaſticitates in
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            pari ratione decreſcant) = A. </s>
            <s xml:id="echoid-s6647" xml:space="preserve">Quæratur ubique altitudo aëris homogenei,
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            qui habeat eandem preſſionem ſeu elaterem cum aëre externo & </s>
            <s xml:id="echoid-s6648" xml:space="preserve">cujus den-
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            ſitas eadem ſit cum aëre interno: </s>
            <s xml:id="echoid-s6649" xml:space="preserve">hæc altitudo ab initio erit {δA/D}, & </s>
            <s xml:id="echoid-s6650" xml:space="preserve">poſt
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            tempus t erit {δA/x}. </s>
            <s xml:id="echoid-s6651" xml:space="preserve">Patet autem velocitatem aëris erumpentis talem ubique
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            fore, quæ reſpondeat differentiæ definitarum altitudinum A & </s>
            <s xml:id="echoid-s6652" xml:space="preserve">{δA/x}; </s>
            <s xml:id="echoid-s6653" xml:space="preserve">eſt itaque
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            poſt tempus t velocitas aëris erumpentis = √(A - {δA/x}).</s>
            <s xml:id="echoid-s6654" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6655" xml:space="preserve">Sunt porro decrementa denſitatum (- d x) proportionalia quantitati-
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            bus aëris erumpentis, quæ rationem habent compoſitam ex velocitate
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            (√(A - {δA/x})) ex denſitate (x) & </s>
            <s xml:id="echoid-s6656" xml:space="preserve">ex tempuſculo (d t): </s>
            <s xml:id="echoid-s6657" xml:space="preserve">ſic igitur eſt - d x
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            = a (√(A - {δA/x})) x d t, ubi a eſt numerus conſtans qui per metho-
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            dum præcedentis paragraphi fit = {1/L}, retenta ſignificatione hujus litteræ
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            ibidem adhibita; </s>
            <s xml:id="echoid-s6658" xml:space="preserve">hocque valore ſubſtituto oritur
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            - d x = {dt/L} X √ (Axx - δAx) ſeu {- dx/√ (xx - δx)} = {dt√A/L}:
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            </s>
            <s xml:id="echoid-s6659" xml:space="preserve">Factaque debita integratione fit: </s>
            <s xml:id="echoid-s6660" xml:space="preserve">
              <lb/>
            log.</s>
            <s xml:id="echoid-s6661" xml:space="preserve">{[√x - √(x - δ)] x [√D + √(D - δ)]/[√x + √(x - δ)] x [√D - √(D - δ)]} = {t√A/L}, aut poſito rurſus, ut in
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            præcedente paragragho, t = 2 n √ s, </s>
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