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

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              <pb o="225" file="0239" n="239" rhead="SECTIO DECIMA."/>
            tate ſuo pondere aërem infimum expellet per foramen, qua aër in vaſe pro-
              <lb/>
            poſito ſua elaſticitate ſe ipſum expellit. </s>
            <s xml:id="echoid-s6610" xml:space="preserve">In priori autem caſu ejicitur veloci-
              <lb/>
            tate quæ debetur ipſi altitudini cylindri, ergo & </s>
            <s xml:id="echoid-s6611" xml:space="preserve">in poſteriori. </s>
            <s xml:id="echoid-s6612" xml:space="preserve">Notandum
              <lb/>
            autem eſt, altitudinem quam pro cylindro finximus, perpetuo eandem eſſe,
              <lb/>
            quia aëris elaſticitas & </s>
            <s xml:id="echoid-s6613" xml:space="preserve">denſitas in eadem ratione diminuuntur, calorem autem
              <lb/>
            non mutari ponimus. </s>
            <s xml:id="echoid-s6614" xml:space="preserve">Igitur ſi altitudo aëris homogenei (quæ à calore aëris
              <lb/>
            interni pendet) dicatur A, effluet aër conſtanter velocitate √ A. </s>
            <s xml:id="echoid-s6615" xml:space="preserve">Nec tamen,
              <lb/>
            quod calculus oſtendit, vas ipſum unquam evacuatur, quia aër effluens fit
              <lb/>
            continue rarior, quod ut æquatione comprehendamus, ponemus denſitatem ſeu
              <lb/>
            quantitatem aëris à fluxus initio = 1; </s>
            <s xml:id="echoid-s6616" xml:space="preserve">denſitatem ſeu quantitatem aëris poſt de-
              <lb/>
            finitum tempus reſidui = x, tempusque ipſum = t, erit, quia velocitas
              <lb/>
            conſtans eſt, - d x = a x d t, ubi per a intelligitur quantitas conſtans defi-
              <lb/>
            nienda ex magnitudine vaſis, amplitudine foraminis & </s>
            <s xml:id="echoid-s6617" xml:space="preserve">altitudine A: </s>
            <s xml:id="echoid-s6618" xml:space="preserve">hinc
              <lb/>
            {- dx/x} = adt & </s>
            <s xml:id="echoid-s6619" xml:space="preserve">log. </s>
            <s xml:id="echoid-s6620" xml:space="preserve">{1/x} = at. </s>
            <s xml:id="echoid-s6621" xml:space="preserve">reperitur autem valor coëfficientis a hoc modo.
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            </s>
            <s xml:id="echoid-s6622" xml:space="preserve">Quia poſitum à nobis fuit - d x = a x d t; </s>
            <s xml:id="echoid-s6623" xml:space="preserve">erit ab initio effluxus - dx = a d t. </s>
            <s xml:id="echoid-s6624" xml:space="preserve">
              <lb/>
            Jam mutetur elementum primum (- d x) in cylindrum foramini ceu baſi ſu-
              <lb/>
            perinſtructum; </s>
            <s xml:id="echoid-s6625" xml:space="preserve">erit autem altitudo iſtius cylindruli = - L d x, ſi L ſit altitu-
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            do cylindri ſuper eodem foramine extructi & </s>
            <s xml:id="echoid-s6626" xml:space="preserve">communem cum vaſe propo-
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            ſito capacitatem habentis: </s>
            <s xml:id="echoid-s6627" xml:space="preserve">hæc porro longitudo - L d x illa eſt, quæ tem-
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            puſculo d t percurritur, & </s>
            <s xml:id="echoid-s6628" xml:space="preserve">quia poni ſolet tempuſculum æquale ſpatio percur-
              <lb/>
            ſo diviſo per velocitatem, erit hic d t = {- L d x/√ A}; </s>
            <s xml:id="echoid-s6629" xml:space="preserve">ſubſtituatur iſte valor in
              <lb/>
            æquatione - d x = a d t & </s>
            <s xml:id="echoid-s6630" xml:space="preserve">habebitur - d x = {- a L d x/√A}, ſive a = {√A/L}. </s>
            <s xml:id="echoid-s6631" xml:space="preserve">Eſt
              <lb/>
            proinde æquatio finalis hæc: </s>
            <s xml:id="echoid-s6632" xml:space="preserve">
              <lb/>
            log. </s>
            <s xml:id="echoid-s6633" xml:space="preserve">{1/x} = {t√A/L}.</s>
            <s xml:id="echoid-s6634" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6635" xml:space="preserve">Si tempus exprimere lubeat per certum minutorum ſecundorum nu-
              <lb/>
            merum, quem vocabimus n, & </s>
            <s xml:id="echoid-s6636" xml:space="preserve">intelligatur per s ſpatium quod mobile ab-
              <lb/>
            ſolvit cadendo libere à quiete intra unum minutum ſecundum, erit ponen-
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            dum t = 2n√s, ſicque fiet
              <lb/>
            log. </s>
            <s xml:id="echoid-s6637" xml:space="preserve">{1/x} = {2n√As/L}.</s>
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