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

Table of Notes

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              <pb o="95" file="0109" n="109" rhead="SECTIO QUINTA."/>
            ſpondentia, ita ut aquæ ex ſuperiori vaſe effluentes omnes in cylindrum ſubje-
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            ctum influant.</s>
            <s xml:id="echoid-s2676" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2677" xml:space="preserve">Incipiant aquæ ex utroque vaſe effluere, ex ſuperiori autem conſtanter ea
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            effluere velocitate ponantur, quam habet ſuperficies aquæ in cylindro ſuppoſito.</s>
            <s xml:id="echoid-s2678" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2679" xml:space="preserve">Ita patet ſatisfieri primæ affuſionis conditioni. </s>
            <s xml:id="echoid-s2680" xml:space="preserve">Jam vero hujus motus
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            phænomena inveſtigabimus, viſuri num cum præcedentibus conveniant.</s>
            <s xml:id="echoid-s2681" xml:space="preserve"/>
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            <s xml:id="echoid-s2682" xml:space="preserve">Conſideremus igitur vas ſuperius eſſe veluti infinitum, ut aquæ per R S
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            effluentes ſingulis momentis habeant velocitatem quæ conveniat altitudini P B
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            ſeu F A: </s>
            <s xml:id="echoid-s2683" xml:space="preserve">ſic fingendum erit eſſe hanc altitudinem P B ab initio infinite parvam,
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            quia tunc aquæ velocitate infinite parva effluere debent, deinde vero ſenſim
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            creſcere, idque continue magis magisque, donec poſt tempus infinitum mo-
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            tus uniformis maneat, quæritur autem an altitudo aquæ P B tandem infinita
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            futura ſit an vero certum terminum non tranſgreſſura. </s>
            <s xml:id="echoid-s2684" xml:space="preserve">Id ſic cognoſcetur.</s>
            <s xml:id="echoid-s2685" xml:space="preserve"/>
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            <s xml:id="echoid-s2686" xml:space="preserve">Sit altitudo G H vel R H (neque enim illas inter ſe differre cenſendum
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            eſt) = a, A F = x, amplitudo orificii L M = n, amplitudo orificii R S = m;
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            </s>
            <s xml:id="echoid-s2687" xml:space="preserve">quia vero, ut manifeſtum eſt, utrumque vas cohærere & </s>
            <s xml:id="echoid-s2688" xml:space="preserve">unum efficere puta-
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            ri poteſt, erit poſt tempus infinitum (per §. </s>
            <s xml:id="echoid-s2689" xml:space="preserve">23. </s>
            <s xml:id="echoid-s2690" xml:space="preserve">Sect. </s>
            <s xml:id="echoid-s2691" xml:space="preserve">III.) </s>
            <s xml:id="echoid-s2692" xml:space="preserve">velocitas
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            aquæ in L M = √a + x, & </s>
            <s xml:id="echoid-s2693" xml:space="preserve">in R S = √ x, (quod poſterius patet, ſi nunc iterum
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            ſeparata vaſa cenſentur, nam utrumque ſine errore fingi poteſt) debent autem
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            velocitates eſſe in inverſa ratione amplitudinum orificiorum: </s>
            <s xml:id="echoid-s2694" xml:space="preserve">eſt itaque
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            √a + x.</s>
            <s xml:id="echoid-s2695" xml:space="preserve">√x:</s>
            <s xml:id="echoid-s2696" xml:space="preserve">:m. </s>
            <s xml:id="echoid-s2697" xml:space="preserve">n, unde a + x. </s>
            <s xml:id="echoid-s2698" xml:space="preserve">x: </s>
            <s xml:id="echoid-s2699" xml:space="preserve">mm. </s>
            <s xml:id="echoid-s2700" xml:space="preserve">nn, vel a.</s>
            <s xml:id="echoid-s2701" xml:space="preserve">x:</s>
            <s xml:id="echoid-s2702" xml:space="preserve">: mm - nn. </s>
            <s xml:id="echoid-s2703" xml:space="preserve">nn, ergo
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            x = {nna/mm - nn} & </s>
            <s xml:id="echoid-s2704" xml:space="preserve">a + x = {mma/mm - nn}, videmus igitur altitudinem, velocitati
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            aquæ in LM debitam, eſſe hoc modo = {mma/mm - nn}, poſtquam ſcilicet infi-
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            nita aquæ quantitas jam effluxit: </s>
            <s xml:id="echoid-s2705" xml:space="preserve">ſuperius autem habuimus eandem altitudinem,
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            ſeu v = {mma/mm - nn} X (1 - c{n
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            - nmm/mmN}x), ubi ſi ponitur x = ∞ (infinito
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            enim tempore infinita quantitas transfluit) evaneſcit terminus exponentialis, ſi
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            modo m major ſit quam n & </s>
            <s xml:id="echoid-s2706" xml:space="preserve">ſic fit pariter v = {mma/mm - nn}. </s>
            <s xml:id="echoid-s2707" xml:space="preserve">Mirabilis eſt iſte con-
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            ſenſus, quia valde diverſæ ſunt viæ, quas ſecuti ſumus. </s>
            <s xml:id="echoid-s2708" xml:space="preserve">Cæterum ſi m non ſit ma-
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            jor quam n motus nunquam fit permanens nequidem poſt tempus infinitum,
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            creſcit enim tunc velocitas in infinitum cum ſecus altitudo velocitatis </s>
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