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

Table of Notes

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            ca ſuperficies extra ſitum æquilibrii, ſupra §. </s>
            <s xml:id="echoid-s4556" xml:space="preserve">19. </s>
            <s xml:id="echoid-s4557" xml:space="preserve">definiti poſita fuerit, fore ut
              <lb/>
            omnes reliquæ motibus reciprocis agitentur, donec poſt tempus infinitum in
              <lb/>
            priſtinum ſitum redierint ſimul.</s>
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          <p>
            <s xml:id="echoid-s4559" xml:space="preserve">§. </s>
            <s xml:id="echoid-s4560" xml:space="preserve">26. </s>
            <s xml:id="echoid-s4561" xml:space="preserve">Sit porro vas ita formatum, ut oſtendit Fig. </s>
            <s xml:id="echoid-s4562" xml:space="preserve">43. </s>
            <s xml:id="echoid-s4563" xml:space="preserve">diviſum ſcilicet
              <lb/>
              <note position="right" xlink:label="note-0171-01" xlink:href="note-0171-01a" xml:space="preserve">Fig. 43.</note>
            in duas partes A B E G & </s>
            <s xml:id="echoid-s4564" xml:space="preserve">L Q N E inter ſe, mediante foramine M communi-
              <lb/>
            cantes; </s>
            <s xml:id="echoid-s4565" xml:space="preserve">ſintque præterea foramina H & </s>
            <s xml:id="echoid-s4566" xml:space="preserve">N per quæ aquæ exiliant, dum in A B
              <lb/>
            totidem affunduntur. </s>
            <s xml:id="echoid-s4567" xml:space="preserve">Sint autem amplitudines in utroque vaſe veluti infinite
              <lb/>
            amplæ ratione foraminum M, H & </s>
            <s xml:id="echoid-s4568" xml:space="preserve">N; </s>
            <s xml:id="echoid-s4569" xml:space="preserve">Hiſque poſitis propoſitum ſit veloci-
              <lb/>
            tates invenire, quibus aquæ tam per H, quam per N ejiciantur ſeu altitudines
              <lb/>
            iſtis velocitatibus debitas. </s>
            <s xml:id="echoid-s4570" xml:space="preserve">Erunt autem velocitates invariabiles, quia vas aquis
              <lb/>
            plenum conſervatur, ſimulque vaſis amplitudines reſpectu foraminum infini-
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            tæ cenſentur.</s>
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          <p>
            <s xml:id="echoid-s4572" xml:space="preserve">Solutio iſtius problematis ex præcedentibus facile colligetur, ſi modo
              <lb/>
            concipiatur foramen M in duas diviſum partes o & </s>
            <s xml:id="echoid-s4573" xml:space="preserve">p, quarum altera o aquas
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            foramini H, altera p foramini N mittat: </s>
            <s xml:id="echoid-s4574" xml:space="preserve">partes autem o & </s>
            <s xml:id="echoid-s4575" xml:space="preserve">p (quia per utram-
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            que eadem fluunt velocitate aquæ) eam habebunt rationem, quam inter ſe ha-
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            bent quantitates aquarum eodem tempore per H & </s>
            <s xml:id="echoid-s4576" xml:space="preserve">N effluentium, id eſt, ra-
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            tionem compoſitam ex ratione amplitudinis H ad amplitudinem N & </s>
            <s xml:id="echoid-s4577" xml:space="preserve">veloci-
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            tatis in H ad velocitatem in N. </s>
            <s xml:id="echoid-s4578" xml:space="preserve">Quibus præmonitis perſpicuum eſt, fi amplitu-
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            dines foraminum M, H & </s>
            <s xml:id="echoid-s4579" xml:space="preserve">N indicentur per α, β, γ, altitudines autem velo-
              <lb/>
            citatibus in H & </s>
            <s xml:id="echoid-s4580" xml:space="preserve">N debitæ deſignentur per x & </s>
            <s xml:id="echoid-s4581" xml:space="preserve">y, ipſæque proinde velocitates
              <lb/>
            per √x & </s>
            <s xml:id="echoid-s4582" xml:space="preserve">√y fore amplitudinem o = {β√x/β√x + γ√y} α & </s>
            <s xml:id="echoid-s4583" xml:space="preserve">amplitudinem
              <lb/>
            p = {γ√y/β√x + γ√y} α.</s>
            <s xml:id="echoid-s4584" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s4585" xml:space="preserve">Ponatur nunc altitudo ſuperficiei A B ſupra orificium H = a, & </s>
            <s xml:id="echoid-s4586" xml:space="preserve">habebi-
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            tur@, ut demonſtratum fuit §. </s>
            <s xml:id="echoid-s4587" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4588" xml:space="preserve">ſi quadratum foraminis o dividatur per ſum-
              <lb/>
            mam quadratorum foraminum o & </s>
            <s xml:id="echoid-s4589" xml:space="preserve">H & </s>
            <s xml:id="echoid-s4590" xml:space="preserve">quod oritur multiplicetur per a; </s>
            <s xml:id="echoid-s4591" xml:space="preserve">ſic
              <lb/>
            igitur fit x = {ααax/ααx + (β√x + γ√y)
              <emph style="super">2</emph>
            }, ex quo oritur hæc æquatio
              <lb/>
            (A) ααx + (β√x + γ√y)
              <emph style="super">2</emph>
            = ααa.</s>
            <s xml:id="echoid-s4592" xml:space="preserve"/>
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            <s xml:id="echoid-s4593" xml:space="preserve">Eodem modo ratione foraminum p & </s>
            <s xml:id="echoid-s4594" xml:space="preserve">N, poſita altitudine A B ſupra
              <lb/>
            N = a + b, obtinetur hæc altera æquatio:</s>
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