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

Table of contents

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[111.] Problema.
[112.] Solutio.
[113.] Scholium.
[114.] Corollarium 1.
[115.] Corollarium 2.
[116.] Scholion Generale.
[117.] HYDRODYNAMICÆ SECTIO SEPTIMA. De motu aquarum per vaſa ſubmerſa, ubi exem-plis oſtenditur, quam inſigniter utile ſit princi-pium conſervationis virium vivarum, veliis in caſibus, quibus continue aliquid de illis perdi cenſendum eſt. PARS PRIMA. De deſcenſu aquarum. §. 1.
[118.] PARS SECUNDA. De aſcenſu aquarum.
[119.] Corollarium.
[120.] Scholium Generale.
[121.] EXPERIMENTA Ad ſect. ſept. referenda. Experimentum 1.
[122.] Experimentum 2.
[123.] Experimentum 3.
[124.] De iſto tubo experimentum ita ſumſi:
[125.] Experimentum 4.
[126.] Experimentum 5.
[127.] HYDRODYNAMICÆ SECTIO OCTAVA. De motu fluidorum cum homogeneorum tum hetero-geneorum per vaſa irregularis & præruptæ ſtru-cturæ, ubi ex theoria virium vivarum, quarum pars continue abſorbeatur, explicantur præcipue Phæno-mena ſingularia fluidorum, per plurima foramina trajecto-rum, præmiſsis regulis generalibus pro motibus fluido-rum ubique definiendis. §. 1.
[128.] Regula 1.
[129.] Regula 2.
[130.] Problema.
[131.] Solutio.
[132.] Scholium 1.
[133.] Scholium 2.
[134.] Corollarium.
[135.] EXPERIMENTA Ad ſectionem octavam pertinentia. Experimentum 1.
[136.] Experimentum 2.
[137.] HYDRODYNAMICÆ SECTIO NONA. De motu fluidorum, quæ non proprio pondere, ſed potentia aliena ejiciuntur, ubi præſertim de Machinis Hydraulicis earundemque ultimo qui da-ri poteſt perfectionis gradu, & quomodo mecha-nica tam ſolidorum quam fluidorum ulterius perſici poſsit. §. 1.
[138.] Definitiones.
[139.] (A) De machinis aquas cum impetu in altum projicientibus. Regula 1.
[140.] Demonſtratio.
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          <head xml:id="echoid-head109" xml:space="preserve">Solutio.</head>
          <p>
            <s xml:id="echoid-s2907" xml:space="preserve">Adhibitis rurſus poſitionibus & </s>
            <s xml:id="echoid-s2908" xml:space="preserve">denominationibus paragraphi tertii & </s>
            <s xml:id="echoid-s2909" xml:space="preserve">
              <lb/>
            duodecimi, invenienda nunc erit æquatio inter x & </s>
            <s xml:id="echoid-s2910" xml:space="preserve">t: </s>
            <s xml:id="echoid-s2911" xml:space="preserve">quia vero, ut vidi-
              <lb/>
            mus §. </s>
            <s xml:id="echoid-s2912" xml:space="preserve">12. </s>
            <s xml:id="echoid-s2913" xml:space="preserve">eſt d t = {γdx/√v}, erit √ v = {γdx/dt}, hicque valor ſubſtituendus
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            erit in æquationibus, quas dedimus §. </s>
            <s xml:id="echoid-s2914" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2915" xml:space="preserve">integratis; </s>
            <s xml:id="echoid-s2916" xml:space="preserve">prior harum æquationum
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            hæc fuit: </s>
            <s xml:id="echoid-s2917" xml:space="preserve">v = {mma/mm - nn} X (1 - c{n
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            - nmm/mmN} x)
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            quæ pro præſecuti inſtituto mutatur in hanc
              <lb/>
            (I) {γγdx
              <emph style="super">2</emph>
            /dt
              <emph style="super">2</emph>
            } = {mma/mm - nn} X (1 - c{n
              <emph style="super">3</emph>
            - nmm/mmN} x)
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            altera ex §. </s>
            <s xml:id="echoid-s2918" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2919" xml:space="preserve">allegatarum æquationum talis fuit
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            v = a X (1 - c
              <emph style="super">{- n/N} x</emph>
            )
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            quæ adeoque ſubminiſtrat in præſenti caſu ſequentem
              <lb/>
            (II) {γγdx
              <emph style="super">2</emph>
            /dt
              <emph style="super">2</emph>
            } = a X (1 - c
              <emph style="super">{- n/N} x</emph>
            )</s>
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            <s xml:id="echoid-s2920" xml:space="preserve">Erunt nunc æquationes (I) & </s>
            <s xml:id="echoid-s2921" xml:space="preserve">(II) integrandæ, quod quidem facile
              <lb/>
            eſt & </s>
            <s xml:id="echoid-s2922" xml:space="preserve">quia prior alteram continet (utraque enim eadem eſt ſi m = ∞)
              <lb/>
            hanc ſolam pertractabimus, eamque nunc ſub hâc forma conſiderabimus.
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            <s xml:id="echoid-s2923" xml:space="preserve">dt = {γ√(mm - nn)/m√a}dx:</s>
            <s xml:id="echoid-s2924" xml:space="preserve">√(1 - c{n
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            - nmm/mmN}x)</s>
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            <s xml:id="echoid-s2925" xml:space="preserve">Ponatur autem ut integrationis modus eo magis pateſcat
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            c{n
              <emph style="super">3</emph>
            - nmm/mmN}x = z, atque proin dx = {mmNdz/(n
              <emph style="super">3</emph>
            - nmm)z},
              <lb/>
            dein brevitatis ergo indice
              <unsure/>
            tur quantitas conſtans
              <lb/>
            {γ√(mm - nn)/m√a} X {mmN/n
              <emph style="super">3</emph>
            - nmm}, ſeu {- γmN/n√(mm - nn) a} per α,
              <lb/>
            & </s>
            <s xml:id="echoid-s2926" xml:space="preserve">habebitur dt = {αdz/z√(1 - </s>
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