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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            tudinem orificii pertinere ad motum aquæ internæ, cujus rei originem jam
              <lb/>
            ſupra (§. </s>
            <s xml:id="echoid-s3741" xml:space="preserve">7.) </s>
            <s xml:id="echoid-s3742" xml:space="preserve">indicavi.</s>
            <s xml:id="echoid-s3743" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3744" xml:space="preserve">In ſequentibus autem demonſtrabimus, non differre hunc motum à
              <lb/>
            ſubſequente motu refluo, hincque oſcillationes fieri tautochronas. </s>
            <s xml:id="echoid-s3745" xml:space="preserve">Prius-
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            quam vero ad alia pergam monendum duxi, in iſto calculo quantitates
              <lb/>
            {c/a} & </s>
            <s xml:id="echoid-s3746" xml:space="preserve">{z/a} non ſolum præ unitate, ſed & </s>
            <s xml:id="echoid-s3747" xml:space="preserve">præ {1/nn} ceu infinite parvas poſitas fuiſ-
              <lb/>
            ſe, ad quod animus probe eſt advertendus in inſtituendis experimentis;
              <lb/>
            </s>
            <s xml:id="echoid-s3748" xml:space="preserve">licet utique theoriam infinite parvorum ad experimenta, ſine notabili erro-
              <lb/>
            re revocare diminuendo admodum quantitates, quæ in theoria ceu infinite
              <lb/>
            parvæ conſideratæ fuerunt, ſed faciendum eſt, ut in experimento omnia
              <lb/>
            huic legi ſint ſubjecta. </s>
            <s xml:id="echoid-s3749" xml:space="preserve">Ita v. </s>
            <s xml:id="echoid-s3750" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s3751" xml:space="preserve">ſi in cylindro omne fundum abſit, poſito
              <lb/>
            n = 1, idque ſubmerſum ponatur ad altitudinem triginta quinque pollicum,
              <lb/>
            ſatis accurate ſumetur experimentum, cum aqua ante oſcillationes elevata
              <lb/>
            tantum fuerit ad altitudinem unius pollicis ſupra ſuperficiem aquæ circum-
              <lb/>
            fluæ nec dum error notabilis erit, ſi vel orificiium inferius ad dimidium
              <lb/>
            obſtruatur exiſtente tunc {c/a} ad {1/nn} ut 1. </s>
            <s xml:id="echoid-s3752" xml:space="preserve">9, quæ ratio in noſtro experimento
              <lb/>
            tuto adhuc negligi poteſt: </s>
            <s xml:id="echoid-s3753" xml:space="preserve">at ſi jam diametrum tubi duplam ponas diame-
              <lb/>
            tri orificii, occluſis tribus quartis aperturæ integræ partibus, jam fiet n = 4
              <lb/>
            & </s>
            <s xml:id="echoid-s3754" xml:space="preserve">{c/a} ad {1/nn} ut 4 ad 9, quæ ratio non ſatis parva amplius erit, ut experimentum
              <lb/>
            conditionibus theoriæ cum ſufficienti præciſione ſatisfacere affirmari poſſit.</s>
            <s xml:id="echoid-s3755" xml:space="preserve"/>
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            <s xml:id="echoid-s3756" xml:space="preserve">Hic itaque jam porro inquirere conveniet, quid de his caſibus ſtatuen-
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            dum ſit, quibus {c/a} & </s>
            <s xml:id="echoid-s3757" xml:space="preserve">{1/nn} notabilem quidem inter ſe habent rationem, utra-
              <lb/>
            que vero quantitas fit admodum exigua, quod nimirum fit, cum cylindrus
              <lb/>
            profundiſſime ſubmergitur, ſimul autem fundum parvulo eſt pertuſum fo-
              <lb/>
            ramine.</s>
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          <p>
            <s xml:id="echoid-s3759" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3760" xml:space="preserve">11. </s>
            <s xml:id="echoid-s3761" xml:space="preserve">Sed iſte, quem modo finximus, caſus melius ex æquatione
              <lb/>
            differentiali paragraphi tertii, quam ex integrali, ut antea factum, deduci-
              <lb/>
            tur: </s>
            <s xml:id="echoid-s3762" xml:space="preserve">poteſt autem pro his circumſtantiis rejici terminus - v d x præ n n v d x,
              <lb/>
            atque ſic aſſumi - x d v + n n v d x = (x - b) d x, in quâ ſi rurſus ponitur
              <lb/>
            a - b = c & </s>
            <s xml:id="echoid-s3763" xml:space="preserve">a - x = z, prodit
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            adv + zdv + nnvdz = (c - z) </s>
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