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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            altitudini v, & </s>
            <s xml:id="echoid-s3610" xml:space="preserve">in ſitu infinite propinquo E F reſpondebit eadem velocitas
              <lb/>
            altitudini v - d v; </s>
            <s xml:id="echoid-s3611" xml:space="preserve">Et cum aſcenſus potentialis aquæ C D M L P I C ſit v, obti-
              <lb/>
            nebitur aſcenſus potent. </s>
            <s xml:id="echoid-s3612" xml:space="preserve">ejusdem aquæ in ſitu proximo E F M L O N P I E, ſi
              <lb/>
            multiplicetur maſſa E F M L P I E (n x - n d x) per ſuum aſcenſum potent.
              <lb/>
            </s>
            <s xml:id="echoid-s3613" xml:space="preserve">(v - d v) ut etiam guttula L O N P (n d x) per ſuum itidem aſcenſum poten-
              <lb/>
            tialem n n v, aggregatumque productorum dividatur per ſummam maſſarum
              <lb/>
            (n x): </s>
            <s xml:id="echoid-s3614" xml:space="preserve">habetur itaque iste aſcenſus potentialis = {(n x - n d x) x (v - d v) + n d x x n n v/nx}
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            ſeu {xv - vdx - xdv + nnvdx/x}.</s>
            <s xml:id="echoid-s3615" xml:space="preserve"/>
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            <s xml:id="echoid-s3616" xml:space="preserve">Eſt proinde incrementum aſcenſus potent. </s>
            <s xml:id="echoid-s3617" xml:space="preserve">= {- vdx - xdv + nnvdx/x}.
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            </s>
            <s xml:id="echoid-s3618" xml:space="preserve">(conf. </s>
            <s xml:id="echoid-s3619" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3620" xml:space="preserve">6. </s>
            <s xml:id="echoid-s3621" xml:space="preserve">ſect. </s>
            <s xml:id="echoid-s3622" xml:space="preserve">3.) </s>
            <s xml:id="echoid-s3623" xml:space="preserve">Iſtud vero incrementum æquale cenſendum eſt cum de-
              <lb/>
            ſcenſu actuali infinitè parvo, qui (per §. </s>
            <s xml:id="echoid-s3624" xml:space="preserve">7. </s>
            <s xml:id="echoid-s3625" xml:space="preserve">ſect. </s>
            <s xml:id="echoid-s3626" xml:space="preserve">3. </s>
            <s xml:id="echoid-s3627" xml:space="preserve">& </s>
            <s xml:id="echoid-s3628" xml:space="preserve">per annotationem modo
              <lb/>
            datam) eſt = {(x - b)dx/x}. </s>
            <s xml:id="echoid-s3629" xml:space="preserve">Habetur itaque talis æquatio
              <lb/>
            - vdx - xdv + nnvdx = (x - b)dx,
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            quæ debito modo integrata mutatur in hanc
              <lb/>
            v = {1/nn - 2} X (x - {x
              <emph style="super">nn - 1</emph>
            /a
              <emph style="super">nn - 2</emph>
            }) - {b/nn - 1} X (1 - {x
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            /a
              <emph style="super">nn - 1</emph>
            }).</s>
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            <s xml:id="echoid-s3631" xml:space="preserve">Ex iſta vero æquatione talia ſequuntur corollaria.</s>
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            <s xml:id="echoid-s3633" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3634" xml:space="preserve">4. </s>
            <s xml:id="echoid-s3635" xml:space="preserve">Fuerit amplitudo cylindri veluti infinita ratione foraminis, & </s>
            <s xml:id="echoid-s3636" xml:space="preserve">
              <lb/>
            erit cènſendum v = {x - b/nn}; </s>
            <s xml:id="echoid-s3637" xml:space="preserve">ipſaque altitudo pro velocitate aquæ, dum
              <lb/>
            effluit, eſt = x - b. </s>
            <s xml:id="echoid-s3638" xml:space="preserve">Unde conſequens eſt, aquam effluere velocitate,
              <lb/>
            quam grave acquirit cadendo ex altitudine ſuperficiei internæ ſupra externam,
              <lb/>
            & </s>
            <s xml:id="echoid-s3639" xml:space="preserve">eo usque effluet, donec ambæ ſuperficies ſint ad libellam poſitæ, tunc-
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            que omnis motus ceſſabit: </s>
            <s xml:id="echoid-s3640" xml:space="preserve">adeoque eadem lege aquæ effluunt, quaſi fun-
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            dum ſitum I M mutaret cum T V.</s>
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            <s xml:id="echoid-s3642" xml:space="preserve">Cum vero foramen non poteſt ceu infinite parvum conſiderari, deſcen-
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            dit ſuperficies aquæ internæ infra externam; </s>
            <s xml:id="echoid-s3643" xml:space="preserve">atque ut innoteſcatad quamnam
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            profunditatem x y ſit deſcenſura ſuperficies C D, facienda eſt v = o, ſeu
              <lb/>
            (nn - 1)(a
              <emph style="super">nn - 1</emph>
            x - x
              <emph style="super">nn - 1</emph>
            a) = (nn - 2) X (a
              <emph style="super">nn - 1</emph>
            b - x
              <emph style="super">nn - 1</emph>
            b),
              <lb/>
            nunquam autem ſuperficies interna tantum deſcendet infra ſuperficiem </s>
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