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

Table of contents

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[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.
[141.] Scholium.
[142.] Regula 2.
[143.] Demonſtratio.
[144.] Scholium.
[145.] Regula 3.
[146.] Demonſtratio.
[147.] Scholium.
[148.] Regula 4.
[149.] Demonſtratio.
[150.] Scholium.
[151.] Regula 5.
[152.] Demonſtratio.
[153.] Regula 6.
[154.] Demonſtratio.
[155.] Scholium.
[156.] Regula 7.
[157.] Scholium.
[158.] Exemplum 1.
[159.] Exemplum 2.
[160.] Digreſſus continens aliquas commentationes in Ma-chinam Hydraulicam quam repræſent at figura 51.
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            v = {(b - β + f - φ) gx - ({bg/2a} + {bgg/2αγ)} xx/ga - gx + {αgg/γ} + {g
              <emph style="super">3</emph>
            /γ} {x/γ} + MN} Q.</s>
            <s xml:id="echoid-s3362" xml:space="preserve">E.</s>
            <s xml:id="echoid-s3363" xml:space="preserve">I.</s>
            <s xml:id="echoid-s3364" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div128" type="section" level="1" n="98">
          <head xml:id="echoid-head128" xml:space="preserve">Corollarium 1.</head>
          <p>
            <s xml:id="echoid-s3365" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3366" xml:space="preserve">8. </s>
            <s xml:id="echoid-s3367" xml:space="preserve">Quia linea mn = mg - nh + gh = h - β + f - m, ponemus
              <lb/>
            mn = c, ſimulque multiplicabimus denominatorem & </s>
            <s xml:id="echoid-s3368" xml:space="preserve">numeratorem per
              <lb/>
            2γγαα: </s>
            <s xml:id="echoid-s3369" xml:space="preserve">Ita vero habebimus
              <lb/>
            v = {2gγγaαcx - (gγγαb + ggγaß)xx/2gγγaaα - 2gγγaαx + 2ggγaαα + 2g
              <emph style="super">3</emph>
            aαx + 2γγaαMN}</s>
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        <div xml:id="echoid-div129" type="section" level="1" n="99">
          <head xml:id="echoid-head129" xml:space="preserve">Corollarium 2.</head>
          <p>
            <s xml:id="echoid-s3370" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3371" xml:space="preserve">9. </s>
            <s xml:id="echoid-s3372" xml:space="preserve">Si fiat v =o, patet tunc valorem x denotare totam fluidi ſuper-
              <lb/>
            ficiei excurſionem in tubo ac, quæ ſic invenitur æqualis {2γaαc/γαb + gαβ}, in altero
              <lb/>
            vero tubo fit = {2gaαc/γαb + gαβ}.</s>
            <s xml:id="echoid-s3373" xml:space="preserve"/>
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            <s xml:id="echoid-s3374" xml:space="preserve">Igitur poterit aqua in tubo ſtrictiori ad quamcunque elevari altitudi-
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            nem, ſi modo ratio amplitudinum g & </s>
            <s xml:id="echoid-s3375" xml:space="preserve">γ ſat magna ſumatur.</s>
            <s xml:id="echoid-s3376" xml:space="preserve"/>
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        <div xml:id="echoid-div130" type="section" level="1" n="100">
          <head xml:id="echoid-head130" xml:space="preserve">Corollarium 3.</head>
          <p>
            <s xml:id="echoid-s3377" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3378" xml:space="preserve">10. </s>
            <s xml:id="echoid-s3379" xml:space="preserve">Pars illa vaſis c A d, quam neutra ſuperficierum unquam attin-
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            gi ponimus, nihil pertinet ad iſtas fluidi excurſiones ſive augendas ſive dimi-
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            nuendas: </s>
            <s xml:id="echoid-s3380" xml:space="preserve">facere tamen poteſt, ut inferius oſtendetur, ad accelerandas retar-
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            dandaſque oſcillationes.</s>
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        <div xml:id="echoid-div131" type="section" level="1" n="101">
          <head xml:id="echoid-head131" xml:space="preserve">Corollarium 4.</head>
          <p>
            <s xml:id="echoid-s3382" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3383" xml:space="preserve">11. </s>
            <s xml:id="echoid-s3384" xml:space="preserve">Ponatur uterque tubus communis amplitudinis, erit, poſito
              <lb/>
            nempe g = γ,
              <lb/>
            v = 2gaαcx - (gαb + gaβ)xx/2gaaα + 2gaαα + 2aαMN}.</s>
            <s xml:id="echoid-s3385" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s3386" xml:space="preserve">In hoc caſu maxima ſuperficiei utriuſque velocitas eſt, cum in medio
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            totius excurſionis poſitæ ſunt, ſecus ac fit, cum tubi ſunt inæqualis amplitu-
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            dinis.</s>
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