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

Table of Notes

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        <div xml:id="echoid-div138" type="section" level="1" n="108">
          <pb o="120" file="0134" n="134" rhead="HYDRODYNAMICÆ"/>
          <p>
            <s xml:id="echoid-s3448" xml:space="preserve">Hæc autem de oſcillationibus finitis. </s>
            <s xml:id="echoid-s3449" xml:space="preserve">Si nunc oſcillationes minimas
              <lb/>
            eſſe cenſeamus, videbimus illas fieri omnes inter ſe tantochronas, manen-
              <lb/>
            te eadem fluidi quantitate, eodemque canali, quæcunque interea ſint cana-
              <lb/>
            lis figura & </s>
            <s xml:id="echoid-s3450" xml:space="preserve">amplitudines. </s>
            <s xml:id="echoid-s3451" xml:space="preserve">Id exponam in ſequenti paragrapho.</s>
            <s xml:id="echoid-s3452" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div139" type="section" level="1" n="109">
          <head xml:id="echoid-head139" xml:space="preserve">Theorema.</head>
          <p>
            <s xml:id="echoid-s3453" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3454" xml:space="preserve">17. </s>
            <s xml:id="echoid-s3455" xml:space="preserve">Oſcillationes minimæ fluidi in quocunque canali oſcillantis,
              <lb/>
            quamvis inæquales inter ſe, ſunt omnes Iſochronæ.</s>
            <s xml:id="echoid-s3456" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div140" type="section" level="1" n="110">
          <head xml:id="echoid-head140" xml:space="preserve">Demonſtratio.</head>
          <p>
            <s xml:id="echoid-s3457" xml:space="preserve">Cum oſcillationes ſunt minimæ, poſſunt illæ canalis particulæ, in qui-
              <lb/>
            bus ſuperficies fluidi agitantur, pro cylindricis haberi, igitur manentibus
              <lb/>
            denominationibus iisdem, manebit valor, quem aſſignavimus litteræ v in
              <lb/>
            §. </s>
            <s xml:id="echoid-s3458" xml:space="preserve">8. </s>
            <s xml:id="echoid-s3459" xml:space="preserve">& </s>
            <s xml:id="echoid-s3460" xml:space="preserve">ex eadem ratione ſequitur, litteras a, b, α, β & </s>
            <s xml:id="echoid-s3461" xml:space="preserve">x ceu infinite parvi
              <lb/>
            valoris negligi poſſe præ {M/g}, ſic ut in præſenti caſu cenſeri debeat
              <lb/>
            v = {2gγaαcx - (gγαb + ggab)xx/2γaαMN}</s>
          </p>
          <p>
            <s xml:id="echoid-s3462" xml:space="preserve">Sunt igitur vi paragraphi duodecimi oſcillationes omnes, quoad mi-
              <lb/>
            nimæ ſunt, inter ſe Iſochronæ. </s>
            <s xml:id="echoid-s3463" xml:space="preserve">Q.</s>
            <s xml:id="echoid-s3464" xml:space="preserve">E.</s>
            <s xml:id="echoid-s3465" xml:space="preserve">D.</s>
            <s xml:id="echoid-s3466" xml:space="preserve"/>
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        <div xml:id="echoid-div141" type="section" level="1" n="111">
          <head xml:id="echoid-head141" xml:space="preserve">Problema.</head>
          <p>
            <s xml:id="echoid-s3467" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3468" xml:space="preserve">18. </s>
            <s xml:id="echoid-s3469" xml:space="preserve">Determinare longitudinem penduli ſimplicis tautochroni cum
              <lb/>
            oſcillationibus minimuis fluidi in canali quocunque agitati.</s>
            <s xml:id="echoid-s3470" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div142" type="section" level="1" n="112">
          <head xml:id="echoid-head142" xml:space="preserve">Solutio.</head>
          <p>
            <s xml:id="echoid-s3471" xml:space="preserve">Quia in omni motu eſt elementum temporis dt = {dx/√v}, erit nunc
              <lb/>
            dt = dx√(2γaαMN/gγαb + ggab}):</s>
            <s xml:id="echoid-s3472" xml:space="preserve">√({2γaαcx/γαb + gaβ} - xx)
              <lb/>
            Igitur vi Paragraphi decimi tertii erit longitudo quæſita penduli cum præ-
              <lb/>
            dictis oſcillationibus tautochroni = {γaαMN/gγαb + ggaβ}. </s>
            <s xml:id="echoid-s3473" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s3474" xml:space="preserve">E. </s>
            <s xml:id="echoid-s3475" xml:space="preserve">I.</s>
            <s xml:id="echoid-s3476" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div143" type="section" level="1" n="113">
          <head xml:id="echoid-head143" xml:space="preserve">Scholium.</head>
          <p>
            <s xml:id="echoid-s3477" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3478" xml:space="preserve">19. </s>
            <s xml:id="echoid-s3479" xml:space="preserve">Quamvis jam paſſim monuerim, quid intelligendum ſit </s>
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