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

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          <pb o="135" file="0149" n="149" rhead="SECTIO SEPTIMA."/>
          <p>
            <s xml:id="echoid-s3884" xml:space="preserve">Idem vero aliter ſic invenitur.</s>
            <s xml:id="echoid-s3885" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3886" xml:space="preserve">Conſideretur ſcilicet guttulæ L O N P quaſi nullam velocitatem fuiſſe,
              <lb/>
            priuſquam influere inciperet, eandem vero ſtatim atque influere incipiat, ac-
              <lb/>
            quirere aſcenſum potentialem, qui ſit = n n v, quamvis mox poſt ſui influxum
              <lb/>
            (per annot. </s>
            <s xml:id="echoid-s3887" xml:space="preserve">ſec. </s>
            <s xml:id="echoid-s3888" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3889" xml:space="preserve">2.) </s>
            <s xml:id="echoid-s3890" xml:space="preserve">cenſenda ſit motum continuare velocitate communi
              <lb/>
            √ v. </s>
            <s xml:id="echoid-s3891" xml:space="preserve">Quo facto ſic erit ratiocinandum. </s>
            <s xml:id="echoid-s3892" xml:space="preserve">Ante influxum guttulæ, eſt aſcenſus
              <lb/>
            potent. </s>
            <s xml:id="echoid-s3893" xml:space="preserve">aquæ c d M L P I c (cujus maſſa = n ξ) = v. </s>
            <s xml:id="echoid-s3894" xml:space="preserve">& </s>
            <s xml:id="echoid-s3895" xml:space="preserve">aſcenſ. </s>
            <s xml:id="echoid-s3896" xml:space="preserve">potent. </s>
            <s xml:id="echoid-s3897" xml:space="preserve">guttulæ
              <lb/>
            L O N P (cujus maſſa = n d ξ) = o; </s>
            <s xml:id="echoid-s3898" xml:space="preserve">ergo aſcenſus potentialis omnis aquæ
              <lb/>
            c d M L O N P I c = {nξv/nξ = ndξ} = {ξv/ξ + dξ}.</s>
            <s xml:id="echoid-s3899" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3900" xml:space="preserve">At vero poſtquam guttula L O N P influxit ſitumque aſſumſit L on P,
              <lb/>
            eſt ejus aſcenſ. </s>
            <s xml:id="echoid-s3901" xml:space="preserve">potent. </s>
            <s xml:id="echoid-s3902" xml:space="preserve">= n n v, reliquæ autem aquæ e f M L o n P I e (cujus
              <lb/>
            quidem maſſa rurſus = n ξ) aſcenſus potent. </s>
            <s xml:id="echoid-s3903" xml:space="preserve">eſt = v + d v; </s>
            <s xml:id="echoid-s3904" xml:space="preserve">igitur aſcenſus
              <lb/>
            potent. </s>
            <s xml:id="echoid-s3905" xml:space="preserve">omnis aquæ hic conſideratæ poſt influxum guttulæ eſt
              <lb/>
            = {ndξ x nnv + nξx(v + dv)/nξ + ndξ} = {ξv + ξdv + nnvdξ/ξ + dξ}, cum ante eundem influ-
              <lb/>
            xum fuerit {ξv/ξ + dξ}: </s>
            <s xml:id="echoid-s3906" xml:space="preserve">cepit igitur incrementum {ξdv + nnvdξ/ξ + dξ}, vel ſimplicius
              <lb/>
            {ξdv + nnvdξ/ξ}. </s>
            <s xml:id="echoid-s3907" xml:space="preserve">Iſtud vero incrementum æquandum eſt cum deſcenſu actuali
              <lb/>
            quem aqua facit mutando ſitum c d M L O N P I c ſitu e f M L O N P I e, qui
              <lb/>
            deſcenſus æqualis eſt quartæ proportionali ad maſſam aquæ internæ n ξ, ad
              <lb/>
            guttulam n d ξ & </s>
            <s xml:id="echoid-s3908" xml:space="preserve">altitudinem V f vel b - ξ, ſic ut præfatus deſcenſus ſit =
              <lb/>
            {(b - ξ)dξ/ξ}: </s>
            <s xml:id="echoid-s3909" xml:space="preserve">unde iterum habetur talis æquatio
              <lb/>
            ξdv + nnvdξ = (b - ξ)dξ;</s>
            <s xml:id="echoid-s3910" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3911" xml:space="preserve">Hujus vero integralis poſt debitæ conſtantis additionem talis fit
              <lb/>
            v = {b/nn} (1 - ({α/ξ})
              <emph style="super">nn</emph>
            ) - {1/nn + 1} (ξ - ({α/ξ})
              <emph style="super">nn</emph>
            α),
              <lb/>
            quam nunc pro diverſis ejus circumſtantiis perpendemus.</s>
            <s xml:id="echoid-s3912" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s3913" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3914" xml:space="preserve">15. </s>
            <s xml:id="echoid-s3915" xml:space="preserve">Et quidem cum fuerit amplitudo tubi infinities major, quam
              <lb/>
            amplitudo foraminis; </s>
            <s xml:id="echoid-s3916" xml:space="preserve">patet fieri v = {b - ξ/nn}, & </s>
            <s xml:id="echoid-s3917" xml:space="preserve">irruere proinde aquam velo-
              <lb/>
            citate quæ debeatur altitudini ſuperficiei externæ fuper internam, neque
              <lb/>
            tunc ultra ſuperficiem aquæ externæ fiet aſcenſus.</s>
            <s xml:id="echoid-s3918" xml:space="preserve"/>
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