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

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        <div xml:id="echoid-div117" type="section" level="1" n="90">
          <p>
            <s xml:id="echoid-s3176" xml:space="preserve">
              <pb o="112" file="0126" n="126" rhead="HYDRODYNAMICÆ"/>
            tiæ paragrapho ſecundo: </s>
            <s xml:id="echoid-s3177" xml:space="preserve">Igitur nihil ad ſolutionem quæſtionis amplius re-
              <lb/>
            ſiduum eſt: </s>
            <s xml:id="echoid-s3178" xml:space="preserve">Neque tamen abs re erit unum alterumve ejus rei exemplum
              <lb/>
            attuliſſe.</s>
            <s xml:id="echoid-s3179" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div118" type="section" level="1" n="91">
          <head xml:id="echoid-head120" xml:space="preserve">
            <emph style="bf">Exemplum 1.</emph>
          </head>
          <p>
            <s xml:id="echoid-s3180" xml:space="preserve">Si v. </s>
            <s xml:id="echoid-s3181" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s3182" xml:space="preserve">canalis B g f C (Fig. </s>
            <s xml:id="echoid-s3183" xml:space="preserve">31.) </s>
            <s xml:id="echoid-s3184" xml:space="preserve">qui figuram habeat coni-truncati; </s>
            <s xml:id="echoid-s3185" xml:space="preserve">in
              <lb/>
            telligatur pars ejus B G F C fluido plena moto verſus g f; </s>
            <s xml:id="echoid-s3186" xml:space="preserve">habeantque parti-
              <lb/>
            culæ fluidi in G F velocitatem debitam altitudini v; </s>
            <s xml:id="echoid-s3187" xml:space="preserve">ac denique pervenerit
              <lb/>
            fluidum in ſitum b g f c: </s>
            <s xml:id="echoid-s3188" xml:space="preserve">His poſitis quæritur velocitas fluidi in g f. </s>
            <s xml:id="echoid-s3189" xml:space="preserve">Voca-
              <lb/>
            bo autem altitudinem velocitati aquæ in g f debitam = V; </s>
            <s xml:id="echoid-s3190" xml:space="preserve">Sit vertex coni
              <lb/>
            in H; </s>
            <s xml:id="echoid-s3191" xml:space="preserve">diameter in B C = n; </s>
            <s xml:id="echoid-s3192" xml:space="preserve">diameter in G F = m: </s>
            <s xml:id="echoid-s3193" xml:space="preserve">longitudo B G = a;
              <lb/>
            </s>
            <s xml:id="echoid-s3194" xml:space="preserve">Gg = b, erit diameter g f = {m a - m b + n b/a}. </s>
            <s xml:id="echoid-s3195" xml:space="preserve">Deinde quia ſolidum B G F C
              <lb/>
            eſt æquale ſolido b g f c erit B C
              <emph style="super">2</emph>
            X B H - G F
              <emph style="super">2</emph>
            X G H = b c
              <emph style="super">2</emph>
            X b H
              <lb/>
            - g f
              <emph style="super">2</emph>
            X g H: </s>
            <s xml:id="echoid-s3196" xml:space="preserve">unde b c
              <emph style="super">2</emph>
            X b H = B C
              <emph style="super">2</emph>
            X B H - G F
              <emph style="super">2</emph>
            X G H
              <lb/>
            + g f
              <emph style="super">2</emph>
            X g H: </s>
            <s xml:id="echoid-s3197" xml:space="preserve">eſt vero b H = {BH/BC} X b c: </s>
            <s xml:id="echoid-s3198" xml:space="preserve">igitur b c
              <emph style="super">3</emph>
            = B C
              <emph style="super">3</emph>
            -. </s>
            <s xml:id="echoid-s3199" xml:space="preserve">
              <lb/>
            {GF
              <emph style="super">2</emph>
            X GH X BC/BH} + {gf
              <emph style="super">2</emph>
            X gH X BC/BH} = B C
              <emph style="super">3</emph>
            - G F
              <emph style="super">3</emph>
            + g f
              <emph style="super">3</emph>
            , ſeu
              <lb/>
            b c = √Cub.</s>
            <s xml:id="echoid-s3200" xml:space="preserve">n
              <emph style="super">3</emph>
            - m
              <emph style="super">3</emph>
            + ({m a - m b + n b/a})
              <emph style="super">3</emph>
            },</s>
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          <p>
            <s xml:id="echoid-s3201" xml:space="preserve">Eſt vero per §. </s>
            <s xml:id="echoid-s3202" xml:space="preserve">3. </s>
            <s xml:id="echoid-s3203" xml:space="preserve">ſect. </s>
            <s xml:id="echoid-s3204" xml:space="preserve">3. </s>
            <s xml:id="echoid-s3205" xml:space="preserve">aſcenſus potent. </s>
            <s xml:id="echoid-s3206" xml:space="preserve">aquæ in ſitu B G F C
              <lb/>
            = {3 m
              <emph style="super">3</emph>
            v/n(mm + mn + nn)}; </s>
            <s xml:id="echoid-s3207" xml:space="preserve">pariterque aſcenſus potent. </s>
            <s xml:id="echoid-s3208" xml:space="preserve">ejusdem aquæ in ſitu b g f c
              <lb/>
            reperitur = {3 α
              <emph style="super">3</emph>
            v;</s>
            <s xml:id="echoid-s3209" xml:space="preserve">/β(αα + αβ + ββ)}, poſito brevitatis ergo α & </s>
            <s xml:id="echoid-s3210" xml:space="preserve">β pro inventis valo-
              <lb/>
            ribus diametrorum g f & </s>
            <s xml:id="echoid-s3211" xml:space="preserve">b c. </s>
            <s xml:id="echoid-s3212" xml:space="preserve">Erit igitur
              <lb/>
            V = {m
              <emph style="super">3</emph>
            X (αα + αβ + ββ) X β X v/α
              <emph style="super">3</emph>
            X (mm + mn + nn) n}.</s>
            <s xml:id="echoid-s3213" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3214" xml:space="preserve">Ex hâc formula facile colligitur, majori continue velocitate moveri
              <lb/>
            particulas anteriores, minori poſteriores, & </s>
            <s xml:id="echoid-s3215" xml:space="preserve">ſic, ut ſi foraminulum g f cen-
              <lb/>
            ſeatur infinite parvum, fiat velocitas aquæ in g f infinita & </s>
            <s xml:id="echoid-s3216" xml:space="preserve">in b c infinite parva.</s>
            <s xml:id="echoid-s3217" xml:space="preserve"/>
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        </div>
        <div xml:id="echoid-div119" type="section" level="1" n="92">
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            <emph style="bf">Exemplum 2.</emph>
          </head>
          <p>
            <s xml:id="echoid-s3218" xml:space="preserve">Fuerit canalis compoſitus ex duobus tubis cylindricis B N & </s>
            <s xml:id="echoid-s3219" xml:space="preserve">O </s>
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