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

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          <pb o="197" file="0211" n="211" rhead="SECTIO NONA."/>
          <p>
            <s xml:id="echoid-s5706" xml:space="preserve">Ut jam determinetur inclinatio plani ad fluidum ſub his circumſtantiis
              <lb/>
            maxime favorabilis ut motus plani in directione B b promoveatur: </s>
            <s xml:id="echoid-s5707" xml:space="preserve">ponemus
              <lb/>
            A B = 1, D E ſeu A C = x, B C = √1 - xx; </s>
            <s xml:id="echoid-s5708" xml:space="preserve">lineam E B, quæ repræ-
              <lb/>
            ſentat motum fluidi, = v, & </s>
            <s xml:id="echoid-s5709" xml:space="preserve">B b ceu menſuram motus plani = V; </s>
            <s xml:id="echoid-s5710" xml:space="preserve">atque
              <lb/>
            ſic inſtituto calculo invenitur
              <lb/>
            ef = xv √ (1 - xx) - (1 - xx) V, atque BN = [xv - V √ (1 - xx]:
              <lb/>
            </s>
            <s xml:id="echoid-s5711" xml:space="preserve">√ (vv + VV); </s>
            <s xml:id="echoid-s5712" xml:space="preserve">unde e f X B N = [xv - V √ (1 - xx)]
              <emph style="super">2</emph>
            X {√ (1 - xx)/√ (vv + VV)}, quæ
              <lb/>
            quantitas maxima erit, cum fit
              <lb/>
            (9v
              <emph style="super">4</emph>
            + 18vvVV + 9V
              <emph style="super">4</emph>
            )x
              <emph style="super">6</emph>
            - (12v
              <emph style="super">4</emph>
            + 30vvVV + 18V
              <emph style="super">4</emph>
            ) x
              <emph style="super">4</emph>
              <lb/>
            + (4v
              <emph style="super">4</emph>
            + 16vvVV + 9V
              <emph style="super">4</emph>
            ) xx - 4vvVV = o.</s>
            <s xml:id="echoid-s5713" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s5714" xml:space="preserve">§. </s>
            <s xml:id="echoid-s5715" xml:space="preserve">40. </s>
            <s xml:id="echoid-s5716" xml:space="preserve">Calculus ratione inclinationis alarum in moletrinis alius eſt,
              <lb/>
            quia velocitates in diverſis alarum locis variæ ſunt; </s>
            <s xml:id="echoid-s5717" xml:space="preserve">ſunt enim proportiona-
              <lb/>
            les diſtantiis à centro, facile autem nunc cuivis erit computum pro mole-
              <lb/>
            trinis inſtituere, huic caſui non ulterius inſiſtam, ſufficiat id notaſſe, quod
              <lb/>
            non ſatis accurate ſtatuatur ab auctoribus x x = {2/3}, & </s>
            <s xml:id="echoid-s5718" xml:space="preserve">quod verus valor ip-
              <lb/>
            ſius x ſemper minor ſit quam √ {2/3}. </s>
            <s xml:id="echoid-s5719" xml:space="preserve">Si fuerit v. </s>
            <s xml:id="echoid-s5720" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s5721" xml:space="preserve">V = v, & </s>
            <s xml:id="echoid-s5722" xml:space="preserve">omnia alæ
              <lb/>
            puncta ſimili velocitate moveri cenſeantur, fiet x = √ {1/2}, quod indicat in-
              <lb/>
            clinandam eſſe alam ad directionem venti ſub angulo ſemirecto. </s>
            <s xml:id="echoid-s5723" xml:space="preserve">Optima
              <lb/>
            alarum conſtructio foret, ſi incurvarentur, ita, ut ſub angulo minori ventus
              <lb/>
            in illas impingat ſuperius quam inferius, aut ſi fiat ut alæ ubique ventum
              <lb/>
            ſub angulo medio quinquaginta præterpropter graduum excipiant.</s>
            <s xml:id="echoid-s5724" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5725" xml:space="preserve">§. </s>
            <s xml:id="echoid-s5726" xml:space="preserve">41. </s>
            <s xml:id="echoid-s5727" xml:space="preserve">Pergo ad alterum caſum, quo omne fluidum à plano, utcun-
              <lb/>
            que id inclinatum ſit, excipi ponitur. </s>
            <s xml:id="echoid-s5728" xml:space="preserve">Hic autem patet; </s>
            <s xml:id="echoid-s5729" xml:space="preserve">quia numerus
              <lb/>
            particularum dato tempore impellentium ſemper idem eſt, nullam eſſe at-
              <lb/>
            tentionem faciendam ad linem B N, atque ſic niſum quem aquæ faciunt ad
              <lb/>
            movendum planum A B in directione B b ſimpliciter repræſentari per e f ſeu
              <lb/>
            xv√1 - xx - (1 - xx) V. </s>
            <s xml:id="echoid-s5730" xml:space="preserve">Igitur niſus iſte maximus obtinebitur ſumendo
              <lb/>
            xx = {1/2} + {V/2√(vv + VV)}, atque erit ipſe niſus tunc = {1/2}√(vv + VV)
              <lb/>
            - {1/2} V, ſi per v intelligatur preſſio directa, quam vena exerit in planum cui
              <lb/>
            perpendiculariter occurrit.</s>
            <s xml:id="echoid-s5731" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5732" xml:space="preserve">§. </s>
            <s xml:id="echoid-s5733" xml:space="preserve">42. </s>
            <s xml:id="echoid-s5734" xml:space="preserve">Conſideremus nunc venam D E B A tanquam immediate ex </s>
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