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

Table of Notes

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            <s xml:id="echoid-s5706" xml:space="preserve">Ut jam determinetur inclinatio plani ad fluidum ſub his circumſtantiis
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            maxime favorabilis ut motus plani in directione B b promoveatur: </s>
            <s xml:id="echoid-s5707" xml:space="preserve">ponemus
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            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-
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            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
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            ſic inſtituto calculo invenitur
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            ef = xv √ (1 - xx) - (1 - xx) V, atque BN = [xv - V √ (1 - xx]:
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            </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
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            quantitas maxima erit, cum fit
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            (9v
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            + 18vvVV + 9V
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            )x
              <emph style="super">6</emph>
            - (12v
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            + 30vvVV + 18V
              <emph style="super">4</emph>
            ) x
              <emph style="super">4</emph>
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            + (4v
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            + 16vvVV + 9V
              <emph style="super">4</emph>
            ) xx - 4vvVV = o.</s>
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            <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,
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            quia velocitates in diverſis alarum locis variæ ſunt; </s>
            <s xml:id="echoid-s5717" xml:space="preserve">ſunt enim proportiona-
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            les diſtantiis à centro, facile autem nunc cuivis erit computum pro mole-
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            trinis inſtituere, huic caſui non ulterius inſiſtam, ſufficiat id notaſſe, quod
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            non ſatis accurate ſtatuatur ab auctoribus x x = {2/3}, & </s>
            <s xml:id="echoid-s5718" xml:space="preserve">quod verus valor ip-
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            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
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            puncta ſimili velocitate moveri cenſeantur, fiet x = √ {1/2}, quod indicat in-
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            clinandam eſſe alam ad directionem venti ſub angulo ſemirecto. </s>
            <s xml:id="echoid-s5723" xml:space="preserve">Optima
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            alarum conſtructio foret, ſi incurvarentur, ita, ut ſub angulo minori ventus
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            in illas impingat ſuperius quam inferius, aut ſi fiat ut alæ ubique ventum
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            ſub angulo medio quinquaginta præterpropter graduum excipiant.</s>
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            <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-
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            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
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            particularum dato tempore impellentium ſemper idem eſt, nullam eſſe at-
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            tentionem faciendam ad linem B N, atque ſic niſum quem aquæ faciunt ad
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            movendum planum A B in directione B b ſimpliciter repræſentari per e f ſeu
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            xv√1 - xx - (1 - xx) V. </s>
            <s xml:id="echoid-s5730" xml:space="preserve">Igitur niſus iſte maximus obtinebitur ſumendo
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            xx = {1/2} + {V/2√(vv + VV)}, atque erit ipſe niſus tunc = {1/2}√(vv + VV)
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            - {1/2} V, ſi per v intelligatur preſſio directa, quam vena exerit in planum cui
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            perpendiculariter occurrit.</s>
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            <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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