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

Table of Notes

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            b a & </s>
            <s xml:id="echoid-s7220" xml:space="preserve">vis gravitatis per verticalem c a compleaturque rectangulum a b e c, erit
              <lb/>
            diagonalis a e ad curvam perpendicularis; </s>
            <s xml:id="echoid-s7221" xml:space="preserve">unde triangulum e c a ſimile eſt tri-
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            angulo a m n & </s>
            <s xml:id="echoid-s7222" xml:space="preserve">ſic d x: </s>
            <s xml:id="echoid-s7223" xml:space="preserve">dy = ec: </s>
            <s xml:id="echoid-s7224" xml:space="preserve">ca = ba: </s>
            <s xml:id="echoid-s7225" xml:space="preserve">ca, vel ut vis centrifuga in
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            puncto a ad vim gravitatis.</s>
            <s xml:id="echoid-s7226" xml:space="preserve"/>
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            <s xml:id="echoid-s7227" xml:space="preserve">Demonſtravit autem Hugenius vim centrifugam corporis in gyrum acti
              <lb/>
            celeritate, quam lapſu libero per altitudinem dimidii radii acquirere poſſit, æqua-
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            lem eſſe vi ſuæ gravitatis: </s>
            <s xml:id="echoid-s7228" xml:space="preserve">quod ſi proinde altitudo reſpondens guttulæ veloci-
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            tati gyratoriæ dicatur V; </s>
            <s xml:id="echoid-s7229" xml:space="preserve">vis gravitalis g: </s>
            <s xml:id="echoid-s7230" xml:space="preserve">erit vis centrifuga = {2gV/y}, unde
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            dx: </s>
            <s xml:id="echoid-s7231" xml:space="preserve">dy = {2gV/y}: </s>
            <s xml:id="echoid-s7232" xml:space="preserve">g, vel dx = {2Vdy/y}.</s>
            <s xml:id="echoid-s7233" xml:space="preserve"/>
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            <s xml:id="echoid-s7234" xml:space="preserve">§. </s>
            <s xml:id="echoid-s7235" xml:space="preserve">3. </s>
            <s xml:id="echoid-s7236" xml:space="preserve">Si ponatur V = {1/2} y, fiet x = y & </s>
            <s xml:id="echoid-s7237" xml:space="preserve">proinde linea E O erit
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            recta conſtituens cum axe G H angulum ſemirectum habebitque cavitas for-
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            mam coni: </s>
            <s xml:id="echoid-s7238" xml:space="preserve">Si vero ſervata eadem proportione velocitatum, quæ nempe ſint
              <lb/>
            ubique radicibus diſtantiarum ab axe proportionales, aquæ celerius tardiuſve
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            circumagantur, fiet angulus E O G eo acutior, quo celerius moventur, ita ut
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            ſi infinita fuerit velocitas, tunc aquæ perpendiculariter fundo inſiſtere debeant
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            inſtar muri, cavitatemque cylindricam interius formare, ſi modo operculum
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            ſit in A D, quod impediat, quominus aquæ omnes ejiciantur.</s>
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            <s xml:id="echoid-s7240" xml:space="preserve">§. </s>
            <s xml:id="echoid-s7241" xml:space="preserve">4. </s>
            <s xml:id="echoid-s7242" xml:space="preserve">Si ponatur paullo generalius 2 V = fy
              <emph style="super">e</emph>
            , fiet dx = fy
              <emph style="super">e - 1</emph>
            dy
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            vel x = {f/e}y
              <emph style="super">e</emph>
            : </s>
            <s xml:id="echoid-s7243" xml:space="preserve">Hinc ſequitur curvam ſemper fore verſus axem conca-
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            vam, ut in figura 65, ſi ſit e major unitate atque convexam, ut in fig. </s>
            <s xml:id="echoid-s7244" xml:space="preserve">66. </s>
            <s xml:id="echoid-s7245" xml:space="preserve">ſi ſit
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            minor. </s>
            <s xml:id="echoid-s7246" xml:space="preserve">In priori cafu eſt angulus E O G ſemper rectus, in altero ſemper nul-
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            lus: </s>
            <s xml:id="echoid-s7247" xml:space="preserve">in ſolo caſu quo e = 1 poteſt angulus iſte eſſe qualiſcunque.</s>
            <s xml:id="echoid-s7248" xml:space="preserve"/>
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            <s xml:id="echoid-s7249" xml:space="preserve">§. </s>
            <s xml:id="echoid-s7250" xml:space="preserve">5. </s>
            <s xml:id="echoid-s7251" xml:space="preserve">Inſervire poſſunt hæc ad dignoſcendam quodammodo ſcalam
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            velocitatum in vortice artificioſe producto: </s>
            <s xml:id="echoid-s7252" xml:space="preserve">ſi enim ſuperficiem videas conca-
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            vam, recte judicabis velocitates majori creſcre ratione, quam diſtantiæ ab axe
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            creſcant, ſi convexam contrarium deduces. </s>
            <s xml:id="echoid-s7253" xml:space="preserve">Si curva non videatur ad parabo-
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            licum genus pertinere, indicium erit velocitates non poſſe comparari cum di-
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            ſtantiarum determinata aliqua potentia. </s>
            <s xml:id="echoid-s7254" xml:space="preserve">Quo major obſervata fuerit linea E M
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            terminata ab horizontali O M, eo major putabitur velocitas particularum ab-
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            ſoluta ſeu littera f.</s>
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