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

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              <pb o="24" file="0038" n="38" rhead="HYDRODYNAMICÆ."/>
            potentias utcunque variabiles, quarum altera ſit ubique ad curvam, altera
              <lb/>
            ad A G perpendicularis: </s>
            <s xml:id="echoid-s642" xml:space="preserve">priorem ponemus in puncto D æqualem A, in
              <lb/>
            puncto E æqualem A + dA, alteram in puncto D = C, in puncto E = C + dC:
              <lb/>
            </s>
            <s xml:id="echoid-s643" xml:space="preserve">Sit porro AB = x, BD = y, AD = s, BC = dx, FE = dy, DE = ds, quod
              <lb/>
            elementum curvæ conſtantis magnitudinis ponatur; </s>
            <s xml:id="echoid-s644" xml:space="preserve">Radius Oſculi in puncto
              <lb/>
            D = R, in puncto E = R + dR. </s>
            <s xml:id="echoid-s645" xml:space="preserve">Dico æquationem ad curvam fore hanc - AdR
              <lb/>
            - R d A = (RdCdx + 2Cdyds + CdxdR) ds, vel poſito CRddx pro Cdyds
              <lb/>
            (eſt enim R = {dyds/ddx}) habebitur - AdR - RdA = (RdCdx + CRdds + Cdyds
              <lb/>
            + Cdx dR): </s>
            <s xml:id="echoid-s646" xml:space="preserve">ds, ſive {-ARds - RCdx/dx} = ſCdy.</s>
            <s xml:id="echoid-s647" xml:space="preserve"/>
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            <s xml:id="echoid-s648" xml:space="preserve">§. </s>
            <s xml:id="echoid-s649" xml:space="preserve">15. </s>
            <s xml:id="echoid-s650" xml:space="preserve">Intelligitur ex præcedente æquatione, quod cum potentiæ,
              <lb/>
            quæ ſunt ad curvam perpendiculares, ſolæ agunt, fiat AR = conſtanti quan-
              <lb/>
            titati, quia nempe ſic fit C = o: </s>
            <s xml:id="echoid-s651" xml:space="preserve">tunc igitur radius oſculi ubique ſequitur ra-
              <lb/>
            tionem inverſam potentiæ reſpondentis. </s>
            <s xml:id="echoid-s652" xml:space="preserve">At ſi potentiæ ad axem perpendi-
              <lb/>
            culares ſolæ adſunt, tunc evaneſcente littera A fit - {RCdx/ds} = ſCdy. </s>
            <s xml:id="echoid-s653" xml:space="preserve">Po-
              <lb/>
            teſt autem hæc æquatio integrari & </s>
            <s xml:id="echoid-s654" xml:space="preserve">ad hanc reduci formam RCdx
              <emph style="super">2</emph>
            = con-
              <lb/>
            ſtanti quantitati; </s>
            <s xml:id="echoid-s655" xml:space="preserve">ex qua apparet potentiam ductam in radium oſculi ubique
              <lb/>
            eſſe in ratione reciproca quadrati ſinus, quem applicata facit cum curva.
              <lb/>
            </s>
            <s xml:id="echoid-s656" xml:space="preserve">Similiter æquatio canonica integrationem admittit, cum potentiæ, quæ ad
              <lb/>
            axem perpendiculares ſunt, omnes inter ſe ſunt æquales ſeu proportionales
              <lb/>
            elemento curvæ d s. </s>
            <s xml:id="echoid-s657" xml:space="preserve">Ita enim poſito d C = o, obtinetur - AdR - RDA =
              <lb/>
            2ndyds + ndxdR, intelligendo per n conſtantem quantitatem, qua æqua-
              <lb/>
            tione recte tractata fit nydy + mmdy - nsds = dsſAdx, ubi m conſtans eſt
              <lb/>
            ab integratione proveniens.</s>
            <s xml:id="echoid-s658" xml:space="preserve"/>
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            <s xml:id="echoid-s659" xml:space="preserve">Si præterea potentiæ ad curvam normales ponantur applicatis y pro-
              <lb/>
            portionales, poterit ulterius reduci poſtrema æquatio ad hanc
              <lb/>
            - dx = (2ff - {gyy/h}) dy: </s>
            <s xml:id="echoid-s660" xml:space="preserve">√(2ny + 2mm)
              <emph style="super">2</emph>
            - (2ff - {gyy/h})
              <emph style="super">2</emph>
            ,
              <lb/>
            cujus conſtantes f & </s>
            <s xml:id="echoid-s661" xml:space="preserve">m caſibus particularibus erunt applicandæ, dum n & </s>
            <s xml:id="echoid-s662" xml:space="preserve">g pen-
              <lb/>
            dent à relatione potentiarum in puncto aliquo: </s>
            <s xml:id="echoid-s663" xml:space="preserve">unde ſi g = o, oritur catenaria, & </s>
            <s xml:id="echoid-s664" xml:space="preserve">
              <lb/>
            ſi n = o prodit elaſtica: </s>
            <s xml:id="echoid-s665" xml:space="preserve">generaliter vero inſervit æquatio ad curvaturam
              <lb/>
            lintei uniformiter gravis, cui fluidum ſuperincumbit, determinandam: </s>
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