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