Full text: Bernoulli, Daniel: Hydrodynamica s. de viribus et motibus fluidorum commentarii

HYDRODYNAMICÆ. potentias utcunque variabiles, quarum altera ſit ubique ad curvam, altera
ad A G perpendicularis: priorem ponemus in puncto D æqualem A, in
puncto E æqualem A + dA, alteram in puncto D = C, in puncto E = C + dC: Sit porro AB = x, BD = y, AD = s, BC = dx, FE = dy, DE = ds, quod
elementum curvæ conſtantis magnitudinis ponatur; Radius Oſculi in puncto
D = R, in puncto E = R + dR. Dico æquationem ad curvam fore hanc - AdR
- R d A = (RdCdx + 2Cdyds + CdxdR) ds, vel poſito CRddx pro Cdyds
(eſt enim R = {dyds/ddx}) habebitur - AdR - RdA = (RdCdx + CRdds + Cdyds
+ Cdx dR): ds, ſive {-ARds - RCdx/dx} = ſCdy.

22.1.

Fig. 7.

§. 15. Intelligitur ex præcedente æquatione, quod cum potentiæ,
quæ ſunt ad curvam perpendiculares, ſolæ agunt, fiat AR = conſtanti quan-
titati, quia nempe ſic fit C = o: tunc igitur radius oſculi ubique ſequitur ra-
tionem inverſam potentiæ reſpondentis. At ſi potentiæ ad axem perpendi-
culares ſolæ adſunt, tunc evaneſcente littera A fit - {RCdx/ds} = ſCdy. Po-
teſt autem hæc æquatio integrari & ad hanc reduci formam RCdx 2 = con-
ſtanti quantitati; ex qua apparet potentiam ductam in radium oſculi ubique
eſſe in ratione reciproca quadrati ſinus, quem applicata facit cum curva. Similiter æquatio canonica integrationem admittit, cum potentiæ, quæ ad
axem perpendiculares ſunt, omnes inter ſe ſunt æquales ſeu proportionales
elemento curvæ d s. Ita enim poſito d C = o, obtinetur - AdR - RDA =
2ndyds + ndxdR, intelligendo per n conſtantem quantitatem, qua æqua-
tione recte tractata fit nydy + mmdy - nsds = dsſAdx, ubi m conſtans eſt
ab integratione proveniens.

Si præterea potentiæ ad curvam normales ponantur applicatis y pro-
portionales, poterit ulterius reduci poſtrema æquatio ad hanc
- dx = (2ff - {gyy/h}) dy: √(2ny + 2mm) 2 - (2ff - {gyy/h}) 2 ,
cujus conſtantes f & m caſibus particularibus erunt applicandæ, dum n & g pen-
dent à relatione potentiarum in puncto aliquo: unde ſi g = o, oritur catenaria, & ſi n = o prodit elaſtica: generaliter vero inſervit æquatio ad curvaturam
lintei uniformiter gravis, cui fluidum ſuperincumbit, determinandam: Ca-

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