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

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4834HYDRODYNAMICÆ. ticulæ aqueæ cdfe ſupra guttulam lonp = x, altitudo centri gravitatis aquæ
efmi a fundo = b, erit altitudo centri gravitatis omnis aquæ in ſitu cdmi
ſupra fundum = b - {ydx/M} X (x - b) & in ſitu efmlonpi erit eadem
altitudo = ({M + ydx/M}) X b; unde differentia altitudinum ſeu deſcenſus actualis
quæſitus = - {ydx/M} X x, quæ æquatio indicat, guttulam quæ effluxerit
multiplicandam eſſe per altitudinem aquæ ſupra foramen, productumque
dividendum per quantitatem aquæ, ut habeatur deſcenſus actualis, qui fit
dum guttula effluit, Q. E. I.

Problema.

§. 8. Determinare motum fluidi homogenei ex vaſe dato per fo-
ramen datum effluentis.

Solutio.

Quoniam per hypotheſin noſtram aſcenſus potentialis ſingulis mo-
mentis æqualis eſt Deſcenſui actuali, erit incrementum prioris dum guttula
effluit æquale incremento poſterioris, quod ſimili tempuſculo oritur. Igi-
tur ſi rurfus ſuperficies aquæ, poſtquam data ejus quantitas effluxit, pona-
tur = y, amplitudo vaſis quocunque in loco ad libitum aſſumta = m, am-
plitudo foraminis = n, altitudo aquæ ſupra foramen = x; ſi præterea
quantitas N ea lege conſtruatur, quæ §. 6. indicata fuit, atque per v in-
telligatur altitudo debita velocitati aquæ in loco aſſumto, ubi nempe am-
plitudo vaſis eſt = m, erit per §. 6. incrementum aſcenſus potentialis =
(Ndv - {mmvydx/nn} + {mmvdx/y}): M, minimusque deſcenſus actualis = {- yxdx/M}
(per præced. §.) ; unde habetur (Ndv - {mmvydx/nn} + {mmvdx/y}): M =
- yxdx: MſeuNdv - {mmvydx/nn} + {mmvdx/y} = - yxdx, quæ æquatio ge-
neraliter integrari poteſt, quandoquidem litteræ N & y ſunt functiones datæ
ipſius x & litera v unius tantum dimenſionis eſt.

Corollarium 1.

§. 9. Quum velocitates ſint in ratione reciproca

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