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

HYDRODYNAMICÆ altitudinem ſupra foramen, exprimat H G amplitudinem vaſis in illo loco. Deinde fiat tertia curva t r u, cujus applicata H r ſit ubique æqualis tertiæ con-
tinue proportionali ad G H & P L ſeu cujus applicata H rſit = P L 2 : G H.

Dicatur ſpatium D C I L = M, ſpatium D t u L = N, & erit aſcen-
ſus potentialis aquæ in vaſe contentæ, poſtquam prædicta quantitas jam efflu-
xit (per §. 2. ſect. 3.) = {N/M}v. Effluere porro intelligatur particula p l o n, ſu-
perficiesque c d deſcendere in e f, erit jam velocitatis altitudo pro particula p l o n
= v + d v; atque ſi nunc conſtruatur parallelogrammum L x y O, cujus latus
L O ſit = l o & alterum L x = P L, erit aſcenſus potentialis ejusdem aquæ
in ſitu e f m l o n p i e æqualis tertiæ proportionali ad ſpatium E F L O N P I E,
(quod rurſus eſt = M, quia P L O N exprimit magnitudinem guttulæ p l o n,
dum C D F E exprimit quantitatem minimam c d f e iſti guttulæ æqualem)
ſpatium w u x y O L F (quod eſt = ſpatio N - D t w F + L x yO, unde ſi
P L ſeu L x ponatur = n, C D = m, L O = lo = dx, erit D t = {nn/m},
D F = {n/m} dx, hinc ſpatiolum D tw F = {n 3 /mm} dx & ſpatium L xy O =
ndx & denique ſpatium w uxy O L F = N - {n 3 /mm} dx + ndx) & altitudi-
nem v + dv. Eſt igitur aſcenſus potentialis modo dictus = (N - {n 3 /mm} dx + ndx) X
(v + dv): M = rejectis differentialibus ſecundi ordinis {N/M} v + {N/M} dv
- {n 3 /mmM} vdx + {n/M}vdx, ſic ut incrementum aſcenſus potentialis, quod aquæ
acceſſit dum guttula plon effluxit, ſit = {N/M}dv - {n 3 /mmM}vdx + {n/M}vdx, ubi
ſpatia N & M ſunt conſtantis magnitudinis ob aquæ continuam affuſionem. Non
conſideramus in hoc caſu primo aſcenſum potentialem guttulæ cdfe, quæ af-
funditur dum altera æqualis plon effluit, quia iſte aſcenſus non generatur vi
interna, neque enim aqua inferior poſt ſe trahere ponitur particulam cdfe,
quin potius hanc vi quadam extrinſeca continue affundi conſideramus, idque
nec ma [?] jori nec minore velocitate quam quæ eſt ſuperficiei ef. Ergo omne
incrementum hic conſiderandum, eſt ut diximus
{N/M}dv - {n 3 /mmM}vdx + {n/M} vdx.

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