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