Bernoulli, Daniel
,
Hydrodynamica, sive De viribus et motibus fluidorum commentarii
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SECTIO QUARTA.
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Geometrico infinitum, non ſolum non fit tempore infinite parvo, prouti in
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caſu foraminis ſimplicis, ſed tempore infinitè magno, intereaque etiam quan-
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titas aquæ infinita effluit, cum per foramen quantitas cæteris paribus infinite
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parva effluat. </
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<
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">Hæc autem ut eruerem, opus habui aliam elicere æquationem
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ex æquatione generali §. </
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echoid-s2005
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xml:space
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">quam ſimpliciſſimam hanc s = x, poſita
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s pro altitudine, quæ velocitati aquæ effluentis reſpondeat & </
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<
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">x pro altitudi-
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ne aquæ ſupra orificium effluxus; </
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">intelliget autem quisque rem pro inſtitu-
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to noſtro ita eſſe efficiendam, ut habeatur ratio incrementorum velocitatis,
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quod antea non requirebatur.</
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<
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(Fig. </
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">is que cenſeatur infinite amplus & </
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">aqua plenus, habeatque tubum
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annexum F M N G finitæ amplitudinis formæ coni truncati, ſive creſcentis
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amplitudine ſive decreſcentis verſus orificium M N, per quod aquæ effluunt:
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">ſit ut ibi altitudo initialis aquæ ſupra foramen M N, nempe N G + H B = a; </
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altitudo ſuperficiei aqueæ in ſitu C D ſupra M N, id eſt, N G + H D = x; </
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longitudo tubi annexi ſeu N G = b, amplitudo orificii M N = n, amplitudo
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orificii F G = g, amplitudo cylindri, quæ eſt infinita, = m; </
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<
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velocitas ſuperficiei aquæ in ſitu C D talis quæ conveniat altitudini v, quæ
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altitudo utique infinite parva erit. </
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<
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">His poſitis vidimus loco citato obtinere
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generaliter hanc æquationem: </
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m(x - b)dv + {bmm/√gn}dv - {m
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/nn}vdx + mvdx = - mxdx
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in quâ patet, poſſe nunc negligi terminum primum m(x - b)dv præ ſe-
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cundo {bmm/√gn}dv, ut & </
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/nn}vdx, atque ſic aſſumi
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{bmm/√gn}dv - {m
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v/nn}dx = - mxdx. </
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in qua æquatione ſi rurſus negligatur primus terminus, quod fieri poteſt,
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niſi mutationes etiam deſiderentur, quæ durante primo deſcenſu, etſi infi-
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nite parvo fiunt, orietur regula vulgaris aſcenſus potentialis aquæ effluentis ad
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altitudinem integram aquæ: </
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<
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">nunc vero pro noſtro negotio, quo mutatio-
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nes illas primas deſideramus, terminus iſte retinendus erit, atque ſic æqua-
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tio ultima in tota ſua extenſione pertractanda.</
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