Bernoulli, Daniel
,
Hydrodynamica, sive De viribus et motibus fluidorum commentarii
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HYDRODYNAMICÆ.
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ſtatim ac x paulo minor eſt quam a. </
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<
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">Regula hæc fallit notabiliter tantum cir-
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ca primum motus initium & </
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<
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xml:space
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">ſi primum iſtud motus elementum conſidera-
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tur (quo nempe altitudo a - x ut infinite parva cenſeri poteſt) indicat æ-
<
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quatio, eſſe tunc v = a - x. </
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<
s
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xml:space
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">Unde ſequitur, in omni cylindro, quodcun-
<
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que fuerit foramen, aquam internam inſtar corporum libere cadentium ac-
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celerari ab initio motus. </
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<
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xml:id
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xml:space
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">Si vero motus aliquantulum continuet, eo minus
<
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fallet hæc Regula, quo majus fuerit foramen, & </
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<
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">quo altior eſt aqua in tubo; </
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<
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porro deſideretur altitudo ea, quæ velocitati aquæ effluentis reſpondeat,
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quam §. </
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<
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">9. </
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<
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xml:space
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">poſuimus = z, erit z = {mm/nn}v, ſeu
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z = {mma/2nn - mm} (({a/x})
<
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style
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">1 - {mm/nn}</
emph
>
- {x/a})</
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<
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<
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">15. </
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<
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xml:space
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">Cum n eſt = m, id eſt, cum nullum eſt fundum, apparet
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ex ipſa rei natura, aquam inſtar corporum gravium libere cadere atque ac-
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celerari, id ipſum autem indicat etiam æquatio; </
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<
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xml:space
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">fit enim in hâc poſitione
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z = a - x. </
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<
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xml:space
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">Si vero foramen eſt veluti infinite parvum ratione amplitudinis
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vaſis, quem caſum jam ſupra conſideravimus, ponendum eſt n = o, & </
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<
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">tunc
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fit z = x, quod indicat, aquam ea conſtantur effluere velocitate, qua ad
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totam aquæ altitudinem aſcendere poſſit. </
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<
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xml:space
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">Denique cum mm = 2nn, pro-
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dit z = {mm/o} (x - x), ex quo valore cum nihil cognoſci poſſit, deſcenden-
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dum eſt ad æquationem differentialem §. </
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<
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</
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<
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xml:space
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">- vdx + xdv = - xdx, vel {xdv - vdx/xx} = {- dx/x},
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quæ integrata cum debitæ conſtantis additione dat {v/x} = log. </
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<
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xml:space
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">{a/x}, vel v =
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xlog.</
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<
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xml:space
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">{a/x}, aut z = 2v = 2xlog.</
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<
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xml:space
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">{a/x}.</
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</
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<
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<
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<
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">Velocitas aquæ effluentis ab initio creſcit poſteaque decreſcit,
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eſtque alicubi maxima, nempe eo in loco, quo aqua deſcendit ad altitudinem
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a:</
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<
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xml:space
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">({mm - nn/nn})
<
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; </
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<
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">id quoque experientia edoctus indicavit Ma-
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riottus in tract. </
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<
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">de motu aquarum part. </
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xima talis eſt, quæ debetur </
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