Bernoulli, Daniel
,
Hydrodynamica, sive De viribus et motibus fluidorum commentarii
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SECTIO QUINTA.
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duodecimi, invenienda nunc erit æquatio inter x & </
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mus §. </
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<
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">eſt d t = {γdx/√v}, erit √ v = {γdx/dt}, hicque valor ſubſtituendus
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erit in æquationibus, quas dedimus §. </
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hæc fuit: </
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xml:space
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">v = {mma/mm - nn} X (1 - c{n
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- nmm/mmN} x)
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quæ pro præſecuti inſtituto mutatur in hanc
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(I) {γγdx
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/dt
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} = {mma/mm - nn} X (1 - c{n
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- nmm/mmN} x)
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altera ex §. </
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v = a X (1 - c
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)
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quæ adeoque ſubminiſtrat in præſenti caſu ſequentem
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(II) {γγdx
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/dt
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} = a X (1 - c
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)</
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">(II) integrandæ, quod quidem facile
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eſt & </
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">quia prior alteram continet (utraque enim eadem eſt ſi m = ∞)
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hanc ſolam pertractabimus, eamque nunc ſub hâc forma conſiderabimus.
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</
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<
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- nmm/mmN}x)</
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">Ponatur autem ut integrationis modus eo magis pateſcat
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c{n
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- nmm/mmN}x = z, atque proin dx = {mmNdz/(n
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- nmm)z},
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dein brevitatis ergo indice
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tur quantitas conſtans
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{γ√(mm - nn)/m√a} X {mmN/n
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- nmm}, ſeu {- γmN/n√(mm - nn) a} per α,
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& </
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