Bernoulli, Daniel
,
Hydrodynamica, sive De viribus et motibus fluidorum commentarii
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SECTIO TERTIA.
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(1 - {mm/nn})vdz + zdv = - zdz - bdz + {mbdz/√gn}
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quæ multiplicata per z
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facit
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(1 - {mm/nn})z
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vdz + z
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dv = - z
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dz - bz
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dz +
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{mbz
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dz/√gn}
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poſt cujus integrationem addita conſtante Coritur
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z
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v = C - {nn/2nn - mm} z
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- {nnb/nn - mm} z
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+ {mnnb/(nn - mm)√gn} z
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in quo valor quantitatis conſtantis C ex eo definitur quod ab initio fluxus
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(cum nempe x = a ſive z = a - b + {mb/√gn}) ſit v = o quia non poteſt motus
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oriri in inſtanti temporis puncto; </
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">hinc igitur fit C =
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[(a - b + {mb/√gn}) X {nn/2nn - mm} + {nnb√gn - mnnb/(nn - mm)√gn}] X (a - b + {mb/√gn})
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Ex his quidem æquationibus definiuntur omnia; </
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">quia verò calculus fit paullo
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prolixior, niſi amplitudo vaſis ſuperioris indicata per m tanta ſit, ut poſſit ra-
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tione amplitudinum g & </
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">n infinita cenſeri, hunc ſolum conſiderabimus caſum,
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idque eo magis quod error notabilis inde non oriatur, etſi mediocris ſit ma-
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gnitudinis numerus {m/n} aut {m/g}</
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">Quod ſi proinde ponamus m = ∞, ſimulque utamur pri-
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mâ æquatione differentiali proximi paragraphi, atque in hâc ponatur
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v = {nn/mm}s, ut ſic inveniatur ex valore litteræ s altitudo ad quam aqua per ori-
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ficium M N effluens ſuâ velocitate aſcendere poſſit, erit primo
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{nn/m} (x - b)ds + {bnn/√gn}ds - msdx + {nn/m}sdx = - mxdx
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& </
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">quia m = ∞ atque facile prævidetur rationem ſore finitam inter s & </
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que inter ds & </
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">dx, hæc eadem æquatio mutabitur rejectis terminis rejiciendis
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rurſus in hanc - msdx = - mxdx vel s = x, quod pariter paragr. </
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