Bernoulli, Daniel - Hydrodynamica, sive De viribus et motibus fluidorum commentarii

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5945SECTIO TERTIA. (1 - {mm/nn})vdz + zdv = - zdz - bdz + {mbdz/√gn}
quæ multiplicata per z{-mm/nn} facit
(1 - {mm/nn})z- {mm/nn} vdz + z1 - {mm/nn} dv = - z1 - {mm/nn} dz - bz- {mm/nn} dz +
{mbz- {mm/nn} dz/√gn}
poſt cujus integrationem addita conſtante Coritur
z{nn - mm/nn} v = C - {nn/2nn - mm} z{2nn - mm/nn} - {nnb/nn - mm} z{nn - mm/nn}
+ {mnnb/(nn - mm)√gn} z{nn - mm/nn}
in quo valor quantitatis conſtantis C ex eo definitur quod ab initio fluxus
(cum nempe x = a ſive z = a - b + {mb/√gn}) ſit v = o quia non poteſt motus
oriri in inſtanti temporis puncto; hinc igitur fit C =
[(a - b + {mb/√gn}) X {nn/2nn - mm} + {nnb√gn - mnnb/(nn - mm)√gn}] X (a - b + {mb/√gn}){nn - mm/nn}
Ex his quidem æquationibus definiuntur omnia; quia verò calculus fit paullo
prolixior, niſi amplitudo vaſis ſuperioris indicata per m tanta ſit, ut poſſit ra-
tione amplitudinum g & n infinita cenſeri, hunc ſolum conſiderabimus caſum,
idque eo magis quod error notabilis inde non oriatur, etſi mediocris ſit ma-
gnitudinis numerus {m/n} aut {m/g}

§. 23. Quod ſi proinde ponamus m = ∞, ſimulque utamur pri-
mâ æquatione differentiali proximi paragraphi, atque in hâc ponatur
v = {nn/mm}s, ut ſic inveniatur ex valore litteræ s altitudo ad quam aqua per ori-
ficium M N effluens ſuâ velocitate aſcendere poſſit, erit primo
{nn/m} (x - b)ds + {bnn/√gn}ds - msdx + {nn/m}sdx = - mxdx
& quia m = ∞ atque facile prævidetur rationem ſore finitam inter s & x, at-
que inter ds & dx, hæc eadem æquatio mutabitur rejectis terminis rejiciendis
rurſus in hanc - msdx = - mxdx vel s = x, quod pariter paragr. 10.

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