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

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117103SECTIO QUINTA.

Solutio.

Adhibitis rurſus poſitionibus & denominationibus paragraphi tertii &
duodecimi, invenienda nunc erit æquatio inter x & t: quia vero, ut vidi-
mus §. 12. eſt d t = {γdx/√v}, erit √ v = {γdx/dt}, hicque valor ſubſtituendus
erit in æquationibus, quas dedimus §. 3. integratis; prior harum æquationum
hæc fuit: v = {mma/mm - nn} X (1 - c{n3 - nmm/mmN} x)
quæ pro præſecuti inſtituto mutatur in hanc
(I) {γγdx2/dt2} = {mma/mm - nn} X (1 - c{n3 - nmm/mmN} x)
altera ex §. 3. allegatarum æquationum talis fuit
v = a X (1 - c{- n/N} x)
quæ adeoque ſubminiſtrat in præſenti caſu ſequentem
(II) {γγdx2/dt2} = a X (1 - c{- n/N} x)

Erunt nunc æquationes (I) & (II) integrandæ, quod quidem facile
eſt & quia prior alteram continet (utraque enim eadem eſt ſi m = ∞)
hanc ſolam pertractabimus, eamque nunc ſub hâc forma conſiderabimus.
dt = {γ√(mm - nn)/m√a}dx: √(1 - c{n3 - nmm/mmN}x)

Ponatur autem ut integrationis modus eo magis pateſcat
c{n3 - nmm/mmN}x = z, atque proin dx = {mmNdz/(n3 - nmm)z},
dein brevitatis ergo indicetur quantitas conſtans
{γ√(mm - nn)/m√a} X {mmN/n3 - nmm}, ſeu {- γmN/n√(mm - nn) a} per α,
& habebitur dt = {αdz/z√(1 -

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