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

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8874HYDRODYNAMICÆ

Ponatur autem pro ſeparandis ab invicem indeterminatis {mm/nn}v - x = s, ſive
v = {nn/mm}(s + x), atque dv = {nn/mm} (ds + dx) ſicque fiet
dx = {- nnbds/nnb - ms√gn},
quæ ita eſt integranda, ut facta x = a, prodeat v = o, hincque s = - a,
ita vero fit
x - a = {nnb/m√gn}log. {nnb - ms√gn/nnb + ma√gn}
& poſito pro s valore ejus aſſumto {mm/nn}v - x, prodit
x - a = {nnb/m√gn}log. {n4b - m3v√gn + mnnx√gn/n4b + mnna√gn}

Hic rurſus in quantitate ſigno logarithmicali involuta poteſt ex nume-
ratore eliminari terminus n4b, infinities nempe minor termino mnnx√gn
nec non ex denominatore terminus n4b infinities pariter minor altero
mnna√gn. Et ſic fit
x - a = {nnb/m√gn}log. {nnx - mma/nna}

Inde habetur, poſito c pro numero cujus logarithmus eſt unitas:
v = {nnx/mm} - {nna/mm} X c {m. (x - a)√gn/nnb}
aut poſita a - x = z, ſic ut z denotet ſpatium, per quod ſuperficies aquæ
jam deſcendit, poterit æquationi hæc conciliari forma:
v = {nn. (a - z)/mm} - {nna/mm}: c{mz/nb}√{g/n}
de qua iterum liquet quod cum z vel minimam habuerit rationem ad b, fiat
denominator alterius termini infinitus & v = {nn. (a - z)/mm} = {nnx/mm}: at vero ali-
ter ſe res habet, quamdiu deſcenſus z infinite parvus eſt, quem caſum nunc
conſideramus.

§. 17. Hiſce præmiſſis facile nunc eſt definire per quantulum ſpatium
deſcendat fluidum, dum maximam velocitatem acquirit, faciendo

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