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

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10793SECTIO QUINTA.

Debet vero iſtud incrementum æquari deſcenſui actuali centri gravitatis;
Atqui iſte deſcenſus, poſita D L = a, eſt per paragraphum ſeptimum ſect. 3.
= {nadx/M}; habetur igitur talis æquatio
{N/M}dv - {n3/mmM}vdx + {n/M}vdx = {nadx/M}, ſeu
dx = Ndv: (na - nv + {n3/mm} v);

Hæc vero ſi ita integretur, ut v & x ſimul evaneſcant, dat
x = {mmN/n3 - nmm} log. {mma - mmv + nnv/mma}
quæ æquatio, poſito c pro numero cujus logarithmus eſt unitas, æquivalet
huic @alteri
v = {mma/mm - nn} X (1 - c{n3 - nmm/mmN} x)

Hæc vero ſolutio quadrat pro caſu primo, ubi aqua ſuperne motu af-
ſunditur communi cum deſcenſu ſuperficiei proximæ.

Caſus II.

Quod ſi jam particula c d f e lateraliter continue affundi ponatur, tunc
propter inertiam ſuam motui aquæ inferioris reſiſtit atque proinde aſcenſus
potentialis ipſius aliter in computum venit. Tunc autem prius conſideran-
dus eſt aſcenſus potentialis maſſæ aqueæ c d m l p i c auctæ guttula mox affunden-
da; deinde indagandus aſcenſus potent. ejusdem aquæ in ſitu c d m l o n p i c,
poſtquam nempe guttula jam effluxit, eorumque differentia eſt æquanda cum
deſcenſu actuali. {nadx/M}. Verum aſcenſus potentialis omnis prædictæ aquæ ante
affuſionem particulæ ejusdemque poſt affuſionem ita invenitur: nempe aſcen-
ſus potentialis aquæ c d m l p i c eſt = {Nv/M}, & aſcenſus potent. particulæ affundi
paratæ nullus eſt, quia lateraliter affuſa motum communem nondum habet
cum maſſa inferiore; Igitur aſcenſus potentialis utriusque aquæ (qui ſcilicet
habetur multiplicando maſſam reſpective per ſuum aſcenſum potentialem, di-
videndoque productorum aggregatum per aggregatum maſſarum) eſt

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