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

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10995SECTIO QUINTA. ſpondentia, ita ut aquæ ex ſuperiori vaſe effluentes omnes in cylindrum ſubje-
ctum influant.

Incipiant aquæ ex utroque vaſe effluere, ex ſuperiori autem conſtanter ea
effluere velocitate ponantur, quam habet ſuperficies aquæ in cylindro ſuppoſito.

Ita patet ſatisfieri primæ affuſionis conditioni. Jam vero hujus motus
phænomena inveſtigabimus, viſuri num cum præcedentibus conveniant.

Conſideremus igitur vas ſuperius eſſe veluti infinitum, ut aquæ per R S
effluentes ſingulis momentis habeant velocitatem quæ conveniat altitudini P B
ſeu F A: ſic fingendum erit eſſe hanc altitudinem P B ab initio infinite parvam,
quia tunc aquæ velocitate infinite parva effluere debent, deinde vero ſenſim
creſcere, idque continue magis magisque, donec poſt tempus infinitum mo-
tus uniformis maneat, quæritur autem an altitudo aquæ P B tandem infinita
futura ſit an vero certum terminum non tranſgreſſura. Id ſic cognoſcetur.

Sit altitudo G H vel R H (neque enim illas inter ſe differre cenſendum
eſt) = a, A F = x, amplitudo orificii L M = n, amplitudo orificii R S = m;
quia vero, ut manifeſtum eſt, utrumque vas cohærere & unum efficere puta-
ri poteſt, erit poſt tempus infinitum (per §. 23. Sect. III.) velocitas
aquæ in L M = √a + x, & in R S = √ x, (quod poſterius patet, ſi nunc iterum
ſeparata vaſa cenſentur, nam utrumque ſine errore fingi poteſt) debent autem
velocitates eſſe in inverſa ratione amplitudinum orificiorum: eſt itaque
√a + x. √x: :m. n, unde a + x. x: mm. nn, vel a. x: : mm - nn. nn, ergo
x = {nna/mm - nn} & a + x = {mma/mm - nn}, videmus igitur altitudinem, velocitati
aquæ in LM debitam, eſſe hoc modo = {mma/mm - nn}, poſtquam ſcilicet infi-
nita aquæ quantitas jam effluxit: ſuperius autem habuimus eandem altitudinem,
ſeu v = {mma/mm - nn} X (1 - c{n3 - nmm/mmN}x), ubi ſi ponitur x = ∞ (infinito
enim tempore infinita quantitas transfluit) evaneſcit terminus exponentialis, ſi
modo m major ſit quam n & ſic fit pariter v = {mma/mm - nn}. Mirabilis eſt iſte con-
ſenſus, quia valde diverſæ ſunt viæ, quas ſecuti ſumus. Cæterum ſi m non ſit ma-
jor quam n motus nunquam fit permanens nequidem poſt tempus infinitum,
creſcit enim tunc velocitas in infinitum cum ſecus altitudo velocitatis

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