Bernoulli, Daniel
,
Hydrodynamica, sive De viribus et motibus fluidorum commentarii
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SECTIO QUINTA.
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ſpondentia, ita ut aquæ ex ſuperiori vaſe effluentes omnes in cylindrum ſubje-
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ctum influant.</
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<
s
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xml:space
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">Incipiant aquæ ex utroque vaſe effluere, ex ſuperiori autem conſtanter ea
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effluere velocitate ponantur, quam habet ſuperficies aquæ in cylindro ſuppoſito.</
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</
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<
s
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xml:space
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">Ita patet ſatisfieri primæ affuſionis conditioni. </
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<
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xml:space
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">Jam vero hujus motus
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phænomena inveſtigabimus, viſuri num cum præcedentibus conveniant.</
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<
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</
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<
s
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xml:space
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">Conſideremus igitur vas ſuperius eſſe veluti infinitum, ut aquæ per R S
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effluentes ſingulis momentis habeant velocitatem quæ conveniat altitudini P B
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ſeu F A: </
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<
s
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xml:space
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">ſic fingendum erit eſſe hanc altitudinem P B ab initio infinite parvam,
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quia tunc aquæ velocitate infinite parva effluere debent, deinde vero ſenſim
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creſcere, idque continue magis magisque, donec poſt tempus infinitum mo-
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tus uniformis maneat, quæritur autem an altitudo aquæ P B tandem infinita
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futura ſit an vero certum terminum non tranſgreſſura. </
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<
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xml:space
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">Id ſic cognoſcetur.</
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<
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</
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<
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<
s
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xml:space
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">Sit altitudo G H vel R H (neque enim illas inter ſe differre cenſendum
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eſt) = a, A F = x, amplitudo orificii L M = n, amplitudo orificii R S = m;
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</
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<
s
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xml:space
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">quia vero, ut manifeſtum eſt, utrumque vas cohærere & </
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<
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xml:space
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">unum efficere puta-
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ri poteſt, erit poſt tempus infinitum (per §. </
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<
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<
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">III.) </
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<
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aquæ in L M = √a + x, & </
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<
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xml:space
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">in R S = √ x, (quod poſterius patet, ſi nunc iterum
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ſeparata vaſa cenſentur, nam utrumque ſine errore fingi poteſt) debent autem
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velocitates eſſe in inverſa ratione amplitudinum orificiorum: </
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<
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xml:space
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√a + x.</
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xml:space
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xml:space
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<
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<
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">mm. </
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<
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xml:space
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<
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xml:space
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">: mm - nn. </
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<
s
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xml:space
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">nn, ergo
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x = {nna/mm - nn} & </
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<
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xml:space
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">a + x = {mma/mm - nn}, videmus igitur altitudinem, velocitati
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aquæ in LM debitam, eſſe hoc modo = {mma/mm - nn}, poſtquam ſcilicet infi-
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nita aquæ quantitas jam effluxit: </
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<
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xml:space
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">ſuperius autem habuimus eandem altitudinem,
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ſeu v = {mma/mm - nn} X (1 - c{n
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- nmm/mmN}x), ubi ſi ponitur x = ∞ (infinito
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enim tempore infinita quantitas transfluit) evaneſcit terminus exponentialis, ſi
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modo m major ſit quam n & </
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<
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xml:space
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">ſic fit pariter v = {mma/mm - nn}. </
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<
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xml:space
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">Mirabilis eſt iſte con-
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ſenſus, quia valde diverſæ ſunt viæ, quas ſecuti ſumus. </
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<
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">Cæterum ſi m non ſit ma-
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jor quam n motus nunquam fit permanens nequidem poſt tempus infinitum,
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creſcit enim tunc velocitas in infinitum cum ſecus altitudo velocitatis </
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