Full text: Bernoulli, Daniel: Hydrodynamica s. de viribus et motibus fluidorum commentarii

HYDRODYNAMICÆ cum ſuperficies aquæ variabilis eſt in h l, fore altitudinem debitam velocitati
aquæ per M transfluentis = B b = x, velocitatemque ipſam = √x, ſi-
milemque altitudinem ratione orificii N = h M = a - x, atque velocita-
tem aquæ per N transfluentis = √a - x; eſt igitur quantitas dato tempu-
ſculo per M in vas B N influentis ad quantitatem eodem tempuſculo ex vaſe
effluentis ut m√x ad n√a - x, harumque quantitatum differentia diviſa
per amplitudinem g dat velocitatem ſuperficiei h l, quæ proinde velocitas,
quam vocabimus v, exprimetur hâc æquatione,
v = {m√x - n√a - x/g}

§. 22. Ut jam innoteſcat tempus, quo ſuperficies fluidi ex H L venit in
h l, vocabimus illud tempus t: quia autem eſt dt = {-dx/v}, erit, poſito
pro v valore modo invento,
dt = {-gdx/m√x - n√a - x}
Poteſt quidem hæc formula immediate rationalis fieri ponendo x = {4aqq/(1 + qq) 2 },
atque deinde debito modo conſtrui: Iſta vero methodus paullo prolixior eſt
hâc altera, qua quantitas reducenda dividitur in duo membra ſeorſim inte-
granda, nempe præmiſſa æquatio non differt ab hâc: dt = {mgdx√x/nna - (mm + nn) x} + {ngdx√a - x/nna - (mm + nn) x}: Et autem ſ{mgdx√x/nna - (mm + nn) x} = - {2mg/mm + nn}√x + {mng√a/(mm + nn)√(mm + nn)} X
log. {n√a + √mm + nn√x/n√a - √mm + nn√x}; alteriusque membri integrale
nempe ſ{ngdx√a - x/nna - (mm + nn) x} fit = {-2ng/mm + nn}√(a - x) +
{mng√a/(mm + nn) X √(mm + nn)} log. {m√a + √mm + nn X √a - x/m√a - √mm + nn X √a - x}; Patet exinde addita debita conſtante fore
t = {2mg√a - b - 2mg√x + 2ng√b - 2ng√a - x/mm + nn} +
{mng√a/(mm + nn) X √(mm + nn)} X

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