## 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

### Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.

powered by Goobi viewer