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

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5137SECTIO TERTIA.

De his quæ pertinent ad effluxum aquarum ex Cy-
lindris verticaliter poſitis, per Lumen quod-
cunque, quod eſt in fundo horizontali.

§. 13.

GEometræ, quibus de aquis ex vaſe erumpentibus ſermo fuit, con-
ſiderare potiſſimum ſolent cylindros verticaliter poſitos: Igitur haud
abs re erit ex theoria noſtra generali conſectaria illa, quæ huc per-
tinent, deducere. Sit amplitudo cylindri ad amplitudinem foraminis ut m
ad n; altitudo aquæ ſupra foramen, cum fluxus incipit = a; altitudo aquæ
reſiduæ = x, altitudo velocitati aquæ internæ debita = v; erit in æquatio-
ne canonica paragraphi octavi y = m, N = mx (per §. 6.) quæ adeoque
abit in hanc æquationem.
mxdv - {m3/nn}vdx + mvdx = - mxdx, vel
(1 - {mm/nn})vdx + xdv = - xdx
multiplicetur hæc poſterior æquatio per x{- mm/nn}, ut habeatur
(1 - {mm/nn})x- {mm/nn} vdx + x1 - {mm/nn}dv = - x1 - {mm/nn}dx.

Poteſt jam hæc æquatio integrari: obſervanda autem eſt in Integratio-
ne conſtantis additio, talis nempe, ut a fluxus initio, id eſt, cum x = a,
ſit velocitas fluidi nulla, ipſaque proin v pariter = o: ita vero oritur:
x1 - {mm/nn} v = {nn/2nn - mm}(a2 - {mm/nn} - x2 - {mm/nn}) vel
v = {nna/2nn - mm}(({a/x})1 - {mm/nn} - {x/a})

§. 14. Ex hâc igitur æquatione cognoſcitur altitudo generans velocita-
tem aquæ internæ; ubi notari meretur, ſi vas ſit ampliſſimum, mox poſſe
cenſeri v = {nn/mm}x, poſtquam ſcilicet vel tantillum deſcendit aqua, id

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