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

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6349SECTIO TERTIA. ubi pro noſtro caſu præſente intelligitur per v altitudo quæſita reſpondens ve-
locitati ſuperficiei aqueæ in ſitu F, per ξ longitudo B D E F & per x altitudo
B G, atque per {m/n} index rationis inter amplitudines tubi & foraminis B: Quod
ſi vero dicatur longitudo B D A = αerit x = {ξ - α/g}, unde nunc habetur
v = ξ{mm/nn} - 1} ſ - ({ξ - α/g}) ξ{- mm/nn} dξ

Indicetur longitudo totius canalis B D E C per β, & erit
ſ - ({ξ - α/g} ξ{- mm/nn} dξ = {nnα/g(nn - mm)} (ξ{nn - mm/nn} - β{nn - mm/nn}})
{- nn/g(2nn - mm)} (ξ{2nn - mm/nn} - β{2nn - mm/nn})
atque proinde
v = {nnα/g(nn - mm)}(1 - ({β/ξ}){nn - mm/nn})
- {nnξ/g(2nn - mm)}(1 - ({β/ξ}){2nn - mm/nn}). Q. E. I.

Scholium.

§. 27. Quoniam hæ æquationes ſunt paullo prolixiores @non immora-
bimur generali earundem contemplationi, conſideraturi potius caſus iſtos
particulares, qui calculum abbreviant, nec ultima iſta æquatione definiri
poſſunt.

Si operculum in B omne abeſſe ponamus, fit m = n & (quod ſeorſim
pro hoc pariter atque @altero caſu mox dicendo erui debet)
v = {b - ξ + αlog. ξ - αlog. β/g}
tuncque velocitas maxima eſt in A, nominatimquæ talis, quæ reſpondet al-
titudini {β - α + αlog. α - αlog. β. /g}

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