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

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148134HYDRODYNAMICÆ

III. Cum deſcenſus incipere intelligatur ab altitudine X Y, ſubſe-
quenſque aſcenſus fieri uſque in CD, fore productum deſcenſus actualis maſſæ aquæ
X Y D C uſque ad T V in maſſam, menſuram rationis utriuſque combinatæ,
quæ, ut §. 2. dictum, aſcenſum à præcedente deſcenſu differre faciunt, & cum
ratio ſecundo loco recenſita evaneſcat, ſi omne auferatur fundum IM, fore
tunc iſtud productum æquale vi vivæ omnis aquæ, durante deſcenſu ejectæ, ita
ut ſine alio calculo, præter hactenus jam poſitos, aſcenſus aquarum in cylin-
dro toto aperto definiri poſſit.

IV. Aſcenſum fore æqualem deſcenſui, cum cylindrus infinite ſub-
merſus intelligitur evaneſcentibus tunc præfatis diminutionis cauſis.

V. Hinc igitur oſcillationes ſine fine fore, quia poſtremæ oſcillatio-
nes ſemper ſint veluti infinite parvæ ratione ſubmerſionis altitudinum: faciunt
autem impedimenta aliena, quorum nullam hucuſque rationem habuimus, ut
omnis motus cito admodum ceſſet.

§. 14. His generatim præmonitis, problema accuratiori calculo ſub-
jiciemus: duplicem autem dabo ſolutionem, alteram ad principia modo ex-
poſita accommodatam, alteram ſpecie quodammodo diverſam.

Igitur retentis tum figura, tum denominationibus §. 3. conſiderabi-
mus aquam ex altitudine X Y deſcendiſſe uſque in x y, & ab hoc termino aſ-
cenſum ſuum inchoare; dicatur M y vel I x = α & poſtquam jam aſcendit uſ-
que ad c d vel e f, ponatur M d = ξ, df = dξ: His ita ad calculum præpa-
ratis, deſignataque rurſus per v altitudine debita velocitati aquæ in c d & per
v + d v ſimili altitudine in ſitu proximo e f, inquiremus in incrementum aſcen.
ſus potentialis aquæ accedens, dum cylindrum ſubit guttula L O N P, ſuperfi-
cieſque ex c d aſcendit in e f; Perſpicuum autem eſt, cum ubique aſcenſus po-
tent. aquæ internæ multiplicatus per ſuam maſſam exprimatur per n ξ v (nec
enim ulla attentio adhibenda eſt ad motum inteſtinum) fore ejusdem produ-
cti incrementum n ξ d v + n v d ξ: Si vero præterea conſideretur aſcenſus po-
tent. n n v - v, (vid. §. 2.) quem guttula influens n d ξ perdit, quique pariter
debetur deſcenſui actuali particulæ aqueæ n d ξ per altitudinem b - x, patet eſſe
ponendum
nξdv + nvdξ + (nnv - v) ndξ = (b - ξ) ndξ, vel
ξdv + nnvdξ = (b - ξ) dξ.

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