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

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143129SECTIO SEPTIMA. expertus ſum cylindrum eodem tempore evacuari, ſive aquæ in aërem eji-
ciantur, ſive fundum aquæ ſtagnanti tantillum ſubmergatur. Docet hæc ex-
perientia parum aut nihil obſtare aërem externum effluxui, cum reſiſtentia
plus quam octingenties major notabiliorem effectum non exerat. Quia adeo-
que iſte caſus nihil particulare habet, quod non loco citato monitum fuerit,
huic non ulterius immorabimur: Inquiremus potius, quid fieri debeat, cum
elevatio aquæ internæ ſuper externam, quanta ab initio deſcenſus eſt, ſumi-
tur valde parva & negligenda præ immerſione cylindri; cui hypotheſi ſatisfit,
cum exceſſus altitudinis a ſuper altitudinem b (quem exceſſum rurſus vocabi-
mus (ut §. 7.) c) eſt admodum parvus.

§. 10. Cum itaque ponitur a - b = c, ponendum etiam erit a - x = z,
tumque utraque quantitas, nempe c & z, erunt negligendæ præ quantitatibus
a & b, ſed ſi a - x = z, erit x = a - z & xnn - 1 = (a - z)nn - 1 =
ann - 1 - (nn - 1)ann - 2z + ({nn - 1. nn -2/2})ann - 3zz
- ({nn - 1. nn - 2. nn - 3/2. 3. })ann - 4 z3 + & c.

Hæc ſeries quantum ad inſtitutum noſtrum ſufficit eſt continuanda;
ſufficiet autem ad tres usque terminos. Igitur in æquatione integrata quam
dedimus §. 3. ponemus, x = a - z &
xnn - 1 = ann - 1 - (nn - 1)ann - 2 z + ({nn - 1. nn - 2/2})ann - 3zz &
ſic erit
v = {1/nn -2} [a - z - a + (nn - 1) z - ({nn - 1. nn -2/2}){zz/a}]
- {b/nn - 1}[1 - 1 + (nn - 1){z/a} - ({nn - 1. nn - 2/2}){zz/aa}]

In qua æquatione ſi termini ſe deſtruentes deleantur, atque ponatur a - c
pro b, rejiciaturque terminus qui affectatur quantitate {czz/aa}, prodit ſimpliciter
v = {2cz - zz/2a}.
ex quâ formula, cum littera n evanuerit, indicium habemus, nihil

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