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

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144130HYDRODYNAMICÆ tudinem orificii pertinere ad motum aquæ internæ, cujus rei originem jam
ſupra (§. 7.) indicavi.

In ſequentibus autem demonſtrabimus, non differre hunc motum à
ſubſequente motu refluo, hincque oſcillationes fieri tautochronas. Prius-
quam vero ad alia pergam monendum duxi, in iſto calculo quantitates
{c/a} & {z/a} non ſolum præ unitate, ſed & præ {1/nn} ceu infinite parvas poſitas fuiſ-
ſe, ad quod animus probe eſt advertendus in inſtituendis experimentis;
licet utique theoriam infinite parvorum ad experimenta, ſine notabili erro-
re revocare diminuendo admodum quantitates, quæ in theoria ceu infinite
parvæ conſideratæ fuerunt, ſed faciendum eſt, ut in experimento omnia
huic legi ſint ſubjecta. Ita v. gr. ſi in cylindro omne fundum abſit, poſito
n = 1, idque ſubmerſum ponatur ad altitudinem triginta quinque pollicum,
ſatis accurate ſumetur experimentum, cum aqua ante oſcillationes elevata
tantum fuerit ad altitudinem unius pollicis ſupra ſuperficiem aquæ circum-
fluæ nec dum error notabilis erit, ſi vel orificiium inferius ad dimidium
obſtruatur exiſtente tunc {c/a} ad {1/nn} ut 1. 9, quæ ratio in noſtro experimento
tuto adhuc negligi poteſt: at ſi jam diametrum tubi duplam ponas diame-
tri orificii, occluſis tribus quartis aperturæ integræ partibus, jam fiet n = 4
& {c/a} ad {1/nn} ut 4 ad 9, quæ ratio non ſatis parva amplius erit, ut experimentum
conditionibus theoriæ cum ſufficienti præciſione ſatisfacere affirmari poſſit.

Hic itaque jam porro inquirere conveniet, quid de his caſibus ſtatuen-
dum ſit, quibus {c/a} & {1/nn} notabilem quidem inter ſe habent rationem, utra-
que vero quantitas fit admodum exigua, quod nimirum fit, cum cylindrus
profundiſſime ſubmergitur, ſimul autem fundum parvulo eſt pertuſum fo-
ramine.

§. 11. Sed iſte, quem modo finximus, caſus melius ex æquatione
differentiali paragraphi tertii, quam ex integrali, ut antea factum, deduci-
tur: poteſt autem pro his circumſtantiis rejici terminus - v d x præ n n v d x,
atque ſic aſſumi - x d v + n n v d x = (x - b) d x, in quâ ſi rurſus ponitur
a - b = c & a - x = z, prodit
adv + zdv + nnvdz = (c - z)

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