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

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129115SECTIO SEXTA.

Si anguli A B C & H C B ſunt recti, qui unicus caſus est; à Newta-
no ſolutus, erit longitudo penduli ſimplicis, quod oſcillanti aquæ iſochronum eſt,
= {1/2} L, ut invenit Newtonus.

§. 5. Hæc ſunt quæ adhuc cum publico communicata fuerunt circa
oſcillationes fluidorum, & quidem primo à Newtono, ut undarum naturam,
à Patre meo, ut fertilitatem principii virium vivarum oſtenderet. Quia vero
noſtrum inſtitutum eſt pleniorem dare de motibus aquarum theoriam, è re
erit iſtud argumenti genus in tota ſua extenſione proſequi: Igitur diſquiram,
quibus modis oſcillationes fluidi inæquales fiant iſochronæ, & quibus non
item? Dein pro prioribus dabo longitudinem penduli ſimplicis tautochroni,
pro alteris tempus durationis indicabo: tubos autem utcunque inflexos & inæ
qualiter amplos conſiderabo.

Lemma.

§. 6. Sit c A d (Fig. 34.) uter ſeu canalis aqua plenus formæ cujuſcun-
11Fig. 34. que datæ deſinens utrobique in duos canales cylindricos a c & f d, utcunque ad
horizontem inclinatos & cujuſcunque amplitudinis, quorum alterum plenum
aqua ponam uſque in a, alterum uſque in f; oporteat determinare altitudinem
centri gravitatis omnis aquæ, ex data altitudine centri gravitatis aquæ in u-
tre c A d contentæ, cæteriſque quantum ſufficit præcognitis.

Solutio.

Fuerit centrum gravitatis aquæ in vaſe c A d contentæ in C, ductaque in-
telligatur per iſtud punctum C verticalis A B, deinde ducantur horizontales
a m, c g, f n, & d h una cum verticalibus c b & d e. Ponatur a c = a: f d = α:
b c = b; e d = β: amplitudo tubi a c = g; amplitudo tubi f d = γ: ſit porro
maſſa aquea ſeu capacitas canalis c A d = M, linea A g = f; A h = φ: A C =m:
Dividantur lineæ m g & n h bifariam punctis D & E & ſic erunt centra gravitatis
aquarum in tubis cylindricis contentarum in altitudinibus punctorum D & E.

His poſitis fit A D = f + {1/2}@b; A E = φ + {1/2}β: maſſa aquæ in a c =
g a: in f d = γ α: Igitur ſi centrum gravitatis quæſitum pro omni aqua a c A d f
intelligatur in altitudine F poſitum, habebitur, ut conſtat in mechanicis, A F
multiplicando maſſam aquæ in a c per D A, maſſam aquæ f d per E A &

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Scan	Original
111	97
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115	101
116	102
117	103
118	104
119	105
120	106
121	107
122	108
123	109
124	110
125	111
126	112
127	113
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135	121
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137	123
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140	126

Scan	Original
111	97
112	98
113	99
114	100
115	101
116	102
117	103
118	104
119	105
120	106
121	107
122	108
123	109
124	110
125	111
126	112
127	113
128	114
129	115
130	116
131	117
132	118
133	119
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135	121
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