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

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117103SECTIO QUINTA.

Solutio.

Adhibitis rurſus poſitionibus & denominationibus paragraphi tertii &
duodecimi, invenienda nunc erit æquatio inter x & t: quia vero, ut vidi-
mus §. 12. eſt d t = {γdx/√v}, erit √ v = {γdx/dt}, hicque valor ſubſtituendus
erit in æquationibus, quas dedimus §. 3. integratis; prior harum æquationum
hæc fuit: v = {mma/mm - nn} X (1 - c{n3 - nmm/mmN} x)
quæ pro præſecuti inſtituto mutatur in hanc
(I) {γγdx2/dt2} = {mma/mm - nn} X (1 - c{n3 - nmm/mmN} x)
altera ex §. 3. allegatarum æquationum talis fuit
v = a X (1 - c{- n/N} x)
quæ adeoque ſubminiſtrat in præſenti caſu ſequentem
(II) {γγdx2/dt2} = a X (1 - c{- n/N} x)

Erunt nunc æquationes (I) & (II) integrandæ, quod quidem facile
eſt & quia prior alteram continet (utraque enim eadem eſt ſi m = ∞)
hanc ſolam pertractabimus, eamque nunc ſub hâc forma conſiderabimus.
dt = {γ√(mm - nn)/m√a}dx: √(1 - c{n3 - nmm/mmN}x)

Ponatur autem ut integrationis modus eo magis pateſcat
c{n3 - nmm/mmN}x = z, atque proin dx = {mmNdz/(n3 - nmm)z},
dein brevitatis ergo indicetur quantitas conſtans
{γ√(mm - nn)/m√a} X {mmN/n3 - nmm}, ſeu {- γmN/n√(mm - nn) a} per α,
& habebitur dt = {αdz/z√(1 -

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103	89
104
105	91
106	92
107	93
108	94
109	95
110	96
111	97
112	98
113	99
114	100
115	101
116	102
117	103
118	104
119	105
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Scan	Original
101	87
102	88
103	89
104
105	91
106	92
107	93
108	94
109	95
110	96
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
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