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

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8066HYDRODYNAMICÆ

§. 9. Fuerit igitur vas cylindricum verticaliter poſitum aqua ple-
num, ſitque altitudo aquæ ab initio fluxus = a, amplitudo cylindri = m, am-
plitudo foraminis = n, Sectio venæ ſolidæ = {n/α} effluxerit jam aqua per tempus
t; ſitque tunc altitudo aquæ reſidua ſupra foramen = x, eodemque temporis
puncto habeat ſuperficies aquæ internæ velocitatem, quæ reſpondeat altitudini
v: erit velocitas ipſa = √ v, eſt autem elementum temporis d t proportio-
nale elemento ſpatii - d x diviſo per velocitatem √v, unde dt = {- dx/√v}.

Determinatus @equidem fuit valor ipſius v in ſect. 3. ubi iisdem denomi-
nationibus uſi ſumus, quibus nunc utimur. At quoniam pro recta aquarum
erogatarum menſura requiritur, ut foramini n ſubſtituatur ſectio venæ con-
tractæ {n/α}, ſequitur, ut in valore ipſius v eadem fiat ſubſtitutio atque ſic ſta-
tuatur v = {nna/2nn - mmαα}(({a/x}){1 - mmαα/nn} - {x/a})

Hic vero valor ſi ſubſtituatur in æquatione
dt = {- dx/√v}, oritur
dt = - dx: √[{nna/2nn - mmαα} (({a/x}){1 - mmαα/nn} - {x/a})]
ope cujus æquationis omnia tempora deſiderata definiri poſſunt per approxi-
mationes, ſeu ſeries, ſi modo in ſingulis punctis valor ipſius α innoteſcat:
Aſſumemus autem eſſe illum conſtantis valoris, quandoquidem in præſenti
caſu nihil ſit, à quo mutari poſſit præter diverſas altitudines & velocitates
fluidi, quæ parum vel nihil quantum ſenſibus percipi poteſt ad id negotii
conferunt.

§. 10. Jam ut æquatio deſiderata per ſeries exhiberi poſſit, conſiderabi-
mus quantitatem.

1: √[{nna/2nn - mmαα} (({a/x}){1 - mmαα/nn} - {x/a})] fub hâc forma
({nnx/mmαα - 2nn})- {1/2} X (1 - ({x/a}){mmαα/nn} - 2) - {1/2}

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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
130	116

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
130	116