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

Table of contents

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[91.] Exemplum 1.
Page: 126 (112)
[92.] Exemplum 2.
Page: 126 (112)
[93.] De oſcillationibus fluidorum in tubisrecurvis. Caſus II.
Page: 128 (114)
[94.] Lemma.
Page: 129 (115)
[95.] Solutio.
Page: 129 (115)
[96.] Problema.
Page: 130 (116)
[97.] Solutio.
Page: 130 (116)
[98.] Corollarium 1.
Page: 131 (117)
[99.] Corollarium 2.
Page: 131 (117)
[100.] Corollarium 3.
Page: 131 (117)
[101.] Corollarium 4.
Page: 131 (117)
[102.] Theorema.
Page: 132 (118)
[103.] Demonſtratio.
Page: 132 (118)
[104.] Problema.
Page: 132 (118)
[105.] Solutio.
Page: 132 (118)
[106.] Corollarium. 1.
Page: 133 (119)
[107.] Corollarium 2.
Page: 133 (119)
[108.] Scholion.
Page: 133 (119)
[109.] Theorema.
Page: 134 (120)
[110.] Demonſtratio.
Page: 134 (120)
[111.] Problema.
Page: 134 (120)
[112.] Solutio.
Page: 134 (120)
[113.] Scholium.
Page: 134 (120)
[114.] Corollarium 1.
Page: 135 (121)
[115.] Corollarium 2.
Page: 135 (121)
[116.] Scholion Generale.
Page: 136 (122)
[117.] HYDRODYNAMICÆ SECTIO SEPTIMA. De motu aquarum per vaſa ſubmerſa, ubi exem-plis oſtenditur, quam inſigniter utile ſit princi-pium conſervationis virium vivarum, veliis in caſibus, quibus continue aliquid de illis perdi cenſendum eſt. PARS PRIMA. De deſcenſu aquarum. §. 1.
Page: 138
[118.] PARS SECUNDA. De aſcenſu aquarum.
Page: 146 (132)
[119.] Corollarium.
Page: 152 (138)
[120.] Scholium Generale.
Page: 153 (139)
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              <pb o="101" file="0115" n="115" rhead="SECTIO QUINTA."/>
            communi menſura idemque ponemus = θ, & </s>
            <s xml:id="echoid-s2848" xml:space="preserve">mutabitur pro vaſis ſeu ca-
              <lb/>
            nalibus cylindricis æquatio (α) in hanc
              <lb/>
            t = {nθ/2√(mm - nn)} X log. </s>
            <s xml:id="echoid-s2849" xml:space="preserve">{m√a + √(mmv - nnv)/m√a - √(mmv - nnv)}
              <lb/>
            altera vera ſignata (β) talis fit
              <lb/>
            t = {nθ/2m} X log. </s>
            <s xml:id="echoid-s2850" xml:space="preserve">{√a + √v/√a - √v},
              <lb/>
            ex quarum utraque apparet, non poſſe non breviſſimo tempore aquas om-
              <lb/>
            nem fere velocitatem acquirere, idque eo citius quo amplior eſt tubus,
              <lb/>
            quo brevior, & </s>
            <s xml:id="echoid-s2851" xml:space="preserve">quo magis verticalis: </s>
            <s xml:id="echoid-s2852" xml:space="preserve">Neque accelerationes ullo modo eſſe
              <lb/>
            perceptibiles, niſi prælongi ſtatuantur aquæ ductus & </s>
            <s xml:id="echoid-s2853" xml:space="preserve">tunc quoque brevi
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            tempore omnes fere accelerationum gradus percurri, quod utrumque nunc
              <lb/>
            exemplo illuſtrabo.</s>
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            <s xml:id="echoid-s2855" xml:space="preserve">(I) Quæritur tempus quo fluidum ex cylindro conſtanter pleno verticali,
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            ſedecim pedes anglicos longo & </s>
            <s xml:id="echoid-s2856" xml:space="preserve">cujus diameter quintupla ſit diametri fo-
              <lb/>
            raminis, velocitatem acquirit quæ debeatur altitudini {99/100}a, idque in hypo-
              <lb/>
            theſi, ad quam æquatio ſecunda pertinet; </s>
            <s xml:id="echoid-s2857" xml:space="preserve">ſic eſt {n/m} = {1/25}, v = {99/100}a,
              <lb/>
            b = a, unde tempus quod corpus inſumit cadendo libere per ſpatium {bb/a},
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            ſeu θ = uni minuto ſecundo; </s>
            <s xml:id="echoid-s2858" xml:space="preserve">hinc fit t = {1/50} log. </s>
            <s xml:id="echoid-s2859" xml:space="preserve">399. </s>
            <s xml:id="echoid-s2860" xml:space="preserve">id eſt, proxime no-
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            næ parti unius minuti ſecundi, quod tempusculum utique imperceptibile
              <lb/>
            eſt; </s>
            <s xml:id="echoid-s2861" xml:space="preserve">Cum vero tempus notabile aſſumitur, fiunt mutationes altitudinum v,
              <lb/>
            inſenſibiles. </s>
            <s xml:id="echoid-s2862" xml:space="preserve">Si tempus ſimile (quo nempe velocitas pariter nonaginta no-
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            vem centeſimis partibus altitudinis, quanta poſt tempus infinitum fit, debi-
              <lb/>
            ta generetur) in prima hypotheſi quæratur, nempe tempus quo obtinetur
              <lb/>
            v = {99/100} X ({mma/mm - nn}) reperitur illud præcedente paullulum majus,
              <lb/>
            ſed exceſſu inſenſibili: </s>
            <s xml:id="echoid-s2863" xml:space="preserve">unde patet in hujusmodi vaſis non poſſe fere aquas
              <lb/>
            ſat celeriter affundi in vas ſuperius, ut hypotheſi ſatisfiat, nec adeoque ratio-
              <lb/>
            ne ejusdem hypotheſeos experimenta alia ſumi poſſe, quam ut exploretur,
              <lb/>
            num revera tanta ſit altitudo B P in figura trigeſima, quanta vi paragraphi
              <lb/>
            quinti eſſe debet, ut punctum e aut f, durante fluxu ſitum ſervet, quem
              <lb/>
            ante fluxum obturato orificio L M, nullaque exiſtente aqua in vaſe ſuperiore
              <lb/>
            habuit.</s>
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