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

Table of Notes

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              <pb o="229" file="0243" n="243" rhead="SECTIO DECIMA."/>
            di viribus motricibus, quarum fortaſſe plures novæ non ſine inſigni mechani-
              <lb/>
            cæ practicæ incremento & </s>
            <s xml:id="echoid-s6721" xml:space="preserve">perfectione excogitari poterunt, movendis.</s>
            <s xml:id="echoid-s6722" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6723" xml:space="preserve">§. </s>
            <s xml:id="echoid-s6724" xml:space="preserve">40. </s>
            <s xml:id="echoid-s6725" xml:space="preserve">Ut incipiamus ab aëre in vacuo, conſiderabimus cylindrum ver-
              <lb/>
            ticaliter poſitum A B C D (Fig. </s>
            <s xml:id="echoid-s6726" xml:space="preserve">62.) </s>
            <s xml:id="echoid-s6727" xml:space="preserve">cum ſuſtentaculo E F, quod omni pon-
              <lb/>
              <note position="right" xlink:label="note-0243-01" xlink:href="note-0243-01a" xml:space="preserve">Fig. 62.</note>
            dere deſtitutum liberrime ſurſum dèorſumque moveri poſſit. </s>
            <s xml:id="echoid-s6728" xml:space="preserve">Sit ſpatio E B C F
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            aër incluſus, totus autem cylindrus in vacuo poſitus fingatur: </s>
            <s xml:id="echoid-s6729" xml:space="preserve">Sit preſſio aëris
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            E B C F tanta qua ſuſtinere poſſit pondus p, quod æquale erit preſſioni colum-
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            næ atmoſphæræ, ſi aër iſte ſit naturalis. </s>
            <s xml:id="echoid-s6730" xml:space="preserve">Superveniat jam aliud pondus P: </s>
            <s xml:id="echoid-s6731" xml:space="preserve">ita
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            fiet ut operculum deſcendat in G H motibuſque reciprocis ad puncta H & </s>
            <s xml:id="echoid-s6732" xml:space="preserve">F
              <lb/>
            agitetur. </s>
            <s xml:id="echoid-s6733" xml:space="preserve">Ut motum definiamus, utemur hypotheſi ordinaria, quod preſſio-
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            nes aëris cæteris paribus ſint denſitatibus proportionales.</s>
            <s xml:id="echoid-s6734" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s6735" xml:space="preserve">Fuerit itaque F C = a, F H = x; </s>
            <s xml:id="echoid-s6736" xml:space="preserve">velocitas ſuſtentaculi in ſitu G H = v,
              <lb/>
            erit preſſio, qua ſuſtentaculum G H ad ulteriorem deſcenſum urgetur = P + p
              <lb/>
            - {a/a - x} p, huicque preſſioni æqualis cenſenda eſt vis, quæ pondus ſuſtenta-
              <lb/>
            culo incumbens animat; </s>
            <s xml:id="echoid-s6737" xml:space="preserve">igitur ſi hanc vim dividas per maſſam habebis vim
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            accelerantem, quæ multiplicata per tempuſculum ſeu per {dx/v}, dabit incre-
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            mentum velocitatis dv, eſt itaque
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            dv = (P + p - {ap/a - x}) X {dx/v}: </s>
            <s xml:id="echoid-s6738" xml:space="preserve">(P + p), vel
              <lb/>
            {1/2} (P + p) vv = (P + p) x - ap log. </s>
            <s xml:id="echoid-s6739" xml:space="preserve">{a/a - x}.</s>
            <s xml:id="echoid-s6740" xml:space="preserve"/>
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            <s xml:id="echoid-s6741" xml:space="preserve">Sed ex deſcenſu ponderis (P + p) per altitudinem x generatur vis viva
              <lb/>
            potentialis (P + p) x, & </s>
            <s xml:id="echoid-s6742" xml:space="preserve">cum ſuſtentaculum eſt in ſitu G H, ineſt corpori (P + p)
              <lb/>
            vis viva actualis {1/2} (P + p) vv, id eſt, (P + p) x - ap log. </s>
            <s xml:id="echoid-s6743" xml:space="preserve">{a/a - x}, quæ à prio-
              <lb/>
            ri deficit quantitate ap log. </s>
            <s xml:id="echoid-s6744" xml:space="preserve">{a/a - x}, hæcque in compreſſionem aëris tranſiit.</s>
            <s xml:id="echoid-s6745" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s6746" xml:space="preserve">Dico itaque non poſſe aërem occupantem ſpatium a condenſari in ſpa-
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            tium a - x, quin vis viva impendatur, quæ generatur ex deſcenſu ponderis
              <lb/>
            p per altitudinem a log. </s>
            <s xml:id="echoid-s6747" xml:space="preserve">{a/a - x} quocunque modo illa compreſsio facta fuerit; </s>
            <s xml:id="echoid-s6748" xml:space="preserve">po-
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            t
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            eſt autem modis fieri infinitis. </s>
            <s xml:id="echoid-s6749" xml:space="preserve">Iſtam vero regulam uno nunc alterove exem-
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            plo illuſtrabo.</s>
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