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

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          <p>
            <s xml:id="echoid-s3849" xml:space="preserve">III. </s>
            <s xml:id="echoid-s3850" xml:space="preserve">Cum deſcenſus incipere intelligatur ab altitudine X Y, ſubſe-
              <lb/>
            quenſque aſcenſus fieri uſque in CD, fore productum deſcenſus actualis maſſæ aquæ
              <lb/>
            X Y D C uſque ad T V in maſſam, menſuram rationis utriuſque combinatæ,
              <lb/>
            quæ, ut §. </s>
            <s xml:id="echoid-s3851" xml:space="preserve">2. </s>
            <s xml:id="echoid-s3852" xml:space="preserve">dictum, aſcenſum à præcedente deſcenſu differre faciunt, & </s>
            <s xml:id="echoid-s3853" xml:space="preserve">cum
              <lb/>
            ratio ſecundo loco recenſita evaneſcat, ſi omne auferatur fundum IM, fore
              <lb/>
            tunc iſtud productum æquale vi vivæ omnis aquæ, durante deſcenſu ejectæ, ita
              <lb/>
            ut ſine alio calculo, præter hactenus jam poſitos, aſcenſus aquarum in cylin-
              <lb/>
            dro toto aperto definiri poſſit.</s>
            <s xml:id="echoid-s3854" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3855" xml:space="preserve">IV. </s>
            <s xml:id="echoid-s3856" xml:space="preserve">Aſcenſum fore æqualem deſcenſui, cum cylindrus infinite ſub-
              <lb/>
            merſus intelligitur evaneſcentibus tunc præfatis diminutionis cauſis.</s>
            <s xml:id="echoid-s3857" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3858" xml:space="preserve">V. </s>
            <s xml:id="echoid-s3859" xml:space="preserve">Hinc igitur oſcillationes ſine fine fore, quia poſtremæ oſcillatio-
              <lb/>
            nes ſemper ſint veluti infinite parvæ ratione ſubmerſionis altitudinum: </s>
            <s xml:id="echoid-s3860" xml:space="preserve">faciunt
              <lb/>
            autem impedimenta aliena, quorum nullam hucuſque rationem habuimus, ut
              <lb/>
            omnis motus cito admodum ceſſet.</s>
            <s xml:id="echoid-s3861" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3862" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3863" xml:space="preserve">14. </s>
            <s xml:id="echoid-s3864" xml:space="preserve">His generatim præmonitis, problema accuratiori calculo ſub-
              <lb/>
            jiciemus: </s>
            <s xml:id="echoid-s3865" xml:space="preserve">duplicem autem dabo ſolutionem, alteram ad principia modo ex-
              <lb/>
            poſita accommodatam, alteram ſpecie quodammodo diverſam.</s>
            <s xml:id="echoid-s3866" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3867" xml:space="preserve">Igitur retentis tum figura, tum denominationibus §. </s>
            <s xml:id="echoid-s3868" xml:space="preserve">3. </s>
            <s xml:id="echoid-s3869" xml:space="preserve">conſiderabi-
              <lb/>
            mus aquam ex altitudine X Y deſcendiſſe uſque in x y, & </s>
            <s xml:id="echoid-s3870" xml:space="preserve">ab hoc termino aſ-
              <lb/>
            cenſum ſuum inchoare; </s>
            <s xml:id="echoid-s3871" xml:space="preserve">dicatur M y vel I x = α & </s>
            <s xml:id="echoid-s3872" xml:space="preserve">poſtquam jam aſcendit uſ-
              <lb/>
            que ad c d vel e f, ponatur M d = ξ, df = dξ: </s>
            <s xml:id="echoid-s3873" xml:space="preserve">His ita ad calculum præpa-
              <lb/>
            ratis, deſignataque rurſus per v altitudine debita velocitati aquæ in c d & </s>
            <s xml:id="echoid-s3874" xml:space="preserve">per
              <lb/>
            v + d v ſimili altitudine in ſitu proximo e f, inquiremus in incrementum aſcen.
              <lb/>
            </s>
            <s xml:id="echoid-s3875" xml:space="preserve">ſus potentialis aquæ accedens, dum cylindrum ſubit guttula L O N P, ſuperfi-
              <lb/>
            cieſque ex c d aſcendit in e f; </s>
            <s xml:id="echoid-s3876" xml:space="preserve">Perſpicuum autem eſt, cum ubique aſcenſus po-
              <lb/>
            tent. </s>
            <s xml:id="echoid-s3877" xml:space="preserve">aquæ internæ multiplicatus per ſuam maſſam exprimatur per n ξ v (nec
              <lb/>
            enim ulla attentio adhibenda eſt ad motum inteſtinum) fore ejusdem produ-
              <lb/>
            cti incrementum n ξ d v + n v d ξ: </s>
            <s xml:id="echoid-s3878" xml:space="preserve">Si vero præterea conſideretur aſcenſus po-
              <lb/>
            tent. </s>
            <s xml:id="echoid-s3879" xml:space="preserve">n n v - v, (vid. </s>
            <s xml:id="echoid-s3880" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3881" xml:space="preserve">2.) </s>
            <s xml:id="echoid-s3882" xml:space="preserve">quem guttula influens n d ξ perdit, quique pariter
              <lb/>
            debetur deſcenſui actuali particulæ aqueæ n d ξ per altitudinem b - x, patet eſſe
              <lb/>
            ponendum
              <lb/>
            nξdv + nvdξ + (nnv - v) ndξ = (b - ξ) ndξ, vel
              <lb/>
            ξdv + nnvdξ = (b - ξ) dξ.</s>
            <s xml:id="echoid-s3883" xml:space="preserve"/>
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