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

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            <s xml:id="echoid-s1020" xml:space="preserve">
              <pb o="38" file="0052" n="52" rhead="HYDRODYNAMICÆ."/>
            ſtatim ac x paulo minor eſt quam a. </s>
            <s xml:id="echoid-s1021" xml:space="preserve">Regula hæc fallit notabiliter tantum cir-
              <lb/>
            ca primum motus initium & </s>
            <s xml:id="echoid-s1022" xml:space="preserve">ſi primum iſtud motus elementum conſidera-
              <lb/>
            tur (quo nempe altitudo a - x ut infinite parva cenſeri poteſt) indicat æ-
              <lb/>
            quatio, eſſe tunc v = a - x. </s>
            <s xml:id="echoid-s1023" xml:space="preserve">Unde ſequitur, in omni cylindro, quodcun-
              <lb/>
            que fuerit foramen, aquam internam inſtar corporum libere cadentium ac-
              <lb/>
            celerari ab initio motus. </s>
            <s xml:id="echoid-s1024" xml:space="preserve">Si vero motus aliquantulum continuet, eo minus
              <lb/>
            fallet hæc Regula, quo majus fuerit foramen, & </s>
            <s xml:id="echoid-s1025" xml:space="preserve">quo altior eſt aqua in tubo; </s>
            <s xml:id="echoid-s1026" xml:space="preserve">ſi
              <lb/>
            porro deſideretur altitudo ea, quæ velocitati aquæ effluentis reſpondeat,
              <lb/>
            quam §. </s>
            <s xml:id="echoid-s1027" xml:space="preserve">9. </s>
            <s xml:id="echoid-s1028" xml:space="preserve">poſuimus = z, erit z = {mm/nn}v, ſeu
              <lb/>
            z = {mma/2nn - mm} (({a/x})
              <emph style="super">1 - {mm/nn}</emph>
            - {x/a})</s>
          </p>
          <p>
            <s xml:id="echoid-s1029" xml:space="preserve">§. </s>
            <s xml:id="echoid-s1030" xml:space="preserve">15. </s>
            <s xml:id="echoid-s1031" xml:space="preserve">Cum n eſt = m, id eſt, cum nullum eſt fundum, apparet
              <lb/>
            ex ipſa rei natura, aquam inſtar corporum gravium libere cadere atque ac-
              <lb/>
            celerari, id ipſum autem indicat etiam æquatio; </s>
            <s xml:id="echoid-s1032" xml:space="preserve">fit enim in hâc poſitione
              <lb/>
            z = a - x. </s>
            <s xml:id="echoid-s1033" xml:space="preserve">Si vero foramen eſt veluti infinite parvum ratione amplitudinis
              <lb/>
            vaſis, quem caſum jam ſupra conſideravimus, ponendum eſt n = o, & </s>
            <s xml:id="echoid-s1034" xml:space="preserve">tunc
              <lb/>
            fit z = x, quod indicat, aquam ea conſtantur effluere velocitate, qua ad
              <lb/>
            totam aquæ altitudinem aſcendere poſſit. </s>
            <s xml:id="echoid-s1035" xml:space="preserve">Denique cum mm = 2nn, pro-
              <lb/>
            dit z = {mm/o} (x - x), ex quo valore cum nihil cognoſci poſſit, deſcenden-
              <lb/>
            dum eſt ad æquationem differentialem §. </s>
            <s xml:id="echoid-s1036" xml:space="preserve">13. </s>
            <s xml:id="echoid-s1037" xml:space="preserve">quæ nunc hæc eſt:
              <lb/>
            </s>
            <s xml:id="echoid-s1038" xml:space="preserve">- vdx + xdv = - xdx, vel {xdv - vdx/xx} = {- dx/x},
              <lb/>
            quæ integrata cum debitæ conſtantis additione dat {v/x} = log. </s>
            <s xml:id="echoid-s1039" xml:space="preserve">{a/x}, vel v =
              <lb/>
            xlog.</s>
            <s xml:id="echoid-s1040" xml:space="preserve">{a/x}, aut z = 2v = 2xlog.</s>
            <s xml:id="echoid-s1041" xml:space="preserve">{a/x}.</s>
            <s xml:id="echoid-s1042" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1043" xml:space="preserve">§. </s>
            <s xml:id="echoid-s1044" xml:space="preserve">16. </s>
            <s xml:id="echoid-s1045" xml:space="preserve">Velocitas aquæ effluentis ab initio creſcit poſteaque decreſcit,
              <lb/>
            eſtque alicubi maxima, nempe eo in loco, quo aqua deſcendit ad altitudinem
              <lb/>
            a:</s>
            <s xml:id="echoid-s1046" xml:space="preserve">({mm - nn/nn})
              <emph style="super">nn: (mm - 2nn)</emph>
            ; </s>
            <s xml:id="echoid-s1047" xml:space="preserve">id quoque experientia edoctus indicavit Ma-
              <lb/>
            riottus in tract. </s>
            <s xml:id="echoid-s1048" xml:space="preserve">de motu aquarum part. </s>
            <s xml:id="echoid-s1049" xml:space="preserve">3. </s>
            <s xml:id="echoid-s1050" xml:space="preserve">diſc. </s>
            <s xml:id="echoid-s1051" xml:space="preserve">3. </s>
            <s xml:id="echoid-s1052" xml:space="preserve">exp. </s>
            <s xml:id="echoid-s1053" xml:space="preserve">5, ipſaque velocitas ma-
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            xima talis eſt, quæ debetur </s>
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