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

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            eſſe = a, cujus ſola gravitas nunc aquam expellat; </s>
            <s xml:id="echoid-s1098" xml:space="preserve">deinde deſcendere ſuper-
              <lb/>
            ficiem aquæ in Cylindro per altitudinem verticalem a - x, ita ut altitudo
              <lb/>
            reſidua ſit = x & </s>
            <s xml:id="echoid-s1099" xml:space="preserve">tunc velocitatem aquæ ejectæ talem eſſe, quæ debeatur al-
              <lb/>
            titudini z. </s>
            <s xml:id="echoid-s1100" xml:space="preserve">His ita poſitis utemur æquatione generali differentiali §. </s>
            <s xml:id="echoid-s1101" xml:space="preserve">9. </s>
            <s xml:id="echoid-s1102" xml:space="preserve">quæ
              <lb/>
            hæc eſt nn N dz - mmzydx + {mmnnzdx/y} = -mmyxdx (ubi rurſus, ut
              <lb/>
            §. </s>
            <s xml:id="echoid-s1103" xml:space="preserve">13. </s>
            <s xml:id="echoid-s1104" xml:space="preserve">indicatum fuit, eſt y = m & </s>
            <s xml:id="echoid-s1105" xml:space="preserve">N = mx) quæque in caſu noſtro particu-
              <lb/>
            lari talis fit
              <lb/>
            (1 - {mm/nn}) zdx + xdz = - {mm/nn}xdx,
              <lb/>
            quæ multiplicata x - {mm/nn} poſteaque ſic integrata, ut poſita x = a, fiat z = α
              <lb/>
            dabit æquationem deſideratam finalem
              <lb/>
            z = ({mm/2nn - mm + {α/a}) a
              <emph style="super">{2nn - mm/nn}</emph>
            X x
              <emph style="super">{mm - nn/nn}</emph>
            - {mm/2nn - mm}x
              <lb/>
            vel z = {mma/2nn - mm}(({a/x})
              <emph style="super">1 - {mm/nn}</emph>
            - {x/a}) + ({x/a})
              <emph style="super">{mm - nn/nn}</emph>
            α
              <lb/>
            quæ altitudo ſi comparetur cum illa, quæ paragrapho 14. </s>
            <s xml:id="echoid-s1106" xml:space="preserve">indicata fuit, in-
              <lb/>
            venitur exceſſus unius ſuper alteram = ({x/a})
              <emph style="super">{mm - nn/nn}</emph>
            α unde jam omnia ea
              <lb/>
            confirmantur Phænomena, quæ modo indicata fuerunt; </s>
            <s xml:id="echoid-s1107" xml:space="preserve">exceſſus enim iſte,
              <lb/>
            cum m numerus eſt multo major quam n, inſenſibilis ſtatim fit, poſtquam
              <lb/>
            aqua vel tantillum deſcendit, id eſt, poſt breviſſimum temporis ſpatium,
              <lb/>
            nunquam tamen omnis evaneſcit, quam diu durat fluxus, & </s>
            <s xml:id="echoid-s1108" xml:space="preserve">denique eo
              <lb/>
            notabilior continue eſt, quo magis ratio numeri m ad n ad æqualitatem ac-
              <lb/>
            cedit. </s>
            <s xml:id="echoid-s1109" xml:space="preserve">Fuerit v. </s>
            <s xml:id="echoid-s1110" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s1111" xml:space="preserve">diameter tubi decies major diametro foraminis, expel-
              <lb/>
            laturque aqua vi tali, ut velocitate ſua aſſilire poſſit ad altitudinem quæ ſit
              <lb/>
            quadrupla altitudinis a ſeu aquæ ſupra foramen, quæritur ad quam altitudinem
              <lb/>
            ſua velocitate aqua effluens aſcendere poterit, poſtquam per milleſimam
              <lb/>
            partem ipſius a ſuperficies aquea deſcenderit in tubo, ſi interea aqua ſola
              <lb/>
            propria gravitate ad effluxum ſolicitetur, dein quænam ſimilis altitudo futu-
              <lb/>
            ra fuiſſet, ſi aqua nullum motum ab initio habuiſſet: </s>
            <s xml:id="echoid-s1112" xml:space="preserve">eſt autem m = 100n,
              <lb/>
            mm = 10000nn, x = {999/1000}a, α = 4a, unde in priori caſu </s>
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