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

Table of Notes

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            (B) ααy + (β√x + γ√y)
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            = αα X (a + b).</s>
            <s xml:id="echoid-s4596" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4597" xml:space="preserve">Subtractâ æquatione (B) ab æquatione (A) prodity = x + b, ex quo
              <lb/>
            ſequitur, ſi venæ ambæ verticaliter ſurſum dirigantur, utramque ad eundem lo-
              <lb/>
            cum aſſilire. </s>
            <s xml:id="echoid-s4598" xml:space="preserve">Deinde ſi in æquatione (A) ſubſtituatur pro y valor ejus x + b,
              <lb/>
            erit
              <lb/>
            (C) ααx + (β√x + γ√x + b)
              <emph style="super">2</emph>
            = ααa,
              <lb/>
            unde deducitur valor ipſius x æquatione quadrata.</s>
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            <s xml:id="echoid-s4600" xml:space="preserve">§. </s>
            <s xml:id="echoid-s4601" xml:space="preserve">27. </s>
            <s xml:id="echoid-s4602" xml:space="preserve">Ex præcedentis paragraphi æquationibus ſequentes fluunt affe-
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            ctiones.</s>
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            <s xml:id="echoid-s4604" xml:space="preserve">I. </s>
            <s xml:id="echoid-s4605" xml:space="preserve">Quia velocitas aquæ per M transfluentis eſt = {β√x + γ√y/α}, eritalti-
              <lb/>
            tudo generans hanc velocitatem = ({β√x + γ√y/α})
              <emph style="super">2</emph>
            ; </s>
            <s xml:id="echoid-s4606" xml:space="preserve">ſed ſi addantur æqua-
              <lb/>
            tiones (A) & </s>
            <s xml:id="echoid-s4607" xml:space="preserve">(B) fit:
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            </s>
            <s xml:id="echoid-s4608" xml:space="preserve">({β√x + γ√y/α})
              <emph style="super">2</emph>
            = {2a + b - x - y/2} = ob(y = x + b)a - x.</s>
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            <s xml:id="echoid-s4610" xml:space="preserve">II. </s>
            <s xml:id="echoid-s4611" xml:space="preserve">Si foramen H ſit valde exiguum ratione foraminum M & </s>
            <s xml:id="echoid-s4612" xml:space="preserve">N, id eſt, ſi
              <lb/>
            β poſſit cenſeri nulla ratione α & </s>
            <s xml:id="echoid-s4613" xml:space="preserve">γ, abit æquatio (C) in hanc
              <lb/>
            ααx + γγx + γγb = ααa, ſeu
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            x = {ααa - γγb/αα + γγ};</s>
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            <s xml:id="echoid-s4615" xml:space="preserve">Id vero egregie convenit cum paragrapho decimo nono, cum manife-
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            ſtum ſit aquam per foramen valde exiguum ad eandem altitudinem aſſilire,
              <lb/>
            quam haberet aqua, ſi hæc laminam L Q tantum deorſum premat, quantum
              <lb/>
            ab aqua interna ſurſum premitur; </s>
            <s xml:id="echoid-s4616" xml:space="preserve">Iſta vero præfata altitudo vi paragraphi 19.
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            </s>
            <s xml:id="echoid-s4617" xml:space="preserve">eſt {ααa - γγb/αα + γγ}; </s>
            <s xml:id="echoid-s4618" xml:space="preserve">Eſt porro in iſta hypotheſi altitudo velocitatis aquarum in N
              <lb/>
            ſeu x + b = {ααa + ααb/αα + γγ}
              <lb/>
            & </s>
            <s xml:id="echoid-s4619" xml:space="preserve">denique altitudo velocitatis aquarum in M, ſeu
              <lb/>
            a - x = {γγa + γγb/αα + γγ}; </s>
            <s xml:id="echoid-s4620" xml:space="preserve">
              <lb/>
            quæ poſteriores æquationes in iſto caſu particulari pariter ex §. </s>
            <s xml:id="echoid-s4621" xml:space="preserve">19. </s>
            <s xml:id="echoid-s4622" xml:space="preserve">immediate
              <lb/>
            colligi aut prævideri potuiſſent.</s>
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