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

Table of handwritten notes

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            cum ſuperficies aquæ variabilis eſt in h l, fore altitudinem debitam velocitati
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            aquæ per M transfluentis = B b = x, velocitatemque ipſam = √x, ſi-
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            milemque altitudinem ratione orificii N = h M = a - x, atque velocita-
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            tem aquæ per N transfluentis = √a - x; </s>
            <s xml:id="echoid-s4469" xml:space="preserve">eſt igitur quantitas dato tempu-
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            ſculo per M in vas B N influentis ad quantitatem eodem tempuſculo ex vaſe
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            effluentis ut m√x ad n√a - x, harumque quantitatum differentia diviſa
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            per amplitudinem g dat velocitatem ſuperficiei h l, quæ proinde velocitas,
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            quam vocabimus v, exprimetur hâc æquatione,
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            v = {m√x - n√a - x/g}</s>
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            <s xml:id="echoid-s4470" xml:space="preserve">§. </s>
            <s xml:id="echoid-s4471" xml:space="preserve">22. </s>
            <s xml:id="echoid-s4472" xml:space="preserve">Ut jam innoteſcat tempus, quo ſuperficies fluidi ex H L venit in
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            h l, vocabimus illud tempus t: </s>
            <s xml:id="echoid-s4473" xml:space="preserve">quia autem eſt dt = {-dx/v}, erit, poſito
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            pro v valore modo invento,
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            dt = {-gdx/m√x - n√a - x}
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            Poteſt quidem hæc formula immediate rationalis fieri ponendo x = {4aqq/(1 + qq)
              <emph style="super">2</emph>
            },
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            atque deinde debito modo conſtrui: </s>
            <s xml:id="echoid-s4474" xml:space="preserve">Iſta vero methodus paullo prolixior eſt
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            hâc altera, qua quantitas reducenda dividitur in duo membra ſeorſim inte-
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            granda, nempe præmiſſa æquatio non differt ab hâc:
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            </s>
            <s xml:id="echoid-s4475" xml:space="preserve">dt = {mgdx√x/nna - (mm + nn) x} + {ngdx√a - x/nna - (mm + nn) x}: </s>
            <s xml:id="echoid-s4476" xml:space="preserve">
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            Et autem ſ{mgdx√x/nna - (mm + nn) x} = - {2mg/mm + nn}√x + {mng√a/(mm + nn)√(mm + nn)} X
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            log.</s>
            <s xml:id="echoid-s4477" xml:space="preserve">{n√a + √mm + nn√x/n√a - √mm + nn√x}; </s>
            <s xml:id="echoid-s4478" xml:space="preserve">alteriusque membri integrale
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            nempe ſ{ngdx√a - x/nna - (mm + nn) x} fit = {-2ng/mm + nn}√(a - x) +
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            {mng√a/(mm + nn) X √(mm + nn)} log. </s>
            <s xml:id="echoid-s4479" xml:space="preserve">{m√a + √mm + nn X √a - x/m√a - √mm + nn X √a - x}; </s>
            <s xml:id="echoid-s4480" xml:space="preserve">
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            Patet exinde addita debita conſtante fore
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            t = {2mg√a - b - 2mg√x + 2ng√b - 2ng√a - x/mm + nn} +
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            {mng√a/(mm + nn) X √(mm + nn)} </s>
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