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

Table of figures

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        <div xml:id="echoid-div212" type="section" level="1" n="166">
          <pb o="179" file="0193" n="193" rhead="SECTIO NONA."/>
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        <div xml:id="echoid-div213" type="section" level="1" n="167">
          <head xml:id="echoid-head215" xml:space="preserve">Demonſtratio.</head>
          <p>
            <s xml:id="echoid-s5163" xml:space="preserve">Nam ſi pondus, quod vocabo A, aſcenderit per altitudinem y, eoque
              <lb/>
            in loco animari ponatur potentia movente variabili P directe applicata, move-
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            rique velocitate v, erit tempuſculum, quo pondus per elementum d y eleva-
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            tur = {dy/v}, quod ductum in potentiam moventem P, ejuſdemque velocitatem
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            v, dat elementum potentiæ abſolutæ (per defin. </s>
            <s xml:id="echoid-s5164" xml:space="preserve">§. </s>
            <s xml:id="echoid-s5165" xml:space="preserve">2.) </s>
            <s xml:id="echoid-s5166" xml:space="preserve">= P d y, ergo ſ P dy dabit
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            totam potentiam abſolutam, ſi poſt integrationem fiat y = a; </s>
            <s xml:id="echoid-s5167" xml:space="preserve">in omni vero motu
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            incrementum velocitatis d v eſt æquale potentiæ animanti ſeu moventi, quæ
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            hîc eſt {P - A/A} ductæ in tempuſculum quod nunc eſt {dy/v}; </s>
            <s xml:id="echoid-s5168" xml:space="preserve">habemus igitur d v =
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            ({P - A/A}) X {dy/v} vel A v d v = P d y - A dy, id eſt, {1/2} A v v = ſ P d y - A y, ſive
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            ſ P d y = {1/2} A v v + Ay, ubi faciendum eſt y = a & </s>
            <s xml:id="echoid-s5169" xml:space="preserve">v = o (per hypoth.) </s>
            <s xml:id="echoid-s5170" xml:space="preserve">ita ut
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            ſit ſ P d y = A a.</s>
            <s xml:id="echoid-s5171" xml:space="preserve"/>
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            <s xml:id="echoid-s5172" xml:space="preserve">Quia autem, ut vidimus, ſ P d y exprimit integram potentiam abſolu-
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            tam in elevandum pondus impenſam @ erit eadem hæc potentia conſtanter
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            eadem, nominatimque æqualis producto ex pondere A & </s>
            <s xml:id="echoid-s5173" xml:space="preserve">altitudine a, ut
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            habet propoſito. </s>
            <s xml:id="echoid-s5174" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s5175" xml:space="preserve">E. </s>
            <s xml:id="echoid-s5176" xml:space="preserve">D.</s>
            <s xml:id="echoid-s5177" xml:space="preserve"/>
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        <div xml:id="echoid-div214" type="section" level="1" n="168">
          <head xml:id="echoid-head216" xml:space="preserve">Corollarium.</head>
          <p>
            <s xml:id="echoid-s5178" xml:space="preserve">§. </s>
            <s xml:id="echoid-s5179" xml:space="preserve">23. </s>
            <s xml:id="echoid-s5180" xml:space="preserve">Ex demonſtratione noſtra apparet, eſſe quoque potentiam abſo-
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            lutam eandem, quoties velocitas in ſummitate eſt eadem, id eſt, quoties
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            altitudo ad quam corpus velocitate ſua reſidua aſcendere poteſt, nempe {1/2} vv
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            eſt conſtans: </s>
            <s xml:id="echoid-s5181" xml:space="preserve">atque ſi altitudo iſta dicatur b, erit potentia abſoluta = A (a + b).
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            </s>
            <s xml:id="echoid-s5182" xml:space="preserve">Igitur patet nunc, quanta pars potentiæ abſolutæ perdatur, cum animus ſit
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            pondus A ad altitudinem a elevare, idemque in ſummitate velocitatem reſi-
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            duam habeat debitam altitudini b; </s>
            <s xml:id="echoid-s5183" xml:space="preserve">erit nempe diſpendium potentiæ ad in-
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            tegram potentiam ut b ad b + a.</s>
            <s xml:id="echoid-s5184" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div215" type="section" level="1" n="169">
          <head xml:id="echoid-head217" xml:space="preserve">Scholium 1.</head>
          <p>
            <s xml:id="echoid-s5185" xml:space="preserve">§. </s>
            <s xml:id="echoid-s5186" xml:space="preserve">24. </s>
            <s xml:id="echoid-s5187" xml:space="preserve">Cavendum itaque eſt, ne machinæ ita ſint conſtructæ, ut ve-
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            hementi motu aquæ ad locum deſtinatum transportentur. </s>
            <s xml:id="echoid-s5188" xml:space="preserve">Parvum autem
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            eſſe ſolet iſtud diſpendii genus in plerisque machinis.</s>
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