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

Table of Notes

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            <s xml:id="echoid-s6345" xml:space="preserve">
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            ſari: </s>
            <s xml:id="echoid-s6346" xml:space="preserve">quia tamen in aëre qui ſit naturali vel quadruplo denſior, hæc proprie-
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            tas, quod nempe elaſticitates ſint in ratione compoſita ex quadrato velocita-
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            tum particularum & </s>
            <s xml:id="echoid-s6347" xml:space="preserve">ſimplici denſitatum experimentis etiamnum ad ſenſus
              <lb/>
            omnino reſpondere viſa fuit, illa ſine ullo ſenſibili errore uti poterimus
              <lb/>
            pro aëre naturali atmoſphæræ mari incumbentis, ſiquidem eo accuratius
              <lb/>
            vera ſit quo rarior eſt aër.</s>
            <s xml:id="echoid-s6348" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6349" xml:space="preserve">His ad calculum præparatis ponemus B F = a, B M = b, B Q = c,
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            B C = x, C c = dx; </s>
            <s xml:id="echoid-s6350" xml:space="preserve">C G = v, C N = z, C R = y, & </s>
            <s xml:id="echoid-s6351" xml:space="preserve">erit y: </s>
            <s xml:id="echoid-s6352" xml:space="preserve">c = vvz:
              <lb/>
            </s>
            <s xml:id="echoid-s6353" xml:space="preserve">aab ſeu y = {cvvz/aab}. </s>
            <s xml:id="echoid-s6354" xml:space="preserve">Quia porro elaſticitatis menſura eſt pondus ſuperin-
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            cumbentis aëris, erit q R (- dy) = ponderi ſtrati aërei intercepti inter C & </s>
            <s xml:id="echoid-s6355" xml:space="preserve">
              <lb/>
            c, quod proportionale eſt aëris denſitati z & </s>
            <s xml:id="echoid-s6356" xml:space="preserve">altitudini ſtrati dx: </s>
            <s xml:id="echoid-s6357" xml:space="preserve">eſt igitur
              <lb/>
            - dy = {zdx/n} ſeu z = {- ndy/dx}, quo valore ſubſtituto in æquatione
              <lb/>
            (y = {cvvz/aab}) habetur y = {cvv/aab} X {- ndy/dx} vel
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            - {dy/y} = {aabdx/ncvv}.</s>
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            <s xml:id="echoid-s6359" xml:space="preserve">§. </s>
            <s xml:id="echoid-s6360" xml:space="preserve">27. </s>
            <s xml:id="echoid-s6361" xml:space="preserve">Si ponatur velocitas particularum aërearum in omni altitudine
              <lb/>
            eadem, nempe = a, fiet {- dy/y} = {bdx/nc}, vel, facta debita integratione,
              <lb/>
            log.</s>
            <s xml:id="echoid-s6362" xml:space="preserve">{c/y} = {bx/nc}; </s>
            <s xml:id="echoid-s6363" xml:space="preserve">Iſtam vero hypotheſin non ſatis experimentis confirmari vi-
              <lb/>
            dimus §. </s>
            <s xml:id="echoid-s6364" xml:space="preserve">24. </s>
            <s xml:id="echoid-s6365" xml:space="preserve">Igitur alia tentata, poſui v = √(aa + mx) vel vv = aa + mx,
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            quæ lex eſt in motibus corporum libere cadentium: </s>
            <s xml:id="echoid-s6366" xml:space="preserve">neque id ſine ſucceſſu;
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            </s>
            <s xml:id="echoid-s6367" xml:space="preserve">ita vero fit
              <lb/>
            {- dy/y} = {aabdx/naac + mncx}
              <lb/>
            vel log. </s>
            <s xml:id="echoid-s6368" xml:space="preserve">{c/y} = {aab/mnc} log. </s>
            <s xml:id="echoid-s6369" xml:space="preserve">{aa + mx/aa}.</s>
            <s xml:id="echoid-s6370" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s6371" xml:space="preserve">In hac æquatione paullo generaliori in qua m & </s>
            <s xml:id="echoid-s6372" xml:space="preserve">n etiamnum arbitra-
              <lb/>
            riæ ſunt, porro periculum feci, num non poſſet poni {aab/mnc} = 1, atque id
              <lb/>
            etiam apte fieri vidi: </s>
            <s xml:id="echoid-s6373" xml:space="preserve">ſic vero obtinui
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            log. </s>
            <s xml:id="echoid-s6374" xml:space="preserve">{c/y} = log. </s>
            <s xml:id="echoid-s6375" xml:space="preserve">{aa + mx/aa} vel {c/y} = {aa + mx/aa} aut {y/c} = {aa/aa + mx}.</s>
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