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

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          <p>
            <s xml:id="echoid-s5472" xml:space="preserve">
              <pb o="190" file="0204" n="204" rhead="HYDRODYNAMICÆ"/>
            cæ indirectis; </s>
            <s xml:id="echoid-s5473" xml:space="preserve">placet tamen hujus rei demonſtrationem dare ex natura vectis
              <lb/>
            petitam, quia mechanici eo omnia reducere amant.</s>
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          </p>
          <p>
            <s xml:id="echoid-s5475" xml:space="preserve">Helicem conſiderabimus a 1 b ex figura quinquageſima ſecunda ſeor-
              <lb/>
            ſim deſumtam, ad evitandam linearum confuſionem, conſervatis denomi-
              <lb/>
            nationibus art. </s>
            <s xml:id="echoid-s5476" xml:space="preserve">IV. </s>
            <s xml:id="echoid-s5477" xml:space="preserve">adhibitis. </s>
            <s xml:id="echoid-s5478" xml:space="preserve">Sic igitur in Figura 53. </s>
            <s xml:id="echoid-s5479" xml:space="preserve">erit rurſus angulus
              <lb/>
              <note position="left" xlink:label="note-0204-01" xlink:href="note-0204-01a" xml:space="preserve">Fig. 53.</note>
            N M G angulus quem facit nucleus cum horizonte, cujus ſinus = N, ſi-
              <lb/>
            nusque anguli a M H = n; </s>
            <s xml:id="echoid-s5480" xml:space="preserve">a 1 b eſt una ſpiralis circumvolutio: </s>
            <s xml:id="echoid-s5481" xml:space="preserve">baſis nuclei
              <lb/>
            eſt circulus a c M p a; </s>
            <s xml:id="echoid-s5482" xml:space="preserve">ſinus anguli p a l eſt ut ante = m, ejusque coſinus M;
              <lb/>
            </s>
            <s xml:id="echoid-s5483" xml:space="preserve">puncta vero l & </s>
            <s xml:id="echoid-s5484" xml:space="preserve">o ſunt extremitates aquæ in ſpirali quieſcentis & </s>
            <s xml:id="echoid-s5485" xml:space="preserve">in ea-
              <lb/>
            dem altitudine ab horizonte poſita, ex iſtis punctis ductæ ſunt ad periphe-
              <lb/>
            riam baſis rectæ l c & </s>
            <s xml:id="echoid-s5486" xml:space="preserve">o p ad baſin perpendiculares. </s>
            <s xml:id="echoid-s5487" xml:space="preserve">In parte helicis quam
              <lb/>
            aqua occupat ſumta ſunt duo puncta infinite propinqua m & </s>
            <s xml:id="echoid-s5488" xml:space="preserve">n & </s>
            <s xml:id="echoid-s5489" xml:space="preserve">per hæc du-
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            ctæ ſunt rectæ n f & </s>
            <s xml:id="echoid-s5490" xml:space="preserve">m g rurſus ad baſin perpendiculares. </s>
            <s xml:id="echoid-s5491" xml:space="preserve">Denique ex pun-
              <lb/>
            ctis c, f, g, p ductæ ſunt ad diametrum a M perpendiculares c d, f h, g i & </s>
            <s xml:id="echoid-s5492" xml:space="preserve">
              <lb/>
            p q; </s>
            <s xml:id="echoid-s5493" xml:space="preserve">atque centrum baſis ponitur in e, radiusque e a = 1. </s>
            <s xml:id="echoid-s5494" xml:space="preserve">Sit jam arcus
              <lb/>
            ſpiralis l 1 o aqua plenus = c & </s>
            <s xml:id="echoid-s5495" xml:space="preserve">conſequenter arcus circularis eidem reſpon-
              <lb/>
            dens c M p = M c; </s>
            <s xml:id="echoid-s5496" xml:space="preserve">a l = e; </s>
            <s xml:id="echoid-s5497" xml:space="preserve">a c = M e; </s>
            <s xml:id="echoid-s5498" xml:space="preserve">a d (ſeu ſinus verſus arcus ac) = f; </s>
            <s xml:id="echoid-s5499" xml:space="preserve">
              <lb/>
            a q = g; </s>
            <s xml:id="echoid-s5500" xml:space="preserve">pondus aquæ in l s o = p: </s>
            <s xml:id="echoid-s5501" xml:space="preserve">arcus a l n = x; </s>
            <s xml:id="echoid-s5502" xml:space="preserve">n m = d x; </s>
            <s xml:id="echoid-s5503" xml:space="preserve">a c f = M x; </s>
            <s xml:id="echoid-s5504" xml:space="preserve">
              <lb/>
            f g = M d x; </s>
            <s xml:id="echoid-s5505" xml:space="preserve">a b = y; </s>
            <s xml:id="echoid-s5506" xml:space="preserve">h i = d y; </s>
            <s xml:id="echoid-s5507" xml:space="preserve">h f = √2y - yy, erit pondus guttulæ in
              <lb/>
            nm = {p d x/c}; </s>
            <s xml:id="echoid-s5508" xml:space="preserve">ſi vero linea h f multiplicetur per ſinum anguli a M H, divida-
              <lb/>
            turque per ſinum totum, habetur vectis quo particula n m cochleam circum-
              <lb/>
            agere tentat: </s>
            <s xml:id="echoid-s5509" xml:space="preserve">eſtigitur vectis iſte = n √ (2y - yy) qui multiplicatus per præ-
              <lb/>
            fatum guttulæ pondus {p d x/c} dat ejusdem momentum {n p d x/c} √ (2y - y y)}. </s>
            <s xml:id="echoid-s5510" xml:space="preserve">
              <lb/>
            Sed ex natura circuli eſt M d x = {dy√ (2y - yy): </s>
            <s xml:id="echoid-s5511" xml:space="preserve">hoc igitur valore ſubſtituto
              <lb/>
            pro d x, fit idem guttulæ n m momentum = {n p d y/M c}, cujus integralis, ſub-
              <lb/>
            tracta debita conſtante, eſt {n p (y - f)/Mc}, denotatque momentum aquæ in ar-
              <lb/>
            cu l n; </s>
            <s xml:id="echoid-s5512" xml:space="preserve">hinc igitur momentum omnis aquæ in l 1 o eſt = {n p (g - f)/Mc}: </s>
            <s xml:id="echoid-s5513" xml:space="preserve">quod
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            diviſum per vectem potentiæ in f applicatæ ſeu per 1 relinquit potentiam
              <lb/>
            iſtam quæſitam pariter = {n p (g - f)/Mc}. </s>
            <s xml:id="echoid-s5514" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s5515" xml:space="preserve">E. </s>
            <s xml:id="echoid-s5516" xml:space="preserve">I.</s>
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