Full text: Bernoulli, Daniel: Hydrodynamica s. de viribus et motibus fluidorum commentarii

HYDRODYNAMICÆ - dz = {f + φ/F} X {z/x} X √ ({P/p} b) xdt; Ex comparatione harum duarum æquationum oritur
- dz = {f + φ/F - f} X {g/b} X {√b/√Pp} X dv,
quæ cum debitæ conſtantis additione integrata dat
z = g - {f + φ/F - f} X {g/b} X {√b/√Pp} X v. Si jam in æquatione prima ſubſtituatur valor iſte inventus pro z, ſimulque
ponatur {dx/v} pro dt, fiet
vdv = {F - f/F} X {b/x} X P X dx - {f + φ/F} X {√(bP)/x√p} X vdx, ſive
{Fvdv√p/(F - f) X bP√p - (f + φ) X v√ (bP)} = {dx/x},
quæ æquatio poſt debitam ſui integrationem, facta x = a, abit in hanc
log. {a/b} = [-F(f + φ) v√ p - F (F - f) p√ (Pb) X log. (1 - {(f + φ)v/(F - f) √ (bPp)})]: (f + φ) 2 X √Pb.

(VIII) Si jam per experimentum innotuerit valor ipſius v, poterit in-
de deduci valor ipſius P, qui denotat elaſticitatem auræ pulveris pyrii non-
dum expanſæ: Quod ut exemplo illuſtremus, eodem utemur experimento,
quod jam articulo IV. expoſuimus, ut appareat inde, quodnam ab avolatione
auræ elaſticitatis augmentum arguat. Sic igitur ponetur calculus.

Quia pondus globi, quod erat trium librarum, indicavimus per uni-
tatem, erunt quatuor unicæ pulveris adhibitæ exprimendæ per {1/12}: igitur
p = {1/12}. Menſuras aperturarum, quas conſideramus, non accepi: ſolet autem
hiatus à globo r [?] elictus conſtituere in ſimili tormento præterpropter partem
decimam quintam amplitudinis animæ; amplitudinem luminis accenſoriihic
fere negligi poſſe puto; itaque ſtatuam F = 15; f = 1; φ = 0: Deinde
habetur rurſus a = 7, 7; b = 0, 08; altitudo ad quam globus in vacuo
aſcendere poſſit ſeu {1/2} vv = 58750, ſeuv = 343: Igitur æquatio ultima
ſuperioris articuli hæc erit
log. 96 = { - 5251/√P} + 17, 5 log. {√P/√P-300},
cui proxime ſatisfit cum ſumitur √ P = 534 & proinde P = 285156,
quod efficit pondus columnæ mercurialis ejusdem cum anima tormenti am-

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