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

SECTIO SEXTA. v = {(b - β + f - φ) gx - ({bg/2a} + {bgg/2αγ)} xx/ga - gx + {αgg/γ} + {g 3 /γ} {x/γ} + MN} Q. E. I.

98. Corollarium 1.

§. 8. Quia linea mn = mg - nh + gh = h - β + f - m, ponemus
mn = c, ſimulque multiplicabimus denominatorem & numeratorem per
2γγαα: Ita vero habebimus
v = {2gγγaαcx - (gγγαb + ggγaß)xx/2gγγaaα - 2gγγaαx + 2ggγaαα + 2g 3 aαx + 2γγaαMN}

99. Corollarium 2.

§. 9. Si fiat v =o, patet tunc valorem x denotare totam fluidi ſuper-
ficiei excurſionem in tubo ac, quæ ſic invenitur æqualis {2γaαc/γαb + gαβ}, in altero
vero tubo fit = {2gaαc/γαb + gαβ}.

Igitur poterit aqua in tubo ſtrictiori ad quamcunque elevari altitudi-
nem, ſi modo ratio amplitudinum g & γ ſat magna ſumatur.

100. Corollarium 3.

§. 10. Pars illa vaſis c A d, quam neutra ſuperficierum unquam attin-
gi ponimus, nihil pertinet ad iſtas fluidi excurſiones ſive augendas ſive dimi-
nuendas: facere tamen poteſt, ut inferius oſtendetur, ad accelerandas retar-
dandaſque oſcillationes.

101. Corollarium 4.

§. 11. Ponatur uterque tubus communis amplitudinis, erit, poſito
nempe g = γ,
v = 2gaαcx - (gαb + gaβ)xx/2gaaα + 2gaαα + 2aαMN}.

In hoc caſu maxima ſuperficiei utriuſque velocitas eſt, cum in medio
totius excurſionis poſitæ ſunt, ſecus ac fit, cum tubi ſunt inæqualis amplitu-
dinis.

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