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

HYDRODYNAMICÆ cæ indirectis; placet tamen hujus rei demonſtrationem dare ex natura vectis
petitam, quia mechanici eo omnia reducere amant.

Helicem conſiderabimus a 1 b ex figura quinquageſima ſecunda ſeor-
ſim deſumtam, ad evitandam linearum confuſionem, conſervatis denomi-
nationibus art. IV. adhibitis. Sic igitur in Figura 53. erit rurſus angulus
N M G angulus quem facit nucleus cum horizonte, cujus ſinus = N, ſi-
nusque anguli a M H = n; a 1 b eſt una ſpiralis circumvolutio: baſis nuclei
eſt circulus a c M p a; ſinus anguli p a l eſt ut ante = m, ejusque coſinus M; puncta vero l & o ſunt extremitates aquæ in ſpirali quieſcentis & in ea-
dem altitudine ab horizonte poſita, ex iſtis punctis ductæ ſunt ad periphe-
riam baſis rectæ l c & o p ad baſin perpendiculares. In parte helicis quam
aqua occupat ſumta ſunt duo puncta infinite propinqua m & n & per hæc du-
ctæ ſunt rectæ n f & m g rurſus ad baſin perpendiculares. Denique ex pun-
ctis c, f, g, p ductæ ſunt ad diametrum a M perpendiculares c d, f h, g i & p q; atque centrum baſis ponitur in e, radiusque e a = 1. Sit jam arcus
ſpiralis l 1 o aqua plenus = c & conſequenter arcus circularis eidem reſpon-
dens c M p = M c; a l = e; a c = M e; a d (ſeu ſinus verſus arcus ac) = f; a q = g; pondus aquæ in l s o = p: arcus a l n = x; n m = d x; a c f = M x; f g = M d x; a b = y; h i = d y; h f = √2y - yy, erit pondus guttulæ in
nm = {p d x/c}; ſi vero linea h f multiplicetur per ſinum anguli a M H, divida-
turque per ſinum totum, habetur vectis quo particula n m cochleam circum-
agere tentat: eſtigitur vectis iſte = n √ (2y - yy) qui multiplicatus per præ-
fatum guttulæ pondus {p d x/c} dat ejusdem momentum {n p d x/c} √ (2y - y y)}. Sed ex natura circuli eſt M d x = {dy√ (2y - yy): hoc igitur valore ſubſtituto
pro d x, fit idem guttulæ n m momentum = {n p d y/M c}, cujus integralis, ſub-
tracta debita conſtante, eſt {n p (y - f)/Mc}, denotatque momentum aquæ in ar-
cu l n; hinc igitur momentum omnis aquæ in l 1 o eſt = {n p (g - f)/Mc}: quod
diviſum per vectem potentiæ in f applicatæ ſeu per 1 relinquit potentiam
iſtam quæſitam pariter = {n p (g - f)/Mc}. Q. E. I.

180.1.

Fig. 53.
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