118104HYDRODYNAMICÆ.
in quâ ſi præterea fiat 1 - z = qq, ſeu z = 1 - qq, dz = - 2qdq,
oritur
dt = {- 2αdq/1 - qq} = {- αdq/1 + q} {- αdq/1 - q}
cujus integralis eſt
t = - α log. (1 + q) + α log. (1 - q) = α log. {1 - q/1 + q}.
oritur
dt = {- 2αdq/1 - qq} = {- αdq/1 + q} {- αdq/1 - q}
cujus integralis eſt
t = - α log. (1 + q) + α log. (1 - q) = α log. {1 - q/1 + q}.
Nec opus eſt conſtante, quandoquidem ex natura rei t &
x, ſimul
evaneſcere debent, poſito autem x = o, fit z = 1, & q = o, igitur pa-
riter t & q ſimul à nihilo incipere debent, cui conditioni ſatisfacit æquatio
inventa t = α log. {1 - q/1 + q}: Supereſt ut retrogrado ordine valores priſtinos
reaſſumamus, ita vero fit
t = α log. {1 - √(1 - z)/1 + √(1 - z)} vel
t = {γmN/n√(mm - nn)a} X log. {1 + √(1 - z)/1 - √(1 - z)} vel denique
(I) t = {γmN/n√(mm - nn) a} X [log. [1 + √(1 - c{n3 - nmm/mmN} x)]
- log. [1 - √(1 - c{n3 - nmm/mmN} x)]]
Iſtaque æquatio poſito m = ∞ dat alteram æquationem quæſitam
(II) t = {γN/n√a} X [log. [1 + √(1 - c{- n/N} x)]
- log. [1 - √(1 - c{-n/N} x)]] Q. E. I.
evaneſcere debent, poſito autem x = o, fit z = 1, & q = o, igitur pa-
riter t & q ſimul à nihilo incipere debent, cui conditioni ſatisfacit æquatio
inventa t = α log. {1 - q/1 + q}: Supereſt ut retrogrado ordine valores priſtinos
reaſſumamus, ita vero fit
t = α log. {1 - √(1 - z)/1 + √(1 - z)} vel
t = {γmN/n√(mm - nn)a} X log. {1 + √(1 - z)/1 - √(1 - z)} vel denique
(I) t = {γmN/n√(mm - nn) a} X [log. [1 + √(1 - c{n3 - nmm/mmN} x)]
- log. [1 - √(1 - c{n3 - nmm/mmN} x)]]
Iſtaque æquatio poſito m = ∞ dat alteram æquationem quæſitam
(II) t = {γN/n√a} X [log. [1 + √(1 - c{- n/N} x)]
- log. [1 - √(1 - c{-n/N} x)]] Q. E. I.
Corollarium 1.
§.
15.
Si ponatur x = ∞, ut appareat natura rei, cum infinita jam
transfluxit aquæ quantitas aſſumaturque m major quam n, prouti plerumque
eſſe ſolet, evaneſcere cenſenda eſt, in utroque logarithmo affirmative ſum-
to, quantitas exponentialis & habebitur utrobique log. 2. At vero in logarith-
mo negative ſumto ſtatuenda eſt
√(1 - c{n3 - nmm/mmN} x) = 1 - {1/2} c{n3 - nmm/mmN} x &
transfluxit aquæ quantitas aſſumaturque m major quam n, prouti plerumque
eſſe ſolet, evaneſcere cenſenda eſt, in utroque logarithmo affirmative ſum-
to, quantitas exponentialis & habebitur utrobique log. 2. At vero in logarith-
mo negative ſumto ſtatuenda eſt
√(1 - c{n3 - nmm/mmN} x) = 1 - {1/2} c{n3 - nmm/mmN} x &
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