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

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8167SECTIO QUARTA. poſteriorem per regulas ſolitas reſolvemus in hanc ſeriem
1 + {1/2} ({x/a}){mmαα/nn} - 2 + {1. 3/1. 2. 4} - ({x/a}){2mmαα/nn} - 4 + {1. 3. 5/1. 2. 3. 8}({x/a}){3mmαα/nn} - 6
+ & c. unde nunc habetur mutata paullulum æquationis forma:
dt = - {dx√mmαα - 2nn}/n√a} X [({x/a})- {1/2} + {1/2} ({x/a}){mmαα/nn} - {@/z}
+ {1. 3/1. 2. 4} ({x/a}){2mmαα/nn} -{9/2} + {1. 3, 5/1. 2. 3. 8} ({x/a}){3mmαα/nn} - {13/2} + & c. ]
Hæc æquatio ita eſt integranda, ut poſita x = a fiat t = 0; ſic autem oritur
t = [2 + {nn/2mmαα - 3nn} + {3nn/16mmαα - 28nn} + & c. ] X {√(mmαα - 2nn). a/n}
- [2{(x/a)}{1/2} + {nn/2mmαα - 3nn} ({x/a}){mmαα/nn}-{3/2}
+ {3nn/16mmαα - 28nn} ({x/a}) {2mmαα/nn} - {7/2} + & c. ] X
X {√(mmαα - 2nn). a/n},
ubi 2 √ a exprimit tempus quod corpus impendit dum libere delabitur per
altitudinem a. Si vero in iſta æquatione ponatur
x = a: ({mmαα - nn/nn})nn: ({mmαα - 2nn})
quæ eſt altitudo aquæ cum velocitas maxima eſt (per §. 16. ſect. 3. & §. 8. ſect
4.) , tum obtinetur tempus quod à fluxus principio ad punctum maximæ ve-
locitatis usque præterit; & cum ponitur x = o, oritur tempus, quo vas to-
tum depletur, ac denique ſi ponatur x = cuicunque quantitati c, exprimet t
tempus quod ſuperficies inſumit in deſcenſum per altitudinem a - c; Videbi-
mus autem pro his caſibus, quid fieri debeat, cum vas eſt valde amplum,
numerusque m alterum n ſic pluries continet.

§. 11. Fuerit primo {m/n} numerus infinitus, erit altitudo aquæ puncto
maximæ velocitatis reſpondens ſeu

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