m 
MECHANICS. 
Comparing this with the infiance juft pointed cut, we 
deduce thefe obvious corollaries : 
Cor. i. The quantities of water in a prifmatic veffel dis¬ 
charged through an aperture in the bottom decreafe in 
equal times as the Series of odd numbers i, 3, 5, 7, 9, See. 
taken in an inverted order. 
Cor. z. The quantity of water contained in an upright 
prifmatic veffel is half that which would be difeharged in 
the time of the entire gradual evacuation of the veil'd, if 
the water be kept always at the fame altitude. 
Prop. XXIX. To determine the time of emptying a vejfiel 
of Zfiater by an orifice in the bottom of it, or in the fide contigu¬ 
ous to the bottom, the height of the orifice being ’’very fimall com¬ 
pared with the altitude of the fluid. 
Let a — the area ol the aperture ; 
h — the whole height of the fluid above the aperture; 
x=the vertical Space defeended by the upper fur- 
face in any time t ; 
A z= the area of the upper furface ; 
g— 32A as before, the meafure of the force of gravity. 
Then will the velocity of the effluent fluid at any time 
be reprefented, not by fz.gk as in Prop. XXVII. but 
by 1/2 g (h—x). This velocity will vary continually, be¬ 
cause x increafes, and the difference h—x diminilhes con¬ 
tinually ; but it may be regarded as constant during the 
indefinitely Small time t ; fo that in the time t there will 
efcape through the orifice a prifm of the fluid which has 
that orifice a for its bafe, and f z g ( h — x) for its altitude. 
Thus the quantity of fluid difeharged during the inftant t 
is — at f zgfit — x). But during the fame time the upper 
Surface has defeended through the fpace x, and the veffel 
has loft a prifm or cylinder of the fluid wbofe height is x 
and bafe A; that is, a prifm whofe capacity is Ax. 
• -;- • A X 
Hence we have Ax~aty/zg(ji —x); and t =— - 
a \f zgfii — x) 
As the area A will be given in functions of x, by the 
form of the veffel, the fecond member of this equation 
may be confidered as containing only the variable quan¬ 
tity x ; and it will be very eafy in molt cafes, by limply 
finding the fluents, to difeover the fucceffive deprefflons 
and discharges of the fluid in any veffel of known form. 
Suppofe, for inftance, the vefiel be an upright prifm or 
cylinder. Here the area A will be conftant, becaufe every 
horizontal lection of the prifm will be equal to its bale. 
Hence we have 
A x z A - 
t =—-.— . J — -- =-7 —V h —x+C. 
a V*g fk—x a \ /2 3 
Now when the time t is nothing, the depreflion of the 
upper furface A of the fluid is nothing'aifo j thus we have 
at the fame time xz=.o, and t—o. This condition deter- 
. 2 A 
mines the conftant quantity C=—- fh- and gives for 
afi zg 
the time of deprefflng the upper Surface through the 
z A - 
Space .v, t— —;— ( J n — J h — x). 
1 a f zg 
To find the time of completely emptying the veffel, we 
have only to make x—k, in which cafe the preceding ex- 
preflion will become t ———. 
a V g 
Cor. The time juft found is double that in which an 
equal quantity would be difeharged, if the veffel were 
kept conftantly full. For, in Prop. XXVII. Cor. 3. we 
have Q=zat f zgk, where, if Q—Ah, we have t 
A h A zh 
~a zg~zaA g : 
afizgk 
which is half the preceding value of t • 
Let A BCD, fig. 23, be a veffel filled with a fluid, and BC 
be perpendicular to the horizontal plane CL, and upon BC 
let a temicirclt be deicribed, and G 1 I an ordinate perpen¬ 
dicular to BC; then the diflance CR, to which the fluid 
fpouts through a very fmall orifice at G, is rerzGH, The 
velocity at the vena contraSa, which is extremely near to 
the veffel, is that which a body would acquire in falling 
down BG; we are therefore to confider this as the velo¬ 
city with which the fluid is projected, and not the velo¬ 
city at the orifice. Now the curve GR deicribed by the 
fluid is a parabola, and BG is one-fourth of the parameter 
belonging to the point G, which point is the vertex of 
the parabola, the fluid fpouting out horizontally; hence, 
GC is the abfeiffa, and CR its ordinate; and by the pro¬ 
perty of the para bola, aBGxGC^CR 2 ; therefore CR 
== 2 \/ BG X GC'=2G rl, by the property of the circle. 
Cor. It Cg=BG, then g-//=;GH, and the fluid Spouts 
to the fame distance from g as from G. If B C be bi- 
feded in O, then the diflance CT, to which the fluid 
fpouts, is equal to zOP=:CB ; and this is the greateft dis¬ 
tance, OP being the greatelt ordinate. 
If the fluid fpout perpendicularly upwards, it ought to 
rife to the altitude of the Surface of the water in the veffel; 
but it falls a little fhortof this, partly from the fridtion at 
the orifice, partly from the refinance of the air, and partly 
from the falling back of the water. If the water fpouc 
upwards through a pipe, inllead of limply a hole, it does 
not afeend fo high, becaufe, there being no vena contracla , 
the velocity is not increafed immediately after it leaves 
the pipe, as it is when it flows out of a Ample orifice. 
Alfo, there is a certain meafure of the hole, compared 
with the fize of the vefiel, through which the velocity is 
the greateft. For the retardation ariling from the lides of 
the orifice varies as the circumference of the orifice, or as 
its diameter, and the quantity of fluid palling through 
varies as the Square of the diameter; therefore, by diini- 
nifliing the orifice, the retarding force does not decreafe 
fo fait as the quantity ot fluid to oppofe the retardation 
decreafes 5 consequently there is a diminution of velocity 
from this caufe, when you diminifn the hole. On the 
contrary, when you diminifh the hole, the effedt of the 
preffure on a Smaller quantity of fluid appears to be in 
favour of increaftng the velocity Thefe feem to be the 
reafons, why there is a certain fize of the hole which gives 
the greateft velocity to the iffuing fluid. 
Tlie very near agreement of tins theory with experiments* 
proves, very Satisfactorily, that the velocity of projection 
muff be that which is acquired in falling down BG, the 
whole altitude of the fluid. And the agreement of the 
theory of emptying veffels with experiments, fliows, very 
clearly, that the velocity at the orifice mult be that which 
is acquired in falling through half the altitude of the 
fluid. Almoft immediately, therefore, after the fluid gets 
out of the orifice, its velocity is increafed in the ratio 
of 1 : y/ z. 
Experiments on the Motion of Fluids. 
In the preceding fedtion, we have taken notice of the 
contraction produced upon the vein of fluid iffuing from 
an orifice in a thin plate, and have endeavoured to afeer- 
tain its caufe. According to fir Ifaac Newton, the dia¬ 
meter of the vena contrafia is to that of the orifice as 21 
to 25. Polenus makes it as 11 to 13 ; Beniouilli, as 5 to 7 ; 
the' chevalier de Buat, as 6 to 9 ; Bolfut, as 41 to 50 ; 
Michelotti, as 4 to 5 ; and Venturi nearly as 4 to 5. 
This ratio, however, is by mo means conftant. It varies 
with the form and pofition of the orifice, with the thick- 
nefs of the plate in which the orifice is made, and like- 
wife with the form,of the vefiel and the weight of the fu- 
perincumbent fluid. But thefe variations are too trifling 
to be regarded in pradlice. We fhall now lay before the 
reader an account of the refults of the experiments of 
different philofophers, bdt particularly thofe of the abbe 
Boffut, to whom the Science is deeply indebted both for 
the accuracy and extent of his labours. 
In the following experiments, which were frequently 
repeated in various ways, the urifice was pierced in a plate 
of copper about half a line thick. When the orifice is.in 
the 
