Therefore 326 16 cubic inches of water will 
{low from an aperture of two inches in dia- 
meter in one minute, the. orifice being made 
nine inches from the surface, which is sup- 
posed to be kept at that height the whole 
time. 
If a vessel of a prismatic form is filled with 
water, and permitted to empty itself entirely 
through an orifice at the bottom, and the 
time that it consumes in emptying itself is ob- 
served ; and if afterwards, having replenished 
the vessel, the water is made to ilow through 
the same aperture', the vessel being kept lull 
the whole time, there will run out in this 
second instance, during tne same time that 
the vessel took to empty itself at first, a quan- 
tity of water, double that which runs out in 
the first case, in which the abstraction of the 
water diminishes the height, and conse- 
quently the velocity. 
We often perceive water flow through la- 
teral apertures, which, though small in com- 
parison to the width of the reservoirs, cannot 
be regarded as having all their points at an 
equal distance from the surface of the fluid ; 
such, for example, as the apertures through 
which water sometimes flows in mills. I he 
common method of determining the quantity 
discharged is as follows: suppose, in the first 
place, the aperture to be stopped up by a 
plate of metal, which is perforated with a 
number of holes ; if each of these holes is re- 
garded as particular and insulated, the rapi- 
dity of t}ie flow through each will be accord- 
ing to the correspondent height of the fluid ; 
then if the number of holes are multiplied ad 
infinitum, or, which will amount to the same 
thing, if the plate is supposed to be entirely 
taken away, the velocity at each point of the 
supposed orifice will be according to the cor- 
respondent height ot the fluid ; and in esti- 
mating the quantity of water discharged, 
some attention must lie paid to the. inequality 
of the motion; yet it must not be asserted 
that this reasoning is entirely conclusive. In 
proportion as the sum of the small holes 
made in the plate is small in comparison with 
the size of the reservoir, the portions of water 
which flow through each hole are forced out 
by the absolute weight of the column above; 
but the moment that the number of apertures 
augment ad infinitum, anti the streams ot wa- 
ter which run through them become conti- 
guous, it cannot be clearly said that the liquid 
flows in the same manner as through small 
insulated holes; yet as this hypothesis gives a 
result sufficiently conformable to experi- 
ments, it may be useful to preserve it, and 
the more so, as it leads to very simple calcu- 
lations, and in all common questions this 
simplicity may be preferable to the minute- 
ness of fractional operations. 
The quantity of water which issues from 
these apertures in a given time is not so great 
as their size might at first suggest, because 
the stream is contracted by running out ot 
each orifice, and that contraction extends to 
a distance nearly equal to half the diameter 
of the aperture ;' and the diameter of the.con- 
tracted stream is to the diameter of the aper- 
ture a little more th m as three to four, or as 
three and one-sixth to four, or nineteen to 
twenty-four ; so that its area is to that of the 
aperture as ten to sixteen. It is nearly the 
same when water flow’s through lateral aper- 
tures. The contraction of the stream is a 
HYDRAULICS. 
withhrside a vessel the lateral particles direct 
themselves towards the orifice with a motion 
more or less oblique; and this oblique motion 
may be decomposed into two forces, the one 
parallel to the plane of the orifice, and which 
contracts the stream; the other perpendi- 
cular to the same plane, and the only one 
which produces the efflux. 
This contraction occurs also when water is 
made to flow through pipes, and that at the 
9*7 
the reservoirs arc nine and four feet, the 
square roots of which are three and two, and 
it will be found that the two quantities ot wa- 
ter, 8135 cubic inches, 543b cubic inches, 
which run through the same orifice ot one 
inch diameter under the different heights of 
nine feet and four feet, are to each other 
nearly in the proportion of three to two. 
3. That in general the quantities of water 
discharged in the same time through different 
entrance of the water into the pipe, and not* apertures, under different heights of surface 
at its exit, where the stream preserves a cy 
lindrical form. We shall prove that this 
contraction diminishes, in a sensible manner, 
tlup quantity of water which would naturally 
flow. 
In order to ascertain these facts by experi- 
ments, many have been made. In all the 
following instances, the orifices through 
which the water flowed were pierced per- 
pendicularly through plates of copper of 
about one-twentv-fourtli of an inch thick, and 
the time of each experiment is reduced to 
one minute. 
The constant height of the water above the 
centre of each orifice Was 1 1 feet 8 inches 10 
lines. 
No. of cubic inches 
furnished in 1 min. 
E.vpt. 1. Through an horizontal cir- 
cular orifice of -§ inch (6 
lines) diameter - 2,311 
2. Through ditto of 1 inch 
diameter - - 9,2S1 
3. Through ditto of 2 inches 
diameter - - 37,203 
4 . Through an horizontal 
rectangular orifice of 1 inch 
long and -V inch wide 2,933 
5. Through an horizonal 
square orifice of l inch the 
side - - - 11,817 
6. Through ditto of 2 inches 
each side of the orifice 47,361 
Constant height = 9 feet. 
7. Through a lateral circular 
orifice of £ inch diameter 2,01S 
8. Through ditto of 1 inch 
diameter - - 8,135 
Constant height = 4 feet. 
9. Through a lateral circular 
orifice off inch diameter 1,353 
10. Through ditto, of 1 inch 
diameter - - 5,436 
Constant height — T 7 ^ inch. 
1 1 . Through a lateral circular 
orifice of 1 inch diameter 628 
It follows from the preceding table, 
1. That the quantities of water discharged 
in the same time, by different apertures, 
under the same height of surface in the reser- 
voir, are to each other nearly as the areas of 
the apertures. Compare together the results 
of the second and third experiments ofwhich 
the areas of the orifices are in the proport ion 
of one to four, and it will be found that the 
quantities of water afforded, viz. 9281 cub e 
inches, 37203, inches, are very nearly in the 
same proportion. 
2. T hat the quantities of water discharged in 
the same time through the same aperture, 
under different heights of surface in the reser- 
voirs, are to each other nearly as the square 
roots of the corresponding heights of the water 
in the reservoir above the centre of the aper- 
ture. Compare together the results of the 8th 
proVf of what hits been before stated, viz. that and 10th experiments, where the heights of 
*■ 6 D 2 
in the reservoirs, are .o each other as the 
areas of the apertures are to the square roots 
of the heights of water in the reservoirs. 
4. That in consequence of the friction, the 
small apertures furnish a less quantity of wa tor 
iir proportion than the great ones, under the 
same height of water in the reservoir; be- 
cause, comparatively to the extent ot the 
area of each orifice, there are more points of 
friction against the sides of the orifice in the 
small than there are in the great ones; for the 
circumferences do not diminish so much as 
the areas. 
5. That of many apertures of equal areas, 
that of which the circumference is the least, 
will, on account of the friction, furnish more 
water than the others, under the same height 
of the reservoir; circular apertures are, tor 
this reason, the most advantageous of all; tor 
the circumference of a circle is the shortest 
line that can be made use of to inclose a 
given space, therefore there is less surface of 
friction relatively to the size of the area. 
It is easy to perceive, that the quantity of 
water discharged in the table ot experiments 
is not near so great as might be expected 
from the extent of the areas and the heights 
of the reservoirs. The quantity is in tact di- 
minished by the friction, and still more by 
the contraction of the stream; for the velocity 
which is in proportion to the entire altitude 
of the fluid is not sensibly changed. Suppos- 
ing, first, that the area of the stream is the 
same as that of the orifice; and supposing, 
secondly, that the stream is contracted, then 
the difference of the quantities afforded is as 
sixteen to ten; that is, supposing the area of 
the aperture to be diminished in the propor- 
tion of sixteen to ten, the discharge of the? 
fluid out of vessels kept equally full may be 
determined with sufficient exactness. By the 
expression, an inch of water, is understood 
the quantity which flows out ot a circular 
and lateral orifice of one inch diameter, the 
surface of the water being constantly kept 
seven-twelfths of an inch above the centre of 
the orifice. This is the case with the eleventh 
experiment in the preceding table, where it 
appears that the quantity of water furnished 
is 628 cubic-inches. Mr. Mariotte, who made 
the same experiment, found the quantity to 
be a little more; but it is probable that lie 
might commit a small error, because the ex- 
periment just cited was made, M. Brissoq hi- 
forms us, with the utmost care and attention. 
A (french) pint of water, he adds, instead of 
weighing two pounds, as is commonly believ- 
ed, is proved to lall short of that weight con- 
siderably, as will be evident by strictly ex- 
amining’ that experiment. 
When additional pipes are employed, it 
appears, 
First, That the quantity of water discharg- 
ed by different addition '1 pipes, under the 
same height of water in the reservoir, is pro- 
