MECH 
mt below, the velocity of the effluentfluid will be equal to that which 
a heavy body would acquire in falling freely through the height of 
the fuid above the orifice. —Let MNOP (fig. 2I -) reprefent 
a veflel filled with a fluid up to the level GH; MP the 
bottom in which is the aperture CD (very fmall compared 
with MP)i C I K D the column of the fluid Handing di- 
*e£ily above the aperture, and C ABD the lowed plate 
of the fluid immediately contiguous to the aperture. 
Alfo, let v denote the velocity which a heavy body would 
acquire in falling freely through B D, the height of the 
plate, and V the velocity acquired by the fame plate during 
its defcent through the fame fpace until it is difcharged 
by the prefl'ure of the column CIKD. If we fuppofe 
the lowed plate of fluid A CBD to fall as a heavy body 
through the height B D, its moving force will be its own 
weight. Again, fuppofe it to be accelerated by its own 
weight, together with the preffure of the ambient fluid 
about the column CIKD, that is, by the weight of the 
column CIKD, through the fame fpace, that is, while 
it is accelerated from quiefcence until it is aftually dif¬ 
charged : then the velocity in the former cafe will be to 
that in the latter as the moving forces and the times in 
which they aft direftly, and the quantities of matter 
moved inverfely. But the moving forces are to each 
other as the heights BD and KD; the times in which 
they aft are inverfely as the velocities, the (pace through 
which the body is accelerated being given ; and the 
quantities of matter moved are equal 5 therefore, v : V :: 
— : whence, » 2 : V 2 :: BD : KD, or v : V :: 
v V 
i/B D : KD. Now v is the velocity which a heavy body 
would aftually acquire in falling through the fpace BD; 
confequently V, the velocity of the effluent fluid, is that 
which a heavy body would acquire in falling through KD, 
the whole altitude of the fluid above the orifice. 
Cor. 1. In the fame manner it may be fhown, that if 
a pipe be inferted horizontally in the veflel MNOP, 
(fig. 24.) the plate of fluid A CBD will be difcharged 
with the fame velocity as before (if its centre of prelfure 
be of the fame depth), whatever be the thicknefs of the 
plate; this velocity not depending upon a continual ac¬ 
celeration through the length of the tube, otherwife the 
effluent fluid could not attain its full velocity until a co¬ 
lumn had been difcharged whofe bale is equal to the ori¬ 
fice and height equal to the length of the tube: whereas 
we find by experience that this full velocity can be at¬ 
tained by the thinned plate which can be let efcape from 
the aperture. 
Cor. 2. The velocities and quantities difcharged at dif¬ 
ferent depths are as the f'quare roots of the depths. 
Cor. 3. The quantity run out in any time is equal to a 
cylinder, or prifm, whofe bafe is the area of the orifice, 
and its altitude the fpace defcribed in that time by the 
velocity acquired by falling through the height of the fluid. 
So that, if h denote the height of the fluid, 
a the area of the aperture, 
g, 32 J- feet, or 386 inches, 
and t the time of efflux, we (hall have, 
For the quantity difcharged, Q —at f igh ; 
Or, when a and h are in feet, Q:=8'02c8 at f h feet; 
When a and h are in inches, at f h inch. 
If the orifice is a circle whofe 
diameter is d, then '785398 d 2 
mud be fubdituted for a ; 
And, when d and h are in feet, Q— 6'20952 d " 1 1 f h feet; 
When d and h are in inches, Cy=n.vy%t > c ) zd 2 l 3/ h inch. 
And from either of thefe it will be eaiy to find either 
a, t, or k, when the other three quantities are given. 
Cor. 4. The force with which the effluent Water im¬ 
pinges againd any quiefcent body is proportional to the 
altitude of the fluid above the orifice. For the force is 
as velocity x quantity of matter; but the quantify dif¬ 
charged in a given time is as the velocity : therefore the 
force is as the fquare of the velocity; that is, by the 
Vol. XIV. No. 1007. 
A N I C S. 7t3 
demonfiration of the propofition, as the height of the 
fluid. 
Cor. 5. The water fpouts out with the fame velocity 
whether it be downwards, or upwards, or fideways ; be- 
caufe the prefl'ure of fluids is the fame in all directions at 
the fame depth. 
Cor. 6. Hence, if the adjutage be turned upwards, the 
jet will afcend to the height of the furface of the water 
in the veflel. And this is confirmed by experience, from 
which it appears that jets really afcend nearly to the height 
of the refervoir; the fmall quantity abated aridng from 
the friction againd the (ides, the refiftance occafioned by 
the oblique motion of the fluid in the bended pipe, and 
the refidance of the air. 
Scholium. What is faid in this Propofition and Corol¬ 
laries of the velocity of the effluent water, is true only of 
the middle filament of particles which iflue through the 
centre of the aperture, which are fuppofed in theory to 
experience no retardation, and which, in faff, fud’er no 
other retardation than what arifes from the refidance of 
the air, and their mutual adhefion and attrition againd 
each other. But thofe which iffue near the edges of the 
aperture undergo a greater attrition, and therefore fuft'er 
a greater retardation. Hence it follows, that the mean 
velocity of the whole column of effluent fluid will be con- 
fiderably lefs than according to theory. 
Sir Ilaac Newton, who examined every fubjeft that came 
before him with peculiar accuracy, fil'd difcovered a con- 
tra&ion in the vein of effluent water; and found, that at 
the didance of about a diameter of the orifice the feftion 
of the vein contrafted nearly in the fubduplicate ratio of 
2 to 1. Hence he concluded that the velocity of the 
water, after its exit from the aperture, was increafed in 
this proportion, the fame quantity pafling in the fame tint# 
through a narrower fpace. From the quantity of water 
difcharged in a given time through that narrow fefdion, 
he found that its velocity there was that which a heavy 
body would acquire in falling through the height of the 
water above the orifice; and, fince the velocity there was 
greater than immediately in the orifice of the fubduplicate 
ratio of 2 to 1, he concluded that the velocity of the ef¬ 
fluent water in the orifice was equal to that which a heavy 
body would acquire in falling through half the altitude. 
But all this is true only of the mean velocity ; for there is 
no caufe which can aftually accelerate the water after its 
exit from the orifice, whatever caufes may contribute to 
its retardation. The manner in which the mean velocity 
of the water is increafed after its dilcharge, though the 
affual velocity of the feveral particles continues unvaried, 
might be eafily explained : but it need not be dwelt upon 
here. A circumdance of confiderable importance in the 
efcape of fluids through a horizontal orifice, feems to have 
been entirely neglefted by mod writers ; we allude to a 
whirling motion : the fluid will revolve about the aperture, 
and at foftve didance from it, and efcape with a revolving 
motion; the fluid ruflies from all fides in fpiral dreams 
to fupply the continual wade. This circumdance will be 
explained more diftinflly in a fublequent propofition. 
Prop. XXVIII. When a vejfiel is left gradually to difcharge 
itf e f by an orifice in the bottom, if the area of the feElion pa¬ 
rallel to the bottom be every-whcre the fame, the velocity of the 
furface of the f uid, and confquent/y the velocity of the efflux , 
will be uniformly retarded. —For, (Prop. XXVI.) the velo¬ 
city of the defending furface is to the velocity at the 
orifice as the area of the orifice to the area of the furface, 
which is a condant ratio; confequently, the velocity of 
the defcending furface varies as the velocity at the orifice, 
or as H, by Cor. 2 of the lad Prop, that is, the velocity 
of the defcending furface varies as the fquare-root of the 
fpace which it has to defcribe ; fo that this exaftly cor- 
refponds with the cafe of a body projected perpendicularly 
upwards, where the velocity is as the fquare-root of the 
fpace to be defcribed ; whence, as the retarding force is 
condant in the indance referred to, it mud alio be condant 
in the cafe before us, and the retardation uniform. 
8 T Comparing 
