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UNIVERSITY OF MICHIGAN
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者
​EXPERIMENTAL ENQUIRY
CONCERNING THE
Natural Powers of Wind and Water
TO TURN
MILLS AND OTHER MACHINES
DEPENDING ON A
.
CIRCULAR MOTION.
AND AN
EXPERIMENTAL EXAMINATION
OF THE
QUANTITY AND PROPORTION
Of MECHANIC POWER
Neceſſary to be employed in giving different degrees of VELOCITY
to HEAVY BODIES from a STATE of REST.
B
ALSO
NEW FUNDAMENTAL EXPERIMENTS
UPON THE
COLLISION OF BODIES.
.
WITH FIVE PLATES OF MACHINES.
BY THE LATE MR. JOHN SMEATON, F. R. S.
THE SECOND EDITION.
LONDON:
PRINTÉD FOR 1. AND J. TAYLOR,
NO. 56, HIGH-HOLBORN.
.
1796.
1
,
言
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:
;
資
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1
.
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i
5
ADVERTISEMENT.
دد
-
V
iua 3-
THE ſubjects of the following sheets, as branches of
praćtical mechanics, are intereſting; and the manner in
which they are treated render them important.
2
V
Previous to theſe experiments, the ſuperior power of over-
shot water-wheels was not only doubted, but ſome authors
of conſiderable celebrity had given a preference to underſhot
wheels.---The experiments made, and bere fully explained
by Mr. SMEATON (who, as a pratical mechanic, muſt
ever be rated in the firſt claſs), jew the fallacy of former
concluſions, and the great ſuperior power of overſhot wheels:
the lapſe of time ſince the publication of theſe principles, bas
confirmed their accuracy, and has eſtabliſhed a durable re-
putation for their author.
's
Theſe Eſays, publiſhed originally in the Philoſophical
Tranſactions (the parts of which are now become ſcarce,
and of high price on their account), are now offered to the
public at a moderate purchaſe; to render them acceſſible to
every
:
iv
ADVERTISEMENT.
every ingenious enquirer, muft tend to ſpread the very uſeful
information they contain. To a fuperior mind nothing can
be more grateful, than imparting to others the advantages
of that ſuperiority, which bountiful Heaven has beſtowed on
enlightened Genius.
1
**
C O N T E N T S.
Page
OF UNDERSHOT WATER WHEELS
1
2
IS
17
1. That the virtual or effective Head being the
fame, the Effect will be nearly as the
Quantity of Water expended
II. That the Expence of Water being the ſame,
the Effect will be nearly as the Height of
the virtual or effective Head
III. That the Quantity of Water expended being
the ſaine, the Effect is nearly as the Square
of its Velocity
IV. The Aperture being the ſame, the Effect will
be nearly as the Cube of the Velocity of
the Water
19
21
C
1
25
OF OVERSHOT WHEELS
I. Of the Ratio between the Power and Effect
of Overíhot Wheels
II. Of the proper Height of the Wheel in Pro-
portion to the whole Deſcent
29
1
31
III. Of
vi
CONTENTS,
Page
III. Of the Velocity of the Circumference of
the Whcel, in order to produce the greateſt
Effect
32
IV. Of the Load for an Overſhot Wheel, in
order that it may produce a Maximum 34
V. Of the greateſt poſſible Velocity of an
Overſhot Wheel
34
VI. Of the greateſt Load that an Overſhot
Wheel can overcome
35
38
43
50
ON THE CONSTRUCTION AND EFFECT OF
WINDMILL SAILS
I. Of the beſt Form and Poſition of Windmill
Sails
II. Of the Ratio between the Velocity of Wind-
mill Sails unloaded, and their Velocity when
loaded to a Maximum
III. Of the Ratio between the greateſt Load the
Sails will bear without ſtopping, &c.
IV. Of the Effects of Sails, according to the
different Velocity of the Wind
V. Of the Effects of Sails of different Magni-
tudes, the Structure and Poſition being
ſimilar, and the Velocity of the Wind the
fame
VI. Of the Velocity of the Extremities of
Windmill Sails, in reſpect of the Velocity
of the Wind
50
.
51
55
57
VII. Of
.
CONTENTS.
vii
>
Page
VII. Of the abſolute Effect, produced by a given
Velocity of the Wind, upon Sails of a
61
given Magnitude and Conſtruction
VIII. Of HorizONTAL WINDMILLS and Water
Wheels, with oblique Vanes
63
General Propoſition
66
EXPERIMENTAL EXAMINATION OF THE QUAN-
TITY OF MECHANIC POWER NECESSARY TO
MOVE HEAVY BODIES, FROM A STATE
OF REST
71
EXPERIMENTS
BODIES
UPON THE COLLISION OF
}
95
.
Directions to the Binder.
TABLE I. is to face Page 15.
TABLE III. is to face Page 43.
The five Plates to be put at the End.
į
..
.
.
ܬܚܐ. ܃ ܃
:
:
*
EXPERIMENTAL ENQUIRY
CONCERNING THE
NATURAL POWERS
WATER AND WIND
To turn Mills and other MACHINES depending on
A CIRCULAR MOTION :
Read before the Royal Society, May 3 and 10, 1759.
WHAT I have to communicate on this ſubject was origi-
nally deduced from experiments made on working models,
which I look upon as the beſt means of obtaining the outlines in
mechanical enquiries. But in this caſe it is very neceſſary to
diſtinguiſh the circumſtances in which a model differs from a
machine in large ; otherwiſe a model is more apt to lead us from
the truth than towards it. Hence the common obſervation,
that a thing may do very well in a model that will not anſwer
in large. And, indeed, though the utmoſt circumfpection be
uſed in this way, the beſt ſtructure of machines cannot be fully
aſcertained, but by making trials with them, when made of their
proper fize. It is for this reaſon that though the models ry
B
ferred
2
EXPERIMENTAL ENQUIRY, &c.
ferred to, and the greateſt part of the following experiments,
were made in the years 1752 and 1753, yet I deferred offering
them to the Society, until I had an opportunity of putting the
deductions made therefrom in real practice, in a variety of caſes,
and for various purpoſes; ſo as to be able to aſſure the Society
that I have found them to anſwer.
PART I.
Concerning Underſhot Water-Wheels:
PLATE I. Fig. 1. is a perſpective view of the machine for
experiments on water-wheels; wherein
ABCD is the lower ciſtern, or magazine, for receiving the
water, after it has quitted the wheel; and for ſupplying
DE the upper ciſtern, or head; wherein the water being raiſed to
any height required, by a pump, that height is ſhewn by
FG, a ſmall rod, divided into inches and parts; with a float at
the bottom, to move the rod up and down, as the ſurface of
the water riſes and falls.
HI is a rod by which the fluice is drawn, and ſtopt at any
height required, by means of
K a pin, or peg, which fits ſeveral holes, placed in the manner
of a diagonal ſcale, upon the face of the rod H 1.
GL is the upper part of the rod of the pump, for drawing
the water out of the lower ciſtern, in order to raiſe and
keep
EXPÉRIMENTAL ENQUIRY, &c.
3
keep up the ſurface thereof at its deſired height, in the head
DE; thereby to ſupply the water expended by the aper-
ture of the Nuice.
:
MM is the arch and handle for working the pump, which is
limited in its ſtroke by
N, a piece for ſtopping the handle from raiſing the piſton too
high ; that alſo being prevented from going too low, by
meeting the bottom of the barrel.
O is the cylinder, upon which a cord winds, and which,
being conducted over the pullies P and Q, raiſes
R; the ſcale, into which the weights are put, for trying the
power of the water.
ST, the two ſtandards, which ſupport the wheel, are made to
ſlide up and down, in order to adjuſt the wheel, as near as
poſſible; to the floor of the conduit.
W the beam which ſupports the ſcale and pullies; this is re-
preſented as but little higher than the machine, for the lake
of bringing the figure into a moderate compaſs, but in re-
ality is placed 15 or 16 feet higher than the wheel.
Plate II. Fig. 2. is a feétion of the ſame machine, wherein
the fame parts are marked with the ſame letters as in Fig. I.
Beſides which
X X is the pump barrel, being 5 inches diameters and in inches
long.
Y is the piſton; and
Z the fixed valve.
ва
GV
I
54
:
$
4
EXPERIMENTAL ENQUIRY, &c.
>
GV is a cylinder of wood, fixed upon the pump-rod, and
reaches above the ſurface of the water: this piece of wood
being of ſuch a thickneſs, that its ſection is half the area
of that of the pump-barrel, will cauſe the ſurface of water
to riſe in the head, as much while the pifton is deſcending,
as while it is riſing: and will thereby keep the gauge-rod
FG more equally to its height.-Note, The arch and
handle MM is here repreſented on a different fide to
what it is ſhewn in the preceding figures, in order that its
dimenſions
may
the better appear.
a a ſhews one of the two wires which ſerve as directors to
the float, in order that the gauge-rod F G may be kept
perpendicular; for the ſame purpoſe alſo ſerve w, a piece
of wood with a hole to receive the gauge-rod, and keep it
upright.
b is the aperture of the fluice.
cc a kant-board, for throwing the water more directly down
the opening cd, into the lower ciſtern; and
ce is a floping board, for bringing back the water that is
thrown up by the floats of the wheel.
Fig. 3. reprefents one end of the main axis, with a ſection of
the moveable cylinder, marked Q in the preceding figures.
ABCD is the end of the axis; whereof the parts
B and D are covered with ferrules or hoops of brafs.
E is a cylinder of metal; whereof the part marked
F is the pivot or gudgeon.
6C is
Į
EXPERIMENTAL ENQUIRY, &c.
j
cc is the ſection of an hollow cylinder of wood, the diameter
of the interior part being ſomewhat larger than the cylin-
drical ferrule B.
na is the ſection of a ferrule of braſs, driven into the end of
the hollow cylinder, and which is adjuſted to that marked B,
ſo as to ſlide freely thereupon, but with as little ſhake as
poffible.
bb, dd, 88, repreſent the ſection of a braſs ferrule, plate, and
ſocket, fixed upon the other end of the hollow cylinder;
the ſocket d d being adjuſted to ſlide freely upon the cylin-
der E, in the ſame manner as the ferrule a a ſlides upon the
cylinder B: the outer end of the ſocket at
&g is formed into a ſort of button; by puſhing whereof, the
hollow cylinder will move backwards and forwards, or
turn round at pleaſure upon the cylindrical parts of the axis
B and E.
ce, ii, 00, repreſent the ſection of a braſs ferrule, alſo fixed upon
the hollow cylinder : the edge of this ferrule
is cut into teeth, in the manner of a contrate wheel; and
the edge thereof
+
0.0 is cut in the manner of a ratchet.
Of confequence, when the plate bddb is puſhed cloſe
to the ferrule D, the teeth of the ferrule ee will lay
hold of
G, a pin fixed into the axis ; by which means the hollow
cylinder is made to turn along with the wheel and axis :
but being drawn back by the button & 8, the hollow
cylinder is thereby diſengaged from the pin G, and ceaſes
turning.
B3
6
EXPERIMENTAL ENQUIRY, &c.
turning. Note, the weight in the ſcale is prevented from
running back, by a catch that plays in and lays hold of the
ratchet oo.
By this means the hollow cylinder upon which the cord
winds and raiſes the weight, is put in action and diſcharged
therefrom inſtantaneouſly, while the wheel is in motion:
for, without ſome contrivance of this kind, it would not
be eaſy to make this fort of experiments with any tolerable
degree of exactneſs.
The uſe of the apparatus now deſcribed will be rendered more
intelligible, by giving a general idea of what I had in view;
but as I ſhall be obliged to make uſe of a term which has herečo
fore been the cauſe of diſputation, I think it neceſſary to affign
the ſenſe in which I would be underſtood to uſe it; and in which
I apprehend it is uſed by practical Mechanicks.
9
The word Power, as uſed in practical mechanicks, I appre-
hend to ſignify the exertion of ſtrength, gravitation, impulſe, or
preſſure, ſo as to produce motion : and by means of ſtrength,
gravitation, impulſe, or preſſure, compounded with motion, to
be capable of producing an effect : and that no effect is pro-
perly mechanical, but what requires ſuch a kind of power to
produce it,
*
The raiſing of a weight, relative to the height to which it
can be raiſed in a given time, is the moſt proper meaſure of
power ; or, in other words, if the weight raiſed is multiplied by
the height to which it can be raiſed in a given time, the product
is the meaſure of the power raiſing it; and conſequently, all
thoſe powers are equal, whoſe products, made by ſuch multi-
plication, are equal : for if a power can raiſe twice the weight
to the ſame height; or the ſame weight to twice the height, in the
fame time that another power can, the firſt power is double the
ſecond
EXPERIMENTAL ENQUIRY, &că
fecond: and if a power can raiſe half the weight to double the
height; or double the weight to half the height, in the ſame time
that another can, thoſe two powers are equal. But note, all
this is to be underſtood in caſe of now or equable motion of the
body raiſed; for in quick, accelerated, or retarded motions, the
vis inertiæ of the matter moved will make a variation.
In comparing the effects produced by water-wheels with the
powers producing them; or, in other words, to know what
part of the original power is neceſſarily loſt in the application,
we muſt previouſly know how much of the power is ſpent in
overcoming the friction of the machinery, and the reſiſtance of
the air; alſo what is the real velocity of the water at the inſtant
that it ſtrikes the wheel ; and the real quantity of water ex-
pended in a given time,
From the velocity of the water, at the inſtant that it ſtrikes
the wheel, given, the height of head productive of ſuch velo.
city can be deduced, from acknowledged and experimented
principles of hydroſtatics : ſo that by multiplying the quantity,
or weight of water, really expended in a given time, by the
height of a head ſo obtained; which muſt be conſidered as the
height from which that weight of water had deſcended in that
given time; we ſhall have a product, equal to the original
power of the water, and clear of all uncertainty that would
ariſe from the friction of the water, in pafling ſmall apertures ;
and from all doubts, ariſing from the different meaſure of ſpout-
ing waters, aſſigned by different authors. On the other hand,
the ſum of the weights raiſed by the action of this water, and
of the weight required to overcome the friction and reſiſtance of
the machine, multiplied by the height to which the weight can
be raiſed in the time given, the product will be equal to the
effect of that power; and the proportion of the two products,
will be the proportion of the power to the effect: ſo that by
loading the wheel with different weights ſucceſlively, we hall
be
:
В4
.
EXPERIMENTAL ENQUIRY, &c.
...
be able to determine at what particular load, and velocity of the
wheel, the effect is a maximum.
The manner of finding the real velocity of the water, at the
inſtant of its ſtriking the wheel; the mariner of finding the va-
lue of the friction, reſiſtance, &c. in any given cafe; and the
manner of finding the real expence of water, fo far as concerns
the following experiments, without having recourſe to theory;
being matters upon which the following determinations depend,
it will be neceſſary to explain them.
To determine the Velocity of the Water ſtriking the
Wheel.
It has already been mentioned, in the references to the figures,
that weights are raiſed by a cord winding round a cylindrical
part of the axis. Firſt, then, let the wheel be put in motion by
the water, but without any weights in the ſcale ; and let the
number of turns in a minute be 60: now it is evident, that was
the wheel free from friction and reſiſtance, that 60 times the
circumference of the wheel would be the ſpace through which the
water would have moved in a minute : with that velocity where-
with it ſtruck the wheel : but the wheel being encumbered by
friction and reſiſtance, and yet moving 60 turns in a minute, it
is plain that the velocity of the water muſt have been greater
than 60 circumferences before it met with the wheel. Let now
the cord be wound round the cylinder, but contrary to the uſual
way, and put a weight in the ſcale ; the weight fo diſpoſed
(which may be called the counter-weight) will endeayour to afliſte
the wheel in turning the ſame way, as it would have been turn-
ed by the water: put therefore as much weight into the ſcala
as, without any water, will cauſe it to turn ſomewhat fafter
than at the rate of 60 turns in a minute; ſuppoſe 63; let it now
be tried again by the water, affifted by the weight; the wheel
there
.
:
)
EXPERIMENTAL ENQUIRY, &c.
therefore will now make more than 60 turns; ſuppoſe 64:
hence we conclude the water ſtill exerts fome power in giving
motion to the wheel. Let the weight be again increaſed, ſo as
to make 64 turns in a minute without water: let it once more
be tried with water as before; and ſuppoſe it now to make the
fame number of turns with water as without, viz. 64: hence
it is evident, that in this caſe the wheel makes the ſame number
of turns in a minute, as it would do if the wheel had no friction
or reſiſtance at all; becauſe the weight is equivalent thereto;
for was it too little the water would accelerate the wheel be.
yond the weight: and if too great, retard it; ſo that the water
now becomes a regulator of the wheel's motion; and the velo-
city of its circumference becomes a meaſure of the velocity of,
the water.
In like manner, in ſeeking the greateſt product, or maximum,
of effect; having found by trials what weight gives the greateſt
product, by ſimply multiplying the weight in the ſcale by the
number of turns of the wheel, find what weight in the ſcale,
when the cord is on the contrary ſide of the cylinder, will cauſe
the wheel to make the ſame number of turns the ſame
way,
with-
out water; it is evident that this weight will be nearly equal to
all friction and reſiſtance taken together; and conſequently, that
the weight in the ſcale, with twice * the weight of the ſcale, add-
ed to the back or counter-weight, will be equal to the weight
that could have been raiſed, ſuppoſing the machine had been
without friction or reſiſtance; and which multiplied by the
height to which it was raiſed, the product will be the greateſt
effect of that power.
3
.
• The weight of the ſcale makes part of the weight both ways.
}
The
:
EXPERIMENTAL ENQUIRY, &c.
The Quantity of Water expended is found thus :
The pump made uſe of for repleniſhing the head with water
was ſo carefully made, that, no water eſcaping back by the lea-
thers, it delivered the ſame quantity of water at every ſtroke,
whether worked quick or Now; and as the length of the
ſtroke was limited, conſequently the value of one ſtroke
(or, on account of more exactneſs, 12 ſtrokes) was known, by
the height to which the water was thereby raiſed in the head;
which, being of a regular figure, was eaſily meaſured. The
juice, by which the water was drawn upon the wheel, was
made to ſtop at certain heights by a peg; ſo that when the pey
was in the ſame hole, the aperture for the effluent water was
the fame. Hence the quantity of water expended by any given
head, and opening of the ſluice, may be obtained: for, by ob-
ſerving how many ſtrokes a minute was ſufficient to keep up
the ſurface of the water at the given height, and multiplying
the number of ſtrokes by the value of each, the water expended
by any given aperture and head in a given time will be given.
Theſe things will be further illuſtrated by going over the cal-
culus of one ſet of experiments.
Specimen of a Set of Experiments.
The ſluice drawn to the firſt hole.
The water above the floor of the fluice
30 Inches.
Strokes of the pump in a minute 391
The head raiſed by 12 ſtrokes
21 Inches.
The wheel raiſed the empty ſcale, and made turns in a minute 80
With a counter-weight of ilb. 8 oz. it made
85
Ditto tried with water
86
No.
EXPERIMENTAL ENQUIRY, &c;
No.
I
1
4
2
1
1
1
convenient
1
med
Weight. Tuns in a Min. Product.
lb. oz.
4 0
45
180
5
0
42
210
6 0
361 217
331
236
30
240 maximum.
9 0
26
238.1
22
220
16
181
I2
* ceaſed working
1
3
4
5
6
7 0
magamaminen
8 0
1
1
1
1
1
IO
O
Guamas
naman ang
7
8
0
9
Counter-weight, for 30 turns without water, 2 oz. in the ſcale.
N.B. The area of the head was 105,8 ſquare inches.
Weight of the empty ſcale and pulley, 10 oz.
Circumference of the cylinder, 9 inches.
Circumference of the water-wheel, 75 ditto.
Reduction of the above Set of Experiments.
The circumference of the wheel, 75 inches, multiplied by
86 turns, give 6450 inches for the velocity of the water in a
minute; is of which will be the velocity in a ſecond, equal to
107,5 inches, or 8,96 feet, which is due to a head of 15
inches t; and this we call the virtual or effective head.
The
N.B. When the wheel moves ſo ſlow as not to rid the water
fo faſt as fupplied by the ſluice, the accumulated water falls back up-
on the aperture, and the wheel immediately ceaſes moving.
+ This is determined upon the common maxim of hydroſtatics,
that the velocity of ſpouting waters is equal to the velocity that an
heavy
12
!
EXPERIMENTAL ENQUIRY, &c.
***
The area of the head being 105,8 inches, this multiplied by
the weight of water of the inch cubic, equal to the decimal
,579 of the ounce avoirdupoiſe, gives 61,26 ounces for the
weight of as much water, as is contained in the head, upon
1 inch in depth, i, of which is 3,83 pounds; this multiplied
by the depth 21 inches, gives 80,43 lb. for the value of 12
ſtrokes; and by proportion, 397 (the number made in a mi-
nute) will give 264,7 lb. the weight of water expended in a
minute.
Now as 264,7 lb. of water may be conſidered as having de-
fcended through a ſpace of 15 inches in a minute, the product
of theſe two numbers 3970 will expreſs the power of the water
to produce mechanical effects; which were as follows:
The velocity of the wheel at the maximum, as appears above,
was 30 turns a minute; which multiplied by 9 inches, the cir-
cumference of the cylinder, makes 270 inches; but as the ſcale
was hung by a pulley and double line, the weight was only
raiſed half of this, viz. 135 inches.
The weight in the ſcale at the maximum 8 lb. 00Z.
Weight of the ſcale and pulley
O 10
Counter-weight, ſcale, and pulley O 12
Sum of the reſiſtance
9
6
or lb.9,375.
Now as 9,375 lb. is raiſed 135 inches, theſe two numbers being
multiplied together, the product is 1266, which expreſſes the
effect produced at a maximum: ſo that the proportion of the
power to the effect is as 3970 : 1266, or as 10:3,18.
heavy body would acquire in falling from the height of the reſer-
voir; and is proved by the riſing of jets to the height of their
reſervoirs nearly.
But
.
EXPERIMENTAL ENQUIRY, &c.
13
7
But though this is the greateſt ſingle effect producible from
the power mentioned, by the impulſe of the water upon an un-
derſhot wheel; yet, as the whole power of the water is not ex-
hauſted thereby, this will not be the true ratio between the power
of the water, and the ſum of all the effects producible therefrom:
for as the water muft neceſſarily leave the wheel with a ve-
locity equal to the wheel's circumference, it is plain that ſome
part of the power of the water muſt remain after quitting the
wheel.
The velocity of the wheel at the maximum is 30 turns a
minute; and conſequently its circumference moves at the rate
of 3,123 feet a ſecond, which anſwers to a head 1,82 inches ;
this being multiplied by the expence of water in a minute, viz.
264,7 lb. produces 481 for the power remaining in the water
after it has paſſed the wheel: this being therefore deducted from
- the original power 3970, leaves 3489, which is that part
of the power which is ſpent in producing the effect 1266; and
conſequently the part of the power ſpent in producing the
effect, is to the greateſt effect
the greateſt effect producible' thereby as
3489 : 1266 :: 10 : 3,62, or as 11 to 4.
The velocity of the water ſtriking the wheel has been deter-
mined to be equal to 86 circumferences of the wheel per mi-
nute, and the velocity of the wheel at the maximum to be 30;
the velocity of the water will therefore be to that of the wheel
as 86 to 30; or as 10 to 3,5, or as 20 to 7.
The load at the maximum has been ſhown to be equal to 9 lb.
6. oz. and that the wheel ceafed moving with 12 lb. in the
ſcale : to which if the weight of the fcale is added, viz. 10
ounces *, the proportion will be nearly as 3 to 4 between
* The reſiſtance of the air in this caſe ceaſes, and the friction is
not added, as 12 lb. in the ſcale was ſufficient to ſtop the wheel af.
ter it had been in full motion ; and therefore ſomewhat more than
a counterbalance to the impulſe of the water.
the
14
EXPERIMENTAL ENQUIRY, &c.
2
the load at the maximum and that by which the wheel is
ſtopped.
It is ſomewhat remarkable, that though the velocity of the
wheel in relation to the water turns out greater than of the
velocity of the water, yet the impulſe of the water in the caſe of a
maximum is more than double of what is aſſigned by theory;
that is, inſtead of of the column, it is nearly equal to the
whole column.
:
It muſt be remembered, therefore, that in the preſent caſe,
the wheel was not placed in an open river, where the natural
current, after it has communicated its impulſe to the Aoat, has
room on all fides to eſcape, as the theory ſuppoſes; but in a
conduit, or race, to which the float being adapted, the water
cannot otherwiſe eſcape than by moving along with the wheel.
It is obſervable, that a wheel working in this manner, as ſoon as
the water meets the float, receiving a ſudden check, it riſes up
againſt the float, like a wave againſt a fixed object; inſomuch
that when the ſheet of water is not a quarter of an inch thick
before it meets the float, yet this ſheet will act upon the whole
furface of a float, whoſe height is 3 inches; and conſequently
was the float no higher than the thickneſs of the ſheet of water,
as the theory alſo ſuppoſes, a great part of the force would have
been loft, by the water daſhing over the float*.
In
:
. Since the above was written, I find that Profeſſor Euler, in
the Berlin Acts for the year 1748, in a memoir entitled Maxims
pour aranger le plus avantageuſement les machines deſtinees a elever.
de l'eau par le moyen de pompes, page 192. $ 9. has the following
paſſage; which ſeems to be the more remarkable, as I do not find
he has given any demonſtration of the principle therein contained,
either from theory or experiment; or has made any uſe thereof in
his calculations on this ſubject : Cependant dans ce cas puiſque
“ l'eau eſt reſiechie, & qu'elle decoule ſur les aubes vers les cotés,
" elle y exerce encore une force particuliere, dont l'effet de l'ima
“ pulfion
.
魔
​.
하
​that is or it res . . . . . diere is . . . . . . ."
13
T A B L Ê 1.
7
In
.
02.
I 33
21 30
lb. oz. lb.
13 IOIO
I2 109
II 21 8
9
8
1266 10:32
6 243,
In.
88 15,85 30,
86 15,0
30,
3 27
821327
28,
4 24 78 12,3 27,7
5 21 75 | 1194
25,9
6) 18
70 9,95 23,5
7 15 65
8,54 23,4
8 12 bo
7,29 22,
9 9 52 5947 19
IO
3,55 16,
IO 7
IO 6
6 10 5
9 275, 4358
6 264,7 3970
3329
5) 235, 2890
5) 214, 2439
51992 1970
4 178,51524
5 161, 1173
733
404,7
1411 10:3,24 10:34 10:7,75
10:3,5 10:7,4
1044 10:3,15 10:34 10:7,5
901,4 10:3,12 10:3,55 10:7,53 At
735,7 10:3,02 10:3,45 10:7,32 the
561,8 10:2,85 10:3,36 10:8,02 iſt
442,5 10:2,9 10:3,6 10:8,3 hole.
328 10:2,8 10:3,77 10:9,1
213,7 10:2,9 10:3,6510:9,1
117 10:2,82 10:3,8 10:9,3
24
5
3
2
10 3
6 | 42
1 2 2
I 2 I
81 1342
IO 1142
I
81 13,5
62972
13/ 18
10 8
11 24 84 14,2 30,75 13 IOIO 141 342, 4890
I2 21
29, II IO
4009
72 10,5 26,
9
7 285,
2993
14 15 69 9,6 25, 7
10 6 141 277,
15 12 63 8,0 25, 5 10 4 141234,
1872
16 56 6,37 23,
4 0 3 13 201,
1280
17
4,25 21,
2 8) 2 4 167,5 712
2659
1505 10:3,075 10:3,66 10:7,9
1223 10:3,01 10:3,62 10:8,05
975 10:3,25 10:3,6 10:8,75 At
774 10:2,92 10:3,62 10:9, the
549 10:2,94 10:3,97 10:8,7 2d.
390 10:3,05 10:4,1 10:9,5
212 10:2,98 10:4,55 10:9,
46
18) 15
I 210
72 10,5 29, II
191 12
66 8,75 26,75 | 8
20
91 58
6,8
24,5 5
21
6 48 437 23,5 | 3
1019
IO 7
85
21 3
61 357)
61 330,
0 255
o 228,
9,3
21 8
61 359,
22) 12
68
23
9 / 58
24
27,
26,25
24,5
6,8
3748
10:3,23 10:4,02 10:8,05
2887 878 10:3,05 10:4,05 10:8,1 The
1734 541 10:3,01 10:4,22 10:9,1 3d.
1064 317 10:2,99 10:4,9 10:9,6
3338 1006 10:3,02 10:3,97|10:9,17
2257
686 10:3,04 10:4,52 10:9,5 4th.
1231 385 10:3,13 10:5,1 10:9,35
2588
783 10:3,03 10:495510:9,45 sth.
1544 450 10:2,92 10:4,9 10:9,3
9
6
3
6 48
21 5
1 2 3
13 332,
8 262,
437
3
25)
26
9
6
60
50
7,29 27,3
5,03 24,6
6 1216
4
6 355,
1 307,
4 6
|
27
6
50
5,03 26,
4 15) 4
9 360,
1811
534 10:2,95 10:5,2 10:9,256th.
I.
2.
3
4
.
5.
6.
7.
8.
9.
IO.
II.
I 2.
13
To face page 15.
EXPERIMENTAL ENQUIRY, &c.
IS
In further confirmation of what is already delivered, I have
adjoined a table, (TABLE I.) containing the reſult of 27 ſets of
experiments, made and reduced in the manner above ſpecified.
What remains of the theory of underſhot wheels, will naturally
follow from a compariſon of the different experiments to-
gether.
Maxims and Obſervations deduced from the foregoing
Table of Experiments.
:
Maxim I. That the virtual or effe Etive head being the ſame,
the effect will be nearly as the quantity of water expended.
This will appear by comparing the contents of the columns
4, 8, and 10, in the foregoing ſets of experiments; as for
Example ift, taken from Nº. 8. and 25, viz.
No.
8.
Virtual Head. Water expended.
7,29
161
7,29
355
Effect
328
785
25
Now the heads being equal, if the effects are proportioned to
the water expended, we ſhall have by maxim iſt, 161 : 355 : :
328 : 723; but 723 falls ſhort of 785, as it turns out in ex-
periment, according to Nº. 25, by 62; the effect therefore of
“ pulfion ſera augmenté; & experience jointe a la theorie a fait
“ voir que dans ce cas, la force eſt preſque double : de forte qui'il
« faut prendre le double de le ſection du fil d'eau pour ce qui repond
“ dans ce cas a le ſurface des aubes, poârvu qu'elles ſoient aſſez
“ larges pour recevoir ce ſupplement de force. Car fi les aubes
« nétoient plus larges que le fil, on trait d'eau on ne devroit prendre
que ne ſimple ſection, tout comme dans le premier cas, on l'aube
toute entire eſt pappee par l'eau.”
NO
1
3
3
16
EXPERIMENTAL ENQUIRY, &c.
.:
Hence
Examples.
No Table I.
Virtual
Head.
Expence of
Water.
Effect,
Care
Compariſon
Variation.
Proportional
Variation.
Inch.
Iſt Š 81 7,29
i
Nº.25, compared with Nº. 8, is greater than according to the
preſent maxim in the ratio of 14 to 03:54
The foregoing example, 'with four fimilar ones, are ſeen at
one view. in the following Table.
225
7,29
62+
14 :
13
2d 131 10,5
owiu
b
161
355
285
357
255
332
228
262
II .
I2I : 122
i
3d
22
23
23
328
161
785)
: 355:: 328 : 723
1275 285: 357 :: 975: 1221
546 } 255: 332 : : 541 : 704
317?
385)
228 : 262 :: 317: 364 | 21+
$34} 307: 360 :: 450 : 531
3+
18.
38 :
10,5
6,8
6,8
4,7
4,7
5,03
5,03
39
4th
18 :
17
5th
307
366
178 : 177
$
HC
EXPERIMENTAL ENQUIRY, &c.
37
Hence therefore, in comparing different experiments, as ſome
fall ſhort, and others exceed the maximum, and all agree therewith,
as near as can be expected, in an affair where ſo many different
circumſtances are concerned, we may, according to the laws of
reaſoning by induction, conclude the maxim true; viz. that the
effects are nearly as the quantity of water expended.
Maxim II. That the expence of water being the ſame, the
effe it will be nearly as the height of the virtual or effective heads
This alſo will appear by comparing the contents of columns
4, 8, and 10, in any of the ſets of experiments.
Example ift, of Nº.2, and Nº. 24; viz.
NO
Virt. Head.
2
15
* 4:7
Expence.
264,7
262
Effect.
1266
385
24
Now as the expenses are not quite equal, we muſt proportion
one of the effects accordingly: thus
by maxim iſt, 262 : 264,7 : : 385 : 389
and by max. 2d, 15 : 4,7 : 1 1266 : 397
Difference
8
The effect therefore of Nº. 24, compared with Nº. 2, is lefs
than according to the preſent maxim in the ratio of 49:50.
c
The
S
Maxim
Compariſon.
Variation.
Proportional
Variation.
EXPERIMENTAL ENQUIRY, &c.
The foregoing, and two other ſimilar examples, are compriſed
in the following Table:
18
ift{24
2
4,71 262
!
2d s
LI
15 264,7 1266 Max. Ift
, 262 : 26427 385 : 319} 8 – 49: 50
385 Max. 2d, 15 : : : 397
1 15,85 275 14117 Max. iſt, 114 : 275 :: 117 : 282
114
3,55
117 ) Max. 2d, 15,85 : 3,55 :: 1411 : 316 $ 34–
} 34–8:
14,2 342 1505 Max. Ift, 167,5 : 342 ::
4,25 / 1675 212] Max. 2d, 1432 : 4.25 :: 1505 : 455 } 17–25: 26
8: 9
3d
II
17
:
:
EXPERIMENTAL ENQUIRY, &c.
19
Maxim III. That the quantity of water expended being the
fame, the effect is nearly as the ſquare of its velocity.
This will appear by comparing the contents of columns 3,
8, and 10, in any of the ſets of experiments; as for
Example ist, of Nº. 2, with Nº. 24, viz.
Nº.
2
Turns in a min.
- 86
48
Expence.
264,7
262
Effect.
1266
385
24
The velocity being as the number of turns, we ſhall have,
by max. iſt,
: 385 1 389
and by max. 3d, { 1396 : 2304
262 : 264,7
862 : 482
: 2304
}
: 1266 : 394
Difference
5
The effect therefore of NÓ. 24, compared with Nº. 2, is leſs
than by the preſent maxim in the ratio of 78 : 79.
The foregoing, and three other ſimilar examples, are com-
priſed in the following Table :
Examples
20
EXPERIMENTAL ENQUIRY, &c.
Maxim
Examples.
Compariſon
Variation.
Proportional
Variation.
2 86 264,7 1266
241 48 262
385}} {Max, it
Max. 3d, {-1396:2304
262 : 264,7
86° : 487
7396
: : 385 : 389
: : 1266: 394
} 5–78:79
}
{
Max. iſt,
2d
:: 117
I14: 275
i 88
42
275
114
1411
117
}{Max. In
Max
. 3a, [1994: 1964
: 28313394758
4 }
}:: 1411: 321
:: 212
{li
3d
11 84 | 342 150512 S Max. Ift, 167,5 : 342
842
171 46 167,5 212
,
{ Max
. ift, 4625 : 342
: 433}:
Max. 3d, 2056 : 2116
}18–24 : 25
} :: 1505 :
:: 1505 : 451
18172
,
228 : 357
72
21 48
4th
357
228
499}
131;} {Max
. It
Max. 3d, {5732348
:: 317 : 496
: 1210: 538
}42–12:
42-12 : 13
5184 : 2301
}
ber
EXPERIMENTAL ENQUIRY, &c.
21
ܕ
Maxim IV. The aperture being the ſame, the effe&t will be
nearly as the cube of the velocity of the water.
This alſo will appear by comparing the contents of columns
3, 8, and 10; as for
Example iſt, of Nº. 1, and Nº. 10, viz.
N
Effect.
Turns.
88
I
Expence.
275
114
1411
117
10
42
Lemma. It muſt here be obſerved, that if water paſſes out
of an aperture, in the ſame ſection, but with different velocities
the expence will be proportional to the velocity; and therefore
converſely, if the expence is not proportional to the velocity, the
ſection of the water is not the ſame.
Now comparing the water diſcharged with the turns of Nº.
I, and 10, we ſhall have 88:42 :: 275 : 131,2; but the
water diſcharged by Nº. 10, is only 114 lb. therefore, though
the ſluice was drawn to the ſame height in Nº. 10, as in Nº. 1,
yet the ſection of the water paſſing out, was leſs in Nº. 10,
than Nº. 1, in the proportion of 114 to 131,2; conſequently
had the effective aperture or ſection of the water been the ſame
in Nº. 10, as in Nº. 1, ſo that 131,2 lb. of water had been
diſcharged inſtead of 114, the effect would have been increaſed
in the ſame proportion; that is
:
by the Lemma, 88
42:
by maxim iſt, 114 : 131,2 :
.
: 275
131,2
117 : 134,5
:: 1411 : 153,5
and by max. 4th, {681472 : 74088}
:
423
Difference
19
C3
Thc
22
EXPERIMENTAL ENQUIRY, &c.
Compariſon.
Variation.
Proportional
Variation.
Obſervations,
:
:
:
131,27
Iſt Ş 1 88
Lemma.
88
s
42
275 1411
Max. I. 114: 131,2
114 117 Max. 4. 883:
:
: 275
117
: 1411
134,5
IO 42
423
.
:
153,55119-17:8
The effect therefore of 10, compared with Nº. I, is leſs than
it ought to be by the preſent maxim in the ratio of 7: 8.
The foregoing, and three other ſimilar examples, are con-
tained in the following Table.
.
187,3
2d
S11 84 342 1505
2 17 46 167,5 212
S Lemma. 84:
84 : 46 :
Max. I. 167,5 : 187,3 :
Max. 4. 843 : 463 :
: 342
: 2 12
; 1505
237
247
10–23:24
*
181 72
23
3d
357 1210 S Lemma.
Lemma. 72 : 48
Max. I. 228 : .238 :
317||
Max. 4. 723 : 48
: 357
•
317
: 1210
:
21 48
228
331
355
24--14:15
3
:
:
4th
359 1006
22 68
24 48
Lemma. 68 : 48 :
Max. 1. 262 : 253,4 :
: 483
359 :
: 385 :
262 385 Max. 4. 683
253,4
372
354
18+ 20:19
0:19
.
: 1006
:
ha
EXPERIMENTAL ENQUIRY, &c.
23
Obſervations.
Obſerv. ift. On comparing column 2d and 4th, Tab. I. it
is evident that the virtual head bears no certain proportion to
the head of water; but that when the aperture is greater, or the
velocity of the water iſſuing therefrom leſs, they approach
nearer to a coincidence; and conſequently in the large openings
of mills and ſluices, where great quantities of water are diſ-
charged from moderate heads, the head of water, and virtual
head determined from the velocity, will nearly agree, as ex-
perience confirms,
f
Obſerp. 2d. Upon comparing the ſeveral proportions be-
tween the power and effe&t in column 11th, the moſt general is
that of 10 to 3; the extremes 10 to 3,2 and 10 to 2,8; but as
it is obſervable, that where the quantity of water, or the ve-
locity thereof; that is, where the power is greateſt, the 2d
term of the ratio is greateſt alſo: we may therefore well allow
the proportion fubfiſting in large works, as 3 to 1.
Obferv. 3d. The proportions of velocities between the water
and wheel in column 12, are contained in the limits of 3 to i
and 2 to 1; but as the greater velocities approach the limit of
3 to 1, and the greater quantity of water approach to that of
2 to I, the beſt general proportion will be that of 5 to 2.
Obſerv. 4th. On comparing the numbers in column 13, it
appears, that there is no certain ratio between the load that the
wheel will carry at its maximum, and what will totally ſtop it;
but that they are contained within the limits of 20 to 19, and
of 20 to 15; but as the effect approaches neareſt to the ratio of
20 to 15, or of 4 to 3, when the power is greateſt, whether by
increaſe of velocity, or quantity of water, this ſeems to be the
moſt applicable to large works; but as the load that a wheel
ought to have, in order to work to the beſt advantage, can be
affigned,
C4
24
EXPERIMENTAL ENQUIRY, &c.
aſſigned, by knowing the effect, it ought to produce, and the
velocity it ought to have in producing it; the exact knowledge of
the greateſt load it will bear, is of the leſs conſequence in
practice.
It is to be noted, that in all the examples under the three laſt
of the four preceding maxims, the effect of the leffer power
falls ſhort of its due proportion to the greater, when compared
by its maxim; except the laſt example of maxim 4th: and
hence, if the experiments are taken ſtrictly, we muſt infer,
that the effects increaſe and diminiſh in an higher ratio than thoſe
maxims ſuppoſe: but as the deviation is not very conſiderable,
the greateſt being about 1-8th of the quantity in queſtion; and
as it is not eaſy to make experiments of ſo compounded a nature
with abſolute preciſion; we may rather fuppoſe, that the leſſer
power is attended with ſome friction, or works under fome dif-
advantage, which has not been duly accounted for; and therefore
we may conclude, that theſe maxims will hold very nearly,
when applied to works in large.
After the experiments above mentioned were tried, the wheel,
which had originally 24 floats, was reduced to twelve; which
cauſed a diminution in the effect, on account of a greater
quantity of water eſcaping between the floats and the floor; but
a circular ſweep being adapted thereto, of ſuch a length, that
one foat entered the curve before the preceding one quitted it,
the effect came fo near to the former, as not to give hopes of
advancing it by increaſing the number of floats beyond 24 in
this particular wheel.
i
PART
:
EXPERIMENTAL ENQUIRY, &c.
25
PART II..
Concerning OVERSHOT Wheels.
Read before the Royal Society, May 24, 1759.
IN
N the former part of this eſſay, we have conſidered the im-
pulſe of a confined ſtream, acting on Underſhot Wheels. We
now proceed to examine the power and application of water, when
acting by its gravity on Overſpot Wheels.
*
In reaſoning without experiment, one might be led to ima-
gine, that however different the mode of application is; yet that
whenever the fame quantity of water deſcends through the ſame
perpendicular ſpace, that the natural effective power would be
equal : ſuppoſing the machinery free from friction, equally cal-
çulated to receive the full effect of the power, and to make the
moſt of it: for if we ſuppoſe the height of a column of water to
be 30 inches, and reſting upon a baſe or aperture of one inch
fquare, every cubic inch of water that departs therefrom will
acquire the ſame velocity, or momentum, from the uniform pref-
fure of
30
cubic inches above it, that one cubic inch let fall from
the top will acquire in falling down to the level of the aperture;
viz. ſuch a velocity as, in a contrary direction, would carry
it to the level from whence it fell *; one would therefore
ſuppoſe, that a cubic inch of water, let fall through a ſpace of
30 inches, and there impinging upon another body, would be
capable of producing an equal effect by colliſion, as if the ſame
cubic inch had deſcended through the ſame ſpace with a flower
motion, and produced its effects gradually: for in both caſes
gravity acts upon an equal quantity of matter, through an equal
* This is a conſequence of the riſing of jets to the height of their
reſervoirs nearly,
ſpace;
26
EXPERIMENTAL ENQUIRY, &c.
ſpace * ; and conſequently, that whatever was the ratio between
the power and effect in underſhot wheels, the fame would obtain
in overſhot, and indeed in all others : yet, however concluſive
this reaſoning may ſeem, it will appear in the courſe of the follow-
ing deductions, that the effect of the gravity of deſcending bodies
is very different from the effect of the ſtroke of ſuch as are non
elaſtic, though generated by an equal mechanical power.
The alterations in the machinery already deſcribed, to accom-
modate the fame for experiments on overſhot wheels, were prin-
cipally as follows:
Plate II. Fig. 2. The ſluice I b being ſhut down, the rod
HI was unſcrewed and taken off.
The underſhot water-wheel was taken off the axis, and in-
ſtead thereof an overſhot wheel of the ſame diameter was put
into its place. Note, This wheel was two inches in the ſhroud
or depth of the bucket; the number of the buckets was 36.
The ſtandards S and T, Fig. 1. were raiſed half an inch, ſo
that the bottom of the wheel might be clear of ſtagnant water.
A trunk, for bringing the water upon the wheel, was fixed
according to the dotted lines f g, Fig. 2. The aperture was
adjuſted by a fhuttle hi, which alſo cloſed up the outer end of
the trunk, when the water was to be ſtopped.
Fig. 3. The ratchet oog not being of one piece of metal
with the ferrule e e, i i (though ſo deſcribed before, to prevent
unneceſſary diſtinctions), was with its catch turned the contrary
fide ; conſequently the moveable barrel would do its office
equally, notwithſtanding the water-wheel, when at work, moved
the contrary way.
• Gravity, it is true, acts a longer ſpace of time upon the body
that deſcends ſlow than upon that which falls quick; but this can-
not occaſion the difference in the effect : for an elaſtic body falling
through the ſame ſpace in the ſame time, will, by colliſion upon
another elaſtic body, rebound nearly to the height from which it
fell; or, by communicating its motion, cauſe an equal one to aſ-
cend to the fame height.
Specimen
2 게
​EXPERIMENTAL ENQUIRY, &c.
27
4
I
2
I
2
s
III
111
Specimen of a Set of Experiments.
Head 6 inches.
141 ſtrokes of the pump in a minute, 12 ditto = 80 lb.
Weight of the ſcale (being wet) 10 oz.
Counterweight for 20 turns, beſides the ſcale, 3 oz.
Weight in
No. the Scale. Turns. Product. Obſervations.
oilb. 60
Threw moſt part
56
of the water out
3
52
of the wheel.
4
3 49
147
Received the water
5
4 47
188 more quietly.
6
5 45
225
7
6.
421 255
8
7
41
287
9 8
38.1
IO
9 361 3281
II
IO 352 355
I 2
32 3602
13
314
375
14 13 28-1 3705
15
14 272 385
16
15
26
390
17
16
241 392
18
17 22 386
19
18 213
20 19 203 3946?
maximum.
21
20
193
395
22
21
184
3887
23
22
18 396 Worked irregular.
24 23 Overſet by its load.
308
|||||
||||||
3
II
I 2
3911
}
* The ſmall difference, in the value of 12 ſtrokes of the pump,
from the former experiments, was owing to a ſmall difference in the
length of the ſtroke, occaſioned by the warping of the wood.
Reduction
20
EXPERIMENTAL ENQUIRY, &c.
Reduetion of the preceding Specimen.
In theſe experiments the head being 6 inches, and the height
of the wheel 24 inches, the whole deſcent will be 30 inches :
the expence of water was 14 i ſtrokes of the pump in a minute,
whereof 12 contained 80 lb.; therefore the water expended in a
minute was 96 lb. which multiplied by 30 inches, gives the
power = 2900.
If we take the 20th experiment for the maximum, we fhalt
have 20 turns in a minute, each of which raiſed the weight
4 inches, that is, 93,37 inches in a minute. The weight in
the ſcale was 19 lb. the weight of the ſcale po oz. ; the coun-
terweight 3 oz. in the ſcale, which, with the weight of the ſcale
10 oz. makes in the whole 20 lb. which is the whole refift-
ance or load: this, multiplied by 93,37 inches, makes 1914 for
the effect.
The ratio therefore of the power and effect will be as
2900 : 1914, or as 10:6,6, or as 3 : 2 nearly.
But if we compute the power from the height of the wheel
only, we ſhall have 96 lb. multiplied by 24 inches = 2320 for
the power, and this will be to the effect as 2320 : 1914, or as
10: 82, or as 5:4 nearly,
The reduction of this ſpecimen is ſet down in Nº. 9, of the
following Table; and the reſt were deducted from a ſimilar ſet
of experiments, reduced in the ſame manner.
TABLE
EXPERIMENTAL ENQUIRY, &C.
29
10
:
8
TABLE II. containing the Reſult of Sixteen Sets of
Experiments on Overſhot Wheels.
1
I
2
Inch. lb.
lb.
27 30 19 6 810720 556 10:6,910: 7,7
27 56 | 1614715301360106010:6,910:7,8
3 27 5620
20124153013601167 10:7,610:8,4
427 31 204 13117101524124510:7,310: 8,2
5 27 76 21: 15520701840 150010: 7,3 10:8,23
6 28.73 181 1742090 1764 147610:710: 8,4)
7 281 96 205 20427552320 1868 10:6,810:8,
10:8,2 | Medium 10:8,1 | Mean ratio.
ܕ܀
8 30
20
90 19.2700 2160 1755 10:6,510:8,1
9 30 96.3 2020 2900 2320 1914 10:6,610: 8,2
10 30 113 21
10:8,5 10:8,5
I.
2.
4.
IO.
II
11 33 56 204 13118701360123010:6,610:9,
12 33 1063 22 2143520 2560 215310:6,110:8,4
13 33 146 | 23 | 27748403520 284610:5,910:8,1
14 35 65 197| 16422751560 146610: 6,5 10:9,4
15 35 120
21 | 25+4200 28802467 10:5,9 10:8,6
16 35 1637| 25 265572839242981 10:5210:7,6
3.
5.
16.
7.
8.
9.
Obſervations and Deductions from the foregoing
Experiments.
1. Concerning the Ratio between the Power and Effect
of Overſhot Wheels.
The effective power of the water muſt be reckoned upon the
whole deſcent; becauſe it muſt be raiſed that height, in order to
be in a condition of producing the ſame effect a ſecond time.
The
है
30
EXPERIMENTAL ENQUIRY, &c.
The ratios between the powers fo eſtimated, and the effects
at the maximum deduced from the ſeveral ſets of experiments,
are exhibited at one view in column 9. of Table II.; and
from hence it appears, that thoſe ratios differ from that of 10
to 7,6 to that of 10: 5,2, that is, nearly from 4:3 to 4: 2. In
thoſe experiments where the heads of water and quantities ex-
pended are leaſt, the proportion is nearly as 4:3; but where
the heads and quantities are greateſt, it approaches nearer to
that of 4 : 2; and by a medium of the whole, the ratio is that of
3:2 nearly. We have ſeen before, in our obſervations upon the
effects of underſhot wheels, that the general ratio of the power
to the effect, when greateſt was 3: 1; the effect therefore of
over
hot wheels, under the ſame circumſtances of quantity and
fall, is at a medium double to that of the underſhot : and, as a
conſequence thereof, that non-elaſtic bodies, when atting by their
impulſe or colliſion, communicate only a part of their original
power; the other part being ſpent in changing their figure in
conſequence of the ſtroke.
The powers of water computed from the height of the wheel,
only compared with the effects, as in column 10, appear to ob-
ſerye a more conſtant ratio: for if we take the medium of each
claſs, which is ſet down in column 11, we ſhall find the extremes
to differ no more than from the ratio of 10: 8,1 to that of
10:8,5; and as the ſecond term of the ratio gradually increaſes
from 8,1 to 8,5, by an increaſe of head from 3 inches to II, the
exceſs of 8,5 above 8,1 is to be imputed to the ſuperior impulſe
of the water at the head of 1 inches above that of 3 inches :
ſo that if we reduce 8,1 to 8, on account of the impulſe of the
3 inch head, we hall have the ratio of the power, computed
upon the height of the wheel only, to the effect at a maximum as
10:8, or as 5:4 nearly: and from the quality of the ratio
between power and effect, ſubſiſting where the conſtructions
are fimilar, we muſt infer, that the effects, as well as the powers,
are as the quantities of water and perpendicular heights multi-
plied together reſpectively,
II. Cena
EXPERIMENTAL ENQUIRY, &c.
31
-
II. Concerning the moſt proper Height of the Wheel in
Proportion to the whole Deſcent.
We have already ſeen, from the preceding obſervation, that
the effect of the ſame quantity of water, deſcending through the
fame perpendicular ſpace, is double, when acting by its gravity
upon an overſhot wheel, to what the ſame produces when acting by
its impulſe upon an underſhot. It alſo appears, that by increaſing
the head from 3 inches to IT, that is, the whole deſcent, from 27
inches to 35, or in the ratio of 7 to g nearly, the effect is advanced
no more than in the ratio of 8,1 to 8,4 that is, as 7:7,26; and
conſequently the increaſe of effect as not 1-7th of the increaſe of
perpendicular height. Hence it follows, that the higher the wheel
is in proportion to the whole deſcent, the greater will be the effett;
becauſe it depends leſs upon the impulſe of the head, and more
upon the gravity of the water in the buckets: and if we conſider
how obliquely the water iſſuing from the head muſt ſtrike the
buckets, we ſhall not be at a loſs to account for the little ad
vantage that ariſes from the impulſe thereof; and ſhall immedi-
ately ſee of how little conſequence this impulſe is to the effect of
an overſhot wheel. However, as every thing has its limits, ſo
has this: for thus much is deſirable that the water ſhould have
fomewhat greater velocity, than the circumference of the wheel, in
coming thereon; otherwiſe the wheel will not only be retarded,
by the buckets ſtriking the water, but thereby daſhing a part of
it over, ſo much of the power is loft.
The velocity that the circumference of the wheel ought to
have, being known by the following deductions, the head requi-
fite to give the water its proper velocity is eaſily computed from
the common rules of hydroſtatics; and will be found much leſs
than what is generally practiſed.
HII. Con
32
EXPERIMENTAL ENQUIRY, &c.
III. Concerning the Velocity of the Circumference of tbe
Wheel, in order to produce the greateſt Effect.
If a body is let fall freely from the ſurface of the head to the
bottom of the defcent, it will take a certain time in falling;
and in this caſe the whole action of gravity is ſpent in giving
the body a certain velocity: but if this body in falling is made
to act upon ſome other body, ſo as to produce a mechanical
effect, the falling body will be retarded; becauſe a part of the
action of gravity is then ſpent in producing the effect, and the
remainder only giving motion to the falling body: and therefore
the power a body defcends, the greater will be the portion of the
action of gravity applicable to the producing a mechanicaleffe£t;
and in conſequence the greater that effect may be.
If a ſtream of water falls into the bucket of an overſhot
wheel, it is there retained until the wheel by moving round
diſcharges it: of conſequence the flower the wheel moves, the
more water each bucket will receive: ſo that what is loft in
ſpeed, is gained by the preſſure of a greater quantity of water
acting in the buckets at once: and, if conſidered only in this
light, the mechanical power of an overſhot wheel to produce effects
will be equal whether it moves quick or Now: but if we attend
to what has been juſt now obſerved of the falling body, it will
appear that ſo much of the action of gravity, as is employed in
giving the wheel and water therein a greater velocity, muſt be
fubtracted from its preſſure upon the buckets; ſo that, though
the product made by multiplying the number of cubic inches
of water acting in the wheel at once by its velocity will be the
ſame in all caſes; yet, as each cubic inch, when the velocity is
greater does not preſs ſo much upon the bucket as when it is
lefs, the power of the water to produce effects will be greater in
the leſs velocity than in the greater: and hence we are led to
this general rule, that, cæteris paribus, the leſs the velocity of the
wheel, the greater will be the effect thereof. A confirmation of
this
EXPERIMENTAL ENQUIRY, &c.
33
this doctrine, together with the limits it is ſubject to in practice,
may be deduced from the foregoing ſpecimen of a ſet of expe-
riments.
From theſe experiments it appears that, when the wheel made
about 20 turns in a minute, the effect was, near upon, the
greateſt. When it made 30 turns, the effect was diminiſhed
about to
zó part; but that when it made 40, it was diminiſhed
about 4 when it made leſs than 18., its motion was irregular;
and when it was loaded ſo as not to admit its making 18 turns,
the wheel was overpowered by its load.
It is an advantage in practice, that the velocity of the wheel
ſhould not be diminiſhed further than what will procure fome
folid advantage in point of power: becauſe, cæteris paribus, as
the motion is ſlower, the buckets muſt be made larger ; and the
wheel being more loaded with water, the ſtrefs upon every part
of the work will be increaſed in proportion : The beſt velocity for
practice therefore will be ſuch, as when the wheel here uſed
made about 30 turns in a minute ; that is, when the velocity of
the circumference is a little more than 3 feet in a ſecond.
Experience confirms, that this velocity of 3 feet in a ſecond
is applicable to the higheſt overſhot wheels, as well as the loweſt;
and all other parts of the work being properly adapted thereto,
will produce very nearly the greateſt effect poſſible: however
this alſo is certain from experience, that high wheels may deviate
further from this rule, before they will loſe their power, by a given
aliquot part of the whole, than low ones can be admitted to do ;
for a wheel of 24 feet high may move at the rate of ſix feet.
per ſecond without loſing any conſiderable part of its power*;
* The 24 feet wheel going at 6 feet in a ſecond, ſeems owing to
the ſmall proportion that the head (requiſite to give the water the
proper velocicy of the wheel) bears to the whole height.
D
and
:
6
34
EXPERIMENTAL ENQUIRY, &c.
1
and on the other hand, I have ſeen a wheel of 33 feet high,
that has moved very ſteadily and well with a velocity but little
exceeding 2 feet.
IV. Concerning the Load for an Overſhot Wheel,
in Order that it may produce a Maximum.
The maximum load for an overſhot wheel, is that which re-
duces the circumferences of the wheel to its proper velocity; and
this will be known, by dividing the effect it ought to produce in
a given time by the ſpace intended to be deſcribed by the circum-
ference of the wheel in the ſame time: the quotient will be the
reſiſtance overcome at the circumference of the wheel; and is
equal to the load required, the friction and reſiſtance of the ma-
chinery included.
-
V. Concerning the greateſt poſible Velocity of an
Overſhot Wheel.
The greateſt velocity that the circumference of an overſhot
wheel is capable of, depends jointly upon the diameter or height
of the wheel, and the velocity of falling bodies ; for it is plain
that the velocity of the circumference can never be greater than
to deſcribe a ſemi-circumference, while a body let fall from the
top of the wheel will deſcend through its diameter ; nor indeed
quite ſo great, as a body deſcending through the fame perpen-
dicular ſpace cannot perform the ſame in ſo ſmall a time when
paſſing through a ſemi-circle, as would be done in a perpendi-
cular line. Thus, if a wheel is 16 feet i inch high, a body
will fall through the diameter in one ſecond : this wheel there-
fore can never arrive at a velocity equal to the making one
turn in two ſeconds; but, in reality, an overſhot wheel can never
come near this velocity; for when it acquires a certain ſpeed,
the greateſt part of the water is prevented from entering the
buckets;
EXPERIMENTAL ENQUIRY, &c.
35
buckets; and the reſt, at a certain point of its deſcent, is thrown
out again by the centrifugal force. This appears to
This appears to have been
the cafe in the three firſt experiments of the foregoing fpecimen;
but as the velocity, when this begins to happen, depends upon
the form of the buckets, as well as other circumſtances, the ut-
moſt velocity of overſhot wheels is not to be determined generally:
and, indeed, it is the leſs neceſſary in practice, as it is in this
circumſtance incapable of producing any mechanical effect, for
reaſons already given.
VI. Concerning the greateſt Load that an Overſhor
Wheel can overcome.
The greateſt load an overſhot wheel will overcome, conſidered
abftractedly, is unlimited or infinite : for as the buckets may be
of any given capacity, the more the wheel is loaded, the flower
it turns; but the flower it turns, the more will the buckets be
filled with water ; and conſequently though the diameter of the
wheel and quantity of water expended, are both limited, yet no
reſiſtance can be aſſigned, which it is not able to overcome: but
in practice we always meet with ſomething that prevents our
getting into infiniteſimals; for when we really go to work to
build a wheel, the buckets mutt neceſſarily be of fome given
capacity; and conſequently ſuch a reſiſtance will ſtop the wheel,
as is equal to the effort of all the buckets in one ſemi-circumfe-
rence filled with water.
The ſtructure of the buckets being given; the quantity of
this effort may be aſſigned; but is not of much conſequence to
the practice, as in this caſe alſo the wheel Icfes its power; for
though here be the exertion of gravity upon a given quantity of
water, yet being prevented by a counterbalance from moving,
is capable of producing no mechanical effect, according to our
definition. But, in reality, an overſhot wheel generally ceaſes
to be uſeful before it is loaded to that pitch; for when it meets with
Da
such
36
EXPERIMENTAL ENQUIRY, &c.
ſuch a reſiſtance as to diminiſh its velocity to a certain degree, its
motion becomes irregular , yet this never happens until the velo-
city of the circumference is leſs than 2 feet per ſecond, where the
reſiſtance is equable, as appears not only from the preceding
ſpecimen, but from experiments on larger wheels.
SCHOLIUM.
Having now examined the different effects of the power of
water, when acting by its impulſe, and by its weight, under the
titles of underſhot and overſhot wheels; we might naturally pro-
ceed to examine the effects when the impulſe and weight are
Combined, as in the ſeveral kinds of breaſt wheels, & c. but,
what has been already delivered being carefully attended to,
the application of the ſame principles in theſe mixt caſes will be
eaſy, and reduce what I have to ſay on this head into a narrow
compaſs: for all kinds of wheels where the water cannot deſcend
through a given ſpace, unleſs the wheel moves therewith, are to
be conſidered of the nature of an overfhot wheel, according to
the perpendicular height that the water deſcends from; and all
thoſe that receive the impulſe or ſhock of the water, whether
in an horizontal, perpendicular, or oblique direction, are to be
conſidered as underſhots. And therefore a wheel, which the
water ſtrikes at a certain point below the ſurface of the head,
and after that deſcends in the arch of a circle, preſſing by its
gravity upon the wheel; the effect of ſuch a wheel will be
equal to the effe Et of an underſhot, whoſe head is equal to the
difference of level between the ſurface of the water in the rea
ſervoir and the point where it ſtrikes the wheel, added to that
of an over ſhot, whoſe height is equal to the difference of level
between the point where it ſtrikes the wheel and the level of
the tail-water. It is here ſuppoſed, that the wheel receives the
ſhock of the water at right angles to its radii ; and that the ve-
locity of its circumference is properly adapted to receive the
utmoſt advantage of both theſe powers; otherwiſe a reduction
muſt be made on that account,
Many
Š
1
:
EXPERIMINTAL ENQUIRY, &c.
37
Many obvious and conſiderable improvements upon the com-
mon practice naturally offer themſelves, from a due confidera-
tion of the principles here eſtabliſhed, as well as many popular
errors ſhow themſelves in view: but as my preſent purpoſe ex-
tends no farther than the laying down ſuch general rules as
will be found to anſwer in practice, I leave the particular
application to the intelligent artiſt, and to the curious in theſe
matters.
Š
pe
&
A
The
3
4
:
PART
D 3
.
3
:
38
EXPERIMENTAL ENQUIRY, &c.
*
PART III.
On the CONSTRUCTION and Effects of
WINDMILL Sails.
Read before the Royal Society 31 May and 14 June 1759.
IN
N trying experiments on windmill fails, the wind itſelf is too
uncertain to anſwer the purpoſe, we muſt therefore have recourſe
to an artificial wind.
This may be done two ways; either by cauſing the air to
move againſt the machine, or the machine to move againſt the
air. To cauſe the air to move againſt the machine, in a ſufficient
volume, with ſteadineſs and the requiſite velocity, is not eaſily
put in practice: To carry the machine forward in a right line
againſt the air, would require a larger room than I could conve-
niently meet with. What I found moſt practicable, therefore,
was to carry the axis, whereon the fails were to be fixed, pro-
greſſively round in the circumference of a large circle. Upon
this idea * a machine was conſtructed, as follows.
PLATE
* Some years ago Mr. Rouſe, an ingenious gentleman of Har-
borough, in Leiceſterſhire, ſet about trying experiments on the ve-
locity of the wind, and force thereof upon plain ſurfaces and wind-
mill-fails; and, much about the ſame time, Mr. Ellicott contrived
a machine for the uſe of the late celebrated Mr. B. Robins, for try-
ing the reſiſtance of plain ſurfaces moving through the air. The
machines of both theſe gentlemen were much alike, though at that
time totally unacquainted with each other's inquiries. But it often
happens, that when two perſors think juftly upon the ſame ſubject,
their experiments are alike. This machine was alſo built
upon
the
fame idea as the foregoing ; but differed in having the hand for the
firit mover, with a pendulum for its regulator, inſtead of a weight,
as
EXPERIMENTAL ENQUIRY, &c.
39
Plate III. Fig. 1.
ABC is a pyramidical frame for ſupporting the moving parts.
DE is an upright axis, whereon is framed
FG, an arm for carrying the ſails at a proper diſtance from the
centre of the upright axis.
H is a barrel upon the upright axis, whereon is wound a
cord; which, being diawn by the hand, gives a circular
motion to the axis, and to the arm F G; and thereby
carries the axis of the fails in the circumference of
a circle, whoſe radius is DI, cauſing thereby the fails to
ſtrike the air, and turn round upon their own axis.
At L is fixed the end of a ſmall line, which paſſing through
the pullies MNO, terminates upon a ſmall cylinder or
barrel upon the axis of the fails, and, by winding there-
on, raiſes
:
P the ſcale, wherein the weights are placed for trying the
power of the fails. This ſcale, moving up and down
in the direction of the upright axis, receives no diſtur-
bance from the circular motion.
QR two parallel pillars ſtanding upon the arm F G, for the
purpoſe of ſupporting and keeping ſteady the ſcale P;
which is kept from ſwinging by means of
..
as in the former; which was certainly beſt for the purpoſes of mea-
furing the impulſe of the wind, or reſiſtance of plains ; but the lat-
ter is more applicable to experiments on windmill fails; becauſe every
change of poſition of the ſame fails will occaſion their meeting the
air with a different velocity, though urged by the fame weight.
D4
ST,
܀
40
EXÞERIMENTAL ENQUIRY, &c.
ST, two ſmall chains, which hang looſely round the two
pillars.
.
W is a weight for bringing the centre of gravity of the move-
able part of the machine into the centre of motion of the
axis DE.
V X is a pendulum, compoſed of two balls of lead, which are
moveable upon a wooden rod, and thereby can be fo ad-
juſted, as to vibrate in any time required. This pendu-
lum hangs upon a cylindrical wire, whereon it vibrates,
as on a rolling axis.
Y is a perforated table for ſupporting the axis of the pen-
dulum.
A
Note, The pendulum being ſo adjuſted, as to make two vi-
brations in the time that the arm F G is intended to make
one turn; the pendulum being fet a vibrating, the exi-
perimenter pulls by the cord Z, with ſufficient force to
make each half revolution of the arm to correſpond with
cach vibration, as equal as poſſible, during the number
of vibrations that the experiment is intended to be con-
tinued. A little practice renders it eaſy to give motion
thereto with all the regularity that is neceſſary.
Specimen of a Set of Experiments.
Radius of the fails
21 inches
Length of ditto in the cloth
18
Breadth of ditto
ŞAngle at the extremity
10 degrees
Ditto at the greateſt inclination
25
11
Il
5,6
1
20
• In all the following experiments the angle of the fails is ac.
counted from the plain of their motion; that is, when they ſtand
at
::
EXPERIMENTAL ENQUIRY, &c.
41
11,3 inches.
20 turns of the ſails raiſed the weight
Velocity of the centre of the fails, in the cir-
cumference of the great circle, in a fe- > 6 feet o inches.
cond
Continuance of the experiment
for
-
52 feconds
No.
Product.
I
2
3
4
5
6
Wt. in the ſcale.
olb.
6
6.
7
7를
​8
||||
Turns.
108
85
81
78
73
65
llll
510
526
546
547 maximum
520
O
7
9
O
N. B. The weight of the ſcale and pulley was 3 oz.; and that
I oz. ſuſpended upon one of the radii, at 12 inches from the
centre of the axis, juſt overcame the friction ſcale and load of
7lb.; and placed at 14' inches, overcame the ſame refift.
ances with g lb. in the ſcale.
;
Reduction of the preceding Specimen.
Nº. 5. being taken for the maximuin, the weight in the ſcale
was 7 lb. 8 oz. which, with the weight of the ſcale and pulley
3 oz. makes 7 lb. 1 1 oz. equal to 12302.; this added to the fric-
tion of the machinery, the ſum is the whole reſiſtance*. The
at right angles to the axis, their angle is denoted 0°, this notation
being agreeable to the language of practitioners, who call the an-
gle ſo denoted, the weather of the fail; which they denominate
greater or leſs, according to the quantity of this angle.
* The reſiſtance of the air is not taken into the account of
refiftance, becauſe it is inſeparable from the application of the
power.
friction
42
EXPERIMENTAL ENQUIRY, &c.
friction of the machinery is thus deduced: Since 20 turns of the
fails raiſed the weight 11,3 inches, with a double line, the radius
of the cylinder will be .18 of an inch; but had the weight been
raiſed by a ſingle line, the radius of the cylinder being half the
former, viz. .09, the reſiſtance would have been the ſame: we
ſhall therefore have this analogy; as half the radius of the cylin-
der, is to the length of the arm where the ſmall weight was ap-
plied; ſo is the weight applied to the arm, to a fourth weight, which
is equivalent to the ſum of the whole reſiſtance together; that is,
.09 : 12,5 :: 1 oz. : 1390z. this exceeds 123 oz. the weight in
the ſcale, by 16 oz. or ilb. which is equivalent to the friction ;
and which, added to the above weight of 7 lb. 1 1 oz. makes 8 lb.
11 OZ. = 8,69 lb. for the ſum of the whole reſiſtance; and this,
multiplied by 73 turns, makes a product of 634, which may be
called the repreſentative of the effect produced.
In like manner, if the weight glb. which cauſed the fails to
reft after being in motion, be augmented by the weight of the
ſcale and its relative friction, it will become 10,37 lb. The re-
fult of this ſpecimen is ſet down in Nº. 12. of Table III. and
the reſult of every other ſet of experiments therein contained
were made and reduced in the ſame manner.
:
TABLE III.
"!-
ܕܠ
ܕ
;
;
.ܝܙܝܝ̈ܐ܆ ܇ ܀܆ - :": ": !"'ܬ܀ ܕܠܐܕ܀mx.:i܀".s܀
ܟܙ:11 ܕ݂ܺܝ: ܃ ܃ :::܇ ܀ܕ cit:
..;-ܪ܂' ܀,; ܂ . .'-
** ...'n..v,ܙܙܙܙܐܙܫܙܙܢܘܪ ܐܙܪܘܐܚ ܐܢ. r،،، ܀ ܙ ܙ ܐ.'F:$ ܗ: ܟ݂ ܀
܀
TABLE III. Containing Nineteen Setts of Experiments on Windmill Sails
of various Structures, Poſitions, and Quantities of Surfaces.
The kind of fails
made uſe of.
Angle at the ex-
tremities.
Nº.
Greateſt angle.
Turns of the fails
unloaded.
Turns of ditto at
the maximum.
Load at the
maximum.
Greateſt load.
Quantity of
ſurface.
Ratio of greateſt
velocity to the ve-
locity at maximm
Ratio of greateſt
load to the load
at maximum.
Product.
Ratio of ſurface
to the product.
O
Plain fails at an
angle of 55
{
lb. Ib. Sqin
427,56 12,59 318 404 10:7
I 35 35
66
10:6
10:7,9
I 2
Plain fails weather'd
according to the
common practice.
2 12
3 15
15
18
70 6,3 7,56 441 404
10:8,3 10:10,1
105 69 6,72 8,12 464 404 10:6,6 10:8,3 10:10,15
7,0
9,81 462 404 10:7, 10:7,1| 10:10,15
41 18
96 66
5
926
10:11,4
Weathered accord-
ing to Maclaurin's
theorem.
6 12
291
66
7,0
702 7035
637 8,3
462 404
518 404
527 404
10:12,8
7 15 32
10:13
0
8
9
IO
1201 93
1201 79
20
Sails weathered in
the Dutch mail-
ner, tried in vari-
ous poſitions.
15
4,75 5,31 442 404 10:7,7 10:8,9 10:11,
3
18
7,0 8,12 553 404 10:6,6 10:8,6 10:13,7
5
78 7,5 8,12 585 404
10:9,2 10:14,5
7 22 113 77 8,3 9,81 639 404 10:6,8 10:8,5 10:15,8
25 108 73 8,69 10,37 634 404 10:6,8 10:8,4 10:15,7
27 100 66 8,41 10,94 580 404 10:6,6 10:7,7 10:14,4
II
12 IO
13 12
Sails weathered in
the Dutch manner,
but enlarged to-
wards the extre-
mities.
14 77 22 123 75 10,6512,59 799 505 10:6,1 10:8,5 10:15,8
15) IO
25 117 74 11,08 13,69 820 505 10:6,3 10:8,1 10:16,2
27 114 66 12,09 14,23 799 505 10:5,8 10:8,4) 10:15,8
17) 15 30 96 63 12,09 14,78 762 505 10:6,6 10:8,2 10:15,1
161 12
18 12
22
:
8 fails being ſeEtors
of ellipſes in their
beſt poſitions.
105 647 16,42 27,87 1059 854 10:6,1 10:5,9 10:12,4
99 64,18,06 11651146 10:5,9
10:10, I
19 12
22
I.
2.
3.
4.
5.
6.
7.
8.
9.
IO.
II.
I 2.
To face page 43
EXPERIMENTAL ENQUIRY, &c.
1
43
2
Obſervations and Deductions from the preceding
Experiments.
**
I. Concerning the beſt Form and Poſition of
Windmill Sails.
In Table III. N°. 1. is contained the reſult of a ſet of expe-
riments upon fails fet at the angle which the celebrated Monf.
Parint, and ſucceeding geometricians for many years, held to be
the beſt; viz. thoſe whoſe planes make an angle 55° nearly with
the axis ; the complement whereof, or angle that the plane of
the fail makes with the plane of their motion, will therefore be
35°, as ſet down in col. 2. and 3. Now if we multiply their
number of turns by the weight they lifted, when working to the
greateſt advantage, as ſet down in columns 5. and 6. and com-
pare this product (col. 8.) with the other products contained in
the ſame column, inſtead of being the greateſt, it turns out the
leaſt of all the reſt. But if we ſet the angle of the fame planes
at ſomewhat leſs than half the former, or at any angle from 15°
to 18°, as in Nº. 3. and 4. that is, from 72° to 75º with the axis,
the product will be increaſed in the ratio of 31:45; and this is
the angle most commonly made uſe of by practitioners, when
the ſurfaces of the fails are planes.
If nothing more was intended than to determine the moſt effi.
cacious angle to make a mill acquire motion from a ſtate of reft,
or to prevent it from paffing into reſt from a ſtate of motion, we
ſhall find the poſition of N°. 1. the beſt; for if we conſult col.
7. which contains the leaſt weights, that would make the fails
paſs from motion to reſt, we ſhall find that of Nº. 1. (relative
to the quantity of cloth) the greateſt of all But if the fails are
intended, with given dimenſions, to produce the greateſt effect
poſſible in a given time, we muſt entirely reject thoſe of
NO
44
EXPERIMENTAL ENQUIRY, &c.
Nº. 1. and, if we are confined to the uſe of planes, conform our-
ſelves to fome angle between Nº:3. and 4. that is, not leſs than 72°,
or greater than 75°, with the axis.
2 a
The late celebrated Mr. Maclaurin has judiciouſly diſtinguiſh-
ed between the action of the wind upon a fail at reſt, and a fail
in motion; and, in conſequence, as the motion is more rapid
near the extremities than towards the centre, that the angle of
the different parts of the fail, as they recede from the centre,
ſhould be varied. For this purpoſe he has furniſhed us with
the following theorem*, « Suppoſe the velocity of the wind to
« be repreſented by a, and the velocity of any given part of the
« fail to be denoted by c; then the effort of the wind upon that
part of the fail will be greateſt when the tangent of the angle,
u in which the wind ſtrikes it, is to radius as
9°C 30
2 ++-- to 1." This theorem then affigns the
4 aa
law, by which the angle is to be varied according to the ve-
locity of each part of the fail to the wind: but as it is left unde-
termined what velocity any one given part of the ſail ought to
have in reſpect to the wind, the angle that any one part of the
fail ought to have, is left undetermined alſo; ſo that we are ſtill
at a loſs for the proper data to apply the theorem. However,
being willing to avail myſelf thereof, and conſidering that any
angle from 15° to 18° was beſt ſuited to a plane, and of confe-
quence to the beſt mean angle, I made the fail, at the middle
diſtance between the centre and the extremity, to ſtand at an
angle of 15° 41' with the plane of the motion; in which caſe the
velocity of that part of the fail, when loaded to a maximum,
would be equal to that of the wind, or c=a. This being de
termined, the reſt were inclined according to the theorem, as
follows:
t
• Maclaurin's account of Sir Iſaac Newton's philoſophical diſ-
coveries, p. 176, art. 29.
Angle
+
V
EXPERIMENTAL ENQUIRY, &c.
45
3
6
6
Angle with Angle of
the axis. weather.
ca--63° 26' - - 26° 34'
ca --69 54 - - 20 6
74 19--15 41 middle.
c=1a - 77 20 40
c=15a - 79 27
33
c=2a --81
O-- 9
o extremity.
Parts of the
radius from
the centre.
. com
en a
weile
I2
olu
1
IO
6
The reſult hereof was according to Nº. 5. being nearly the
ſame as the plane fails, in their beſt poſition: but being turned
round in their ſockets, ſo that every part of each fail ſtood at an
angle of 3°, and afterwards of 6°, greater than before, that is,
their extremities being moved 9° to 12° and 15°, the pro-
ducts were advanced to 518 and 527 reſpectively. Now from
the ſmall difference between thoſe two products, we may con-
clude, that they were nearly in their beſt poſition, according to
Nº. 7. or ſome angle between that and Nº.6: but from theſe,
as well as the plane fails and others, we may alſo conclude, that
a variation in the angle of a degree or two makes very little dif-
ference in the effect, when the angle is near upon the beſt.
*
It is to be obſerved, that a fail inclined by the preceding rule
will expofe a convex ſurface to the wind: whereas the Dutch,
and all our modern mill-builders, though they make the angle
to diminiſh, in receding from the centre towards the extremity,
yet conſtantly do it in ſuch manner, as that the ſurface of the
fail
may
be concave towards the wind. In this manner the fails
made uſe of in Nº.8, 9, 10, 11, 12, and 13, were conſtructed; the
middle of the fail making an angle with the extreme bar of 12°;
and the greateſt angle (which was about of the radius from the
centre) of 150 therewith. Thofe fails being tried in various po-
fitions, the beſt appears to be that of Nº: 11. where the extre-
mities ſtood at an angle of 70 with the plane of motion, the
product
46
EXPERIMENTAL ENQUIRY, &c.
product being 639: greater than that of thoſe made by the
theorem in the ratio of 9:11, and double to that of Nº. 1; and
this was the greateſt product that could be procured without an
augmentation of ſurface. Hence it appears, that when the wind
falls upon a concave ſurface, it is an advantage to the power of
the whole, though every part, taken ſeparately, hould not be diſ-
poſed to the beſt advantage *.
Having thus obtained the beſt poſition of the ſails, or manner
of weathering, as it is called by the workmen, the next point
was to try what advantage could be made by an addition of ſur-
face upon the fame radius. For this purpoſe, the fails made uſe
of had the ſame weather as thoſe Nº. 8. to 13, with an addition
to the leading ſide of each of a triangular cloth, whoſe height
was equal to the height of the fail, and whoſe baſe was equal
to half the breadth: of conſequence the increaſe of ſurface upon
the whole was one-fourth part, or as 4:5. Thoſe fails, by
being turned round in their fockets, were tried in four different
poſitions, ſpecified in Nº. 14, 15, 16, and 17; from whence it
appears, that the beſt was when every part of the fail made a
greater angle by 2, with the plane of the motion, than thofe
without the addition, as appears by Nº. 15. the product being
* By ſeveral trials in large I have found the following angles to
anſwer as well as any. The radius is ſuppoſed to be divided into
6 parts and 1-5th, reckoning from the centre, is called 1, the ex-
tremity being denoted 6.
Angle with
No.
the axis.
Angle with the plane
of the motion.
18°
I
720
2
2
3
1
4
71
72
74
771
83
19
18 middle.
16
121
7 extremity.
5
6
820 :
EXPERIMENTAL ENQUIRY, &c.
47
820: this exceeds 639 more than in the ratio of 4: 5, or that
of the increaſe of cloth. Hence it appears, that a broader fail
requires a greater angle; and that when the fail is broader at
the extremity, than near the centre, this mape is more advan.
tageous than that of a parallelogram *.
Many have imagined, that the more fail, the greater the ad-
vantage, and have therefore propoſed to fill up the whole area:
and by making each fail a ſector of an ellipfis, according to Mon-
ſieur Parint, to intercept the whole cylinder of wind, and there-
by to produce the greateſt effect poſſible.
.
$
We have therefore proceeded to inquire how far the effect
could be increaſed by a further enlargement of the ſurface, upon
the ſame radius of which Nº. 18 and 19 are ſpecimens. The
ſurfaces indeed were not made planes, and ſet at an angle of
35°, as Parint propoſed; becauſe, from Nº. 1. we learn, that
this poſition has nothing to do, when we intend them to work
to the greateſt advantage. We therefore gave them ſuch an
angle as the preceding experiments indicated for ſuch fort of fails,
viz. 12° at the extremity, and 22° for the greateſt weather. By
Nº. 18. we have the product 1059, greater than Nº. 15. in the
ratio of 7:9; but then the augmentation of cloth is almoſt
7:12. By N°. 19. we have the product 1165, that is greater
than Nº. 15. as 7:10; but the augmentation of cloth is nearly
as 7:16; conſequently had the ſame quantity of cloth as in
* The figure and proportion of the enlarged fails, which I have
found beſt to anſwer in large, are repreſented in the figure, Plate III.
where the extreme bar is 1-3d of the radius (or whip, as it is called
by the workmen), and is divided by the whip in the proportion of
3 to 5. The triangular or leading fail is covered with board from
the point downwards 1-3d of its height, the reſt with cloth as uſual.
The angles of weather in the preceding note are beſt for the enlarged
fails allo; for in practice it is found, that the fails had better have
too little than too much weather,
Nº.
48
EXPERIMENTAL ENQUIRY, &c.
!
Nº. 18, been diſpoſed in a figure ſimilar to that of Nº. 15,
inſtead of the product being 1059, we ſhould have had the product
1386; and in Nº. 19, inſtead of the product 1165, we ſhould
have had a product of 1860; as will be further made appear in
the courſe of the following deductions. Hence it appears, that
beyond a certain degree, the more the area is crouded with fail,
the leſs effect is produced in proportion to the ſurface: and, by
purſuing the experiments ſtill further, I found, that though in
Nº. 19, the ſurface of all the fails together were not more than
7-8ths of the circular area containing them, yet a further addi-
tion rather diminiſhed than increaſed the effect. So that when
the whole cylinder of wind is intercepted, it does not then pro-
duce the greateft effe Et for want of proper interftices to eſcape.
It is certainly deſirable that the fails of windmills ſhould be
as ſhort as poſſible; but at the ſame time it is equally deſirable,
the quantity of cloth ſhould be the leaſt that may be, to avoid da-
mage by ſudden fquals of wind. The beſt ſtructure, therefore,
for large mills, is that where the quantity of cloth is the greateſt,
in a given circle, that can be : on this condition, that the effect
holds out in proportion to the quantity of cloth; for otherwiſe
the effect can be augmented in a given degree by a lefſer increaſe
of cloth upon a larger radius, than would be required, if the
cloth was increaſed upon the ſame radius. The moſt uſeful
figure, therefore for practice, is that of Nº. 9 or 10, as has
been experienced upon ſeveral mills in large.
1
S
TABLE
EXPERIMENTAL ENQUIRY, &c.
49
TABLE IV. Containing the Reſult of ſix Sets of Experiments, made for determining the
Difference of Effect, according to the different Velocity of the Wind.
N. B. The fails were of the fame ſize and kind as thoſe of Nº. 10, 11, and 12, Tab. III.
Continuance of the Experiment one minute.
177
1
f f. in.
16. lb,
I 5
50 4 41
4 41 95
951 66 4,47 5,37 295
10:6,910:8,3
2 5 89 207 122 16,42 18,06 2003 4,47 180 80510 : 27,3 10:5,9 10:9,1
1
1
4 4
.
3 72
4 72
651 4,62
130 17,52
300
2278 4,62 180 83210 : 27,8
8
9
4
2
510
610
4
9
*
II. Con-
10:6.7110:8,51
91
61 5,03
5,87 307
178 110 18,61 21,34 2047 5,03 158 79510 : :26 10:6,2 10:8,7
8
6
4
13
5
14
9
7
II
10
I 2
2
I
50
EXPERIMENTAL, ENQUIRY, &c.
II. Concerning the Ratio between the Velocity of Windmill
Sails unloaded, and their Velocity when loaded to a
Maximum.
:
Thoſe ratios, as they turned out in experiments upon
different kinds of fails, and with different inclinations (the
velocity of the wind being the ſame) are contained in column 10
of Tab. III. where the extremes differ from the ratio of 10: 7,7
to that of 10:5,8; but the moſt general ratio of the whole will
be nearly as 3:2. This ratio alſo agrees ſufficiently near with
experiments where the velocity of the wind was different, as in
thoſe contained in Tab. IV. col. 13. in which the ratios differ
from 10: 6,9 to that of 10: 5,9. However, it appears in
general, that where the power is greater, whether by an en-
largement of ſurface, or a greater velocity of the wind, that the
ſecond term of the ratio is leſs.
III. Concerning the Ratio between the greateſt Load that
the Sails will bear without ſtopping, or what is nearly
the ſame Thing, between the leaft Load that will ſtop the
Sails, and the Load at the Maximum.
Thoſe ratios for different kinds of fails and inclinations, are
collected in col. 11. Tab. III. where the extremes differ from the
ratio of 10:6 to that of 10:9,2; but taking in thoſe ſets of
experiments only, where the fails reſpectively anſwered beft,
the ratios will be confined between that of 10:8 and of 10:9;
and at a medium about 10:8,3 or of 6:5. This ratio alſo
agrees nearly with thoſe in col. 14. of Tab. IV. However
it appears, upon the whole, that in thoſe inſtances, where
the angle of the fails or quantity of cloth were greateſt, that
the ſecond term of the ratio was leſs.
IV. Con
EXPERIMENTAL ENQUIRY, &c.
$1
을
​IV. Concerning the Effeets of Sails, according to the
different Velocity of the Wind.
Maxim 1. The velocity of windmill fails, whether unloaded, or
loaded, ſo as to produce a maximum, is nearly as the velocity of
the wind, their ſhape and poſition being the ſame.
This appears by comparing together the reſpective numbers of
columns 4 and 5, Tab.IV. wherein thoſe of numbers 2, 4 and 6,
ought to be double of numbers 1, 3 and 5: but as the deviation is
no where greater than what may be imputed to the inaccuracy of
the experiments themſelves, and hold good exactly in numbers
3 and 4; which ſets were deduced from the medium of a number
of experiments, carefully repeated the ſame day, and on that ac-
count are moſt to be depended upon; we may therefore conclude
the maxiin true,
Maxim 2d. The load at the maximum is nearly, but ſomewhat
leſs than, as the ſquare of the velocity of the wind, the shape
and poſition of the ſails being the ſame.
This appears by comparing together the numbers in col. 6.
Tab. IV, wherein thoſe of numbers 2, 4 and 6 (as the velocity
is double), ought to be quadruple of thoſe of numbers 1, 3
and 5; inſtead of which they fall ſhort, number 2 by 1, num-
ber 4 by i' and number 6 by is part of the whole. The
greateſt of thoſe deviations is not more conſiderable than might
be imputed to the unavoidable errors in making the experiments :
but as thoſe experiments, as well as thoſe of the greateſt load, alı
deviate the ſame way; and alſo coincide with ſome experiments
communicated to me by Mr. Rouſe upon the reſiſtance of planes;
I am led to ſuppoſe a ſmall deviation, whereby the load falls
ſhort of the ſquares of the velocity; and ſince the experiments
Nº. 3 and 4, are moſt to be depended upon, we muſt conclude,
that
E 2
52
EXPERIMENTAL ENQUIRY, &c.
that when the velocity is double, the load falls ſhort of its due
proportion by 1, or, for the fake of a round number, by about
z's part of the whole.
Maxim 3d. The effects of the ſame fails at a maximum are
nearly, but ſomewhat leſs than, as the cubes of the velocity of
the wind.
It has already been proved, Maxim iſt, that the velocity of
fails at the maximum, is nearly as the velocity of the wind; and
by Maxim 2d, that the load at the maximum is nearly as the
ſquare of the fame velocity: if thoſe two maximums would
hold preciſely, it would be a conſequence that the effect would
be in a triplicate ratio thereof: how this agrees with experiment
will appear by comparing together the products in col.8 of Tab.IV.
wherein thoſe of Nº. 2, 4 and 6, (the velocity of the wind being
double) ought to be octuple of thoſe of Nº. 1, 3 and 5, inſtead of
which they fall ſhort, Nº. 2, by 4, Nº. 4, by zs, and Nº. 6,
by part of the whole. Now, if we rely on No. 3 and 4, as
the turns of the fails are as the velocity of the wind; and ſince
the load of the maximum falls ſhort of the ſquare of the velo-
city by about a part of the whole: the product made by the
multiplication of the turns into the load, muſt alſo fall ſhort of
the triplicate ratio by about to part of the whole product. .
Maxim 4th. The load of the ſame fails at the maximum is
nearly as the ſquares, and their effect as the cubes of their num-
ber of turns in a given time.
This maxim may be eſteemed a conſequence of the three pre-
ceding; for if the turns of the fails are as the velocity of the
wind, whatever quantities are in any given ratio of the velocity
of the wind will be in the ſame given ratio of the turns of the
fails : and therefore, if the load at the maximum is as the ſquare,
or the effect as the cube, of the velocity of the wind, wanting z's
part
1
EXPERIMENTAL ENQUIRY, &c.
53
part when the velocity is double; the load at the maximum will
alſo be as the ſquare, and the effect as the cube, of the number of
turns of the fails in a given time, wanting in like manner to
part when the number of turns are double in the ſame time. In
the preſent caſe, if we compare the loads at the maximum,
col. 6, with the ſquares of the number of turns, col. 5, of Nº. 1
and 2, 5 and 6, or the products of the fame numbers col. 8, with
the cubes of the number of turns col. 5, inſtead of falling ſhort,
as Nº. 3 and 4, they exceed thoſe ratios : but as the ſets of expe-
riments No. 1 and 2, of 5 and 6, are not to be eſteemed of equal
authority with thoſe of No. 3 and 4, we muſt not rely upon them
further than to obſerve, that in comparing the groſs effeEts of large
machines, the direct proportion of the ſquares and cubes reſpec-
tively, will hold as near as the effects themſelves can be obſer-
ved; and therefore be ſufficient for practical eſtimation, without
any allowance.
->
Maxim 5th. When ſails are loaded ſo as to produce a maximum
at a given velocity, and the velocity of the wind increaſes, the
load continuing the ſame; 1/1, The increaſe of effect, when the
increaſe of the velocity of the wind is ſmall, will be nearly as
the ſquares of thoſe velocities : 2dly, When the velocity of the
wind is double, the effects will be nearly as 10:27: But, 3dly,
When the velocities compared, are more than double of that where
the given load produces a maximum, the effects increaſe nearly
in a ſimple ratio of the velocity of the wind.
It has already been proved, maxim iſt and 2, that when the
velocity of the wind is increaſed, the turns of the falls will in-
creaſe in the ſame proportion, even when oppoſed by a load as
the ſquare of the velocity: and therefore if wanting the oppo-
ſition of an increaſe of load, as the ſquare of the velocity, the
turns of the fails will again be increaſed in a fimple ratio of the
velocity of the wind on that account alſo; that is, the load con-
tinuing the ſame, the turns of the fails in a given time will be as
the
E 3
54
EXPERIMENTAL ENQUIRY, &c.
the ſquare of the velocity of the wind; and the effect, being in
this caſe as the turns of the fails will be as the ſquare of the velo-
city of the wind alſo; but this muſt be underſtood only of the firſt
increments of the velocity of the wind : for,
out
as
3
2dly, As the fails will never acquire above a given velocity in
relation to the wind, though the load was diminiſhed to nothing;
when the load continues the ſame, the more the velocity of the
wind increaſes (though the effect will continue to increaſe) yet
the more it will fall ſhort of the ſquare of the velocity of the
wind; ſo that when the velocity of the wind is double, the
increaſe of effect, inſtead of being as 1 : 4, according to
the ſquares, it turns
10 : 274, as thus appears.
In Tab. IV. col. 9. the loads of Nº. 2, 4 and 6, are the ſame
as the maximum loads in col. 6, of Nº, 1, 3 and 5. The
number of turns of the ſails with thoſe loads, when the vclocity
of the wind is double, are ſet down in col. 10, and the products
of their multiplication in col. 11: thofe being compared with
the products of Nº. 1, 3 and 5, col. 8, furniſh the ratios ſet
down in col. 12, which at a medium (due regard being had to
No.
3
and 4) will be nearly as 10:27 3dly. The load con-
tinuing the fame, grows more and more inconſiderable, reſpect-
ing the power of the wind as it increaſes in velocity; ſo that
the turns of the fails grow nearer and nearer a coincidence with
their turns unloaded ; that is, nearer and nearer to the ſimple
ratio of the velocity of the wind. When the velocity of the
wind is double, the turns of the fails, when loaded to a maxi-
mum, will be double alſo; but, unloaded, will be no more than
triple, by deduction 2d: and therefore the product could not
have increaſed beyond the ratio of 10:30 (inſtead of 10:27
even ſuppoſing the fails not to have been retarded at all by carrying
the maximum load for the half velocity. Hence we fee, that when
the velocity of the wind exceeds the double of that, where a con-
ftant load produces a maximum, that the increaſe of effect, which
follows the increaſe of the velocity of the ſails, will be nearly
as the velocity of the wind, and ultimately in that ratio preciſely.
Hence
六七​中心
​1
EXPERIMENTAL ENQUIRY, &c.
55
Hence alſo we ſee that windmills, ſuch as the different ſpecies for
raiſing water for drainage, &c. loſe much of their full effect,
when acting againſt one invariable oppoſition.
V. Concerning the Effeets of Sails of different Magnitudes
tbe Stručture and Poſition being ſimilar, and the Velocity
of the Wind the fame.
Maxim 6. In fails of a ſimilar figure and poſition, the num-
ber of turns in a given time will be reciprocally as the radius
or length of the fail.
The extreme bar having the ſame inclination to the plane of
its motion, and to the wind; its velocity at a maximum will
always be in a given ratio to the velocity of the wind; and
therefore, whatever be the radius, the abſolute velocity of the ex-
tremity of the fail will be the ſame: and this will hold good re-
ſpecting any other bar, whoſe inclination is the fame, at a pro-
portionable diſtance from the centre; it therefore follows, that the
extremity of all ſimilar fails, with the ſame wind, will have the
fame abſolute velocity, and therefore take a ſpace of time to per-
form one revolution in proportion to the radius; or, which is
the ſame thing, the number of revolutions in the ſame given
time, will be reciprocally as the length of the fail.
Maxim 7. The load at a maximum that ſails of a ſimilar
figure and poſition will overcome, at a given diſtance from the
centre of motion, will be as the cube of the radius.
Geometry informs us, that in ſimilar figures the ſurfaces are
as the ſquares of their fimilar fides; of confequence the quan-
tity of cloth will be as the ſquare of the radius : alſo in ſimilar
figures and poſitions, the impulſe of the wind, upon every
fimilar fection of the cloth, will be in proportion to the ſurface
E 4
of
56
EXPERIMENTAL ENQUIRY, &c.
of that ſection ; and conſequently, the impulſe of the wind upon
the whole, will be as the ſurface of the whole : but as the
diſtance of every ſimilar fection, from the centre of motion,
will be as the radius; the diſtance of the centre of power of the
whole, from the centre of motion, will be as the radius alſo;
that is, the lever by which the power acts, will be as the radius :
as therefore the impulſe of the wind, reſpecting the quantity of
cloth, is as the ſquare of the radius, and the lever, by which it
acts, as the radius fimply; it follows, that the load which the
fails will overcome, at a given diſtance from the centre, will be
as the cube of the radius.
Maxim 8. The effect of ſails of ſimilar figure and poſition,
are as the ſquare of the radius.
By maxim 6, it is proved, that the number of revolutions
made in a given time, are as the radius inverſely. Under
maxim 7, it appears, that the length of the lever, by which the
power acts, is as the radius directly; therefore theſe equal and
oppoſite ratios deſtroy one another: but as in ſimilar figures the
quantity of cloth is as the ſquare of the radius, and the action of
the wind is in proportion to the quantity of cloth, as alſo ap-
pears under maxim 7; it follows that the effect is as the ſquare
of the radius.
COROL. I. Hence it follows, that augmenting the length of
the fail, without augmenting the quantity of cloth, does not
increaſe the power; but becauſe what is gained by the length of
the lever, is loft by the flowneſs of the rotation.
COROL. 2. If fails are increaſed in length, the breadth remain-
ing the ſame, the effect will be as the radius.
VI. Con-
EXPERIMENTAL ENQUIRY, &c.
57
VI. Concerning the Velocity of the Extremities of Wind-
mill Sails, in Repeat to the Velocity of the Wind.
Maxim 9. The velocity of the extremities of Dutch fails, as well
as of the enlarged fails, in all their uſual poſitions when unloaded,
or even loaded to a maximum, are conſiderably quicker than the
velocity of the wind.
The Dutch fails unloaded, as in Tab. III. N°8. made 120 re-
volutions in 52": the diameter of the fails being 3 feet 6 inches,
the velocity of their extremities will be 25,4 feet in a ſecond;
but the velocity of the wind producing it, being 6 feet in the
fame time, we ſhall have 6 : 25,4 :: 1:4,2; in this caſe there-
fore, the velocity of their extremities was 4,2 times greater than
that of the wind. In like mariner, the relative velocity of the
wind, to the extremities of the fame fails, when loaded to a
maximum, making then 93 turns in 52", will be found to be as
I: 3,3; or 3,3 times quicker than that of the wind.
The following table contains 6 examples of Dutch fails, and
4 examples of the enlarged fails, in different poſitions, but with
the conſtant velocity of the wind of 6 feet in a ſecond, from Table
III. and alſo 6 examples of Dutch fails in different poſitions,
with different velocities of the wind from Table IV,
TABLÉ
$8
EXPERIMENTAL ENQUIRY, &c.
.
:
Table V. Containing the Ratio of the Velocity of the
Extremities of Windmill Sails to the Velocity of the
Wind.
N Nº.
Ratio of the velocity
of the wind and ex-
tremities of the ſails,
Nº. of Tab.
III. and IV.
Angle at the
Extremity.
Velocity of
the wind in a
ſecond.
unloaded, loaded.
I
I: 3,3
2
8
9
IO
I: 2,8
3
5
7
IO
6f. oin. 1 : 4,2
6 0 I: 4,2
6 0
6
1:42
6 O
I: 3,8
I : 3,5
3
4
5
6
I : 2,75
1 : 2,7
II
I 2
I: 2,6
13
12
6 0
I : 2,3
From Table III.
7
를
​0
7
IO
6
6
6
I : 2,6
I: 2,6
14
15
16
17
0
I : 4,3
I :4,1
I : 42
I : 3,35
0
9
I
I2
IS
6 0
I: 2,3
I: 2,2
II
I
I : 42
I: 4,3
2
1: 2,8
I: 2,6
I : 2,8
12
13
14
15
16
5
5
7
74 분
​IO
3
4
5
4 42
8
9
43
8
9
4 4
8
9
From Tab. IV.
I: 2,7
1: 3,8
I: 2,6
6
IO
I : 3,4
I : 23
I
2
31
4
5
6
It appears from the preceding collection of examples, that
when the extremities of the Dutch fails are parallel to the plane
of motion, or at right angles to the wind, and to the axis, as they
are made according to the common practice in England, that
their velocity, unloaded, is above 4 times, and loaded to a maxi.
mum, above 3 times greater than that of the wind: but that when
the Dutch fails, or enlarged fails, are in their beſt poſitions, their
velocity
1
EXPERIMENTAL ENQUIRY, &
59
velocity unloaded is 4 times, and loaded to a maximum, at a me-
dium the Dutch fails are 2,7, and the enlarged fails 2,6 times
greater than the velocity of the wind. Hence we are furniſhed
with a method of knowing the velocity of the wind, from obſer-
ving the velocity of the windmill fails; for knowing the radius,
and the number of turns in a minute, we ſhall have the velocity
of the extremities; which, divided by the following diviſors, will
give the velocity of the wind.
Dutch fails in the common poſition undeaded 4.2
-3.3
unloaded 4.0
Dutch fails in their beſt poſition
loaded -2.7
unloaded 4.0
Enlarged fails in the beſt poſition
loaded 2.6
From the above diviſors there ariſes the following compen*
diums; ſuppoſing the radius to be 30 feet, which is the moſt
uſual length in this country, and the mill to be loaded to a
maximum, as is uſually the caſe with corn mills; for every 3
turns in a minute, of the Dutch fails in their common pofition,
the wind will move at the rate of 2 miles an hour; for every 5
turns in a minute, of the Dutch fails in their beſt poſition, the
wind moves 4 miles an hour; and for every 6 turns in a minute,
of the enlarged fails in their beſt poſition, the wind will move
S
miles an hour.
The following table, which was communicated to me by my
friend Mr. Rouſe, and which appears to have been conſtructed
with great care, from a conſiderable number of facts and expe-
riments, and which having relation to the ſubject of this article;
I here inſert it as he ſent it to me: but at the ſame time muft
obſerve, that the evidence for thoſe numbers where the velocity
of the wind exceeds 50 miles an hour, do not ſeem of equal
authority with thoſe of 50 miles an hour and under, It is alſo to
be
60
EXPERIMENTAL ENQUIRY, &c.
be obſerved, that the numbers in col. 3. are calculated accord-
ing to the ſquare of the velocity of the wind, which, in mode-
rate velocities, from what has been before obſerved, will hold
very nearly.
TABLE VI. Containing the Velocity and Force of Wind,
according to their common Appellations.
III
Velocity of
the Wind.
foot area
Perpendicular force on
Miles in one
Hour.
Common appellations of the force of
winds.
Feet in one
ſecond.
pounds avoirdupois.
one
I
2
U AWNA
1
1,47 2005 Hardly perceptible.
2,93 2020
Juſt perceptible.
3 4,40 ,044
4 5,87 3079
Gentle pleaſant wind.
5 7,33 ,123
10 14,67 1492
Pleaſant briſk gale.
15 22,00 | 1,107
20 29,34 1,968
Very briſka
25 36,67 3,075
30 44,01 | 47429
High winds.
35 51,34 6,027
40
58,68 7,873} Very high.
45 66,01 9,963
50 73,35 12,300
A ſtorm or tempeft.
60
88,02 173715
A great ſtorm.
80 117,36 31,490
An hurricane.
100 146,70 49,200 An hurricane that tears up trees, car-
ries buildings before it, &c.
3
I
2
VII.
:
.
EXPERIMENTAL ENQUIRY, &c.
61
$
VII. Concerning the abſolute Effect, produced by a given
Velocity of the Wind, upon Sails of a given Magni-
tude and Conſtruction.
It has been obſerved by practitioners, that in mills with
Dutch fails in the common poſition, that when they make about
13 turns in a minute, they then work at a mean rate: that is, by
the compendiums in the laſt article, when the velocity of the
wind is 8 miles an hour, or 12 feet in a ſecond; which, in
common phraſe, would be called a freſh gale.
I
The experiments ſet down in Tab. IV. N° 4. were tried
with a wind, whoſe velocity was 8 feet in a ſecond; confe-
quently had thoſe experiments been tried with a wind, whoſe
velocity was 12 feet in a ſecond, the effect, by maxim 3:1,
would have been 3 times greater; becauſe the cube of 12 is 3
times greater than that of 81.
From Tab. IV. N° 4. we find that the fails, when the velo-
city of the wind was 8. feet in a ſecond, made 130 revolutions
in a minute, with a load of 17,52 lb. From the meaſures of
the machine, preceding the ſpecimen of a ſet of experiments, we
find, that 20 revolutions of the fails raiſed the ſcale and weight
11,3 inches: 130 revolutions will therefore raiſe the ſcale
73,45 inches, which, multiplied by 17,52 lb. makes a product
of 1287, for the effect of the Dutch fails in their beſt poſition;
that is, when the velocity of the wind is 8. feet in a ſecond:
this product therefore multiplied by three, will give 3861 for
the effect of the fame fails, when the velocity of the wind is 12
feet in a ſecond.
Deſaguliers makes the utmoſt power of a man, when work-
ing ſo as to be able to hold it for ſome hours, to be equal to that
of
62
EXPERIMENTAL ENQUIRY, &c.
5
of raiſing an hogſhead of water 10 feet high in a minute. Now,
an hogſhead conſiſting of 63 ale gallons, being reduced into
pounds avoirdupois, and the height into inches; the product
made by multiplying thoſe two numbers will be 76800; which
is 19 times greater than the product of the fails laſt-mentioned,
at 12. feet in a ſecond: therefore, by maxim 8th, if we mul-
tiply the ſquare root of 19, that is 4,46, by 21 inches, the
length of the fail producing the effect 3861, we ſhall have
93,66 inches, or 7 feet 9 inches for the radius of a Dutch fail
in its beſt poſition, whoſe mean power ſhall be equal to that of
a man: but if they are in their common poſition, their length
muſt be increaſed in the ratio of the ſquare root of 442 to that
of 639, as thus appears ;
The ratio of the maximum products of Nº. 8 and 11. Tab.
III. are as 442 : 639: hut by maxim 8, the effects of fails of
different radii are as the ſquare of the radii ; conſequently the
ſquare roots of the products or effects, are as the radii ſimply;
and therefore as the ſquare root of 442 is to that of 639; fo is
93,66 to 112,66; or 9 feet 4 inches.
If the fails are of the enlarged kind, then from Tab. III. Nº.
II and 15. we ſhall have the ſquare root of 820 to that of
639:: 93,66 : 82,8 inches, or 6 feet 10 inches: ſo that in round
numbers we ſhall have the radius of a fail, of ſimilar figure to
their reſpective models, whoſe mean power ſhall be equal to that
of a man;
9 feet.
The Dutch fails in their common poſition
The Dutch ſails in their beſt poſition
The enlarged fails in their beſt poſition
8
7
Suppoſe now the radius of a fail to be 30 feet, and to be con-
ſtructed upon the model of the enlarged fails, Nº. 14 or 15. Tab.
III. dividing 30 by 7 we Ihall have 4328, the ſquare of which
is
4
EXPERIMENTAL ENQUIRY, &c.
63
is 18,3; and this, according to maxim 7, will be the relative
power of a ſail of 30 feet, to one of 7 feet ; that is, when work-
ing at a mean rate, the 30 feet fail will be equal to the power of
18,3 men, or of 3 horſes; reckoning 5 men to a horſe: whereas
the effect of the common Dutch fails, of the ſame length, being
leſs in the proportion of 820 : 442, will be ſcarce equal to the
power of 10 men, or of 2 horſes.
That theſe computations are not merely ſpeculative, but will
nearly hold good when applied to works in large, I have had an
opportunity of verifying: for in a mill with the enlarged fails of
30 feet applied to the cruſhing of rape ſeed, by means of two
runners upon the edge, for making oil; I obſerved, that when
the fails made 11 turns in a minute, in which caſe the velocity
of the wind was about 13 feet in a ſecond, according to article
6th, that the runners then made 7 turns in a minute : whereas
2 horſes, applied to the fame 2 runners, ſcarcely worked them
at the rate of 3 turns in the ſame time. Laſtly, with regard to
the real ſuperiority of the enlarged fails, above the Dutch fails as
commonly made, it has ſufficiently appeared, not only in thoſe
caſes where they have been applied to new mills, but where they
have been ſubſtituted in the place of the others.
VIII. Concerning horizontal Windmills and Water-
Wheels, with oblique Vanes.
Obſervations upon the effects of common windmills, with
oblique vanes, have led many to imagine, that could the vanes be
brought to receive the direct impulſe, like a ſhip failing before
the wind, it would be a very great improvement in point of
power: while others attending to the extraordinary and even
unexpected effects of oblique vanes have been led to imagine
that oblique vanes applied to water-mills, would as much exceed
the common water-wheels, as the vertical wind-mills are found to
have exceeded all attempts towards an horizontal one. Both theſe
notions,
64
EXPERIMENTAL ENQUIRY, &c.
i
notions, but eſpecially the firſt, have ſo plauſible an appearance,
that of late years there has ſeldom been wanting thoſe, who have
aſſiduouſly employed themſelves to bring to bear deſigns of this
kind : it may not therefore be unacceptable to endeavour to fet
this matter in a clear light.
PLATE III. fig. 2. Let A B be the ſection of a plane, upon
which let the wind blow in the direction CD, with ſuch a ve-
locity as to deſcribe a given ſpace B E, in a given time (ſup-
poſe I ſecond); and let A B be moved parallel to itſelf, in the
direction CD. Now, if the plane A B moves with the fame
velocity as the wind; that is, if the point B moves through the
ſpace B E in the ſame time that a particle of air would move
through the ſame ſpace; it is plain that, in this caſe, there can be
no preſſure or impulſe of the wind upon the plane: but if the
plane moves flower than the wind, in the ſame direction, ſo that
the point B may move to F, while a particle of air, ſetting out
from B at the fame inſtant, would move to E, then BF will
expreſs the velocity of the plane ; and the relative velocity of
the wind and plane will be expreſſed by the line F E. Let the
ratio of F E to B E be given (ſuppoſe 2 : 3); let the line AB
repreſent the impulſe of the wind upon the plane A B, when
acting with its whole velocity B E; but, when acting with its
relative velocity, F E, let its impulſe be denoted by ſome aliquot
part of A B, as for inſtance, AB: then will of the paralle-
logram A F repreſent the mechanical power of the plane ; that
is, AB X BE.
ز
2dly, Let IN be the ſection of a plane, inclined in ſuch a
manner, that the baſe I K of the rectangle triangle I KN may
be equal to AB; and the perpendicular NK=BE; let the
plane IN be ſtruck by the wind, in the direction L M, perpen-
dicular to IK: then, according to the known rules of ob-
lique forces, the impulſe of the wind upon the plane IN, tend-
ing to move it according to the direction LM, or NK, will
be
E
3
BXPERIMENTAL ENQUIRY, &c.
65
be denoted by the baſe İ K ; and that part of the impulle, tending
to move it according to the direction I K, will be expreſſed by the
perpendicular NK. Let the plane IN be moveable in the di-
rection of IK only; that is, the point I in the direction of IK,
and the point N in the direction Nu, parallel thereto. Now
it is evident, that if the point I moves through the line IK,
while a particle of air, ſetting forwards at the fame time from
the point N, moves through the line N K, they will both ar-
rive at the point K at the ſame time; and conſequently, in this
caſe alſo, there can be no preſſure or impulſe of the particle of
the air upon the plane I N. Now let 1 O be to I K as BF to
BE; and let the plane I N move at ſuch a rate, that the point I
may arrive at O, and acquire the poſition IQ, in the ſame time
that a particle of wind would move through the ſpace NK:
as O Q is parallel to IN; (by the properties of ſimilar tri-
angles) it will cut N. K in the point P, in ſuch a manner, that
NP=BF, and PK=FE: hence it appears, that the plane IN,
by acquiring the poſition O Q, withdraws itſelf from the action of
the wind, by the fame ſpace N P, that the plane A B does by acquir-
ing the poſition FG; and conſequently, from the equality of PK
to FE, the relative impulſe of the Wind PK, upon the plane
O , will be equal to the relative impulſe of the wind FE;
upon the plane F G: and ſince the impulſe of the wind upon
AB, with the relative velocity FE, in the direction BE, is
repreſented by , A B; the relative impulſe of the wind upon the
plane IN, in the direction NK, will in like manner be repres
ſented by IK; and the impulſe of the wind upon the plane
IN, with the relative velocity PK, in the direction I K; will
be repreſented by NK: and conſequently the mechanical
power of the plane IN, in the direction I K, will be the
parallelogram IQ: that is IK XNK: that is, from the
equality of IK=A B and NK=BE, we ſhall have I Q==
ABX BE=FABX BE=of the area of the parallelogram
AF. Hence we deduce this
F
ĠENERAL
tom
66
EXPEKIMENTAL ENQUIRY, &c.
GENERAL PROPOSITION,
That all planes, however ſituated, that intercept the ſame
ſection of the wind, and having the ſame relative velocity, in
regard to the wind, when reduced into the fame direction, have
equal powers to produce mechanical effe&ts.
For what is loſt by the obliquity of the impulſe, is gained
by the velocity of the motion.
Hence it appears, that an oblique fail is under no diſadvan-
tage in reſpect of power, compared with a direct one; except
what ariſes from a diminution of its breadth, in reſpect to the
fection of the wind: the breadth I N being by obliquity re-
duced to IK
The diſadvantage of horizontal windmills therefore does not
conſiſt in this; that each fail, when directly expoſed to the wind
is capable of a leſs power, than an oblique one of the ſame dimen-
fions; but that in an horizontal windmill, little more than
one ſail can be acting at once: whereas in the common wind-
mill, all the four act together: and therefore, fuppofing each
vanс of an horizontal windmill, of the fame dimenſions as each
vane of the vertical, it is manifeſt the power of a vertical mill
with four fails, will be four times greater than the power of the
horizontal one, let its number of vanes be what it will : this
diſadvantage ariſes from the nature of the thing; but if we
conſider the further diſadvantage, that ariſes from the difficulty
of getting the fails back again againſt the wind, &c. we need
not wonder if this kind of mill is in reality found to have not
above or of the power of the common fort; as has appeared
in ſome attempts of this kind.
?
In
.
:
EXPERIMENTAL ENQUIRY, &r.
67
.
In like manner, as little improvement is to be expected from
water-mills with oblique vanes: for the power of the ſame fec-
tion of a ſtream of water, is not greater when acting upon an
oblique vane, than when acting upon a direct one: and any
advantage that can be made by intercepting a greater ſection,
which ſometimes may be done in the caſe of an open river, will
be counterbalanced by the ſuperior reſiſtance, that ſuch vanes
would meet with by moving at right angles to the current :
whereas the common floats always move with the water nearly
in the ſame direction.
Here it may reaſonably be aſked, that ſince our geometrical
demonſtration is general, and proves, that one angle of ob-
liquity is as good as another, why in our experiments it appears,
that there is a certain angle which is to be preferred to all the
reft? It is to be obſerved, that if the breadth of the fail IN is
given, the greater the angle KIN, and the leſs will be the baſe
IK: that is, the ſection of wind interfected, will be lefs : on
the other hand, the more acute the angle KIN, the leſs will
be the perpendicular KN: that is, the impulſe of the wind, in
the direction I K being leſs, and the velocity of the fail greater ;
the reſiſtance of the medium will be greater alſo. Hence there.
fore, as there is a diminution of the ſection of the wind inter-
cepted on one hand, and an increaſe of reſiſtance on the other,
there is ſome angle, where the diſadvantage ariſing from theſe
cauſes upon the whole is the leaſt of all; but as the diſadvantage
ariſing from reſiſtance is more of a phyſical than geometrical con-
ſideration, the true angle will beſt be aſſigned by experiments.
:
SCHOLIUM.
In trying the experiments contained in Tab. III. and IV.
the different ſpecific gravity of the air, which is undoubtedly
different at different times, will cauſe a difference in the load,
proportional to the difference of its ſpecific gravity, though its
F 2
velocity
:
68
EXPERIMENTAL ENQUIRY, &c.
velocity remains the ſame; and a variation of ſpecific gravity
may ariſe not only from a variation of the weight of the whole
column, but alſo by the difference of heat of the air concerned
in the experiment, and poſſibly of other cauſes; yet the irregu-
larities that might ariſe from a difference of ſpecific gravity
were thought to be too ſmall to be perceivable, till after the
principal experiments were made, and their effects compared ;
from which, as well as ſucceeding experiments, thoſe variations
were found to be capable of producing a ſenſible, though no
very confiderable effect : however, as all the experiments were
tried in the ſummer ſeaſon, in the day-time, and under cover, we
may ſuppoſe that the principal ſource of error would ariſe from
the different weight of the column of the atmoſphere at different
times : but as this ſeldom varies above t's part of the whole, we
may conclude, that though many of the irregularities contained
in the experiments referred to in the foregoing eſſay, might ariſe
from this cauſe; yet as all the principal concluſions are drawn
from the medium of a confiderable number, many whereof were
made at different times, it is preſumed that they will nearly agree
with the truth, and be altogether fufficient for regulating the
practical conſtruction of thoſe kind of machines, for which uſe
they were principally intended.
M
AN
AN
EXPERIMENTAL EXAMINATION
OF THE QUANTITY AND PROPORTION OF
H
MECHANIC POWER
NECESSARY TO BE EMPLOYED IN GIVING DIFFERENT
DEGREES OF VELOCITY TO
HEAVY BODIES
FROM A
STATE OF REST.
By MR. JOHN SMEATON, F.R.S.
1
1
2
}
A N
1
EXPERIMENTAL EXAMINATION,
&c.
Read before the Royal Society, April 25, 1776.
ABOUT
BOUT the year 1686 Sir ISAAC NEWTON firſt pub-
liſhed his Principia, and, conformably to the language of
mathematicians of thoſe times defined, that “the quantity of
“ motion is the meaſure of the ſame, ariſing from the velocity
" and quantity of matter conjointly." Very ſoon after this pub-
lication, the truth or propriety of this definition was diſputed by
certain philoſophers, who contended, that the meaſure of the
quantity of motion ſhould be eſtimated by taking the quantity
of matter and the ſquare of the velocity conjointly. There
is nothing more certain, than that from equal impelling powers,
acting for equal intervals of time, equal increaſes of velocity
are acquired by given bodies, when unreſiſted by a medium;
thus gravity cauſes a body, in obeying its impulſe during one
ſecond of time, to acquire a velocity which would carry it uni-
formly forward, without any additional impulſe, at the rate
of 32 ft. 2 in. per fecond; and if gravity is ſuffered to act
F4
upon
을
​72
EXPERIMENTAL EXAMINATION, &c.
.
upon it for two ſeconds, it will have, in that time, acquired a
velocity that would carry it, at an uniform rate, juſt double of
the former ; that is, at the rate of 64 ft. 4 in. per ſecond. Now,
if in conſequence of this equal increaſe of velocity, in an equal
increaſe of time, by the continuance of the ſame impelling
power, we define that to be a double quantity of motion, which
is generated in a given quantity of matter, by the action of the
fame impelling power for a double time; this will be co-inci-
dent with Sir ISAAC NEWTON's definition above mentioned ;
whereas, in trying experiments upon the total effects of bodies
in motion, it appears, that when a body is put in motion, by
whatever cauſe, the impreſion it will make upon an uni-
formly reſiſting medium, or upon uniformly yielding ſub-
ſtances, will be as the maſs of matter of the moving body,
multiplied by the ſquare of its velocity: the queſtion,
therefore, properly is, whether thoſe terms, the quantity of
motion, the momenta of bodies in mation, or forces of bodies in
motion, which have generally been eſteemed ſynonymous, are
with the moſt propriety of language to be eſteemed equal,
double, or triple, when they have been generated by an equable
impulſe, acting for an equal, double, or triple time; or that
it ſhould be meaſured by the effects being equal, double, or
triple, in overcoming refiftances before a body in motion can be
ſtopped ? For, according as thoſe terms are underſtood in this or
that way, it will neceſſarily follow, that the momenta of equal
bodies will be as the velocities, or as the ſquares of the velo.
cities, or as the ſquares of the velocities reſpectively; it being
certain, that, whichever we take for the proper definition of the
term quantity of motion, by paying a proper regard to the
collateral circumſtances that attend the application of it, the
fame concluſion, in point of computation, will reſult. I ſhould
not, therefore, . have thought it worth while to trouble the
Society upon this ſubject, had I not found, that not only myſelf
and other practical artiſts, but alſo fome of the moſt approved
writers, had been liable to fall into errors, in applying theſe
doctrines to practical mechanics, by fometimes forgetting or
neglecting
EXPERIMENTAL EXAMINATION, &c.
73
neglecting the due regard which ought to be had to theſe
collateral circumſtances. Some of theſe errors are not only
very confiderable in themſelves, but alſo of great conſequence
to the public, as they tend greatly to miſlead the practical artiſt
in works that occur daily, and which often require very great
fums of money in their execution. I ſhall mention the follow-
ing inſtances.
?
DESAGULIERS, in his ſecond volume of Experimental
Philoſophy, treating upon the queſtion concerning the forces of
bodies in motion, after taking much pains to ſhew that the
diſpute, which had then ſubſiſted fifty years, was a diſpute about
the meaning of words; and that the ſame concluſion will be
brought out, when things are rightly underſtood, either upon the
old or new opinion, as he diſtinguiſhes them; among other
things, tells us, that the old and new opinion may be eaſily
reconciled in this inſtance: that the wheel of an underſhot
water-mill is capable of doing quadruple work when the
velocity of the water is doubled, inſtead of double work only;
« becauſe (the adjutage being the fame), ſays he, we find, that
as the water's velocity is double, there are twice the number
« of particles of water that iflue out, and therefore the ladle-
« board is ſtruck by twice the matter, which matter moving
fr with twice the velocity that it had in the firſt caſe, the whole
effect muſt be quadruple, though the inftantaneous ſtroke of
« each particle is increaſed only in a ſimple proportion of the
$
velocity.” See vol. II. Annotations on lecture 6th, p. 92.
Again, in the ſame volume, lecture 12th, p. 424, referring
to what went before, he tells us, “ The knowledge of the fore-
« going particulars is abſolutely neceſſary for ſetting an under-
6 ſhot wheel to work; but the advantage to be reaped from it
I would be ſtill gueſswork, and we ſhould be ſtill at a loſs to
find out the utmoſt it can perform, if we had not an in-
In genious propoſition of that excellent mechanic M. PARENT,
of
74
EXPERIMENTAL EXAMINATION, &c.
e of the Royal Academy of Sciences, who has given us a
« maximum in this caſe, by ſhewing, that an underſhot wheel
“ can do the moft work, when its velocity is equal to the
" third part of the velocity of the water that drives it, &c.
« becauſe then two-thirds of the water is employed in driving
is the wheel with a force proportionable to the ſquare of its
« velocity. If we multiply the ſurface of the adjutage or open-
« ing by the height of the water, we ſhall have the column of
water that moves the wheel. The wheel thus moved will
“ ſuſtain on the oppoſite ſide only four-ninths of that weight,
« which will keep it in equilibrio; but what it can move with
" the velocity it goes with, will be but one-third of that weight
« of equilibrium; that is, 4ths of the weight of the firſt
« column, &c.-This is the utmoſt that can be expected."
The ſame concluſion is likewiſe adopted by MacLAURIN, in
art. 907. p.728. of his Fluxions, where, giving the Auxionary
deduction of M. PARENT's propoſition, he ſays, " that if a re-
« preſents the weight which would balance the force of the
“ ſtream, when its velocity is a ; and u repreſents the velocity
« of the part of the engine, which it ſtrikes when the motion
« of the machine is uniform, &c.--the machine will have the
“ greateſt effect when u is equal to ; that is, if the weight
" that is raiſed by the engine be leſs than the weight which
“ would balance the power, in the proportion of 4 to 9, and the
momentum of the weight is 444.»
27
Finding that theſe concluſions were far from the truth; and
ſeeing, from many other circumſtances, that the practical theory
of making water and wind-mills was but very imperfectly deli-
vered by any author I had then an opportunity of conſulting*;
in
* BeLIDOR, Architecture Hydraulique, greatly prefers the ap-
plication of water to an underſhot mill, inſtead of an overſhot; and
attempts
}
EXPERIMENTAL EXAMINATION, &c.
75
in the year 1751 I began a courſe of experiments upon this ſub-
ject. Theſe experiments, with the concluſions drawn from
them, have already been communicated to this Society, who
printed them in vol. lI. of their Tranſactions for the year 1759,
and for this communication I had the honour of receiving the an-
nual medal of Sir GODFREY COPLEY, from the hands of our
very worthy Preſident the late Earl of MACCLESFIELD. Thoſe
experiments and concluſions ſtand uncontroverted, ſo far as I
know, to this day; and having ſince that time been concerned
in directing the conſtruction of a great number of mills, which
were all executed upon the principles deduced from them, I have
by that means had many opportunities of comparing the effects
actually produced with the effects which might be expected from
the calculation; and the agreement, I have always found be-
tween theſe two, appears to me fully to eſtabliſh the truth of the
attempts to demonſtrate, that water applied underſhot will do fix
times more execution than the ſame applied overſhot. See vol. I.
p. 286. While DESAGULIERS, endeavouring to invalidate what
had been advanced by BELIDOR, and greatly preferring an over-
Mot to an underſhot, ſays, Annotat. on lecture 12. vol. II. p. 532.
that from his own experience, “ a well-made overſhot mill ground
as much corn in the ſame time with ten times leſs water;" ſo
that betwixt Belidor and DesAGULIERS, here is a difference of
no leſs than 60 to 1.
Again, BelIDOR, vol. II. p. 72. ſays, that the centre of gra-
vity of each ſail of a windmill ſhould travel in its own circle with
one-third of the velocity of the wind; ſo that, taking the diſtance
of this centre of gravity from the centre of motion at 20 feet, as he
ftates it p. 38. art. 849. the circumference will be exceeding 126
feet Engliſh meaſure : a wind, therefore, to make the mill go
twenty turns per minute, which they frequently do with a freſh
wind and all their cloth ſpread, would require the wind to move
above eighty miles an hour; a velocity perhaps hardly equalled in
the greateſt ſtorms we experience in this climate.
princi.
3
76
EXPERIMENTAL EXAMINATION, &c.
principles upon which they were conſtructed, when applied to
great works, as well as upon a ſmaller ſcale in models.
Reſpecting the explanatory deduction of DesAGULIERS in the
firſt example abovementioned, which, indeed, I have found to
be the commonly received doctrine among theoretical me-
chanics, it is ſhewn, in my former Eſſay, page 21, 22, and 247
part 1, maxim 4, that, where the velocity of water is double, the
adjutage or aperture of the fluice remaining the ſame, the effect
is eight times; that is, not as the ſquare but as the cube of the
velocity; and the ſame is inveſtigated concerning the power of
the wind ariſing from difference of velocity, p. 52, being part
3, maxim 4.
The concluſion in the ſecond example abovementioned, adopt-
ed both by DESAGULIers and MACLAURIN, is not leſs wide of
the truth than the foregoing; for if that conclufion were true,
only 24, ths of the water expended could be raiſed back again to
the height of the reſervoir from which it had deſcended, exclu-
fively of all kinds of friction, &c. which would make the actual
quantity raiſed back again ſtill leſs; that is, leſs than one-ſeventh
of the whole; whereas it appears, from Table I. of the preced-
ing eſſay, that in ſome of the experiments there related,
even upon the ſmall ſcale on which they were tried, the work
done was equivalent to raiſing back again about one quarter
of the water expended; and in large works the effect is ſtill
greater, approaching towards half, which ſeems to be the limit
for the underſhot mills, as the whole would be the limit for the
overſhot mills, if it were poſſible to ſet aſide all friction, refift-
ance from the air, &c. ſee p. 29.
The velocity alſo of the wheel, which, according to M. PA-
RENT's determination, adopted by DESAGULIERS and MA-
ÇLAURIN, ought to be no more than one-third of that of the water,
varies at the maximum in the abovementioned experiments of
Table
EXPERIMENTAL EXAMINATION, &c.
77
A
E
Table I. between one-third and one-half; but in all the caſes
there related, in which the moſt work is performed in propor-
tion to the water expended, and which approach the neareſt to
the circumſtances of great works, when properly executed, the
maximum lies much nearer to one-half than one-third; one-half
feeming to be the true maximum, if nothing were loſt by the
reſiſtance of the air, the ſcattering of the water carried up by
the wheel, and thrown off by the centrifugal force, &c. all
which tend to diminiſh the effect more, at what would be the
maximum if theſe did not take place, than they do when the
motion is a little flower.
Finding theſe matters, as well as others, to come out in the
experiments, ſo very different from the opinions and calcula-
tioris of authors of the first reputation, who, reaſoning accord-
ing to the Newtonian definition, muſt have been led into theſe
errors from a want of attending to the proper collateral circum-
ſtances; I thought it very material, eſpecially for the practical
artiſt, that he ſhould make uſe of a kind of reaſoning in which
he ſhould not be ſo liable to miſtakes; in order, therefore, to
make this matter perfectly clear to myſelf, and poſſibly ſo to
others, I reſolved to try a ſet of experiments from whence it
might be inferred, what proportion or quantity of mechanical
power is expended in giving the ſame body different degrees of
velocity. This ſcheme was put in execution in the year 1759,
and the experiments were then ſhewn to ſeveral friends, parti-
cularly my very worthy and ingenious friend Mr. WILLIAM
RUSSELL
In my experimental inquiry concerning the powers of water
and wind before referred to, I have, p. 105, part 1. defined
what I meant by power as applied to practical mechanics, that
is, what I now call mechanical power ; which, in terms fynony-
mous to thoſe there uſed, may be ſaid to be meaſured by multi-
plying the weight of the body into the perpendicular height from,
which
78
EXPERIMENTAL EXAMINATION, &c.
which it can deſcend; thus the ſame weight, deſcending froin
a double height, is capable of producing a double mecha-
nical effect, and is therefore a double mechanical power. A
double weight deſcending from the ſame height is alſo a double
power, becauſe it likewiſe is capable of producing a double ef-
fect; and a given body, deſcending through a given perpendi-
cular height, is the ſame power as a double body deſcending
through half that perpendicular ; for, by the intervention of pro-
per levers, they will counterbalance one another, conformably
to the known laws of mechanics, which have never been con-
troverted. It muſt, however, be always underſtood, that the
deſcending body, when acting as a meaſure of power, is ſuppoſed
to deſcend flowly, like the weight of a clock or a jack; for, if
quickly deſcending, it is fenfibly compounded with another law,
viz. the law of acceleration by gravity.
DESCRIPTION OF THE MACHINE.
AB is the baſe of the machine placed upon a table.
A c is a pillar or ſtandard.
cd is an arm, upon the extremity of which is fixed a plate
fg, which is here ſeen edge-ways, through which is a ſmall
hole for receiving a ſmall ſteel pivot e, fixed in the top of the
upright axis e B; the lower end of this axis finiſhes in a conical
ſteel point, which refts upon a ſmall cup of hard ſteel poliſhed at B.
H I is a cylinder of white fir, which paſſes through a perfora-
tion in the axis, and therein fixes; and, upon the two arms formed
thereby, are capable of ſliding.
KL two cylindric weights of lead of equal fize, which are
capable of being fixed upon any part of the cylindric arms,
from
EXPERIMENTAL EXAMINATION, &c.
79
from the axis to their extremities, by means of two thin wedges
of wood. The two weights, therefore, being at equal diſtan-
ces from the centre, and the axis perpendicular, the whole will
be balanced upon the point at B, and moveable thereupon by an
impelling power, with very little friction.
Upon the upper part of the axis are formed m N, two cylindri-
cal barrels, whereof M is double the diameter of N; they have
a little pin ſtuck into one ſide of each at 0, p.
e is a piece capable of Niding higher or lower, as occafion re.
quires ; and carries
R, a light pulley of about three inches diameter, hung upon
a ſteel axis, and moveable upon two ſmall pivots. The plane of
the pulley, however, is not directed to the middle of the upright
axis, but a little on one ſide, ſo as to point (at a mean) between
the ſurface of the bigger barrel and the leſs.
s is a light ſcale for receiving weights, and hangs by a ſmall
twine, cord, or line, that paſſes the pulley, and terminates
either upon the bigger barrel or the leſs, as may be required; the
Riding-piece e being moved higher or lower for each, that the
line, in palling from the pulley to the barrel, may be nearly
horizontal. The end of the line, that is furtheſt from the ſcale,
is terminated by a ſmall loop, which hangs on upon the pin o,
or the pin p, according as the bigger or the leſſer barrel is to be
uſed.
,
Now, having wound up a certain number of turns of the
line upon the barrel, and having placed a weight in the ſcale s, it
is obvious, that it will cauſe the axis to turn round, and give
motion to its arms, and to the weights of lead placed thereon,
which are the heavy bodies to be put in motion by the impulſe
of the weight in the ſcale; and when the line is wound off to the
ping
80
EXPERIMENTAL EXAMINATION, &c.
work
pin, the loop Nips off, and the ſcale then falling down, the weight
will ceaſe to accelerate the motion of the heavy bodies, and leave
them revolving, equably forward, with the velocity they have
acquired, except fo far as it muſt be gradually leſſened by the
friction of the machine and reſiſtance of the air, which being
ſmall, the bodies will 'revolve ſometime before their velocity is
apparently diminiſhed.
MEASURES OF SOME PARTS OF THE MACHINE.
Inches.
Diameter of the cylinders of lead, or the heavy bodies
Length of ditto
Diameter of the hole therein
1
2,57
1,56
372
.
1
mener
ti
8,25
Weight of each cylinder 3 lbs. Avoirdupois.
Greater diſtance of the middle of each body from the
}
centre of the axis
The ſmaller diſtance of ditto
jo turns of the ſmaller barrel raiſes the ſcale, 5 ditto
of the bigger ditto
am
3,92
} 25,25
+
When the bodies are at the ſmaller diſtance above ſpecified
from the axis of rotation, they are then in effect at half the
greater diſtance from that axis : før, ſince the axis itſelf, and
the cylindric arms of wood, keep an unvaried diſtance from the
centre of rotation, the bodies themſelves muſt be moved nearer
than half their former diſtance, in order that, compounded with
the unvariable parts, they may be virtually at the half diſtance.
In order to find this half diſtance nearly, I put in an arm of the
fame wood, that only went through the axis; without extending
in the oppoſite direction; one of the bodies being put upon the
end of this arm, at the diſtance of 8,25 inches, the whole ma-
chine was inclined till the body and arm became a kind of pen-
duluri,
EXPERIMENTAL EXAMINATION; &c.
81
1
dulum, and vibrated at the rate of 92 times per minute; and as
a pendulum of the half length vibrates quicker in the proportion
of ✓ I to v 2; that is, in the proportion of 92 to 130 nearly;
therefore, keeping the ſame inclination of the machine, the
weight was moved upon the arm till it made 130 vibrations per
minute, which was found to be, when it was at 3,92 inches
diſtance from the centre as above ſtated, which is about oths
of an inch nearer than the half diſtance. The double arm was
then put in, and marked accordingly, and the bodies being
mounted thereon, the whole was adjuſted ready for uſe; and
with it were tried the following experiments, each of which was
repeated ſo many times as to be fully fatisfactory.
TABLE
OF
EX PER I M E N T S.
the
Ounces Avoirdupois
in the Scale.
Barrel uſed, M the
bigger, N
The Arms, W the
whole, H the half-
Number of Turns of
the Line wound
Time of the Deſcent
-on the Barrel.
of the Weight in
ſmaller.
length.
Revolutions with
the Scale.
Time in making 20
equal motions.
Nº.
I
14
29
2
8
8
8
M
N
N
W
W
W
5
IO
24
28.
N
29
Oo
3
147
587
41 32
5 32
õu
14
M
N
N
3
14
7
14
7
IO
2
6 32
28-1
w HHH
3
7
8
8
8
M
N
N
5
IO
21
7
14
7
14
15
30+
9
I
2
3
4
5
6
7
.:
G
The
.
82
EXPERIMENTAL EXAMINATION, &c.
The 51"} in number 3, column 7, was determined in fact
from 29"$, being the time of making 10 equable revolutions
after the weight was dropped off, in order to prevent the ſenſi-
ble retardation that might take place, and affect the obſervation,
if continued for 20 revolutions made fo flowly.
FURTHER DEFINITIONS.
1 have already defined what I mean by mechanic power; buty
before I proceed further, it will be neceſſary alſo to define the
following terms:
Impulſe or Impulſion, By all which, I underſtand the
Impulſive Force or Power, Suniform endeavour that one bo-
Impelling Force or Power,) dy exerts upon another, in order
to make it move; and that, whether it produces or generates
motion by this endeavour or not; and the quantity of this im-
pelling power may be meaſured either by its being a weight of
itſelf, or by being counterbalanced by a weight. It may alſo
act either immediately upon the body to be moved, ſo that if
motion is the conſequence, they move with the fame velocity;
and that, either by a ſimple contact, or by being drawn as by a
cord, or puſhed as by a ſtaff: or it nay act by the intervention
of a lever or other mechanic inftrument, in which the velocity
of the body to be moved may be very different from the velo-
city of the impelling power or mover ; but in comparing them,
the impelling powers muſt be reduced according to the
propor-
tional velocities of the mover and moved; or, in levers of diffe-
rent lengths, they may be compared by a ſtandard length of le-
ver, which is the method taken in the ſubſequent reafoning upon
the preceding experiments. An impelling power, therefore,
conſiſting of a double weight, or requiring a double weight to
counterbalance it, when acting with equal levers, is a double
impelling power, or an impelling power of double the intenſity.
OBSER
:
EXPERIMENTAL EXAMINATION, &c.
83
OBSERVATIONS AND DEDUCTIONS FROM THE PRECEDING
EXPERIMENTS.
iſt
, By the firſt experiment it appears, that the mechanic
power employed, conſiſting of 8 ounces in the ſcale, deliberate-
ly deſcending (by $ turns of the bigger barrel) through a per-
pendicular ſpace 25 inches, will repreſent the quantity of me
chanic power which cauſes the two heavy bodies, from a ſtate of
reſt, tò acquire a velocity, ſuch as to carry them equably
through 20 circumferences of their circle of revolution in the
ſpace of 29"; and that the time in which the mechanic power
produced this effect was 14"}, as appears by column 6th. And
this mechanic power we ſhall expreſs by the number 202, the
product of the number of ounces in the ſcale multiplied by the
inches in its perpendicular deſcent, for 8 * 257=202.
2d, By the ſecond experiment; as 10 turns of the ſmaller bara
rel are equal to the ſame perpendicular height as 5 turns of the
bigger, it follows, that the ſame mechanic power, viz. 202,
acting upon the ſame heavy bodies to accelerate them, produces
the very fame effect in generating motion in the bodies as it did
before, viz. 20 revolutions in 29", the ſmall difference of of
a ſecond being no more than may reaſonably be attributed to the
unavoidable errors ariſing from friction of the machine, want of
perfect accuracy in its meaſures, reſiſtance of the air, and im-
perfections in the obſervations themſelves, which muſt not only
be allowed for in this, but the reſt; but as the impelling power
is acting here upon a lever of but half the length, and, conſe
quently, buthalf the intenſity, when referred to the bodies to be
moved, it takes juſt double the time to generate the ſame velo-
city therein.
Deduction. It appears from hence, that the ſame me
chanic power is capable of producing the fame velocity in a given
G2
body
84
EXPERIMENTAL EXAMINATION, &c.
body, whether it is applied fo as to produce it in a greater
or leſſer time; but that the time taken to produce a given velo-
city, by an uniformly continued action, is in a ſimple inverſe pro-
portion of the intenſity of the impulſive power.
3dly, The third experiment being made with 2 turns of the
leſfer barrel, the ſame weight in the ſcale of 8 ounces deſcending
only 1-quarter part of the former perpendicular, the mechanic
power employed will be only one quarter part of the former, viz.
50; but as one quarter part of the mechanic power produces half
of the former velocity in the heavy bodies; that is, they make 20
revolutions in 58"} that is, nearly to revolutions in 29"; we
may conclude, in this inftance, that the mechanic power, em-
ployed in producing motion, is as the ſquare of the velocity pro-
duced in the ſame body; and that the velocity produced is as the
time that an impelling power, of the fame intenſity, continues to
act upon it, as appears by the near agreement of numbers 2 and
3, column 6th.
4thly, In the fourth experiment, the apparatus is the ſame as
the firſt, only here the weight in the ſcale is 32 ounces; that is,
the impelling power is the quadruple of the firſt, and hereby a
double velocity is given to the bodies; for they make 20 revolu-
tions in 14", which is a ſmall matter leſs than half the time taken
up in making 20 revolutions in the firſt experiment. It alſo ap-
pears, that the velocity acquired is ſimply as the impelling power
compounded with the time of its action; for a quadruple impul-
fion acting for y" inſtead of 14" generares a double velocity, while
the mechanic power employed to generate it is quadruple, for
32 x 251808. And here the mechanic power employed being
four times greater than the firſt, it holds here alſo, that the me-
chanic power, to be neceſſarily employed, is as the ſquare of the
velocity to be generated; that is, in the ſame proportion as turned
out in the third experiment, where the mechanic power employ-
ed was only a quarter part of the firſt.
5thly,
EXPERIMENTAL EXAMINATION, &c.
85
Sthly, The fifth and ſixth experiments were made with a mea
chanic power four times greater than thoſe employed in numbers
2 and 3 reſpectively; and ſince the ſame deductions reſult from
hence as from numbers 2 and 3, they are additional confirma-
tions of the conclufions drawn from them and from the laſt
article,
.
6thly, In the ſeventh experiment, the diſpoſition of the appa-
ratus is the ſame as number 1, only here the bodies are placed
upon the arms at the half-length; from whence it appears, that
the ſame mechanic power ſtill produces the fame velocity in the
fame bodies; for though 20 revolutions were performed in 14"}
(ſee column 7) which is nearly half the time that 20 revolutions
were performed in the firſt experiment; yet, ſince the circles in
which the bodies revolved in the ſeventh are only of half the
circumference as thoſe of number 1, it is obvious, that the abſo-
lute velocity acquired by the moving bodies in the two caſes is
equal. But, by column 6th, the time in which it was generated
is only half; yet, notwithſtanding, this will coincide with the
former concluſions, if the intenſity of the impelling power is
compounded therewith; for, though the barrel was the fame
with the ſame number of turns as in number 1, and therefore the
lever the ſame, by which the impelling power acted, yet, as the
bodies, upon which this lever was to act, were placed upon a
lever of only half the length from the centre, the impelling power
acting by the firſt lever, would act upon the ſecond with double
the intenſity, according to the known laws of mechanics; that
is, it would require a double weight oppoſing the bodies, to pre-
vent their moving, in order to balance it. An impulſive power,
therefore, of double the intenſity, acting for half the time, pro-
duces the ſame effect in generating motion, as an impulſiva
power, of half the intenſity, acting for the whole time,
7thly, The eighth and ninth experiments afford the ſame de-
ductions and confirmations relative to the ſeventh experiment,
that
G3
26
EXPERIMENTAL EXAMINATION, &c.
that the fifth and fixth do reſpecting the fourth, and that the
ſecond and third do reſpecting the firſt; and from the near agree-
ment of the whole, when the neceſſary allowances before men
tioned are made, together with ſome ſmall inequality ariſing from
the mechanical power loft by the difference of the motion given
by gravity to the weight in the ſcale : I ſay, from theſe agree-
ments, under the very different mechanical powers applied,
which were varied in the proportion of 1 to 16, we may ſafely
conclude, that this is the univerſal law of nature, reſpecting the
capacities of bodies in motion to produce mechanical effects, and
the quantity of mechanic power neceſſary to be employed to pro,
duce or generate different velocities (the bodies being ſuppoſed
equal in their quantity of matter); that the mechanic powers to
be expended are as the ſquares of the velocities to be generated,
and vice verſa; and that the ſimple velocities generated are as
the impelling power compounded with, or multiplied by, the
time of its action, and vice verſa.
We ſhall, perhaps, form a ſtill clearer conception of the re-
lation between velocities produced and the quantity of mechanic
power required to produce them; together with the collateral
circumſtances attending, by which theſe propoſitions, ſeemingly
two, are reconciled and united, by ſtating the following popular
elucidation, which indeed was the original idea that occurred
to me on conſidering this fubject; to put which to an expe-
rimental proof gave birth to the foregoing apparatus and expen
fiments,
Suppoſe then a large iron ball of 10 feet diameter, turned
truly ſpherical, and ſet upon an extended plane of the fame
metal, and truly level. Now, if a man begins to puſh at it,
he will find it very reſiſting to motion at firſt; but, by. con-
ținuing the impulſe, he will gradually get it into motion, and
having nothing to reſiſt it but the air, he will, by continuing
his efforts, at length get it to rolì almoſt as faſt as he can run,
Şup-
EXPERIMENTAL EXAMINATION, &c.
87
f
Suppoſe now, in the firſt minute he gets it rolled through a
fpace of one yard; by this motion, proceeding from reft (ſimilar
to what happens to falling bodies) it would continue to roll
forward at the rate of two yards per minute, without further
help; but fuppofing him to continue his endeavours, at the end
of another minute he will have given it a velocity capable of.
carrying it through a ſpace of two yards more, in addition to
the former, that is, at the rate of four yards per minute; and at
the end of the third minute, he has again added an equal in-
creaſe of velocity, and made it proceed at the rate of fix yards
per minute ; and ſo on, increaſing its velocity at the rate of two
yards in every minute. The man, therefore, in the ſpace of
every minute exerts an equal impulſe upon the ball, and ge-
nerates an equal increaſe of movement correſpondent to the
definition of Sir Isaac NewTON. But let us ſee what hap-
pens befides: the man, in the firſt minute, has moved but one
yard from where he ſet out; but he muſt in the fecond minute
move two yards more, in order to keep up with the ball; and
as he exerted an impulſe upon it, ſo as at the end of the ſecond
minute to have given it an additional velocity of the two yards,
he muſt alſo in this time. have gradually changed its velocity
from the rate of two yards per minute to that of four, and the
ſpace, that he will of conſequence have actually been obliged to
go through in the ſecond minute, will be according to the mean
of the extremes of velocity at the beginning and end thereof, that
is, three yards in the ſecond minute; ſo that being one yard
from his original place at the beginning of the ſecond minute, at
the end of it he will have moved the ſum of the journies of the
firſt and ſecond minute, that is, in the whole four yards from his
original place. As he has now generated a velocity in the ball
of four yards per minute, in the third minute he muſt travel
four yards to keep up with the ball, and one more in generating
the equal increment of velocity; ſo that in the third minute,
he muſt travel five yards to keep up the fame impelling power
G4
upon
88
EXPERIMENTAL EXAMINATION, &c.
:.
:
upon the ball that he did in the firſt minute in travelling one,
ſo that theſe five yards in the third minute added to the four yards
that he had travelled in the two preceding minutes, fets him at
the end of the third minute nine yards from whence he ſet out,
having then given the ball a velocity capable of carrying it
uniformly forward at the rate of fix yards per minute, as be-
fore ſtated. We may now leave the further purſuit of theſe
proportions, and ſee how the account ſtands. He generated a
velocity of two yards per minute in the firſt minute, the ſquare
of which is four, when he had moved but one yard from his
place; and he had generated a velocity of fix yards per minute,
the ſquare of which is thirty-ſix, at the end of the third
minute, when he had travelled nine yards from his place. Now,
ſince the ſquare of the velocity, generated at the end of the firſt
minute, is to that of the velocity generated at the end of the
third minute, as 4 : 36, that is, as 1 : 9; and fince the
ſpaces, moved through by the man to communicate theſe velo-
cities, are alſo as 1:9, it follows, that the ſpaces through
which the man muſt travel, in order to generate thefe velocities
reſpectively (preſerving the impelling power perfectly equal),
muſt be as the ſquares of the velocities that are communicated
to the ball; for, if the man was to be brought back again to his
original place by a mechanical power, equally exerted upon the
man equally refifting, this would be the meaſure of what the
man has done in order to give motion to the ball. It therefore
directly follows, conformably to what has been deduced from the
experiments, that the mechanic power that muſt of neceſſity be
employed in giving different degrees of velocity to the ſame
body, muſt be as the ſquare of that velocity; and if the con-
verſe of this propoſition did not hold, viz. that if a body in mo-
tion, in being ſtopped, would not produce a mechanical effect
equal or proportional to the ſquare of its velocity, or to the
mechanical power employed in producing it, the effect would not
correſpond with its producing cauſe,
the
Thus
EXPERIMENTAL EXAMINATION, &c.
89
Thus the conſequences of generating motion upon a level plane
exactly correſpond with the generating of motion by gravity:
viz. that though in two ſeconds of time the equal impulſive
power of gravity gives twice the velocity to a body that it does
in one ſecond, yet this collateral circumſtanee attends it, that at
the end of the double time, in confequence of the velocity ac-
quired in the firſt half, the body has fallen from where it ſet for-
ward through four times the perpendicular; and therefore, though
the velocity is only doubled, yet four times the mechanical
power has been conſumed in producing it, as four times the
mechanical power muſt be expended in bringing up the fallen
body to its firſt place,
N
This then appears to be the foundation, not only of the
diſputes that have ariſen, but of the miſtakes that have been
made, in the application of the different definitions of quantity
of motion; that while thoſe, that have adhered to the definition
of Sir Isaac Newton, have complained of their adverſaries, in
not conſidering the time in which effects are produced, they
themſelves have not always taken into the account the ſpace that
the impelling power is obliged to travel through, in producing
the different degrees of velocity. It ſeems, therefore, that,
without taking in the collateral circumſtances both of time and
ſpace, the terms, quantity of motion, momentum, and force of
bodies in motion, are abſolutely indefinite; and that they cannot
be ſo eaſily, diſtinctly, and fundamentally compared, as by having
recourſe to the common meaſure, viz. mechanic power,
From the whole of what has been inveſtigated, it, therefore
appears, that time, properly ſpeaking, has nothing to do with
the production of mechanical effects, otherwiſe than as, by
equally flowing, it becomes a common meaſure; ſo that,
whatever mechanical effect is found to be produced in a given
time,
.
90
EXPERIMENTAL EXAMINATION, &c.
time, the uniform continuance of the action of the ſame me.
chanical power will, in a double time, produce two ſuch effects,
or twice that effect. A mechanical power, therefore, properly
ſpeaking, is meaſured by the whole of its mechanical effect
produced, whether that effect is produced in a greater or a leſſer
time; thus, having treaſured up 1000 tuns of water, which
I can let out upon the overſhot wheel of a mill, and deſcending
through a perpendicular of 20 feet, this power applied to proper
mechanic inſtruments, will produce a certain effect, that is, it
will grind a certain quantity of corn; and that, at a certain rate
of expending it, it will grind this corn in an hour, But fup-
poſe the mill equally adapted to produce a proportionable effect,
by the application of a greater impulſive power as with a leſs;
then, if I let out the water twice as faſt upon the wheel, it will
grind the corn twice as faſt, and both the water will be expended
and the corn ground in half an hour. Here the ſame mechanical
effect is produced; viz. the grinding a given quantity of corne
by the fame mechanical power, viz. 1000 tuns of water deſcen-
ding through a given perpendicular of 20 feet, and yet this effect
is in one caſe produced in half the time of the other, What time,
therefore, has to do in the buſineſs is this: let the rate of doing
the buſineſs, or producing the effect, be what it will, if this
rate is uniform, when I have found by experiment what is done
in a given time, then, proceeding at the ſame rate, twice the
effect will be produced in twice the time, on fuppoſition that I
have a ſupply of mechanic power to go on with. Thus 1000
tuns of water, deſcending through 20 feet of perpendicular,
being, as has been ſhewn, a given mechanic power, let me ex-
pend it at what rate I will, if when this is expended, I muſt wait
another hour before it be renewed, by the natural flow of a
river or otherwiſe, I can then only expend twelve ſuch quan-
tities of power in 24 hours; but if, while I am expending 1000
tuns in one hour, the ftream renews me the ſame quantity, then
I can
.
1.
EXPERIMENTAL EXAMINATION, &c.
97
44
I can expend 24 ſuch quantities of power in 24 hours; that is,
I can go on continually at that rate, and the product or effect
will be in proportion to time, which is the common meaſure;
but the quantity of mechanic power ariſing from the flow of the
two rivers, compared by taking an equal portion of time, is
double in the one to the other, though each has a mill, that, when
going, will grind an equal quantity of corn in an hour.
NEW
美
​注​:
:
:
县
​子
​:
NEW
..
FUNDAMENTAL EXPERIMENTS
UPON THE
COLLISON OF BODIES.
BY MR. JOHN SMEATON, F.R. S.
:;
.
:
1
.
주
​EXPERIMENTS
UPON THE
COLLISION OF BODIES.
Read before the Royal Society, April 18, 1782.
T is univerſally acknowledged, that the firſt ſimple principles
of fcience cannot be too critically examined, in order to their
being firmly eſtabliſhed; more eſpecially thoſe which relate to
the practical and operative parts of mechanics, upon which much
of the active buſineſs of mankind depends. A ſentiment of this
kind occafioned my tract upon Mechanic Power which is the
fecond paper of this volume. What I have now to offer was
intended as a ſupplement thereto, and the experiments were then,
in part, tried; but the completion thereof was deferred at that
time, partly from want of leiſure; partly to avoid too great a
length of the paper itſelf, and partly to avoid the bringing for
ward too many points at once.
My preſent purpoſe is to ſhew, that the true doctrine of the
colliſion of bodies hangs as it were upon the ſame hook, as the
doctrine of the gradual generation of motion from reft, confi-
dered
96
EXPERIMENTS UPON COLLISION.
dered in that paper : that is, that whether bodies are put into
gradual motion, and uniformly accelerated from reft to any given
velocity; or are put in motion, in an inſtantaneous manner, when
bodies of any kind ſtrike one another; the motion, or fum of the
motions produced, has the ſame relation to mechanic power
therein defined, which is neceſſary to produce the motion deſired.
To prove this, and at the ſame time to fhew ſome capital miſtakes
in principle, which have been aſſumed as indiſputable trụths by
men of great learning, is the reaſon of my now purſuing the
fame ſubject.
I do not mean to point out the particular miſtakes which
have been made by particular men, as that would lead me into
too great a length : I ſhall therefore content myſelf with obſer-
ving, that the laws of collifion, which have been inveſtigated
by mathematical philoſophers, are principally of three kinds;
viz. thoſe relating to bodies perfectly elaſtic; to bodies perfectly
unelaftic, and perfectly ſoft; and to bodies perfectly unelaſtic,
and perfectly hard. To avoid prolixity, I ſhall conſider in each,
only the fimple caſe of two bodies which are equal in weight or
quantity of matter ſtriking one another. Reſpecting thoſe which
are perfectly elaſtic, it is univerſally agreed, that when two ſuch
bodies ſtrike one another, no motion is loft; but that in all caſes,
what is loft by one is acquired by the other: and hence, that if
an elaſtic body in motion ſtrikes another at reſt, upon the ſtroke
the former will be reduced to a ſtate of reft, and the latter will
fly off with an equal velocity.
A
In like manner, if a non-elaſtic ſoft body ſtrikes another at
reſt, they neither of them remain at reſt, but proceed together
from the point of colliſion with exactly one half the velocity
that the firſt had before the ſtroke; this is alſo univerſally allow-
ed to be true, and is fully proved by every good experiment upon
the ſubject.
Reſpecting the third fpecies of body, that is, thoſe that are
non-elaſtic and yet perfectly hard: the laws of motion relating
੫
<
i
EXPERIMENTS UPON COLLISION.
97
to them, as laid down by one ſpecies of Philoſophers, have been
rejected by another ; the latter alledging, that there are no ſuch
bodies to be found in nature whereon to try the experiment;
but thoſe who have laid down and aſſigned the doctrine that
would attend the colliſion of bodies, of this kind (if they could
be found) have univerſally agreed, that if a non-elaſtic hard body
was to ſtrike another of the ſame kind at reſt, that, in the ſame
manner as iš agreed concerning non-elaſtic ſoft bodies, they
neither of them would remain at reſt, but would in like manner
proceed from the point of colliſion, with exactly one half of the
velocity that the firſt had before the ſtroke: in ſhort, they lay it
down as a rule attending all non-elaſtic bodies, whether hard or
ſoft, that the velocity after the ſtroke will be the ſame in both,
viz, one half of the velocity of the original ſtriking body.
Here is therefore the aſſumption of a principle, which in reality
is proved by no experiment, nor by any fair deduction of reaſon
that I know of, viz. that the velocity of non-elaſtic hard bodies
after the ſtroke muſt be the ſame as that reſulting from the ſtroke
of non-elaſtic ſoft bodies į and the queſtion now is, whether it is
true or not?
Here it may be very properly aſked, what ill effects can re-
ſult to practical men, if philoſophers ſhould reaſon wrong con-
cerning the effects of what does not exiſt in nature, ſince the
practical men can have no ſuch materials to work upon, or miſ.
judge of? But it is anſwered, that they who infer an equality
of effects between the two ſorts, may from thence be miſled
themſelves, and in confequence miſlead practical men in their
reaſonings and concluſions concerning the fort with which they
have abundant concern, to wit, the non-elaſtic ſoft bodies, of
' which water is one, which they have much to do with in their
daily practice.
Previous to the trying my experiment on mills, I never had
doubted the truth of the doctrine, that the ſame velocity reſulted
H
from
98
EXPERIMENTS UPON COLLISION.
from the ſtroke of both forts of non-elaſtic bodies; but the trial
of thoſe experiments made me clearly fee at leaſt the inconclu-
ſiveneſs, if not the falſity of that doctrine: becauſe I found a
reſult which I did not expect to have ariſen from either fort; and
for the which, when it appeared from experiment, I could fee
a ſubſtantial reaſon why it ſhould take place in one fort, and that
it was impoſſible that it could take place in the other; for if it
did, the bodies could not have been perfectly hard, which would
be contrary to the hypotheſis: of this deduction I have given no-
tice in my faid tract on mills, page 30. The effect therefore of
overſhot wheels, &c.
It may alſo be ſaid, that fince we have no bodies perfectly
elaſtic, or perfectly unelaſtic and ſoft, why ſhould we expect
bodies perfectly unelaſtic and hard? Why may not the effects be
ſuch as ſhould reſult from a ſuppoſition of their being imperfe&tly
elaſtic joined with their being imperfe&tly hard? But here I muſt
obſerve, that the ſuppoſition appears to be a contradiction in
1
terms.
We have bodies which are ſo ncarly perfectly elaſtic, that the
laws may be very well deduced and confirmed by them; and the
fame obtains with reſpect to non-elaſtic ſoft bodies; but con-
cerning bodies of a mixed nature, which are by far the greateſt
number, ſo far as they are wanting in elaſticity, they are ſoft
and bruiſe, yield or leave a mark in colliſion; and fo far as
they are not perfectly foft they are elaſtic, and obſerve a mixture
of the law relative to each; but imperfectly elaſtic bodies,
imperfectly hard, come in reality under the ſame deſcription as
the former mixed bodies : for ſo far as they are imperfectly
hard they are ſoft, and either bruiſe and yield, or leave a mark
in the ſtroke; and ſo far as they want perfect elaſticity, they are
non-elaſtic; that is to ſay, they are bodies imperfectly elaſtic,
and imperfectly ſoft; and in fact I have never yet feen any
bodies but what come under this deſcription. It ſeems, there-
fore, that reſpecting the hardneſs of bodies they differ in de-
grees
EXPERIMENTS UPON COLLISION.
99
grees of it, in proportion as they have a greater degree of tena-
city or coheſion ; that is, are further removed from perfect
ſoftneſs, at the ſame time that their elaſtic ſprings, ſo far as they
reach, are very ſtiff; and hence we may (by the way) conclude,
that the ſame mechanic power that is required to change the
figure in a ſmall degree of thoſe bodies that have the popular
appellation of hard bodies, would change it in a great degree in
thoſe bodies that approach towards ſoftneſs, by having a ſmall
degree of tenacity or coheſion. In the former kind we may
rank the harder kinds of caſt iron, and in the latter ſoft tempered
clay.
While the philoſophical world was divided by the diſpute
about the old and new opinion, as it was called, concerning the
powers of bodies in motion, in proportion to their different
velocities : thoſe who held the old opinion contending, that it
was as the velocity ſimply, aſked thoſe of the new, How, upon
their principles, they would get rid of the concluſions ariſing
from the doctrine of unelaſtic perfectly hard bodies ? They
replied, they found no ſuch bodies in nature, and therefore
did not concern themſelves about them. On the other hand, ,
thoſe of the new opinion aſked thoſe of the old, How they
would account for the caſe of non-elaſtic foft bodies, where,
according to them, the whole motion loft by the ſtriking body
was retained in the two after the ſtroke (the two bodies moving
together with the half velocity,) though the two non-elaſtic
bodies had been bruiſed and changed their figure by the ſtroke;
for, if no motion was loft, the change of figure muſt be an
effect without a cauſe? To obviate this, thoſe of the old opi-
nion ſeriouſly ſet about proving, that the bodies might change
their figure, without any loſs of motion in either of the ſtrik-
ing bodies.
$
Neither of theſe anſwers have appeared to me ſatisfactory,
eſpecially ſince my mill experiments: for with reſpect to the
firſt, it is no proper argument to urge the impoffibility of find-
H2
ing
0
6
I
100
EXPERIMENTS UPON COLLISION.
ing the proper material for an experiment, in anſwer to a con-
clufion drawn from an abſtract idea. On the other hand, if it
can be ſhewn, that the figure of a body can be changed, with-
out a power, then, by the ſame law, we might be able to make
a forge hammer work upon a maſs of ſoft iron, without any
other power than that neceſſary to overcome the friction, refift-
ance, and original vis inertiæ of the parts of the machine to be
put in motion : for, as no progreſſive motion is given the maſs of
iron by the hammer (it being ſupported by the anvil), no power
can be expended that way; and if none is loft to the hammer
from changing the figure of the iron, which is the only effect
produced, then the whole power muſt reſide in the hammer, and
it would jump back again to the place from which it fell, juſt
in the fame manner as if it fell upon a body perfectly elaſtic, upon
which, if it did fall, the caſe would really happen: the power
tủerefore, to work the hammer would be the ſame, whether it
fell upon an elaſtic or non-elaſtic body; an idea ſo very contrary
to all experience, and even apprehenſion, of both the philo-
ſopher and vulgar artiſt, that I ſhall here leave it to its own con-
demnation.
As nothing, however, is ſo convincing to the mind as experi-
ments obvious to the ſenſes, I was very deſirous of contriving
an experiment in point, and as I ſaw no hopes of finding matter
to make a direct experiment, I turned my mind towards an in-
direct one: ſo circumſcribed, however, as to prove inconteſta-
bly, that the reſult of the ſtroke of two non-elaſtic perfectly
hard bodies could not be the ſame as would reſult from the colli.
fion of two ſoft ones; that is, if it can be bona fide proved,
that one half of the original power is loſt in the ſtroke of ſoft
bodies by the change of figure (as was very ſtrongly ſuggeſted
by the mill experiments); then fince no ſuch loſs can happen in
the colliſion of bodies perfectly hard, the reſult and conſequence
of ſuch a ſtroke muſt be different.
The conſequence of a ſtroke of bodies perfectly hard, but
void of elasticity, muſt doubtleſs be different from that of bodies
perfectly
EXPERIMENTS UPON COLLISION.
IOI
'
perfectly elaſtic : for having no ſpring the body at reſt could not
be driven off with the velocity of the ſtriking body, for that is.
the conſequence of the action of the ſpring or elaſtic parts be-
tween them, as will be fhewn in the reſult of the experiments;
the ſtriking body will therefore not be ſtopped, and as the mo-
tion it loſes muſt be communicated to the other, from the equa-
lity of action and re-action, they will both proceed together,
with an equal velocity, as in the caſe of non-elaſtic ſoft bodies :
thie queſtion, therefore, that remains is, what that velocity muſt
be? It muſt be greater than that of the non-elaſtic ſoft bodies,
becauſe there is no mechanical power loſt in the ſtroke. It muſt
be leſs than that of the ſtriking body, becauſe, if equal, inſtead
of a loſs of motion by the colliſion it will be doubled. If,
therefore, non-elaſtic ſoft bodies loſe half their motion, or me-
chanical power, by change of figure in colliſion, and yet pro-
ceed together with half the velocity, and the non-elaſtic hard
bodies can loſe none in any manner whatever; then, as they
muſt move together, their velocity muſt be ſuch as to preſerve
the equality of the mechanic power, unimpaired, after the ſtroke
the ſame as it was before it.
A
For example, let the velocity of the ſtriking body before the
ſtroke be 20, and its maſs or quantity of matter 8; then
according to the rule deduced from the experiments in the Tract
on Mechanic Power (ſee exp. third and fourth) that power will
be expreſſed by 20 x 20=400, which x 8=3200 ; and if half
of it is loſt in the ſtroke, in the caſe of non-elaſtic ſoft bodies,
it will be reduced to 1600; which + 16 the double quantity of
matter will give 100 for the ſquare of their velocity; the
ſquare root of which being 10, will be the velocity of the two
non-elaſtic ſoft bodies after the ſtroke, being juſt one half of the
original velocity, as it is conſtantly found to be. But in the
non-elaſtic hard bodies, no power being loft in the ſtroke, the
mechanic power will remain after it as before it, = 3200 ; this,
in like manner, being divided by 16, the double quantity of
matter, will give 200 for the ſquare of the velocity, the ſquare
root
H 3
I02
EXPERIMENTS UPON COLLISION. .
3
root of which is 14.14, &c. for their velocity after the ſtroke,
which is to 10, the velocity of the non-elaſtic ſoft bodies after
the ſtroke, as the ſquare root of 2 to 1, or as the diagonal of a
ſquare to its fide.
It remains, therefore, now to be proved, that preciſely half
of the mechanic power is loft in the colliſion of no-elaſtic ſoft
bodies; for which purpoſe my mind ſuggeſted the following re-
flections. In the colliſion of elaſtic bodies the effect ſeemingly
inſtantaneous, is yet performed in time; during which time the
natural ſprings reſiding in elaſtic bodies, and which conſtitute
them ſuch, are bent or forced, till the motion of the ſtriking
body is divided between itſelf and the body at reſt ; and in this
ſtate the two bodies would then proceed together, as in the caſe of
non-elaſtic ſoft bodies; but as the ſprings will immediately reſtore
themſelves in an equal time, and with the ſame degree of impul-
five force, wherewith they were bent in this re-action, the motion
that remained in the ſtriking body will be totally deſtroyed, and
the total exertion of the two ſprings, communicated to the origi-
nal reſting body, will cauſe it to fly off with the fame velocity
wherewith it was ſtruck.
Upon this idea, if we could conſtruct a couple of bodies in
ſuch a way that they ſhould either act as bodies perfectly elaſtic
or that their ſprings ſhould at pleaſure be hooked up, retained,
or prevented from reſtoring themſelves, when at their extreme
degree of bending; and if the bodies under theſe circumſtances
obſerved the laws of colliſion of non-elaſtic ſoft bodies, then it
would be proved, that one half of the mechanical power, reſiding
in the ſtriking body, would be loſt in the action of colliſion;
becauſe the impulſive force or power of the ſpring in its reftitu-
tion being cut off, or ſuſpended from acting, which is equal to
the impulſive force or power to bend it (and which alone has
been employed to communicate motion from one body to the
other) it would make it evident, that one half of the impulſive
force is loſt in the action, as the other half remains locked up in
the
EXPERIMENTS UPON COLLISION.
103
the ſprings, it alſo follows, as a collateral circumſtance, that be
the impulſive power of the ſprings what it may from firſt to laſt,
yet as one half of the time of the action is by this means cut off,
in this ſenſe alſo it will follow, that one half of the mechanic
power is deſtroyed; or rather, in this caſe, remains locked up
in the ſprings, capable of being re-exerted whenever they are
ſet at liberty, and of producing a freſh mechanical effect, equiva-
lent to the motion or mechanical power of the two non-elaſtic
ſoft bodies after their colliſion.
Hence we muſt infer, that the quantity of mechanical power
expended in diſplacing the parts of non-elaſtic ſoft bodies in
colliſion, is exactly the ſame as that expended in bending the
ſprings of perfectly elaſtic bodies; but the difference in the ulti-
mate effect is, that in the non-elaſtic ſoft bodies, the power
taken to diſplace the parts will be totally loft and deſtroyed, as
it would require an equal mechanic power to be raiſed a-freſh,
and exerted in a contrary direction to reſtore the parts back again
to their former places; whereas, in the caſe of the elaſtic bodies,
the operation of half the mechanic power is, as obſerved already,
only locked up and ſuſpended, and capable of being re-exerted
without a further original acceſſion.
Theſe ideas aroſe from the reſult of the experiments tried upon
the machine deſcribed in my faid tract upon mechanic power, and
were alſo communicated to my very worthy and ingenious friend
WILLIAM RUSSEL, Eſq. F.R. S. at the ſame time that I ſhewed
him thoſe experiments in 1759; but the mode of putting this
matter to a full and fair mechanical trial has ſince occurred; and
though ſome rough trials, ſufficient to ſhew the effect, were made
thereon, prior to the offering the Paper on Mechanical Power to
the Society in 1776, yet the machine itſelf I had not leiſure to
complete to my fatisfaction till lately; which I mention to apo-
logize for the length of time that theſe ſpeculations have taken in
bringing forward.
H 4
DESCRIP
Š
104
EXPERIMENTS UPON COLLISION,
DESCRIPTION OF THE MACHINE FOR COLLISION,
FIG. I. Thews the front of the Machine as it appears at reſt
when fitted for uſe.
ܪܪ
A is the pedeſtal, and A B the pillar, which ſupports the whole,
CD are two compound bodies of about a pound weight each, but
as nearly equal in weight as may be. Theſe bodies are alike in
conſtruction, which will be more particularly explained by fig. 2.
Theſe bodies are ſuſpended by two white fir rods of about half an
inch diameter, ef and g h being about four feet long from the
point of ſuſpenſion to the centre of the bodies; and their ſuſpen-
ſion is upon the croſs piece II, which is mortiſed through, to let
the rods paſs with perfect freedom; and they hang upon two finall
plates filed to an edge on the under ſide, and paſs through the
upper part of the rods. Their centers are at k and l, and the
edges being let into a little notch, on each ſide the mortiſe,
the rods are at liberty to vibrate freely upon their reſpective points
(or rather edges) of ſuſpenſion, and are determined to one plain
of vibration. MN is a flat arch of white wood, which may be
covered with paper, that the marks thereupon may be more con-
ſpicuous.
The croſs piece II is made to project ſo far before the pillar,
that the bodies in their vibrations may paſs clear of it, without
danger of ſtriking it; and alſo the arch MN is brought ſo far
forward as to leave no more than a clearance, ſufficient for the
rods to vibratę freely without touching it,
FIG. 2. thews one of the compound bodies, drawn of its full
fize, AB is a block of wood, and about as much in breadth
as it is repreſented in height, through a hole in which the wood
rod CC paſſes, and is fixed therein,
DO
EXPERIMENTS UPON COLLISION.
105
3
!
DB repreſents a plate of lead about three-eights of an inch
thick, one on each ſide, ſcrewed on by way of giving it a com-
petent weight. d Bef g repreſents the edge of a ſpringing plate
of braſs, rendered elaſtic by hard hammering; it is about five
eights of an inch in breadth, and about one twentieth of an
inch thick. It is fixed down upon the wooden block at its end
dB by means of a bridge plate, whoſe end is ſhewn bi, and is
ſcrewed down on each ſide of the ſpring plate by ſcrews, which
being relaxed the ſpring can be taken out at pleaſure, and ada
juſted to its proper ſituation. kl is a light thin ſlip of a plate,
whoſe under edge is cut into teeth like a fine ſaw or ratchet, and
is attached to the ſpring by a pin at k, which paſſes through it,
and alſo through a ſmall ftud rivetted into the back part of the
ſpring, and upon which pin, as a center, it is freely moveable.
mn fhews a ſmall plate or ftud ſeen edgeways raiſed upon the
bridge plate, through an hole in which ſtud the ratchet paſſes ;
and the lower part of the hole is cut to a tooth ſhaped properly
to catch the teeth of the ratchet, and retain it together with the
spring at any degree to which it may be ſuddenly bent; and for
this intent it is kept bearing gently downward, by means of a
wire-ſpring opg, which is in reality double, the bearing part ato
being ſemi-circular; from which branching off on each ſide the
rod C C, paſſes to P, and fixes at each end into the wood at q.
However to clear the ratchet, which is neceſſarily in the middle
as well as the rod, the latter is perforated; and alſo the block is
cut away, fo far as to ſet the mainſpring at e free of all obſtacles
that would prevent its play from the point B. The part f g is
ſhewn thicker than the reſt, by being covered with thin kid
leather tight fewed on, to prevent a certain jarring that other.
wiſe takes place on the meeting of the ſprings in colliſion,
Let us now return to fig. 1. the marks upon the arch MN
are put on as follows. Op is an arch of a circle from the centre
h, and qr an arch of a circle from the centre k, interſecting
each other at S. Now the middle line of the marks t, v, are:
at the fame diſtance from the middle line at $ that the centres del
are;
106
EXPERIMENTS UPON COLLISION.
are; ſo that when each body hangs in its own free poſition, with
out bearing againſt the other, the rod e f will cover the mark at
t; and the rod g h will cover the mark at v. From the point S
upon the arches S p and Sq reſpectively, ſet off points at an equal
and competent diſtance from S each way, which will give the
middle of the mark w and x: and upon the arch Sp find the
middle point between the mark v and w, which let be y; and on
the other ſide, in like manner, upon the arch Sq find a middle
point for the mark Z; then ſet off the diſtance Sv or S t from y
each way, and from z each way; and from theſe points, drawing
lines, to the reſpective centres i and k, they will give the place
and poſition of the marks a, b, and cod; and thus is the machine
prepared for uſe.
FOR TRIALS ON ELASTIC BODIES.
FOR this uſe take out the pins and ratchets from each re-
ſpectively, and the ſprings being then at liberty, with a ſhort
bit of ſtick (ſuppoſe the fame fize as the rods) turn aſide the rod
gh with the right hand, carrying the body D upwards till the
ſtick is upon the mark w, as ſuppoſe at 0; there hold it, and
with the left ſet the body C perfectly at reſt; in which caſe the
rod e f will be over the mark t; then ſuddenly withdraw the
ſtick, in the direction that the rod g h is to follow it, and the
ſpring of the body D, impinging upon that of the body C, they
will be both bent, and alſo reſtored; and the body C will fly off,
and mount till its rod e f covers the mark *; the rod of the
ſtriking body D remaining at reſt upon its proper mark of reſt v,
till the body C returns, when the body D will fly off in the
fame manner; the two bodies thus rebounding a number of
times, loſing a part of their vibration each time; but fo nearly
is the theory of elaſtic bodies fulfilled hereby, that the ſingle ad-
vantage of originally puſhing the rod g h beyond the mark w,
by the thickneſs of the ſtick, or its own thickneſs, is ſufficient
to carry the rod of the quieſcent body C completely to its
mark *.
There
representante
EXPERIMENTS UPON COLLISION.
107
?
There are ſeveral other experiments which may be made with
this apparatus, in confirmation of the doctrine of the colliſion of
elaſtic bodies; which being univerſally agreed upon, and well
known, it is needleſs further to dwell upon here; but reſpecting
the application to non-elaſtic ſoft bodies, it is far more difficult
to come at a fitneſs of materials for this kind of experiment,
than it is for thoſe ſuppoſing perfectly elaſticity. The conclu .
fions, however, may be attained with equal certainty.
FOR TRIALS ON NON-ELASTIC SOFT BODIES.
3
FOR this purpoſe the ratchets muſt be applied and put in
order as before deſcribed, and the ſprings being both put to their
point of reft, let the body D be put to its mark w in the ſame
manner as before deſcribed, and the body C to reſt. The body
D being let go, and ſtriking the body C at reſt, in conſequence
of the ſtroke, the ſprings being hooked up by the ratchets, they
both move from their reſting marks t, v, reſpectly towards
M: Now if they both moved together, and the rod e f covered
the mark c, and the rod g h covered the mark d at their utmoſt
limit, then they would truly obey the laws of non-elaſtic ſoft
bodies; becauſe their medium aſcent would be to the mark z,
which is juſt half the angle of aſcent to the mark x; but as in
this piece of machinery, though the main or principal ſprings are
hooked up, yet every part of them, and all the materials of which
they are compoſed, and to which they are attached, have a degree,
or more properly ſpeaking, a certain compaſs of elaſticity, which,
as ſuch, is perfect, and no motion loft therehy.
.
We muſt not, therefore, expect the two compound bodies after
the ſtroke to ſtick together without ſeparating, as would be the
caſe with bodies truly non-elaſtic and ſoft; but that from the elaf-
ticity they are poſſeſſed of, they will by rebounding be ſeparated;
but that elaſticity being perfect, can occaſion no loſs of motion to
the ſum of the two bodies; ſo that if the body C aſcends as much
above its mark e as the body D falls ſhort of its mark d, then
it
1
108
EXPERIMENTS UPON COLLISION,
it will follow, that their medium afcent will ſtill be to the mark
%, as it ought to have been, had they been truly non-elaſtic ſoft
bodies; and this, in reality, is truly the caſe in the experiment,
as nearly as it can be diſcerned.
After a few vibrations, by the rubbing of the ſprings
againſt one another, they are foon brought to reſt; and here
they would always reft had they been truly and properly perfect
non-elaſtic ſoft bodies; but here, as in the caſe of theſe bodies,
by a change of the figure and ſituation of the component parts,
there is expended one half of the mechanical power of the firſt
mover, yet in this caſe the other half is not loft but ſuſpended
ready to be re-exerted whenever it is ſet at liberty; and that it is
really and bona fida one half, and neither more or leſs appears
from this uncontroverted fimple principle, that the power of
reftitution of a perfect ſpring is exactly equal to the power
that bends it. And this may, in a certain degree, he ſhewn
to be fact by experiment, if there were any need of ſuch a proof;
for if, when the bodies are at reft after the laſt experiment the two
rods are laſhed together at the bottom with a bit of thread, and
then the ratchets unpinned and removed; on cutting the thread
with a pair of ſcifrars they will each of them rebound, C towards
M and D towards N, and if they rebounded reſpectively to
z and y, the mechanical power exerted would be the ſame as it
was after the ſtroke, when the mean of their two afcents was up
to the mark Z ; but here it is not to be expected, becauſe not
only the motion loft by the friction of the ratchets is to be
deducted, becauſe it had the effect of real non-elaſticity; but
alſo the elaſticity that ſeparated them in the ſtroke, which was
loft in the vibrations that ſucceeded; neither of which hindered
the mean aſcent to be to z; but yet, under all theſe diſadvan-
tages in the machine (if not unreaſonably ill made) the rod ef
will aſcend to d, and g h to a : and hence I infer, as a poſitive
truth, that in the colliſion of non-elaſtic ſoft bodies, one half
of the mechanic power reſiding in the ſtriking body is loft in
the Aroke.
Reſpecting
1
.:
EXPERIMENTS UPON COLLISION.
109
Reſpecting bodies unelaſtic and perfectly hard, we muſt infer,
that ſince we are unavoidably led to a concluſion concerning them,
which contradicts what is eſteemed a truth capable of the ſtrict-
eft demonſtration ; viz. that the velocity of the centre of gra-
vity of no ſyſtem of bodies can be changed by any collifion be-
twixt one another, ſomething muſt be aſſumed that involves
a contradiction. This perfectly holds, according to all the
eſtabliſhed rules, both of perfectly elaſtic and perfectly non-
elaſtic ſoft bodies ; rules which muſt fail in the perfectly non-
elaſtic hard bodies, if their velocity after the ſtroke is to the
velocity of the ſtriking body as one is to the ſquare root of 2;
for then the centre of gravity of the two bodies will by the
ftroke acquire a velocity greater than the centre of gravity the
two bodies had before the ſtroke in that proportion, which is
proved thus.
At the outſet of the ſtriking body, the centre of gravity of
the two bodies in our caſe will be exactly in the middle between
the two; and when they meet it will have moved from their
half diſtance to their point of contact, ſo the velocity of the
centre of gravity before the bodies meet will be exactly one half
of the velocity of the ſtriking body; and, therefore, if the velo-
city of the ſtriking body is 2, the velocity of the centre of gravity
of both will be 1. After the ſtroke, as both bodies are ſup-
poſed to move in contact, the velocity of the centre of gravity
will be the ſame as that of the bodies; and as their velocity is
proved to be the ſquare root of 2, the velocity of their centre of
gravity will be increafed from 1. to the ſquare root of 2; that is,
from 1. to 1.414, &c.
The fair inference from theſe contradictory concluſions there-
fore is, that an unelaſtic hard body (perfectly fo) is a repugnant
idea, and contains in itſelf a contradiction ; for to make it agree
with the fair conclufions that may be drawn on each ſide, from
clear premiſes, we ſhall be obliged to define its properties thus :
That in the ſtroke of unelaſtic hard bodies they can not poſibly
Loje
3
IIO
EXPERIMENTS UPON COLLISION.
.
loſe any mechanic power in the ſtroke; becauſe no other im-
preſſion is made than the communication of motion; and yet
they muſt loſe a quantity of mechanic power in the ſtroke; be-
cauſe, if they do not, their common centre of gravity, as above
ſhewn, will acquire an increaſe of velocity by their ſtroke upon
each other.
In a like manner the idea of a perpetual motion perhaps, at
firſt ſight, may not appear to involve a contradiction in terms;
but we ſhall be obliged to confeſs that it does, when, on exa-
mining its requiſites for execution, we find we ſhall want bo-
dies having the following properties; that when they are made
to aſcend againſt gravitation their abſolute weight ſhall be leſs ;
and when they deſcend by gravitation (through an equal ſpace)
their abſolute weight ſhall be greater ; which, according to all
we know of nature, is a repugnant or contradictory idea.
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>
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London Published by I. &JIalor, Holborn.
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​6
EXPERIMENTAL
EN QUIRIES
CONCERNING THE PRINCIPLE
OF THE
LATERAL COMMUNICATION OF MOTION
IN
FLUIDS;
>
APPLIED TO THE EXPLANATION OF
VARIOUS HYDRAULIC PHENOMENA.
BY CITIZEN J. B. VENTURI,
Profeſſor of Natural Philoſophy at Modena, Member of the Italian
Society, of the Inſtitute of Bologna, of the Agrarian
Society of Turin, &c.
}
TRANSLATED FROM THE FRENCH.
:
London:
PRINTED FOR J. TAYLOR, AT THE ARCHITECTURAL
LIBRARY, HIGH-HOLBORN.
1799.
*
!
ADVERTISEMENT.
.
This intereſting Work (lately publiſhed at
Paris, with the approbation of the French Na-
tional Inſtitute, after the examination and re-
port of Boſſut, Coulomb, and Prony) was ſent
to me ſome time ago, by the Author. I peruſed it
with much pleaſure and inſtruction, and with
the intention of publiſhing an abſtract of its
contents in my Journal of Natural Philofo-
phy, Chemiſtry, and the Arts*; but found
it impoſſible to do juſtice to the novelty and
importance of the ſubject, in the way of
abridgment: for which reaſon I gave an
entire tranſlation of the whole. The Public
will no doubt be glad to have it in a ſeparate
form. It is now printed in the ſame ſize and
type as the original. The tranſlation is very
cloſe, and almoſt verbal; no liberties having
* This work, which I hope and believe is at prefent
well known to the ſcientific world, is publiſhed monthly,
in quarto.-Two volumes have already appeared, and the
third is in progreſs.
A 2
in
(iv)
in
any
inſtance been taken, except ſuch as
the different ſtructure of the two languages,
and the clear expreſſion of the ſenſe, demanded.
The proof ſheets in the Journal were exa-
mined and corrected by the work itſelf, and
all the parts where reference is made to the
engravings were at the ſame time literally com-
pared with the plates. The ſame proceſs was
alſo repeated with this ſeparate publication,
which
may
therefore be concluded to poſſeſs
all that fidelity and accuracy which are ne-
ceſſary to inſure the confidence of the reader,
and to render a tranſlated work of equal uti-
lity with the original.
WILLIAM NICHOLSON.
LONDON,
April 5, 1799.
TABLE
TABLE OF CONTENTS.
DESCRIPTION of the Apparatus
Page
It is proper to miſtruſt all Theories of Hydraulics, not excepting
that which is exhibited in the preſent Work, excepting ſo far
as thoſe Theories may be fupported by Experiment 3
In any Fluid, thoſe Parts which are in Motion carry along with
them the lateral Parts which are at Reſt. Prop. I. Exper. I 5
I call this Phenomenon the latcral Communication of Motion ; and
I conſider it as a Principle of Experiment, or elementary
Fact, without explaining its Cauſe
7
If Water be drawn out of a Veſſel by an horizontal cylindrical
Pipe, of which the Part neareſt the Veſſel is contracted accord-
ing to the Form of the contracted Vein of Water which
flows through an equal Orifice in a thin Plate, the Expenditure
will be increaſed by this Pipe, in the ſame Mamer as if there
had been no Contraction. Prop. II. Exper. 3 and 4 8
The Velocity of the Stream, within this Tube, is greater than
that of the Jet through a thin Plate
9
The Increaſe of Expenditure of Water through an horizontal
cylindrical Pipe, whether it be of uniform Diameter through-
out or contracted at the End next the Reſervoir, is cauſed by
the Preffure of the Atmoſphere. Prop. III. Exper. 5, 6, 7 IL
This Increaſe of Expenditure, through a Pipe, does not take
place in the Vacuum of the Air Pump. Ibid. Exper. 8 - 13
When Water is drawn through a deſcending cylindrical Tube,
of which the upper Part is of a divergent Form, anſwerable
to that of the contracted Vein, the Expenditure will be
that which is anſwerable to the Height of the Charge above
the lower Orifice of the Tube. Rectification of the Theory
aſſerted on this Subject by Guillielmini, and adopted by
various Philoſophers.--Experiments. Prop. IV. Exper. 9, 10,
15
The
1
!
Il, 12
(vi)
24
C
The lateral Communication of Motion in Fluids is the Cauſe
which excites the Preſſure of the Atmoſphere to increaſe the
Expenditure and internal Velocity irr horizontal conical Tubes
of a certain Form. Prop. V.
Page 21
Experiments relative to this Augmentation--Their Reſult al-
ways falls ſhort of the Theory--Cauſe of this Defect. Exper.
13, 14, 15
24
Limits of the Augmentation of Expenditure in the faine horizon-
tal conical Tubes. Exper. 16, 17
26
In horizontal cylindrical Tubes, the Increaſe of Expenditure
does not approach the Maximum fo nearly as in conical Tubes,
Prop. VI. Exper. 18
Caſe of the fimplecylindrical Tube more particularly examined 30
The Velocity of the fluid Stream iſſuing through a Tube is leſs
than that which flows through an Orifice in a thin Plate.
Cauſe of this Difference. Exper. 19
GI
The ſame Law and the fame Cauſes alſo deterinine the Ex-
penditure through aſcending and defcending Tubes. Ex-
per. 20, 21
32
The Effect of the lateral Communication of Motion is produced
in a very ſhort Portion of the Length of the inner Cavity of
the cylindrical Tube. Exper. 22
34
The Effect itſelf is greater than could have ariſen from the mu-
tual Attraction of the Particles of the Fluid
ibiu,
The Expenditure which takes place through a cylindrical Tube
of given Dimenſions, and under the fame Charge, may be.
increaſed to nearly double by proper Adjutages. Prop. V. 35
Roman Law relative to this Object. Ibid.
34
Application of the fame Law to the Flues of Chimnies. Ib. 36
How far Elbows and Sinuofities diminiſh the Expenditure
through Tubes. Exper. 23
37
Lors of Expenditure occaſioned by Enlargements or inflated
Parts in Tubes. It is neceſſary to have regard to this in the
Conſtruction of hydraulic Machines. Exper. 24 38
In the Machine for blowing by means of a Fall of Water, the
Air is afforded to the Furnace by the accelerating Force of
Gravity, and the lateral Communication of Motion com -
bined together. Prop. VIII. -
40
Wind
:
(vii)
Wind produced by Falls of Water in the internal Parts of
Mountains. Ibid.
- Page 4
The Wind of the water-blowing Machine is not produced by
the Decompoſition of Water. Exper. 25
43
Quantity of Wind which one of theſe Machines is capable of
affording in a given Time. Ibid. -
46
It is poffible, in certain Circumſtances, by means of a Fall of
Water, to drain a Piece of Ground, without the Help of Ma-
chines, even though the Ground mould be on a lower Level
than the eſtabliſhed Current below the Fall. Prop. IX. 47
Application of the fame Principle to the Tail Water of
Mills. Ibid.
47
The Eddies of Water in Rivers are produced by Motion com-
inunicated from the more rapid Parts of the Stream to the
lateral Parts which are leſs rapidly moved. Prop. X. 49
Vertical Eddies at the Surface and at the Bottom of the Stream
of Rivers. lbid.
50
Theſe circular Motions conſtitute one of the principal Cauſes
of the Loſs of active Force, and the Retardation of the Cur.
rent of Rivers. Ibid.
51
In a River of which the Courſe is permanent, and the Sections
of its Bed unequal, the Water continues more elevated than
it would have done, if the whole River had been equally
contracted to the Dimenſions of its ſmalleſt Section. Ibid. 53
Whirling Motions or Eddies formed in a Reſervoir from which
the Water iſſues by an horizontal Aperture. Theory de.
duced from the Doctrine of central Forces.
55
The Cavity of theſe Whirls is convex on the Side next the Axis.
Ibid.
55
Phenomena relative to theſe Whirls, and their Explanation.
Exper. 26, 27, 28, 29, 30
58
The lateral Communication of Motion takes place in the Air
as well as in Water. Prop. XII.
60
The Excitation of Sound in Organ-pipes is effected by this
Communication. Ibid.
6r
The ſame Cauſe augments the Force of Sound in conical di-
vergent Pipes. Ibid.
63
Remarkable
5
Prop. XI.
w
1
(viii)
Remarkable Differences between the reſonant Vibrations of the
Air in a Pipe, and the Pulſations propagated in the Atmo-
ſphere. Ibid. -
Page 64
The Contraction of the fluid Vein which iſſues through a thin
Plate, is not the Newtonian Cataract
65
The Velocity of the contracted Stream through a thin Plate is
nearly the fame as that of an heavy Body which may have
fallen through the Height of the Charge
69
Singular Form of the Stream which is emitted through a long
Hole or Cieft. Exper. 31
70
In right-lined Orifices the sides of the contracted Stream an-
ſwer to the Angles of the Orifice, and the contrary. Cauſe of
this Phenomenon
71
The Contraction of the fluid Vein is made at a greater Diſtance
under a ſtrong Charge than when the Charge is weak. Ex-
per. 32
792
Other Varieties in the Figure and Velocity of the contracted
Stream. Exper. 33 and 34
72
Expenditure through a Tube, the inner Extremity of which is
thruſt into the Cavity of the Reſervoir itſelf. Exper. 35 74
5.
INTRO
INTRODUCTION.
THE apparatus made uſe of in moſt of the fol-
lowing experiments is the ſame as that of Poleni*.
It is repreſented at fig. 1, pl. I. The reſervoir X,
of a conical form, is forty inches in diameter at
CE, and thirty at OP. FP is a broad plate of
copper, the plane of which is perpendicular to
the horizon ; it is applied to the inſide of the
reſervoir. The valve or flap FS, movcable by
the handle K, is drawn up againſt the ſide of the
veſſel above F, in order that it may not impede the
courſe of the particles of the fluid, contained in the
reſervoir, to the aperture P. I have applied dif-
ferent ajutages to this aperture, according to the
exigence of the caſe. The tubes which I applied
were made of tinned iron of the beſt quality; the
longitudinal junction of the edges was made by
immediate contact, and not by overlapping, and
i
* De Caſtellis. This treatiſe is reprinted in the Third Vo-
lume of Hydraulic Treatiſes, publiſhed at Parma. V.
B
thc
( 2 )
:
the whole of thc workmanſhip was executed with
great care. When the aperture was fimply a
hole through a thin plate, the thickneſs of its
edge did not exceed one fourth of a line.
The
upper veſſel Z ſerves to maintain the water
of the rcfervoir X at the conſtant height of the
line CE, while it flows out through P. The
plug A B is drawn more or leſs back, in order to
regulate the introduction of the ſupply. The box
or ſhelf D L prevents this water from exciting by
its fall any agitation which might influence the
emiſſion at P. The opening at Q diſcharges the
ſuperfluous water which might riſe above the line
CE. The height of the ſurface CE above the
centre of the orifice at P was 32,5 inches, in all
caſes where it is not otherwiſe expreſſed.
Moſt of the experiments here deſcribed were
made in public at the Philoſophical Theatre of
Modena; various men of ſcience were preſent at
the reſt; and the different departments of experi-
ment were performed by ſeveral perſons at the ſame
time. One of theſe operators repeated the ſeconds
audibly from the clock; another drew back the
valve SF; a third regulated by the means of the
plug B the introduction of the ſupply of water,
ſo that a very thin ſheet of water conſtantly
flowed at Q. At the inſtant agreed upon, the
paſſages of the water were again cloſed. Every
experiment was repeated ſucceſſively for a num-
ber of times, until the agreement of the reſults
had
I
( 3 )
be re-
had removed every ſuſpicion of error.
I am aí.
ſured that even in the moſt complicated caſes,
the quantity of error could not exceed one fortieth
part of the reſult.
The meaſures indicated in the courſe of theſe
experiments were taken from a toiſe adjuſted by
that of the Academy, which Citizen Lalande fent
me in 1783. Theſe meaſures, as well as all the
others of the cighteenth century, will undergo
the fate which is prepared for them by the eſta-
bliſhment of the new metre. They may
duced to this new ſtandard, by obſerving that the
foot is to the metre as 100 to 308 *.
The wiſeft philoſophers have their doubts with
regard to every abfiract theory concerning the
motion of fluids; and even the greateſt geometers
avow that thoſe methods, which have afforded
them ſuch ſurpriſing advances in the mechanics
of folid bodies, do not afford any conclufions
with regard to hydraulics but ſuch as ara toa
general and uncertain for the greater number of
particular caſes. Impreſſed with a conviction of
this truth, I have attended to theory only, when
it combined with the facts, and was neceſſary
* To avoid fractions, and for other obvious reaſons, I have left
all the numbers as they ſtand in the original. But, for the con-
venience of thoſe who cannot readily refer to the proportional
magnitudes, I take the opportunity of remarking that the Paris
foot royal is to the Engliſh foot as 1065 is to 1000; that it is
divided into twelve inches, and the inch into twelve lines. T.
B 2
to
1
( 4 )
to unite them under a ſingle point of view.
Even this ſmall portion of theory may, if the
reader pleaſes, be rejected; and he may confider
the following propofitions ſimply as the reſults of
experiment.
When I quote the eſtimable work of Citizen
Boſſut, on hydrodynamics, I refer to the edition
of 1786 *
.
* I conſider this treatiſe as ſuperior to all which before were
extant. It is founded on a combination of the principles of
experiment and of theory. I have profited by theſe principles, and
ſeveral particular remarks which the ſame Citizen Boſſut and
Citizen Prony have been ſo good as to communicate after peru.
fal of my memoir. V.
!
EXPE-
EXPERIMENTAL
RESEARCHES
Concerning the Principle of the lateral Com-
munication of Motion in Fluids, applied to
the Explanation of various hydraulic Phe-
nomena.
.
PROPOSITION I.
The Motion of a Fluid is communicated to the
lateral Parts which are at Reft.
NEWTON has affirmed, that when motion is
propagated in a fluid, and has arrived beyond
the aperture B C, fig. 2, the motion diverges from
that opening, as from a centre, and is propagated
in right lines towards the lateral parts N K, as
well as towards S. The ſimple and immediate
application of this theorem cannot be made to
a jet which iſſues from the aperture BC at the
ſurface of ſtill water. Circumſtances enter into
this
2
4
:
( 6 )
$
this caſe which transform the reſult of the prin-
ciple into particular motions. It is nevertheleſ;
true, that the jet B C communicates its motion
to the lateral parts NK; but it does not repel
them towards P and Q, but, on the contrary,
tranſports them along with its own ftream to-
wards s.
Experiment I. The horizontal cylindric pipe
AC, fig. 3, is introduced into the veſſel DEFB,
which is filled with water as high as D B. Op-
poſite, and at a ſmall interval from the aperture
C, commences a ſmall rectangular channel of
tinned iron, SMBR, which is open at top SR;
the inclined bottom M B reſis on the edge of the
vefſel B. It is twenty-four lines broad ; the dia-
meter of the tube AC is 14,5 lines; the extre-
mity A is applied to the aperture P of fig. 1.
The water of the reſervoir being ſuffered to flow
through the tube AC, the jet riſes along the
ſmall channel MB, and flies out of the veſſel in
the fircam BV. By this means a current is pro-
duced in the fluid of the veſſel DEFB; this
fluid enters into the channel SR, and iſſues by
MB V along with the jet AC, ſo that in a few
feconds the water DB falls to MH.
Exper. II. Bring ſome very light or moveable
bodies near the jet of water P Y, fig. 1, which
iſſues from the aperture P, and falls from a cer-
tain height into the inferior veffel RT. It is
ſeen that theſe bodies are carried along by the air
which
A
를
​>
1
(7)
which defcends with the jet P Y. Part of this air
is carried along and plunged into the water of the
inferior veſſel.
Theſe experiments clearly prove that the fluid
which iſſues by B C, fig. 2, impreſſes its motion
on the lateral parts NK; not by impelling them
towards P Q, but by carrying them along with
itſelf towards s. I call this the lateral communica-
tion of motion in fluids. Newton was acquainted
with this communication, and has deduced from
it the propagation of rotatory motion from the
interior to the exterior ſtrata of a whirlpool. Is
this lateral communication of motion occaſioned
by the viſcidity or mutual adheſion of the parts
of the fluid, or their mutual engagement or inter-
mixture, or the divergency of thoſe parts which
are in motion ? We may perhaps to ſay a few
words on this ſubject when we ſhall have ſeen the
effects; but in the mean time, whatever
may
be
the cauſe, let us take the effect as experience
points it out; let us conſider it as a principle, and
endeavour to apply it to ſome particular caſes, in
order to aſcertain the reſult.
The firſt circumftance to which I propoſe to
apply this principle, is the increaſe of expenditure
of fluid iſſuing out of an orifice fitted with addi-
tional tubes.
PRO.
.
7
( 8 )
PROPOSITION II.
If that Part of an additional cylindric Tube which is
neareſt the Side of the Reſervoir be corilracted, ac-
cording to the form of the contrasted Vein of Fluid
which iſues through a Hole of the fame Diameter
in a thin Plate, the Expenditure will be the ſame
as if the Tube were not contraEted at all.
It is well known, that when the water of a re-
fervoir is ſuffered to flow through a circular orifice
in a thin plate, the fluid vein which forms the jet
becomes contracted at a ſhort diſtance from the
orifice; and the diameter of the contracted vein
is nearly 0,8 of the diameter of the orifice. Poleni
firſt obſerved, that by applying an additional cy-
lindric pipe to the orifice, of the fame diameter as
the orifice itſelf, and from two to four times that
length, the expenditure is increaſed from 100 to
133. To account for this augmentation, he ſup-
poſes that the fluid vein is leſs contracted in pipes
than after paſſing through the thin plate. The
fuppofition was not unreaſonable ; but it could
not apply to the caſe announced in this propofi-
tion. I ſhall proceed to give the particulars in
the following experiment.
Exper. III. To the aperture P, of fig. 1, I ap-
plied a circular orifice 18 lines in diameter, pierced
through a thin plate. Four cubical feet of water
flowed into the veſſel Y in 41 ſeconds.
I then
I
(9)
I then applied to the orifice a cylindric tube
of the ſame diameter, and fifty-four lines long. The
four cubic fect flowed out in thirty-one ſeconds.
Inſtead of this ſimple cylindric tube, I ap-
plied the compound tube of fig. 5; the parts of
which have the following dimenſions in fines :
AC=GI=MN= 18; DF= 14,5; AB=11;
BG = 10; GM= 37; AM = 58. With this
compound tube the expenditure of four cubic feet
of water was made in thirty-one ſeconds, as with
the ſimple cylindric tube.
The form of the conical portion A CDF was
nearly the ſame as that of the contraction of the
vein which iſſues through a thin plate. The vein
muſt therefore have paſſed through a contraction
nearly equal to that of the contracted vein from a
thin plate; the expenditure nevertheleſs was more
abundant, in the ſame proportion as through the
ſimple cylindric tube. It follows, therefore, that
the velocity of the ſection DF, and of the whole
conoid ACDF, muſt have been greater than
that of the contracted vein from a thin plate; and
it remains to be ſhown what was the cauſe of this
augmentation of velocity which takes place within
the tube, and does not manifeſt itſelf externally.
That the conical tube ACDF does not itſelf
cauſe any augmentation of expenditure is evinced
by the following
Exper. IV. The conical tube ACDF, from
which the remaining part DGMNIF was ſepa-
rated,
Cara
( 10 )
rated, was applied to the orifice P. The four
cubic feet were emitted in forty-two ſeconds,
which is the time of the expence through the ori-
fice itſelf AC in the thin plate, with the differ-
ence of one ſecond only. This flight variation
ariſes from its being almoſt impoſſible to make
the tube ADCF perfectly of the form of the
natural contracted vein.
PROPOSITION III.
The Preſſure of the Atmoſphere increaſes the Expence
of Water through a ſimple cylindric Tube, when
compared with that which ilues through a Hole
in a thin Plate, whatever may the Direction of
the Tube.
IT has long been known, that a heavy fluid
which moves in a deſcending cylindric pipe tends
to accelerate its motion. The inferior parts tend
to ſeparate themſelves from the ſuperior, and by
that means cauſe the preſſure of the atmoſphere
to increaſe the velocity of the ſuperior parts. This
ſucceſſive acceleration of gravity cannot take place
in an horizontal or afcending pipe. We ſhall
nevertheleſs find that the preſſure of the atmo-
ſphere acts even in theſe laſt ſituations to increaſe
the velocity of the fluid within the pipe. Certain
queſtions of legal right, which aroſe in my coun-
try, reſpecting the quantity of water ſupplied by
a pipe
2
3
:
.
(1)
a pipe for watering lands (canal d'arroſement), di-
rected my attention to this object ; and in the
year 1791 I made the following experiments pub-
Jicly in the Theatre of Natural Philoſophy at
Modena :
Exper. V. To the aperture P, fig. 1, I applied
a cylindrical pipe fifty-four lines in length and
eighteen in diameter. At the diſtance of nine
lines from the interior orifice P, twelve ſmall
holes were made in its circumference.
When
theſe ſmall holes were open, the four cubic fect
iſſued out in forty-one ſeconds, in the ſame man-
ner as through a thin plate. Not a ſingle drop
paſſed through any of the holes, and the ſtream
did not fill the tube. The holes were then cloſed
one after the other with wet ſkin. As long as
there was one hole open the expence continued
the ſame; but when, at laſt, all the twelve lioles
were well cloſed, the fluid ftream iſſued out in a
body which filled the pipe, and the four cubic feet
were emitted in thirty-one ſeconds.
Exper. VI. To the cylindric tube K LV, fig. 6,
cighteen lines in diameter and fifty-ſeven lincs
long, was joined the glaſs tube QRST, at tlie
diſtance of eight lines from the interior orifice K.
The glaſs tube was plunged in coloured water
contained in the veſſel T. When this
When this apparatus
was applied to the aperture P, fig. 1, the four
cubic feet of water flowed out in thirty-one fè-
conds.
we
C 2
( 12 )
conds. The coloured liquid T roſe in the tube
TR as high as S, at the heigh: of twenty-four
inches above the ſurface T.
The branch RT of the glaſs tube was ſhortened,
ſo that RT was only fix inches longer than RQ.
The efflux being then permitted to take place,
the coloured liquor of the veſſel T roſe through
the tube RT, and mixed with the water which
flowed from the reſervoir through KV, both of
which flowed out at V, and in a ſhort time the
veffel T was emptied.
I repeated this experiment with the compound
tube fig. 5, and the reſults were the ſame.
Exper. VII. The cylindrical pipe KLV, fig. 6,
was applied in an aſcending and nearly vertical
ſituation to the orifice R, fig. 8, of the veſſel HI,
of which the end H communicated by an opening
of conſiderable extent with the water of the refer-
voir X, fig. 1. The charge on the upper extre-
mity V of the tube was 27,5 inches. I inclined
the tube a little from the vertical direction, in
order that the jet might not fall back upon itſelf,
The glaſs tube QRT, fig. 6, in this new ſitua-
tion was fo diſpoſed that its lower extremity was
immerſed as before in the coloured liquid of the
veffel T. When the efflux was permitted, the
expenditure of four cubic feet was made in thirty-
four ſeconds; and the coloured liquid roſe in the
tube RT to the height of ncar twenty inches.
With
ter
( 13 )
8,3
With the ſame charge of 27,5 inches the orifice of
eighteen lines in a thin plate would have afforded
the four cubic feet in forty-five ſeconds.
Exper. VIII. A cylindrical veſſel of 4,5 inches
diameter had in its vertical fides ncar the baſe
a circular opening of 4,5 lines in diameter,
opened in a thin plate of tinned iron. The ſur-
face of the water contained in this vefſel was
inches above the centre of the aperture. The
water was then ſuffered to flow out of this
aper-
ture in the thin plate, and its ſurface was de-
preſſed feven inches in the veſſel in 27,5 ſeconds
of time.
To the fame aperture was applied a cylindric
tube of the ſame diameter, and in length eleven
lines. The veſſel was filled to the ſame height
as before, and, the water being ſuffered to flow
out, its ſurface was depreſſed ſeven inches in
twenty-one ſeconds.
The ſame experiment was afterwards fepeated
in the receiver of the air-pump, under which the
mercurial gauge ſtood at no more than ten lines
in height. The ſurface of the water in the veſſel
was depreſſed ſeven inches in 27,5 ſeconds, whe-
ther the aperture was made in a thin plate, or
whether it was provided with an additional cylin-
dric tube.
The height of the coloured water in the tube
of glaſs (in Experiment VI. and VII.) meaſures
the active quantity of the preſſure of the atmo-
ſphere
( 14 )
ſphere which is exerted on the ſurface of the water
in the refervoir X to increaſe the expenditure.
For example, in the fixth experiment we have
32,5 + 24 inches charge on the orifice P; and
we have nearly V 32,5 N 56,5:: 31":41", as
is
required by the common theory of the motion
of fluids which iffue out of veſſels by a fmall aper-
ture. The ſame obtains in Experiment VII.
Daniel Bernoulli made the feventh experiment
in deſcending tubes, and in diverging conical
tubes, and explained the reſult merely by his
theory of conſervation of living forces. Euler
and d'Alembert obſerved to him, that the preflure
of the atmoſphere was concerned in the effect *.
Though the caſe of the deſcending tube be differ-
ent from that of the horizontal or aſcending tube,
the knowledge of the firſt of thefe two caſes may
nevertheleſs facilitate the knowledge of the ſecond.
Beſides which, the cauſes which act in both caſes
are often combined together, and it is neceſſary
to be well acquainted with both, in order to dif-
tinguiſh the reſults. On this account it is, that
in the following propoſition I have turned from
my principal ſubject for a moment, to conſider
the firſt caſe, after which I ſhall return to the
ſecond.
A
* D'Alembert, Traité des Fluides, Art. 1
149.
PRO.
( 15 )
PROPOSITION IV.
In deſcending cylindrical Tubes, the upper Ends of
which polleſs the Form of the contracted Vein, the
Expence is ſuch as correſponds with the Height of
the Fluid above the inferior Extremity of the
Tube.
THE ancients remarked, that a deſcending tube
applied to a reſervoir increaſes the expenditure *.
Mariotte eſtimated that the water ifſues through
CQ, fig. 7, with a velocity nearly the mean pro-
portional between the velocities ariſing from the
two heights A B, A Cf Guillielmini fought for
the cauſe of this augmentation in the weight of
the atmoſphere, and determined the velocity at C
to be the ſame as would ariſe from the whole
height A C*. In his reaſoning he ſuppoſes that
the preſſure at C is the ſame for the ſtate of mo-
tion as for that of reſt; which is not true. In the
experiments he made upon this object, he paid
no regard either to the diminution of expenditure
produced by the irregularity of the inner ſurface
of the tubes, nor the augmentation occaſioned
* Calix devexus amplius rapit. Frontin, de Aquæduct. Art. 36.
See alſo the Pneumatics of Hero, in the Mathem. Vet. ed. 1693,
page 157
+ Mouvement des Eaux, part iii. diſc. 2.
Epift. hydroſtatic. Oper. tom. i. page 212.
by
i
( 16 )
.
by the form of the tubes themſelves. By a fingu-
Iar accidental concurrence, one of theſe errors com-
penſated for the other. I know of no other deci .
five experiment on this head fince Guillielmini.
I ſhall therefore proceed to eſtabliſh the propofi-
tion upon the principle of virtual aſcenſion com-
bined with the preſſure of the atmofphere, and
that in a manner which ſhall be clear of every ob-
jeflion, of theory as well as of experiment.
Let B LKO, fig. 7, repreſent a conical tubo
adapted to the form of the contracted vein * ; the
cylindrical tube LCQK is of the fame diameter as
the contracted part. The fluid ftratum, L K, con-
tinuing to deſcend through L C, tends to accelerate
its motion, according to the laws of gravitation;
and conſequently when it paſſes from LK to
MN, it tends to detach itſelf from the ſtratum
which follows, or, in other words, it tends to
produce a vacuum between L K and M N; and
the ſame effect takes place through the-whole
length of the tube LC. The preſſure of the at-
moſphere becomes active as far as is neceſſary to
prevent the vacuum; and its action is alike both
at the ſurface of the fluid at A, and at the inferior
extremity of the tube at C. At A it increaſes the
expenditure, and at C it deſtroys the ſum of the
* When I ſpeak of the form of the contracted vein, I always
mean to expreſs the conoid formed by the fluid iſſuing from an
orifice through a thin plate.
acce,
1
( 17 )
:
accelerations which would be produced along
LC, ſo that the fluid reinains continuous in the
tube.
Let T repreſent the time which the continuous
column of fluid LCR K employs to paſs through
the tube LC, whatever may be the velocity at L,
and the ſucceſſive acceleration from L to C. And
if we ſuppoſe this fame column to return upwards
from D to E, it will paſs through the ſpace D E =
LC in the ſame timc 7, during which it will loſe
all the acceleration it acquired from I to C. The
preſſure of the column ED, continued for the
time T, is therefore the quantity required to de-
ſtroy the ſucceſſive acceleration from L to C, and to
prevent the fluid from ceaſing to be continuous in
the tube L C; conſequently, that part of the pref-
ſure of the atmoſphere which is exerted at CQ to
deſtroy the ſum of the accelerations through LC,
is equal to the preſſure of a column E D of a fluid,
homogeneous to that of the reſervoir A B. And
ſince the ſame preſſure muſt alſo be exerted on
the ſurface A of the reſervoir, if we take FA=
LC, the fluid at LK will poffefs the velocity
which is proper to the height F L = AC; with-
out conſidering the retardation which the in-
ternal inequalities of the tube LCQK muſt
produce.
Exper. IX.
1. The orifice P (fig. 1.) through
a thin plate is circular, and eighteen lines in dia-
meter. The charge of fluid above the centre of
the
$
D
( 18 )
1
the orifice is forty inches. Four cubic feet of
water were emitted in thirty-eight ſeconds.
2. To the orifice P, fig. 1, I applied the tube
ACD, fig. 4, the upper end of which AC had
the form of the contracted vein. The diameter
at A was eighteen lincs, its length, AD, thirty-one
inches, and the ſituation of the tube horizontal.
The expenditure of four cubical fcet was made in
forty-eight ſeconds.
3. The ſame orifice and the ſame tube were
applied to the horizontal bottom of the reſervoir
fig. 7, ſo that the tube was vertical, and AC =
forty inches, or the height of the charge in the
two former experiments. The four cubic feet
flowed out in forty-eight ſeconds, as in the ſecond
experiment.
Exper. X. The laſt-deſcribed experiment was
repeated with a circular aperture of 11,2 lines in
diameter. The extremity A C of the tube fig. 4,
had the form of the contracted vein ; the end A
having the ſame diameter as that of the orifice.
The other circumſtances were as in the preceding
caſes. In the difpofition, according to the firſt
caſe, four cubical feet of water flowed out in
ninety-eight ſeconds; in the ſecond caſe the time
was 130 ſeconds; and in the third cafe 129 fe-
conds.
In each of theſe two experiments the tubes and
the expence of water were the ſame for the ſecond
and the third caſes; whence it follows, that the
force
( 19 )
3
force by which the expenditure was governed was
the ſame in both caſes. Now the force which
acts in the fecond caſe is the ſame as in the firft;
and conſequently the ſame force likewiſe acts in
the firſt and third cafes. All the difference of
the reſult between the firſt caſe and the two fol-
lowing ariſes from the retardation produced by
the inequalities of the internal ſurface of the tubes.
Exper. XI.
The height AB, fig. 7, being
conſtantly 32,5 inches, and orifice B O eighteen
lines in diameter, the tube BOCQ was applied
to the orifice itſelf, the ſuperior extremity of this
tube having the form of the contracted vein.
When the length of the tube was varied, the
times of the efflux of four cubic feet of water were
as in the following table.
41
3
12
4011
15,3
38"
35",2
31",2
I"
2",8
3",8
373
4
24
35"
5".
The fifth column of this table is calculated
from the proportion of retardation produced by
the irregularities of the internal ſurface of the
tubes in the following experiment. Citizen Boſſut
has
D 2
:
.
.
( 20 )
has obſerved, that theſe retardations * increaſe
rather in a leſs ratio than the velocity of the ſtream.
This is perhaps the reaſon of the difference ob-
ſerved between the fourth and fifth columns.
Exper. XII. I applied to the orifice P, fig. 1,
the ſame tubes as in the foregoing experiment one
after another in an horizontal ſituation, the height
of the charge being conſtantly 32,5 inches above
the centre of the orifice. The times of emiſſion
were as in the following table.
Length of the
tube B C in
inches.
Time of efflux
of four cubic
feet.
Differences.
O
41"
0
3
12
82"
"
854'5
8735
24
4811
"
I muſt here obſerve, that the viſcidity or mu-
tual adheſion of the particles of the water p is of
very little conſequence to the increaſe of expend-
iture through the orifice B 0, fig. 7, by the addi-
tional tube BC. For as ſoon as a ſmall hole is
opened at K, the increaſe of expenditure dimi-
niſhes or entirely ceaſes, and the fluid is no
longer continuous in the tube.
* Hvdrodyn. Art. 622.
f Graveſande and others have attributed the increaſe of ex-
penditure, through defcending tubes, to the natural coheſion of
the particles of water. V.
Wc
( 21 )
We will now return to tubes in the horizontal
and aſcending ſituations.
PROPOSITION V.
In an additional conical Tube, the Preſſure of the
Atmoſphere increaſes the Expenditure in the Pro.
portion of the exterior Section of the Tube to the
Section of the contracted Vein, whatever be
the Poſition of the Tube, provided its internal
Figure be adapted throughout to the lateral Com-
munication of Motion.
may
WE have ſeen (Propoſition III.) that the pref-
ſure of the atmoſphere incrcaſes the expenditure
through additional tubes, whatever may be their
poſition. We ſhall, in the next place, examine
the inode of action by which the atmoſphere pro-
duces this augmentation, and determine the reſult
from its cauſe. I ſhall begin with the caſe beſt
adapted to favour the action of the atmoſphere,
which is, that of conical diverging tubes of a cer-
tain form, which 'we have not yet conſidered.
Let the extremity AB, fig. 10, of the tube
ABEF be applied to an orifice formed in a thin
plate. The part ABCD is nearly of the figure
of the contracted vein, which form has been
Thown to make no perceptible alteration in the
expenditure (experiment IV.). The fluid which
iſſues through CD is diſpoſed to continue its
courſe
2
( 22 )
courſe in the cylindrical form CDHG. But if
the lateral parts of the diverging conical tube
CEG, DFH, contain a maſs of the fluid at reli,
the cylindrical ſtream CDIIG will communicate
its motion to the lateral parts (by Prop. I.) ſuc-
ceflively from part to part. And provided the
divergence of the fides CE, DF, be ſuch as is
beſt adapted to the ſpeedy and complete lateral
communication of motion, all the fluid contained
in the truncated cone CDEF will at length ac-
quire the ſame velocity as that of the ſtream which
continues to iſſue through CD. On this ſuppo-
fition, while the fluid fi ratum CDQR, preſerving
its velocity and thickneſs, would paſs into
RQTS, a vacuum would be forined in the ſolid
zone Rmr $ Qont. Or otherwiſe, if it be ſup-
poſed that the ſtratum CDQR, preſerving its
progreflive velocity, 1hould enlarge in RQTS,
this cannot happen without its becoining thinner,
and detaching itſelf from the ſtratum which fol-
lows, and by that means leaving a vacuum cqual
in magnitude to the zone laſt mentioned. A
ſimilar effect would take place through the whole
of the tube CE; and if the quantity Cm be
ſuppoſed to be invariable, the ſum of all theſe
void ſpaces will be equal to the folid zone
V Ex GÖYF H.
From this conſideration, we ſee that the lateral
communication of motion cauſes the ſame effect
in a conical tube, whether horizontal or vertical,
3
as
A
.
.
( 23 )
}
as gravity produces in the deſcending tube of Pro-
poſition IV. The atmoſphere in this caſe alſo
renders part of its preſſure active on the reſervoir,
and at EF. If the action of the atmoſphere upon
the reſervoir increaſes the velocity of the ſection
CD, this velocity will communicate itſelf likewiſe
to the whole fluid CDFE, and the tendency to
a vacuum will take place as before ; but ſince the
action of the atmoſphere is exerted equally at
EF, it will take away at E Fall the velocity which
it added at CD; ſo that, being deducted from
the ſame maſs, and in the ſame time, at EF, the
fluid will not ceaſe to be continuous in the pipe.
It is found by computation that this will happen
when the velocity of CD is increaſed in the ratio
of C D: to EF2.
By applying the general laws of motion to the
lateral fluid filaments of the ſtream which iffucs
through A B, it is found that they tend to de-
ſcribe a curve which commences within the refer-
voir, for example, at A, and continues towards
CSE. To determine the nature of this curve, it
is requiſite to know, and to combine together by
calculation, the mutual convergency of the fluid
filaments in AB, the law of the lateral commu-
nication of motion between the filaments them-
ſelves and their divergent progreſſion from C to
E. Theſe combinations and calculations are per-
haps beyond the utmoſt efforts of analyſis. While
the tube AB FE poffefſes a different figure from
this
( 24 )
this natural curve, the reſults of experiment will
always differ more or leſs from the theory.
Exper. XIII. The compound tube A BFE of
the ſame fig. 10, having the following dimenſions
in lines AB=EF= 18; AC=11; CD=15,5;
CG=49; and this tube being applied to the ori-
fice P, fig. 1, under a charge of 32,5 inches, the
four cubical feet of water were emitted in 27”,5.
We have ſeen that, in the third experiment,
under like circumſtances, the orifice through it
thin plate afforded four cubic feet of water in 41
The contracted vein was 0,64 of the orifice. Con-
fequently, by following the enunciation of the
theorem, the expence through the pipe ABFE
ought to be made in 26",24. The experiment
falls fort in the quantity 1",26.
Exper. XIV. Between the two conical tubes of
the preceding experiment is interpoſed a cylin-
drical tube three inches long and 15,5 lines in
diameter. The interpoſition of the cylinder bc-
tween the two cones was made as in fig. 13. This
addition retarded the expenditure 1", the time now
being 28”,5.
Exper. XV. The charge of the reſervoir being
conſtantly 32,5 inches, the portion of the tube
ABCD, fig. 11, had the ſame dimenſions as
before ; the tube CDFE was ſeventy-eight lines
in length, and its diameter E Ftwenty-three lines.
To this horizontal tube I added three glaſs tubes;
the firſt D Xat CD; the ſecond NY at the diſtance
of
( 25 )
of twenty-fix lines from the firſt; and the third O Z
at twenty-ſix lines diſtance from the ſecond. The
lower extremities of theſe three tubes were plunged
in the mercury of the veſſel Q. When the water
was ſuffered to flow through the tube A EFB,
the mercury roſe fifty-three lines in the tube D X;
20,5 in NY, and ſeven in OZ. Theſe quantities
correſpond with fixty two inches height of water
in D X; twenty-four inches in NY; and 8,1 in
OZ. The expenditure of four cubic feet was ef-
fected in 25".
I cut off the portion PNFE of the tube, and
the remaining pipe ABNP emitted the ſame
quantity in 31":
In the truncated conical tube ACPBD N, the
fection P N is to the ſection of the contracted vein
(namely 0,64 of the ſection A B) as 41" to 30".
In the experiment with this laſt truncated tube
the retardation is conſequently no more than
1" leſs than the theory.
In the entire tube CDFEwe have v 62 +32,5:
✓ 32,5 = 41": 24". The difference of thirty-eight
inches elevation of water in the two tubes DX, NY,
muft ariſe from the motion of the fluid from C to
P; it is 1-13th leſs than by the theory. The loſs
is ſucceſſively greater in the two portions PQ,
QE. The reaſon of this is, that the ſtream de-
ſcends as it moves from CD, ſo that the lateral
communication not being made uniformly through
the whole of any one ſection, the different parts of
the
E
.
( 26 )
}
the current acquire irregular motions, and even
eddies within the tube; whence the jet comes
forth by leaps and with irregular ſcattering...Theſe
uncertain motions cannot be reduced to the theory,
and manifeft themſelves the more, the longer or
the more diverging the ſides of the tube. The
effects confequently remain to be aſcertained by
experiment.
Exper. XVI. I conſtructed a tube CDFE as
before (fig. 11), 148 lines long, and twenty-ſeven
lines in diameter at EF, the reſt of the apparatus
being the ſame as in the foregoing experiment. The
expenditure of four cubical feet was effected in
21"; the inequality and irregularity of motion in
the ſtream were greater in this experiment than in
the foregoing.
It was uſeleſs to prolong the tube CDFE be-
yond 148 lines; for the ſtream did not in that
caſe fill the portion of tube added beyond that
length, and the expenditure remained conſtantly
at 21". This expenditure is nearly double what
took place through the ſimple aperture in a thin
plate; and it is the greateſt I have been able to
obtain by additional tubes, the axis of which had
an horizontal poſition under a charge of 32,5
inches.
It is true, that by prolonging the tube CDFE
to the length of 204 lines in the horizontal poſi-
tion, the four cubic feet flowed out in 19". But
to obtain this effect, I found it neceſſary to fix a
pro-
( 27 )
prominence with the tube at 0, which forced the
fluid to fly upwards, and by that means to fill the
whole tube.
Exper. XVII. In this experiment the horizon-
tal tube CDFE, fig. Il, was more divergent
than in the foregoing trials. It was 117 lines long,
and thirty-ſix lines in diameter at EF. The reſt
of the apparatus was the ſame as before. The
expenditure was made in 28"; the fiream did not
fill the whole ſection EF. The reſult was the
fame when ſucceſſive portions of the pipe were
cut off, until CE was no longer than twenty lines,
and the external diameter cighteen lines. In this
caſe the ſtream filled the pipe, and the expenditure
was alſo made in 28".
When the length CE was twenty lines, its ex-
ternal diameter E F was increaſed to twenty lincs.
In this caſe the ſtream was detached from the ſides
of the tube, and the expence of four feet took
place in forty-two ſeconds, as in the Vith
expe-
riment.
Theſe experiments teach us, that by varying
the divergence of the ſides of tubes, the lateral
communication of motion has a minimum and a
maximum of effect. The minimum is ſeen in the
laſt experiment. It appears that the lateral com-
munication ceaſes to produce its effect when the
angle made by the ſides of the tube with each
other exceeds fixteen degrees. The XIIIth experi-
ment nearly determines the maximum of the effect
when
( 28 )
when the fame angle is about three degrees. Theſe
limits may alſo, perhaps, in a ſmall degree de-
pend upon ſome function of the velocity.
PROPOSITION VI.
In cylindrical Pipes the Expenditure is leſs than
through conical Pipes, which diverge from the
Place of the contracted Vein, and have the fume
exterior Diameter.
THE general theory is the ſame for both theſe
forms of tubes; but the loſs of living force is
greater in the cylinder, and the effect of the
communication of motion in theſe tubes can-
not approach its maximum as in the cone.
Let the tube ACNM, fig. 5, Pl. I have
the form of the contracted vein in ACFD;
the cylindrical part GINM has its diameter
MN, greater than DF. By the reaſoning
inade uſe of in the preceding Propofition, it is
proved, that the lateral communication of motion
tends to produce a vacuum in the ſolid zone
ROYSXQTZ. If the communication of mo.
tion in this tube were completely made, it would
follow, that the preſſure of the atmoſphere would
increaſe the velocity of the contracted vein in the
ratio of DF2 to MN2.
But the form itſelf of the cylindrical pipe al-
ways deſtroys a notable part of the effect: for the
fluid
!
"
( 29 )
Muid filaments AD, in turning through the curve
DR, proceed briſkly to ſtrike the ſides of the
tube GM at R, where they loſe part of their
motion. In the space DGR, eddies, or circular
whirls, are produced, as in a baſin which receives
water by a channel. Theſe eddies produce, to a cer-
tain extent, a failure of the effect, and retard the
effux of the ſtream. A much leſs increaſe of the
expenditure takes place in the cylindrical tube
than would anfwer to the ratio of DF to MN2.
Exper. XVIII. A notion may be formed of
theſe internal ſhocks and eddies in the cylindric
tube, and their effects on the efflux of the fluid,
if attention be paid to the following table of the
expenditure through the different additional tubes
in the horizontal poſition. All theſe tubes have
the diameter of their two extremities = 18 lines ;
they were all provided with the conical tube of
the form of the contracted vein at their inner ex-
tremity, excepting that of fig. 6. The charge was
always 32,5 inches above the centre of the ori-
fice.
Table of the Times employed in diſcharging Four
Cubic Feet of Water through the different Ad-
jutages.
Through the orifice in a thin plate
Through the fimple tube of fig. 6
Through the tube of the form of fig. 5
After
1
41"
31"
1
1
31"
(30)
30"
32",5
After having amended (adouci) the coni-
cal divergent part, DFIG, of the ſame
tube
Through the tube fig. 9
Through the conical tube of the form
fig. 10
Through the tube fig. 5, the portion
GINM being 23,5 lines in diameter,
and eighty-four in length, the reſt as
before
27",5
27"
It may perhaps be demanded, whether, in the
internal part of the ſimple cylindric tube K L V of
fig. 6, there be the ſame augmentation of velo-
city, and the fame contraction of the ſtream, as
in the compound tube of fig. 5? By reaſoning
according to the principles we have eſtabliſhed,
I think, 1. That in the ſection K L of fig. 6, there
is the ſame increaſe of velocity as we have ſeen
(Prop. II.) take place in the ſection A C of fig. 5.
The direction of the fluid particles which paſs
through theſe two ſections muſt be the ſame in
both caſes, becauſe this direction can depend only
on the impulſe received within the reſervoir, which
is the ſame in both. 2. In fig. 6, the fluid parti-
cles, after having paſſed through the ſection KL,
begin immediately to experience the effect of the
lateral communication of motion. They muſt
therefore deviate laterally through the curve L x %,
before
:
}
( 31 )
3
before they arrive at the place of contraction which
they aſſume at DF, fig. 5, and which they like-
wife affume when the orifice is made in a thin
plate. If we imagine a tube of glaſs y K, one
extremity of which is applied at K, fig. 6, and the
other extremity open in the interior part of the
reſervoir, it will be ſeen that the preſſure of the
atmoſphere, which is exerted upon the coloured
fluid T, muſt likewiſe act on the ſurface of the
reſervoir, and join the preſſure of the fluid in the
reſervoir to preſs the water into the tube y K, as
it preſſes the coloured liquor into TS. The pref-
ſure of the atmoſphere muſt, in the ſame manner,
augment the impulſe of all the fluid particles which
arrive at KL, and conſequently muſt increaſe the
expenditure.
Since the checks and eddies in an additional
cylindric tube muſt always deſtroy a part of the
active force of the fluid, it follows, that the fluid
column iſſuing out of the tube can never acquire
the whole velocity which is due to the actual
charge, and is obſerved nearly entire in the ori-
fices through a thin plate; and the diminution of
velocity correſponds with the increaſe of the time
beyond that indicated by the theory, as may be
ſeen in the following
Exper. XIX. The orifice P, fig. 1, being made
through a thin plate, and the vertical height P M,
being fifty-four inches, the diſtance MN of the
jet was 81,5 inches. Having applicd to the ſame
orifice
2
F
:
( 32 )
orifice the cylindrical tube of fig. 5, and the
pera
pendicular PM being let fall from the external
orifice of the tube, the diſtance M N was found
to be fixty-nine inches. According to the theory,
the expenditure of four cubical feet through this
tube ought to have taken place in 26",24, but
it really employed 31". And the proportions
31": 26",24 = 81,5: 69 nearly.
The ſame obfervation may be made on an ex-
periment of Michelotti (tom. ii. pages 22 and 23),
PM being 19,33 feet, and the water iſſuing
through an orifice in a thin plate M N was 23,2
feet; it was no more than twenty when an addi-
tional cylindric tube was applied which had not
even the proper length.
It is evident, that the theory of the lateral com-
munication of motion muſt likewiſe apply in the
fame manner to defcending and aſcending tubes,
whenever their form admits of this lateral com-
munication. In defcending tubes, we muſt add
the increaſe of expenditure occafioned by this
cauſe to that which is produced by the accele-
ration of gravity, and which we have eſtimated in
Propofition IV. In aſcending tubes, gravity acts
in a contrary direction, and conſequently its ef-
fect muſt be deducted from that of the lateral
communication. Experiment VII. relates to af-
cending tubes. The following relate to other
poſitions.
Exper. XX. The tube A BFE of fig. 11, ex-
periment
है
:
A
( 33 )
periment XV. was applied in the place of the tube
BCQO, in fig. 7. The height of the water in
the reſervoir above the lower extremity of the tube
was 41,5 inches. The four cubical feet of water
were emitted in 22".
I applied the fame conical tube AB FE, fig. 11,
to the orifice R, fig. 8, to form an aſcending jet a
little inclined from the perpendicular. The height
of the water of the reſervoir above the upper ex-
tremity of the tube was twenty-three inches. The
expenditure of four cubical feet was made in 30".
The time of the expenditure in experiment XV.
was 251". And by comparing it with the preſent,
we find nearly v 41,5: V32,5 = 25": 22". And
✓ 23: N 32,5 = 25": 30.
Exper. XXI. The orifice R, fig. 8, was circu.
lar, and 4,5 lines in diameter; the charge was
31,7 inches, and the jet declined a little from the
perpendicular. The orifice being through a thin
plate, afforded a cubical foot of water in 161".
With an additional cylindrical tube of the ſame
diameter, and ten lines in length, the cubical foot
of water was emitted in 121".
Under a charge of fifty-ſix inches, the ſame
orifice afforded, by the vertical jet, a cubi-
cal foot in 123" through the thin plate, and in
91 with the ſame additional tube.
Theſe two reſults being combined, give for the
expenditure of vertical jets a mean ratio, between
the thin plate and the cylindrical adjutage, of 100
to
:
.
( 34 )
to 134, which is alſo the ratio between the hori-
zontal jets.
Exper. XXII. I applied the glaſs tube QRT,
fig. 6, to the point S, fig. 5, of the compound
tube ACMN, the diſtance BS being twenty-
four lines. In this ſituation the fluid T no longer
riſes in the tube. This proves that the lateral
tranſlation of the fluid in the cylindrical tube is
made very near the place where the vein is con-
tracted, and that conſequently D R-muſt briſkly
ſtrike the ſide GM.
By this experiment we ſee that the diſtance B R,
at which the oblique filaments ſtrike the ſides of
the tube, does not amount to twenty-four lines.
Suppoſing DO = 20 lines, the time which the
particle D employs to paſs through the ſpace DO
in my experiments is leſs than 0",01. Let us de-
compoſe the curve-lined motion D R according to
the lines DO, OR. Let us ſuppoſe the accelera-
tion through O R to be uniform, and it will be
found that this acceleration is at leaſt five times
as great as that of heavy bodies. If the lateral
force through OR were ſimply the mutual attrac-
tion of the particles of the water, this attraction
in the particle D muſt not only overcome the in-
ertia of the particle itſelf, but likewiſe that of the
other particles nearer the axis, which follow Din
its deviation through DR, and impreſs upon them
a much greater ſum of acceleration than that of
gravity. Now the force of attraction of one par-
ticle
à
:
A
1
( 35 )
:
ticle of water is not greater than the natural gra-
vity.of a thread of water of the length of one line
at moſt. The lateral communication of motion,
which is the cauſe of the acceleration through OR,
is therefore much greater than could have been
produced by the mutual attraction of the particles
of water,
PROPOSITION VII.
By means of proper Adjutages applied to a given
cylindric Tube, it is poſſible to increaſe the Expen-
diture of Water through that Tube in the Propor-
tion of Twenty-four to Ten, the Charge or Height
of the Reſervoir remaining the ſame.
I SHALL here give an account of the different
precautions neceſſary to be taken when the expen-
diture of water through a cylindrical tube of a
given length is required to be the greateſt poſ-
fible.
1. The inner extremity of the tube AD (fig. 13)
muſt be fitted at AB with a conical picce of the
form of the contracted vcin *; this increaſes the
expenditure as 12,1 to 10. Every other form will
afford leſs. If the diameter at A. be too great,
the contraction will be made beyond B, and the
ſection of the vein will be ſmaller than the ſection
of the tube.
2. At the other extremity of the pipe B Cap-
1
* Boſſut, Art. 509
ply
.
( 36 )
ply a truncated conical tube CD, of which let the
length be nearly nine times the diameter C, and
its external diameter D muſt be 1,8 C. This ad-
ditional piece will increaſe the expenditure as 24
to 12,1. (Experiment XVI.) By this means the
quantity of water will be increaſed by the two
adjutages AB, CD, in the proportion of twenty-
four to ten.
At Rome, the inhabitants purchaſe the right of
conveying water from the public refervoirs into
their houſes. The law prohibits them from
making the pipe of conveyance larger than the
aperture granted them at the reſervoir, as far as
the diſtance of fifty feet *. The legiſlature was
therefore aware, that an additional pipe of greater
diameter than the orifice would increafe the ex-
penditure; but it was not perceived that the law
might be equally evaded by applying the conical
fruftum CD beyond the fifty feet.
From this ſecond rule we learn, that it is not
proper to make the flues of chimnies too large in
the apartments; but that it will be ſufficient if
they be enlarged at their upper terminations, ac-
cording to the form CD, fig. 13. This diver-
gency of the upper part will carry off the fmoke
very well, even when it is not practicable to af-
ford chimnies of ſufficient length to the upper
apartments. The ſame obſervation is applicable
to chemical furnaces for ſtrong fire.
* Fontin. de Aquæduct. art. 205. 106 et 112.
3
1
( 37 )
3. The pipe B C ought to be firaight, without
elbows or curvatures. To the experiments which
Boſſut has made on this head *, I ſhall add the
following
Exper. XXIII. The two tubes ABC, DEF,
fig. 14, are fifteen inches long; their diameter
is 14,5 lines. The conical portions A, D, have
the form of the contraction of the vein of
fluid, and are applied to the orifice P, fig. 1,
which is eighteen lines in diameter, with 32,5
inches depth, or charge of ſuperincumbent fluid.
The elbows, or flexures, BC, EF, are made
in the plane of the horizon. Theſe two pipes
are made of copper foldered with ſilver, and
the workmanſhip carefully executed. The cur-
vature B C was drawn out, or bended, into the
form of a quarter of a circle, by filling the tube
with melted lead, in order that it might preſerve
its diameter during the act of bending. The el-
bow D E F is conſtructed in a right angle. The
expenditure through theſe two tubes was com-
pared with that afforded through a right-lined
cylindrical tube of ſimilar dimenſions, and in like
circumſtances. The four cubical feet of water
flowed out of the cylindrical tube in 45"; out of
the curved tube A B C in 50"; and out of the an-
gular tube D E F in 70".
It is of importance that the tube B C, fig.
13, Plate I. ſhould be of an equal diameter
* Art. 631 et ſeq.
through-
:
( 38 )
throughout. It is not enough that care be taken
that there ſhall be no contraction; it is alſo necef-
fary that it ſhould not be enlarged at any part.
For ſuch enlargements have nearly the fame baď
effect in the expenditure as contractions. The
pipe A 0, fig. 12, affords a much lefs quantity of
fluid with the dilatations DE, HI, than if it
were of a diameter equal to that at B throughout
its whole length. The following experiment agrees
with the theory.
Exper. XXIV. The circular orifice A, fig. 12,
has the form of the contraction of the vein, and
the remaining part of the tube is interrupted by
various enlargements of its diameter. This tube
is applied to the aperture P, fig. 1. The dimen-
ſions of its parts meaſured in lines are as follows:
Diameter at A=11,2. Diameter at B, C, F, G,
&c. = 9. Length of BC = FG, &c. = 20. Length
of CD= EF=G H, &c. = 13. Diameter of the
enlarged parts = 24. The length of each of the
enlarged parts was variable. The firſt time of
trial it was thirty-eight lines, the ſecond ſeventy-
fix, and the reſult of the experiment was the fame
in both caſes.
Number of enlarged parts.
Time during which four cubical
feet ifſued out.
но
109"
147"
192"
3
5
240"
I after-
( 39 )
I afterwards applied to the ſame orifice a tube,
having the ſame form, and the ſame diameter, as
AB C, but cylindrical throughout, without any
enlargements, and its length was thirty-ſix inches,
the ſame as that of the tube with five enlarged
parts: in this caſe the expenditure of four cubical
fcet was made in 148".
When the fluid paffes from C to the middle of
the enlarged part D E, part of the motion is di-
verted from the direction CF towards the lateral
parts of the enlargement. This part of the mo-
tion is conſumed in eddies, or againſt the ſides.
Conſequently there remains ſo much the leſs motion
in the following branch FG. This is alſo the
cauſe which deſtroys or weakens the pulſe in the
arteries beyond an aneuriſm.
From this confideration we are juſtified in con-
cluding, that if the internal roughneſs of a pipe
diminiſhes the expenditure, the friction of the
water againſt theſe aſperities does not form any
conſiderable part of the cauſe. A right-lined tube
may have its internal ſurface highly poliſhed
throughout its whole length; it may every where
poffeís a diameter greater than the orifice to which
it is applied; but, nevertheleſs, the expenditure
will be greatly retarded if the pipe ſhould have
enlarged parts, or ſwellings. This is a very in-
tereſting circumſtance, to which, perhaps, ſufficient
attention has not been paid in the conſtruction of
hydraulic machines. It is not enough that elbows
and
( 40 )
and contractions are avoided; for it may happen,
by an intermediate enlargement, that the whole
advantage may be loſt, which may have been
pro-
cured by the ingenious diſpoſitions of the other
parts of the machine.
PROPOSITION VIII.
In the Machine for blowing by means of a Fall of
Water, the Air is afforded to the Furnace by the
accelerating Force of Gravity, and the lateral
Communication of Motion, combined together.
THE academy of Toulouſe, in the year 1791,
invited philoſophers to determine the cauſe and
the nature of the ſtream of air which is produced
by the fall of water in certain forges. I propoſe,
in this place, to develope the complete action of
this kind of blowing apparatus, and to aſcertain
the beſt form of conſtruction. Kircher is the firſt
I know of, who has explained the production of
wind by a fall of water *. Barthes, the father,
has given a theory which appears to me to be de-
fective in many reſpects of Dietrich was of opi.
nion, that this wind is produced by the decom-
pofition of water * Fabri had a ſimilar notion
* Mundus Subterr. lib. xiv. cap. 5, edit. 1662.
it Mémoires des Savans étrangers, vol. iii. p. 378.
+ Gites de Minerai des Pyrénées, p. 48, 49.
in
( 41 )
1
.
in the laſt century *. Moſt philoſophers are well
acquainted with this kind of engine t.
I ſhall begin with an idea, the foundation of
which did not eſcape the penetration of Leonardo
da Vinci. Suppoſe a number of equal balls to
move in contact with each other along the hori-
zontal line AB, fig. 16, Plate II. Imagine them
to paſs with an uniforin motion, at the rate of
four balls in a ſecond. Let us take BF, equal
to fixteen feet Engliſh. During each ſecond, four
balls will fall from B to F, and their reſpective
diſtances in falling will be nearly BC= 1, CD
= 3, DE = 5, EF = 7.
7. We have here a very
evident repreſentation of the ſeparation, and fuc-
ceſſive elongation, which the accelerating force
of gravity produces between bodies which fall af.
ter each other.
The rain-water flows out of gutters by a con.
tinued current; but during its fall it ſeparates into
portions in the vertical direction, and ſtrikes the
pavement with diſtinct blows. The water like-
wife divides, and is ſcattered in the horizontal
direction. The ſtream which iſſues out of the
gutter may be one inch in diameter, and ſtrike
the pavement over the ſpace of one foot. The
air which exifts between the vertical and hori-
zontal ſeparations of the water which falls, is im-
* Phyſic. Tract. i. lib. ii. prop. 243.
+ Art des Forges, partii.Mariotte des Eaux, part i.
diſc. iii.---Tranſact. No. 473, &c.
pelled
G
*
I
(42)
pelled and carried downwards. Other air luc-
ceeds laterally; and in this manner a current of
air or wind is produced round the place firuck by
the water. I went to the foot of the caſcades
which fall from the Glacière of La Roche-Mêlon,
on the naked rock at La Novalêle, towards
Mount Cenis, and found the force of the wind to
be ſuch as could ſcarcely bc withflood. If the
caſcade fall into a baſin, the air is carried to the
bottom, whence it riſes with violence, and dif-
perſes the water all round in the form of a miſt.
The water which is precipitated in the hollow
internal parts of mountains carries the air with it,
which afterwards illuing forth from apertures at
the foot of the mountain, produces thoſe natural
blaſts, thoſe veniaroli *, which are moſt frequently
obſerved in the volcanic mountains, becauſe theſe
mountains arc moli commonly hollow within.
Let BCDE, fig. 16, repretent a pipc, through
which the water of a caual AB falls into the
lower receiver MN. The ſides of the tube have
openings all round, through which the air freely
enters to fupply what the water carries down in
its fall. This mixture of water and air proceeds
* Theſe venturoli are ſometimes produced by the difference
of temperature between the air of the cavern and the external
air. V.--From the effects they ſeem to be oftener produced
by this laſt cauſe than by a fall of water. On this ſubject in
general, namely, the cold winds which iſſue out of the earth,
fee Nicholſon's Philoſ. Journal, i. 229.-T.
to
t.
$
.
A
( 43 )
...
to ſtrike a inafs of ſtone Q; whence rebounding
through the whole width of the receiver MN, the
water feparates from the air, and falls to the bot-
tom at XZ, whence it is diſcharged into the
lower channel or drain, by one or more openings,
T, V. The air, being leſs heavy than the water,
occupies the upper part of the recciver, whence
being urged through the upper pipe 0, it is con-
veyed to the forge.
Exper. XXV. I formed one of theſe artificial
blowing engines of a finall fize. The pipe BD
was two inches in diameter, and four feet in
height. When the water accurately filled the
fection B C, and all the lateral openings of the
pipe BDEC were cloſed, the pipe O no longer
afforded any wind.
It is, therefore, evident that in the open pipes
the whole of the wind comes from the atmo.
ſphere, and no portion is afforded by the decom-
poſition of water. Water cannot be decompoſed,
and transformed into gas, by the ſimple agitation
and mechanical percuſſion of its parts. The opi-
nions of Fabri and Dietrich have no foundation in
nature, and are contrary to experiment.
It remains, therefore, to determine the circum-
ſtances proper to drive into the receiver, MN,
the greateſt quantity of air, and to meaſure that
quantity. The circumſtances which favour the
moſt abundant production of wind are the fol-
lowing
G 2
1. It
f
( 44 )
1. It is known that in the parabola, if dx be
aſſumed as conſtant, dy will decreaſe in the ratio
of The ſeparation of the balls, in fig. 15, is
✓ x
more rapid in the upper ſpaces of the fall than in
the lower. In order, therefore, to obtain the
greateſt effect from the acceleration of gravity, it
is neceſſary that the water ſhould begin to fall at
BC, fig. 16, with the leaſt poſſible velocity; and
that the height of the water FB ſhould be no
more than is ncceffary to fill the ſection BC. I
ſuppoſe the vertical velocity of this ſection to be
produced by an height or head equal to B C.
2. We do not yet know, by direct experiment,
the diſtance to which the lateral communication
of motion between water and air can extend it-
ſelf; but we may admit, with confidence, that it
can take place in a ſection double that of the
original ſection, with which the water enters the
pipe. Let us ſuppoſe the ſection of the pipe
BDEC to be double the ſection of the water at
BC; and in order that the ſtream of fluid may
extend and divide itſelf through the whole double
ſection of the pipe, ſome bars, or a grate, are
placed in B C, to diſtribute and ſcatter the water
through the whole internal cavity of the pipe.
3. Since the air is required to move in the pipe
O with a certain velocity, it muſt be compreſſed
in the receiver. This compreſſion will be pro-
portioned to the ſum of the accelerations, which
thall have been deſtroyed in the inferior part KD
of
$
( 45 )
of the pipe. Taking KD = 1,5 fcet, we ſhall
have a preſſure fufficient to give the requiſite ve-
locity in the pipe 0. The ſides of the portion
KD, as well as thoſe of the receiver M N, muſt
be exactly cloſed in every part.
4. The lateral openings in the remaining part
of the pipe B K, may be fo diſpoſed and multi-
plied, particularly at the upper part, that the air
may have free acceſs within the tube. I will ſup-
poſe them to be ſuch that 0,1 foot height of water
might be ſufficient to give the neceſſary velocity
to the air at its introduction through the apertures.
All theſe conditions being attended to, and
ſuppoſing the pipe BD to be cylindrical, it is
required to determine the quantity of air which
paſſes in a given time through the circular ſection
KL. Let us take in feet KD=1,5; BC=BF
= a; BD = b. By the common theory of fall-
ing bodies, the velocity in KL will be 7,76 ~
(a+b-1,4); the circular ſection KL= 0,785a%.
Admitting the air in KL to have acquired
the fame velocity as the water, the quantity of
the mixture of the water and air which paſſes in
a ſecond, through K L is = 6,1 a? (a+b--1,4).
We muſt deduct from the quantity(a+b1,4) that
height which anſwers to the velocity the water muſt
lofe by that portion of velocity which it communi-
cates to the new air laterally and conſtantly intro-
duced; but this quantity is ſo ſmall that it may
be neglected in the calculation. The water which
paſſes
I
( 46 )
+
pafles in the ſame time of one ſecond througlı
BC is = 0,4 d: v(a +0,1). Confequently, the
quantity of air which paffcs in one fecond through
KL, will be = 6,122 V (at1--1,4-0,4 a? ✓
(a+0,1), taking the air itſelf, even in its ordinary
ftate of compreſſion, under the weight of the at-
moſphere. It will be proper, in practical appli-
cations, to deduct one-fourth from this quantity;
1. on account of the ſhocks which the ſcattered
water ſuſtains againſt the interior part of the tube,
which deprive it of part of its motion ; and, 2. be-
cauſe it muſt happen that the air in L K will not,
in all its parts, have acquired the fame velocity as
the water.
If the pipe O do not diſcharge the whole quan-
tity of air afforded by the fall, the water will de-
fcend at XZ; the point K will riſe in the pipe,
the afflux of air will diminiſh, and part of the
wind will iſſue out of the lower lateral apertures
of the pipe B K.
I ſhall not here examine the greater or leſs de-
gree of perfection of the different forms of water-
blowing machines, which are uſed at various iron
forges, ſuch as thoſe of the Catalans, and elſe-
where. Thefe points may be eaſily determined
from the principles here laid down.
PROPO-
:
:
( 47 )
PROPOSITION IX.
It is poſſible, hy means of a Fall of Water, to drain
a Piece of Ground, without the Help of Ma-
chines ; even though the Ground ſhould lie on a
lower Level than the eſtabliſhed Current below the
Fall.
THE means of doing this are pointed out in the
firſt experiment of this treatiſe. We have ſeen
that the water contained in the veſſel D EFB,
fig: 3, Plate I. iſſues through the channel MBV,
which is higher than the ſurface of the water
itſelf, becauſe the fluid which paſſes through AC
carries with it the water contained in the veffel.
In the artificial fall, which is procured in
channels to give motion to mills, when the wa-
ter ruſhes down by a rectangular trunk of wood,
DBCF, fig. 17, placed nearly horizontal in the
middle of the lower channel, the ſurface of the
water at K is one or two feet beneath the inferior
current (or back-water) FL* The water at F
tends to return and deſcend along FK; but the
current, by its lateral action, conſtantly carries it
away, and docs not permit it to flide down to K.
If an opening G be made in the lateral ſides of
* This depreſſion of the level has already been noticed in K.
Guilielmini, della Natura de Fiumi, cap. vii. fig. 46. Boſſut,
art. 721. The wheel alluded to in the text muft, I preſume,
be of that kind which we call a breaſt-wheel.--N.
the
( 48 )
1
the trunk, the waters from lands lower than the
current of the inferior ſtream FL may be drained
off. In a commiſſion with ſeveral of my col .
leagues, I once propoſed, that this principle
ſhould be applied to a cafe in practice. The
project was adopted, and the drainage fucceeded
very well.
'
The rectangular conduit DBFC muft be
pro-
tonged to a certain extent along the lower chan-
nel, otherwiſe the water might flow back from F
to K, and oppoſe the drainage through G. The
mill-wrights are aware of the utility of this pro-
Tongation. Experience has taught them, that
it prevents the water from returning back fo
readily in the time of floods, which might check
the motion of the water-wheel. For this purpofe,
they make the upper part D Fat the height of
the waters, which the mill-ſtream can reſiſt or
fupport. The town of Final, in the territory of
Modena, having charged me with the direction
of changing the courſe of part of the waters of
the Panaro, which the circumſtances of the town
required to be done; I availed myſelf of this
prolongation of the tail-pipe D F, together with
other contrivances, to maintain the action of mills
in the new channel; and I ſucceeded not only be-
yond the expectation of the inhabitants, but even
beyond my own hopes.
PROPO-
We+
!
( 49 )
PROPOSITION X.
The Eddies of the Water in Rivers are produced by
Motion, communicated from the more rapid Parts
of the Stream, to the lateral Parts, which are
leſs rapidly moved.
FEW authors have examined the cauſe and
the effects of the eddies of water in rivers ; and
thoſe who have undertaken this inveſtigation, do
not appear to have been very happy in their re-
ſearches,
The water which moves in the channel MNH,
fig. 19, meets the obſtacle BA, which impedes
its courſe, and cauſes it to riſe and diſcharge itſelf
in the direction AC with an increaſed velocity.
Suppoſe the water in BDCA to be dormant, the
current A C communicates its motion to the late-
ral particles E (Prop. I.), and conveys them for-
ward; the ſurface of the dormant water becomes
depreſſed at E, and the moſt remote particles to-
wards D are urged, according to the laws of the
equilibrium of fluids, to fill the depreſſion. The
current AC continues to carry them off, and the
ſpace BDCA continues to be exhauſted. The
water of the current AC, by virtue of the ſame
laws, is acted upon by a conſtant force which urges
it towards the cavity E, while its natural courſe or
projection carries it towards A C. Under the agen-
су of theſe two forces, the water AC acquires a
curve-
H
( 50 )
curve-lined motion in CD, and deſcends as it
were through an inclined plane, becoming retro-
grade in DE, whence it would proceed to firike
the obſtacle B A, and the current A C, and after.
wards it would undergo ſeveral oſcillations premo
vious to acquiring a ſtate of equilibrium and re-
poſe. But the current A C continues its lateral
action: a ſecond time it draws away the water
through CD into E, and forces it to renew its
motion through the curve CDE; in which man-
ner the eddy continues without ceaſing.
If the river ſhould paſs through a contraction of
its bed at N, eddies will be produced on both ſides
at P and at Q, ſimilar to thoſe we have contem-
plated at DC.
Suppoſe the ſtream of water, after having ſtruck
the bank GH, to be reflected into a new direc-
tion HS, the lateral communication of motion
will excite eddies in the angle of reflexion R.
When two currents of unequal velocity meet
obliquely in the middle of the river, the moſt
rapid current will produce eddies in that which
is the leaſt rapid.
Suppoſe a ſtream of water to flow over a bed of
unequal depth. If the longitudinal ſection of the
inequalities of the bottom exhibit a gentle ſlope,
as at ABC, fig. 20, the ſuperior water will im-
preſs its motion by lateral communication upon
the inferior water, which is near the bottom, be-
acath the line AC, and a current will take place
through
}
( 51 )
.
through the whole depth of the ſection MB.
The current which is formed near the bottom at
B, is turned out of its courſe by the ſlope B C,
and proceeds to riſe above the ſurface at Q; fome-
times in the form of a curling wave or vertical
whirlpool. If the extremities of the hollow place
form an abrupt angle, as DE, FG, eddies will
be produced even at the bottom, in the vertical
direction at D, and ſometimes alſo at G. Theſe
phenomena may be obſerved in an artificial chan-
nel with glaſs fides.
Every eddy deſtroys a part of the moving force of
the current of the river. For the water which de-
ſcends by a retrograde motion, in the inclined
plane CDE, fig. 19, cannot be reſtored in the
direction of the current of the river, but by a new
impulſe. It is, as it were, a ball which is forced
to riſe on an inclined plane, whence it continually
falls back again to receive new impulſions. It is
the labour of Siſyphus.
Hence I deduce, as a primary confequence,
that in a river, of which the courſe is permanent, and
the ſections of its bed unequal, the water continues
more elevated than it would have done, if the whole
river had been equally contraéled to the dimenſions of
its ſmalleſt ſection. The caụſe of this phenomenon
is the ſame as that which retards the expenditure
through the tube with enlarged parts. (Prop. VII.
No.4.) The water which defcends from the eleva.
tion above the contracted part N, into the baſin
K2
P Q
***
.
( 52 )
PQ, fig. 19, loſes nearly the whole of the velo.
city it acquired by deſcending from it ; becauſe
the narrow part has a curved ſlope towards the
lower part of the river, which directs the velo-
city of the ſtream in an horizontal direction.
Guilielmini has well remarked, that a fall does
not influence the velocity of the lower ſtream,
becauſe the eddies of the water in the baſin P Q
deſtroy the velocity produced by the fall. This
velocity increaſes the depth, and enlarges the
width of the channel at P Q. Eddies are formed
on each fide, at the bottom, and at the ſurface;
both in the horizontal and vertical directions. It
would be to no purpoſe to attempt to prevent
this hollowing out and enlargement of the chan-
nel by ſuch a fall, by adopting the means of
cloſe walls; for the baſin would then obtain its
enlargement, where theſe conſtructions might
end.
If the channel have a number of ſucceſſive
contractions and dilatations, M N, without caſcade
or dam, there will ſtill be formed, at each dilata-
tion, ecldies which will diminiſh the velocity
morc ihan if the channel had an uniform ſection
equal to that in M or N. It will, therefore, fol.
low, that the ſurface of the water after each di-
latation, muſt riſe, in order to recover the velocity
it loft by the eddies. If we call the height to
which the water muſt riſe above the elevation ne-
ceſſary
( 53 )
ceſſary to have overcome the retardations of a bed
of uniform fection, Fa, and that the number of equal
and ſucceſſive alternate dilatations and contractions
be = m, the height of the riſe in the ſtream thus
alternately dilated beyond that of the ſame river
uniformly contracted will be = am.
I here ſup-
poſe the bottom of the river to be uniform. If this
bottom bc of ſuch a nature as to be attacked by
the current, the contracted parts will be hollowed
out, and the matter will be depoſited in the en-
larged parts.
The ſecond conſequence which I draw from
the principle here eſtabliſhed, reſpecting the loſs
of force cauſed by the eddies, is of confiderable
importance in the theory of rivers, and appears
to have been neglected by thoſe who have treated
on this ſubject. The friction of the water along
the wet banks, and over the bottom of rivers, is
very far from being the only cauſe of the retarda-
tion of their courſe, which, confequently, requires
a continued deſcent to maintain its velocity. One
of the principal and moſt frequent cauſes of re-
tardation in a river, is alſo produced by the eddies
which are inceſſantly formed in the dilatations of
the bed, the cavities of the bottom, the inequa-
lities of the banks, the flexures or windings of its
courſe, the currents which croſs each other, and
the ſtreams which ſtrike each other with different
pelocities. A confiderable part of the force of the
current
1
ty
( 54 )
current is thus employed to reſtore an equilibrium
of motion, which that current itſelf does conti.
nually derange.
PROPOSITION XI.
If the Water of a Reſervoir, which flows through an
horizontal Aperture, be influenced by any foreign
Motion, it will form an hollow Whirl above the
Orifice itſelf.
.
CITIZEN Boffut has given a very good de-
fcription of this kind of eddy *. It is of a
different nature from thoſe conſidered in the fore-
going Propoſition; but the cauſes are, in ſome
reſpects, fimilar, for which reaſon I propoſe to
attend to them more particularly in this place.
Let D Q, fig. 18, Plate II. repreſent an hori-
zontal plane near the orifice EF, through which
the fluid of the reſervoir MN flows. A fluid
particle D, ſituated in this plane, has a motion
DB, inclined to the axis A B. This motion
may
be decompoſed into two, DC, CB; let us ſup-
poſe that plane DQ to deſcend parallel to itſelf
along the axis, with the motion CB; the motion
DC of the particle D on the plane D Q, remains
to be examined. This motion impreſſes upon all the
* Hydrodyn. No. 432,
particles,
*
( 55 )
particles, ſituated in the plane D Q, a centripetal
force towards the centre C.
Let
any
other horizontal motion whatever, not
coincident in direction with DC, be impreſſed
upon the ſame particles : under the government of
theſe two forces, the particles will deſcribe round
the centre C, arcas proportional to the tinies, and
by the equilibrium of theſe motions they may
affume an horizontal circular rotation.
Let us imagine, that during this horizontal
circulation, the particle D, in its approach toward
the centre C, as in a ſpiral, ſhall deſcribe circular
orbits, of which the diameter is ſucceſſively dimi-
niſhed ; let us call the velocity of rotation of the
particle D=v; its diſtance from the centre =r;
the time of one revolution=t; and ſince the areas
muſt be as the time, we ſhall have nearly v==;
I=r?; and the centrifugal force of the particle D
will be=-. When we attentively obſerve the par-
ticles which revolve at the ſurface of the funnel, at
MN, we ſee that the effect which really takes
place in nature, is nearly t=r2. Since, therefore,
the centrifugal force, in approaching the centre C,
increaſes as it will become equal to forming an
equilibrium againſt the upper preſſure SD, which
produces the centripetal force DC.
A cavity
KRTHPV, will therefore be formed, round
which
I
r
L
r3
I
73
wein
( 56 )
3
Å
which the whirling fluid will ſupport itſelf by the
centrifugal force of its rotation.
Let DQPR repreſent a circular fluid zone,
the particles of which turn round the cavity RP,
according to the law here indicated. Let the
gravity of a fluid particle be =g; CR=a; RD
=b; D X=2; X2=dz; and the velocity of the
particle D=v. If the centrifugal force of the par-
ticle D were equal to its gravity, its velocity, by
the theorems of Huyghens, would be equal to that
of a body falling by gravity alone, through the
{pacea, And ſince an heary body falls in one
ſecond through the ſpace of 181 inches=S; the
velocity of the particle D, on the ſame ſuppofition,
would be=v (2 S (a+b)). The centrifugal force
in the circle is as v2; the centrifugal force of D
will therefore really be =
v2g
And ſince the
2 S (a+b)
centrifugal force is =-;;; taking
(a+b)3' (a+b_z) 3
-v?g
: a fourth term, we ſhall have the
2 S (a+b)
centrifugal force of the element of DX in X=
veg (a+b)?dz
2 S (a+67x)}; and that of the filament D X= A +
v2g (a+b) 2
When =0 the integral is=0;
4 S(a + b )2
whence A=
mo. ,
v²g
Taking x=b, the centrifu-
4S
bgaa
gal force of the filament DR will be =-
(2 a+b). The quantity bg is the gravity itſelf of
the filament DR. The gravity of this filament is
there
ܪܪ
I
I
4 a2
1
KW
...MY
( 57 )
thercfore to its centrifugal force =v* (2 a+b)
4 a? S.
When the fluid zone, DRP Q, is nearer the
aperture EF, the preſſure SD increaſes; whence
the centrifugal force of the zone muſt alſo be in-
creaſed by diminiſhing the radius of the cavity
RC: hence we may determine the nature of the
curve which forms the perpendicular ſection of the
cavity KRT. For greater fimplicity, let us ſup-
poſe that the ſides of the veſſel have the fame form
MD as that of the cavity itſelf, ſo that D R=b
may be conſtant. Let A C=s; and CR= y. Let
us ſubſtitute y inſtead of a in the preceding
formula. And ſince the gravity of the filament
DR is to the gravity of the filament SD=b: x,
we ſhall have by compoſition of ratios, the cen-
trifugal force of the filament DR, to the preſſure
SD=bu? (2 y+b); 4 x y2 S. Theſe quantities muſt
be equal, in order to afford an equilibrium. We
b o²,
have therefore xyzm
- for the equa-
4 S
tion of the curve KRT. This is the ſixty-fourth
ſpecies in the enumeration of lines of the third
order, by Sir Iſaac Newton. Its convexity is
turned towards the axis; it has two aſymptotes, one
of which is the axis A Y, and the other is in MN,
ſuppoſing the two points M, N to be infinitely
diftant.
If the aſſumed poſitions in this theory do not
abſolutely coincide with nature, they approach
its effects very nearly. It is not only poſſible,
but
2
22 22
2 S
I
( 58 )
but there does exiſt in nature a whirling ſtream,
of which the cavity turns its convex part to the
axis, and in which t=r? very nearly, as is ſhowny
by experiment.
Exper. XXVI. Let the orifice E F be opened,
and any motion 'whatever be impreffed on the
fluid, independent of that which its gravity, and
the preſſure of the circumambient particles tend
to produce; the turning immediately begins, and
is teen to be more rapid in thoſe parts of the fluid ,
which are neareſt the bottom. The cauſe of this
is, that the motion D B is more convergent and
perceptible in thoſe parts which are neareſt the
orifice EF*. The centripetal force DC, pro-
duces its effect there rather than at the upper parts.
Theſe laſt afterwards fall into the cavity which
begins to be formed below, by which means they
alſo acquire a centripetal force, and the funnel or
cavity opens to a much greater Light, than that
in which the convergence of fluid filaments is ob-
ferved towards the orifice EF, in water which is
þeſs agitated.
Exper. XXVII. Place a floating body at the
furface of the fluid, of ſufficient magnitude to
prevent the formation of the cavity. If the fluid
be much agitated, the cavity will take place at the
lower part, and air will introduce itſelf through
the opening EF. Whence it follows that the
* Bernouilli, Hydrod. fect. 4. § 3. Boflut, art. 427.
4
preffure
( 59 )
1
preſſure of the atmoſphere on the upper ſurface of
the fluid is not the cauſe of the cavity, which
aſſumes the ſhape of a funnel. The air does not
enter but becauſe it finds an empty ſpace formed
by the centrifugal force.
Exper. XXVIII. When the fluid remains in a
ſtate of tranquillity without eddies, the veſſel
empties itself in forty feconds; but when the cir-
cular motion takes place, the evacuation is accom-
pliſhed in fifty feconds, more or leſs. It cannot,
therefore, be ſaid in general terms, that the
whirling ſtream abſorbs and draws down bodics
through the opening EF, with more force than if
no ſuch circulation took place.
Exper. XXIX. Pour a ſtratum of oil upon the
water of the veſſel. As ſoon as the funnel forms
itſelf, the oil ruſhes down, and iſſues out before
the greateſt part of the lower water, upon which
it reſted. The portions of oil partake leſs of the
rotation of the lower water ; having leſs denſity,
they likewiſe recede leſs from the axis than the
water ; in conſequence of which, as they occupy
the interior part of the funnel, and are unſupport-
ed, they flow out firſt.
Exper. XXX. Every other ſmall body which
floats on the water in the veſſel acts in the ſame
manner as the oil, provided its dimenſions be very
ſmall. If the volume of the body be ſomewhat
greater, while it approaches the cavity, to fall
therein, its extremity, which is neareſt the axis,
comes
-
I 2
( 60 )
i
comes into a place where the circulation is more
rapid. This rapidity of motion impreſſed at one
extremity of the floating body, is tranſported, by
the laws of mechanics, to its centre of gravity,
which is more remote from the axis in a ſituation
where the circular motion is ſlower; conſequently
the body recedes from the edge of the cavity into
which it was about to fall. It returns a ſhort
time afterwards, is again repelled, and theſe alter-
nate motions continue as long as the circumſtances
which produced them. Laſtly, if the body which
floats at the ſurface of the liquor after the funnel
has been formed, be of ſufficient ſize to cover the
whole cavity, it deſtroys the funnel in the upper
part, and ſometimes alſo in the lower. The rea-
fon is, that the body itſelf cannot turn round its
centre but according to the law v=r; it therefore
deſtroys by friction the law v
of the fluid in contact with it, and conſequently it
deſtroys the funnel itielf.
I
in the parts
:
PROPOSITION XII.
The lateral Communication of Motion takes place, in
the Air as well as in later.
THE fireain of air which moves in the midſt of
a body of air at reſt, produces vinculations and
cddies round its current in the ſame manner as in
Water. Theſe may be obſerved in the ſmoke
which
(61)
which riſes froin a furnace, and produces a re-
markable aſpect, when it iſſues like a dark tree
from an agitated volcano. They may likewiſe be
ſeen, in the particles which float in an obſcure
chamber, 'when a ray of the ſun ſhines in, and
the obſerver blows through them.
If the general wind comes, for example, from
the ſouth, it frequently happens that the north
fide of a mountain is at the ſame time ſtruck by a
north wind. This partial and local wind is no-
thing but the eddy produced by the mountain
itſelf acting as an obſtacle againſt the principal
wind, from the ſouth. It is probably from the
ſame cauſe, that the wind ſometimes acts in the
contrary direction on the fails of a veffel, when
they are too obliquely preſented to its ſtream.
The vapour of water which iſſues from the
eolipile carries the ſurrounding air with it, and
drives it againſt the burning coals oppoſite to the
ſtream of aqueous vapour. It muſt not, there-
fore, be concluded that the aqueous vapour is
itſelf in this caſe decompoſed to maintain the com-
buſtion of the charcoal.
It is known that the flues of chimnies aſſiſt the
riſing of ſmoke by their figure; concerning which,
we have drawn ſome inductions, in the ſeventh
Propofition.
In
organ-pipes, the air which iſſues out of
the ſide opening (lumiere) rubs laterally againft
the extremity of the column of the air included in
the
(62)
the pipe. It rubs it on one ſide in the longitudi-
pal direction, and is, as it were, an elaſtic file,
acting upon an elaſtic ſurface. Though the
column of air be fluid, its parts are, however, ſo
far intermixed together, that the tremulous motion
excited at the place of friction is foon communi-
cated laterally through the whole thickneſs of the
column, which receives vibrations of ſuch a kind,
that they are an equilibrium with each other, and
with the velocity of the ſtream which affords the.
friction. For this effect, it is requiſite that the
column ſhould divide itſelf at different points or
nodes diſtributed through the length of the tube *
It is by repeated actions that the wind which iſſues
from the ſide aperture, impreſſes at length upon
the whole column contained in the pipe, a move-
ment of vibration greater than that which the laws
of impulſe, and of the lateral communication,
would permit it to make by a ſingle impulſe. In
the hautboy, and other ſimilar inſtruments, having
a mouth-piece, or rced, the cauſe which excites the
tremulous motions does not act fideways on the
air contained in the pipe; but ſtrikes the column
directly in the middle: for which reaſon it com-
municates its vibrations with ſo much the more
effect to the whole mafs.
In like circumſtances, the force of ſound, which
is propagated in the atmoſphere, depends on the
magnitude of the ſection of the air which is at the
* Memoires de l'Acad. an 1762, page 431.
extre-
( 63 )
}
ķ
1
extremity of the pipe, and the amplitude of the
vibrations of this ſection. It is this ſurface which
ftrikes the atmoſphere, and communicates the
pulſations *. For this reaſon, conical divergent
pipes afford a ſtronger found than thoſe which are
cylindrical; and theſe laſt afford a ſtronger found
than pipes which are conically convergent. The
firſt cauſe of the found which acts at the mouth
end of the pipe would never, of itſelf, excite ſuch
ſtrong pulſations in the atmoſphere, as it does
excite by the lateral communication in the air
contained in a divergent conical pipe.
The explanation of this phenomenon may be
underſtood by obſerving, ift, That if a number of
elaſtic bodies be diſpoſed in progreſſion, the firſt
will impreſs upon the laſt by the intermedium of
the others, more velocity than would be commu-
nicated by the immediate ftroke. 2. The vibra-
tions excited in the pipe have a certain perma-
nence, which permits them to receive an increaſe
of force by the united effect of ſucceſſive impul-
ſions; whereas, in the open atmoſphere, every
pulſation is tranſient and ſingle.
Is not the augmentation of ſound in the ſpeak.
ing-trumpet, in part owing to the ſame cauſe of
the lateral communication of motion, rather than
to the mere reflexion of the fonorous lines from
the ſides of the tube itſelf?
* It is known that the material, of which a pipe is made, does
not perceptibly affect the found.-7.
I call
( 64 )
I call thoſe reſonant vibrations, which take
place in a tube when ſound is excited; and I call
thoſe propagated vibrations, or pulſations, which
tranſmit the found through the atmoſphere. I
have already pointed out a difference, which ap-
pears to me to take place between theſe two kinds
of vibrations; namely, that the former have a cer-
tain
permanence and connexion with each other,
ſo that each ſucceeding impulſe excites, ſupports,
and reinforces the former ; whereas, thoſe pulfa-
tions which ſucceed each other in the atmoſphere
by the repeated action of the reſonant body, are
fingle, and independent of each other.
But the following is a much more remarkable
difference between theſe two kinds of vibrations.
When at the extremity of a pipe ABC, a re-
fonant vibration is made in the ſection of air, BC,
fig. 2, Plate I.
experience ſhows that this
vibration becomes the centre of pulſations propa-
gated all round in PSQ. For on whatever fide
the obſerver is placed, whether at P or at Q, he
will hear the ſound of the pipe A B C nearly as
much as at S. But when there is no pipe, and
the vibration at CB is a fimple pulſation propa-
gated through the open air from A to B; in this
caſe the pulſation is not propagated laterally and
completely to P and Q like the reſonant vibration;
but is contained almoſt entirely in the limits BZ
and CY, with a divergence of between 15 and 20
degrees. This fact has been diſputed by various
philo
į
( 65 )
W
philoſophers; but it cannot be queſtioned, ſince
it is well known that we do not hear the echo, or
reflected ſound, from the plain ſurface, unleſs we
place ourſelves in the line of reflexion, or very
near it. If the pulſation of the echo were propa-
gated all round, before the reflecting ſurface, di-
verging from thence as centre, ought we not to
hear the echo in every ſituation whatever, before
that reflecting furface? We muſt therefore ad-
mit, with regard to ſonorous pulſations propa-
gated in the atmoſphere, certain exceptions, and
even limits, with regard to the lateral communi-
cation of motion which we have pointed out in
the firſt Propofition, and in the fifth, with regard
to water.
Addition reſpecting the contračica Vein.
MUCH has been written reſpecting the con-
vergent directions aſſumed by the particles of a
fluid contained in a veſſel, previous to their be-
ing emitted through an aperture in the ſide of the
veſſel itſelf, and concerning the form of the con-
tracted vein which is thus produced. The re-
flections and experiments, which I ſhall proceed
to give, may afford ſome farther explanation in
this reſpect.
I ſhall begin by defending the fundamental
doctrine of hydraulics againſt the opinion of a
learned man, diſtinguiſhed by his labours and his
zeal
K Қ
.
( 66 ).
zcal for the advancement of ſcience: Lorgna,
the founder of the Italian ſociety. He pre-
tends * that the contracted vein is nothing elſe
but the continuation of the Newtonian cataract,
and that the celerity of the fluid iffuing from an
orifice in a thin plate, is much leſs than that of a
body which falls from the height of the charge.
Let MD, figi 22, Plate II. repreſent the axis
of the vein which iſſues from B. The radius
of the circular orifice BC=BD=1; MB=a.
Lorgna pretends that 0,472 a =H B, is the height
which would produce, in an heavy body, the ve-
locity of efflux in BC: he ſupports this propofi-
tion by computations deduced from the mufual
action of the particles of the fluid contained in
the veffel. But after having ſeen the failure
of the efforts of the greateſt geometers on
this very ſubject, we ought to miſtruſt all theſe de-
monſtrations founded on mechanical principles,
very true in themfelves, but of which the
ap-
plication to an infinity of bodies, which move
and are preſſed in every direction, becomes ex-
tremely difficult, if not impoſſible. Let us ſee
whether the theory of Lorgna agrees with experi-
ment. Suppoſing the velocity of the fluid at B,
ariſing from the elevation H B= 0,472 a, the
velocity of the ſame fluid in D will be increafed
in the ratio of VHB: „HD; and the vein in D
* Mem. della Società Italiana, vol. iv.
will
( 67 )
!
will be contracted inthe ſame fame ratio. Whence
DE=v* (104772a); which is the formula of
the hyperbolic conoid of Newton. If this be
the fole cauſe of the contraction, the dimenſions
of D E ought, very nearly, to agrce with this
figure when examined by experiment. But they,
in reality, differ from it very much, as may be
ſeen in the following table,
Value of D EI Value of DE
Authors of the Experiments. tual Meaſure- the preceding
found by ac. calculated by
ment.
Formula.
Poleni (de Caſtellis, $ 35)
0,79
0,97
Michelotti ; Sperim. Idraul.
Tom. I. Exper. 46;
Tom. II. Exper. 4
0,80
0,99
Boſſut (Hydrodyn. art. 437.
Exper. 5)
0,818 ,
0,99
Myſelf with 35 inches charge
and an horizontal circular
orifice of 18 lines in diameter 0,798 0,984
.
It is evident, that the contraction of the vein,
as found by experiment, is incomparably greater
than can be produced by the acceleration of gra-
vity, even in deſcending ſtreams. But what can
we ſay of horizontal and aſcending jets, in which
aſſuredly the acceleration of gravity does not
take place, but in which, nevertheleſs, the con-
traction is obſerved nearly in the ſame manner
as in deſcending currents? The contraction of the
ſtream is therefore very different from the New-
tonian hyperboloid.
Defirous
?
K 2
( 68 )
Deſirous of proving that the vein does not pof-
ſeſs the whole velocity ariſing from the height of
the fluid above the centre of the orifice, Lorgna
relates the experiments of Kraft *, which are not
applicable to the queſtion, becauſe they were
made with cylindrical pipes; and we have ſeen
that ſuch pipes always deſtroy part of the velocity
of the fluid; conſequently, we cannot eſtabliſh
any rule from them which ſhall apply to orifices
through thin plates up. He wiſhes not to deter-
mine the velocity of aſcending jets by the height
to which they riſc, becauſe he is apprehenſive that
the preceding part of the ſtream or jet is urged,
and ſupported by the ſucceeding part nearly to
the height of the charge. Nevertheleſs, if we
interrupt the jet all at once, the laſt portions of
water fly to the ſame height as thoſe which pre-
ceded them, without having any continued co-
lumn of the fluid below to follow and ſupport
them : theſc laſt portions muſt conſequently have
received, at their paſſage through the orifice, all
the velocity which was neceſſary to raiſe them
nearly to the ſurface of the fluid in the reſervoir.
Let us confine ourſelves, if it be thought
proper, to horizontal jets; the experiment which
I have related, as a term of compariſon, appears
* A&ta Petrop. vol. viii.
+ Toricelli took notice of this difference at page 168 of his
works, “ quotieſcumque autem aqua per tubum latentem de-
currens per anguſtias tranfire debuerit, falfa omnia reperies."
to
( 69 )
i
to me to be deciſive. Under the charge of 32,5
inches, the vertical line PM, fig. 1, Plate I.
being 54 inches, the horizontal line M N was
always 81,5 inches, which was only two inches
leſs than it would have been if the jet had pre-
ſerved in the direction of the horizon, all the ve-
locity which a heavy body acquires in falling from
the height of 32,5 inches. The diameter of the
contracted vein was 14,3 lines very nearly. Since
the quantity of 81,5 inches in M N ſuppoſes in
the contracted vein a velocity of 149,5 inches
per ſecond ; this number multiplied by the area of
the contracted vein itſelf, gives the expenditure of
four cubic feet in forty-one ſeconds of time, which
is alſo the reſult of experiment. We have, there-
fore, three meaſures determined by experiment,
which agree, and mutually confirm each other ;
namely, the quantity MN, the contraction of
the ſtream, and the time of expenditure. And
ſince the quantities obſerved by Boſſut, Michel-
lotti, and Poleni, give nearly the ſame reſults, it
can no longer be doubted, i. That the contrac-
tion of the ſtream is nearly 0,64 of the orifice;
2. That the velocity of the contracted vein is
nearly the ſame as that of a heavy body which
may have fallen through the height of the charge.
Theſe two experimental principles are true in
all caſes where the orifice is conſiderably ſmall
in proportion to the fection of the reſervoir, where
that orifice is made through a thin plațe, and the
internal
i
( 70 )
internal afflux of the fluid filaments is made in
an uniform manner round the orifice itſelf. But
what would be the conſequence if this internal
afflux ſhould be modified in a manner different
from what uſually happens ? The following ex-
periments were made with the intention of afcer-
taining ſome of the moſt remarkable effects of
theſe particular modifications in the direction of
the fluid filaments which preſs each other in order
to pafs through the orifice.
Exper. XXXI. In the orifice ACBD, fig. 21,
Plate II. the two fides A, B, are parallel to
the horizon; the extremities CD are rounded ;
the width of this aperture is leſs than two lines,
its length eighteen lines, and the charge 32,5 inches.
The ſection of the ſtream which ifſues from this
orifice, firſt aſſumes the form EF; after which,
the two extremities E F approaching nearer and
nearer to enlarge the middle part of the ſection of
the ſtream, at 4,5 inches difiance from the orifice,
acquire the quadrangular form GH. The ſtream
afterwards extends itſelf in the perpendicular di-
rection in the form of a large fan K L.
I have repeated the experiment by placing the
longitudinal axis of the orifice CD vertically.
In this caſe the ſame phenomena were produced;
EF becoming vertical, and KL horizontal, both
preſerving their form.
The fluid filaments, which, iſſuing out of the
orifice, paſs near the two oppoſite borders A, B,
:
ara
.
.
(70)
are very near each other; and being convergent, they
tend to unite at a very ſhort diſtance from the orifice
itſelf. The filaments C, D, are more remote, and,
perhaps, leſs convergent; they cannot unite but
at a greater diſtance than the two former. In this
caſe, therefore, there are mnovements which tend
to form two contractions, the one nearer, and the
other more remote from the orifice; theſe two
contractions counterbalance each other in part.
Their mutual oppoſition carries the effect G H to
a diſtance five times greater than that of the con-
tracted vein of a circular orifice, having a dia-
meter of the ſame breadth as that of this orifice.
In this experiment we ſee the cauſe of a phe-
nomenon which has been obſerved in ſome par-
ticular caſes by Poleni and others, without giving
the explanation. In every orifice of a right-lined
figure through a thin plate, the angles of the
contracted vein anſwer to the ſides of the orifice,
and the contrary. When the quadrangular ori-
fice has the ſituation MNOP, the greateſt con-
traction of the ſtream is made at a greater diſtance
than in a circular aperture ; it aſſumes the form
and ſituation QRST. The reaſon is, that the
oppoſite angles MP are more remote from each
other than the ſides 1, V, whence the ſame thing
happens as in the long orifice ACBD. ' In the
ſame manner the triangular orifice in the ſituation
X produces a contraction of the form and in the
ſituation Z, &c.
Exper.
2
( 72 )
Exper. XXXII. The orifice being the hori-
zontal cleft CD, fig. 21, the place G H, or moſt
contracted point of the ſtream, was found to be
diſtant from that orifice as in the following table :
Height of the charge above the
orifice CD
Inches.
32,5
18
10
6
Diftance of the greateſt con-
traction GH.
Lines.
53
48
40
36
The long orifice CD exhibits to us,, under an
enlarged dimenſion, the diſtance of the contracted
vein from the orifice. By this means, the fore-
going table ſhows us, in a very ſenſible manner,
that the contraction of the ſtream takes place at
a greater diſtance, under ſtrong charges, than in
thoſe which have but little elevation.
Exper. XXXIII. To the centre of the circular
orifice AB, fig. 23, formed in a thin plate, I
diſpoſed within the reſervoir, the cone of metal
DGE, with a cylindrical part CFGD, in ſuch
a manner that the cone was moveable along its
own axis IV, and its ſummit E could be pro-
truded more or leſs through the orifice AB, ap-
proaching or receding from the point V. The
meaſures in lines were AB = 18; IE = 24;
DG = 27; CD= 8. This apparatus was ap-
plied to the orifice P, fig. 1, Plate I.; the
charge
( 73 )
charge being 32,5 inches. The reſults were as
in the following table.
Quantity EP,
by which the
fummit of the
cone projects
beyond the
line AB.
Diſtance of the
contracted
vein.
Diſtance of
MN.
Time of ex-
penditure of
four cubic
feet.
Lines.
Lines.
II. I
9, 1
6,6
12,3
14
1493
Inches.
76
77,5
78,5
81,5
85"
53
43
The cone re-
41
moved.
I intend to repeat and vary this experiment, in
order to diſcover the cauſe of the fingular pheno-
menon which it preſents.
Exper. XXXIV. The orifice being a ſemi-
circle, Plate II. fig. 24, having the diameter AB,
11,2 lines, I applied within the veſſel a plane
QAB, perpendicular to the plate in which the
orifice was made. The line AB was perpendi-
cular to the horizon, and the charge 32,5 inches.
The jet deviated in the horizontal direction in
PFG, departing from the axis CE towards the
ſide on which the plane Q P was placed. The
angle FCE was 9° 5', and the angle FCG was
36°. The vertical ſection of the jet bad the form
KL, ſo that the largeſt part of the jet was in F.
The four cubical feet of water isſued out in 206
ſeconds.
The
I
A
( 74 )
3
4
The rclults of this experiment are analo-
gous to thoſe of the experiments XXXI and
XXXIII.
Exper. XXXV. Citizen Borda, in a very
in-
tereſting memoir*, relates a peculiar phenome-
non, of which he has given a very ſimple demon-
ſtration, from the principle of the equality of
preſſure, which fluids exert in every direction.
It is, that, if the extremity of a cylindrical tube
be puſhed into the interior part of the reſervoir,
the contraction of the vein is greater, and the
expenditure leſs, than if the ſame tube be applied
to the ſide of the veſſel. I have repeated this
experiment, and obſerved a ſimilar reſult when
the tube was cylindrical from one end to the
other, like that made uſe of by the author, and
when the water was made to flow out in a full
ſtream. I afterwards gave to the interior extre-
mities of the pipe the form AC, fig. 4, Plate I.
of the common contracted vein; in this caſe,
there was no longer any remarkable difference
between the two expenditures, in the two fitua-
tions of the tube. For when the end AC was
puſhed into the interior part of the reſervoir, the
full tube afforded, in eighty-one ſeconds, the
ſame quantity of water as was furniſhed in eighty
ſeconds, when it was applied to the ſide of the
t
:
* Memoires de l'Acad. 1766.
veffel.
A
( 75 )
veffel. It may be preſumed, that if the part AC
had more perfectly poſſeſſed the form of the con-
tracted vein, the flight difference of one ſecond
would have diſappeared,
.
THE END.
The following Books were lately publiſhed
By J. TAYLOR,
At the Architectural Library, No.59, High Holborna
$
Am
th
I. A GENERAL HISTORY of INLAND NA.
VIGATION, foreign and domeſtic : containing a com-
plete Account of the Canals already executed in England,
with Conſiderations on thoſe projected. To which are
added, Practical Obſervations. The whole illuſtrated
with a large Map, coloured, ſhowing the Lines of all the
Canals executed, thoſe propoſed, and the navigable
Rivers ; with other uſeful Plates. By J. PHILLIPS.
. A
new Edition, with two Addenda, which complete the
Hiſtory to 1795. Quarto. Price 1l. 8s. in boards.
II. A TREATISE on the IMPROVEMENT of
CANAL NAVIGATION, exhibiting the numerous
Advantages to be derived from Small Canals and Boats of
two to five Feet wide, containing from two to five Tons
Burden ; with a Deſcriprion of the Machinery for faci.
litating Conveyance by Water, through the 'moſt moun-
tainous Countries, independent of Locks and Aqueducts;
including Obſervations on the great Importance of Water
Communications; with Thoughts on, and Deſigns for,
Aqueducts and Bridges of Iron and Wood.
FULTON, Engineer. With 17 Plates. Quarto. Boards,
18s,
III. OBSERVATIONS on the various SYSTEMS
of CANAL NAVIGATION, with Inferences practical
and mathematical, in which Mr. Fulton's Plan of Wheel.
Boats, and the Utility of ſubterraneous and ſmall Canals
are particularly inveſtigated ; including an Account of
the Canals and inclined Planes of China. With 4 Plates,
By W. CHAPMAN, Civil Engineer. Quarto. 6s. ſewed,
IV. EXPERIMENTAL ENQUIRY concerning
the natural Powers of Wind and Water to turn Mills
and other Machines depending on a circular Motion.
And an experimental Examination of the Quantity and
Proportion of mechanic Power neceſſary to be employed
in giving different Degrees of Velocity to heavy Bodies
from a State of Reſt. Alſo new fundamental Experi-
ments upon the Colliſion of Bodies. With 5 Plates of
Machines. By the late Mr. John SMEATON, F. R. S.
Octavo. Price 4s. 6d. boards.
By R.
1
3
Tenturis L.rperiments l'late 1.
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