FLOW OF WATER

RIVERS AND CANALS
* (-e BY =
Č.
D; FARRAND HENRY,
*
CHIEF ENGINEER DETROIT WATER Works.
-------------e. *tº ºr ~~~~---—-
D E T R O | T :
WM. GRAHAM's STEAM PRESSEs, 52 BATEs STREET, (BAGLEY's Block.)
1873.
Jr. º. Inst., Poz. LXI. ON THE FLOW OF WATER IN RIVERS &c. - Plate /.
HENRY'S
Telegraphic Current Meter
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l)rawn by E. Molitor.C.E. Detroit. Mich.
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34 . ~.
*3 the Journal of the Franklin Institute.] "…"
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H 52
ON THE FLOW OF WATER IN CANALS AND RIVERS,
BY D. FARRAND HENRY, PH.B.
(Late Assistant on the United States Lake Survey.)
THE velocity of water in streams was probably first measured by
means of bodies floating upon its surface; and even to the present
time, floats are often used to determine surface velocities; although
Boileau says:* “One can in certain cases obtain the surface velocity
exactly by means of floats; but to obtain this exactitude a great
number of often impossible conditions must be fulfilled.” And
after naming some of the conditions, he concludes (p.269): “Tastly,
even with floats sunk to the surface of the water, it is essential that
observations be made in a calm time; for the molecules of a fluid
current at the surface are so unstable that a breeze apparently in-
significant causes a notable variation in the velocity.” And D'Au-
buisson, after describing the best kind of floats, says: “In this man-
ner by repeating the operation two or three times, we expect to ob-
tain the velocity of the swiftest current with sufficient exactness; but
for the fillets contained between this and the sides, this mode will
not answer, the float will not maintain the necessary direction.”f
The double float was first constructed by Leonardo da Vinci,
(though the idea is attributed to Mariotte.) It was composed of
two balls of wax connected by a thread, one loaded so as to sink
to the required depth; the other being partly immersed. The ob-
jections to surface floats apply with greater force to the double float,
which besides has errors of its own which will hereafter be noticed.
In 1777, Mr. T. H. Mann, in a paper read before the British As-
sociation, recommended floating rods, loaded at one end so as to
sink nearly to the bottom of the stream ; the upper end projecting
slightly above the surface.
These have been used by Buffon in his velocity measurements
of the Tiber; by Krayenhoff in rivers in Holland; and more re-
* Traité de la mesure des eaux courants, par P. Boileau, Paris. Page 267.
† Treatise on Hydraulics, by J. F. D'Aubuissons de Voisins. Translated by J.
Bennett, Boston, 1852, page 156,
2
cently by Mr. J. B. Francis in the Lowell canals. Although more
valuable than any other system of floats, they are only applicable
to small streams and canals; and even then, many precautions and
a large number of observations are necessary to eliminate their un-
avoidable errors.
The float-wheel which could be held in a current, the velocity
being given by the number of revolutions it made in a certain time,
is also quite ancient. It was used by Borda and by Dupuit for de-
termining surface velocities.
The latter engineer used a fir-wheel, over two feet in diameter,
having on its axle a pinion which engaged with an indexed cog-
wheel by means of which the number of revolutions could be di-
rectly observed. Woltmann, in 1790, modified this meter so that it
could be used beneath the surface. He constructed a helicoidal
wheel, with an endless screw on its axle, and a train of two gear-
wheels, so hinged to the frame of the meter, that when raised the
teeth of the first wheel would engage the screw.
This meter could be run down a pole driven into the bed of the
stream, a cord being fastened to an arm connected with the train,
so that the gear-wheels could be raised for any required time, and
thus the number of revolutions of the meter-wheel recorded. This
was a great improvement on the floats, but it is liable to errors
arising from the friction of the wheel on its axle, and of the record-
ing train; from the shock the revolving-wheel receives when the
train is raised; from the inertia of the gears causing them to revolve
after being released; and from the retardation of the train by dirt
and vegetable debris suspended in the water.
There is also great difficulty in using this instrument in deep
water, as it has to be raised to be read after each observation.
Lapointe raised the recording apparatus above the surface by
connecting the axle with a vertical rod by beveled gear. This, of
course, obviated the difficulty of clogging the train by dirt, and of
raising the meter to read the recording apparatus; but the friction
was increased, and the meter could only be used in shallow water.
Baumgarten, Saxton and others have modified the form of this
meter, but they have not altered it so as to materially lessen its
errors.
Dr. Brewster made the axle of the meter a long fine screw; the
hub of the wheel having a female screw cut in it; so that the num-
ber of revolutions were indicated by the distance the wheel trav-
3
eled; and Mr. Laignel modified this meter by fixing the wheel on
the screw, and indicating the number of revolutions by a nut which
moved an index along a scale. A detent held the wheel in these
meters until the commencement of the observations. Although the
friction is much reduced in this instrument, the screw could have
no great length; and the wheel must be stopped before the full
distance is traveled, while the Woltmann meter may be allowed to
run more than one revolution of the second wheel of the train.
Mr. Gaunthey, in 1779, invented the pressure plate, which was
improved by Brunnings in his tachometer, and used by him on the
Rhine; and also by Mr. Racourt in his velocity observations on the
Neva. This instrument consisted simply of a disk of metal opposed
to the current, the velocity being measured by the amount of weight
required to keep it vertical. This meter was further improved by
Capt. Boileau; but though it might determine surface velocities
with considerable accuracy, it would be very difficult to manipulate
in deep water, and as it only shows the velocity at the time of read-
ing, and not the mean of the varying velocities, it would give but
little better results than floats. The hydrometric tube, which was
improved by Capt. Boileau, is simply a glass tube suspended hori-
Zontally in a frame, having a full size opening at one end, and a
small orifice at the other. The tube is filled with water, and the
large end closed; a small bubble of air being allowed to enter the
orifice, and the apparatus is placed in the current, with the small
end up stream. Then, opening the large end, the time required for
the bubble of air to traverse the tube is noted. The velocity of the
stream will be greater than the velocity of the bubble of air by the
ratio of the areas of the tube and the orifice.
Besides these meters there have been many instruments proposed
for the measurement of the velocity of water, such as Castelli's
quadrant, which consisted of a loaded ball, connected by a thread
with the centre of a graduated circle, the velocity being shown by
the number of degrees the ball was carried from a perpendicular
when placed in a current. Dr. Leslie proposed using a thermome-
ter; the temperature of water in motion being higher than when
at rest.
Pitot's tube, which was lately improved by Mr. Darcy, is quite
an accurate instrument for shallow streams. It consists of two tubes;
one of which is drawn to a fine point, and bent at right angles, so that
it can be opposed to the current, and the other has a hole of the same
4.
size at the lower end. The velocity is shown by the difference of
the heights of the water in the two tubes, and by partially exhaust-
ing the air from both tubes, the columns of water can be raised high
enough to be conveniently read. M. Darcy remarks that it will
give an accurate determination of the velocity if the columns be
watched for a short time, and the mean taken of the highest and
lowest stages.
Among these instruments the preference is generally given to
Woltmann's meter, especially in Germany; and if the friction of
the parts could be reduced to zero, the danger of retardation by
clogging be removed, and if it could be run for any required time
at any depth, it would be a perfect instrument for the measurement
of velocities.
The telegraphic meter fulfils most of these conditions, and as it
was fully tested in the determination of the outflow of the lakes,
and has not yet been described in any work on hydraulics, a de-
tailed description will be given.
This meter is shown in Plate I., Figs. 3 and 4, and the register
in Figs. 1 and 2; these parts being separate in this instrument.
Fig. 3 is a float meter, consisting of hemispherical cups attached
by arms to an axle, which runs between adjustable pivots in a frame
B. An independent short arm, C, comes in contact with the fine
spiral platinum wire, D, at every revolution of the cups. This wire
is insulated from the frame, and is connected with one pole of a
magnetic battery, the other pole being connected with the frame, B.
At each revolution of the meter, the battery circuit is made and
broken by the short arm coming in contact with the insulated pla-
tinum wire. If now a Morse's paper register be placed in the cir-
cuit, at each revolution a dot will be made on the moving paper,
and the number of these dots recorded in a given time will give the
number of revolutions of, and thus the distance traveled by the
cups, from which the velocity of the current can be calculated. The
ordinary register used is shown in Figs. 1 and 2.
This consists of a Morse sounder, the armature arm, N, being ex-
tended, and carrying an escapement which engages with the teeth
of a wheel, G. At each revolution of the meter the armature is at-
tracted to the magnet, M, and the escapement moves forward the
wheel, G, one tooth. Any number of wheels can be geared to the
wheel, G, so that the revolutions can be recorded for any re-
quired time.
5
Fig. 4 shows another form of meter, the wheel being helicoidal,
having an eccentric on its hub, which raises the ivory arm I at
each revolution. This arm has a wire passing through it, connect-
ing with the insulated hinge H at one end, and at the other with the
platinum point, P. A light spring, S, serves to keep this point in
contact with a platinum plate on the axle. The hinge, H, is con-
nected with one pole of the battery by an insulated wire, and the
axle with the other pole. At each revolution the eccentric raises
the ivory arm, and thus breaks the battery circuit. Both of these
meters have vanes at right angles to each other, so as to keep the
wheels opposed directly to the current.
The method of using this apparatus is shown in Fig. 5. A boat
is anchored in the river at the place where observations are to be
made. A lead weight of about 50 lbs., having a strong copper wire
fastened to it, is lowered over the stern. This weight is also con-
nected with the anchor by a rope of the proper length to keep it
exactly under the stern of the boat. The spring-pole, which runs
fore and aft is bent down, and the copper wire fastened to its after
end. This serves to keep the wire taut, and also to take up the
small motions of the boat. The yoke, in which the frame of the
meter hangs, has a swivel ring at top and, bottom ; to the lower a
weight is fastened, and to the upper a measured cord. It has also
a spring-clasp which is passed over the wire. The cord has spring
clips every five feet, which are also clasped on the wire to keep the
cord from bowing down stream. The meter being put on the wire,
it can be lowered to any required depth, the wire being connected
with one pole of the battery, and the insulated wire, through the
register, with the other.
By means of a switch the register can be quickly thrown in or
out of circuit, without, of course, affecting or being affected by the
revolutions of the meter. In the float meter, therefore, the friction
is reduced to that of the wheel on the axle, and the contact of the
arm with the fine spiral wire; the register being moved by an in-
dependent power, all retardation from clogging the train is obvia.
ted; and it can be run for any length of time required.
Co-efficient.—By none of these methods can the velocity be di-
rectly obtained; a co-efficient of correction being required. This
can be found for meters by drawing them at different velocities
through still water; by observing the number of revolutions in
currents whose speed is known, and by comparison with other me.
6
ters and with floats. Some engineers have thought that the rela-
tion between the number of revolutions of the meter and the velo-
city of the current could be expressed by the simple formula:
v = A + B n ; in which
v = the velocity of the current.
n = the number of revolutions,
and A and B are constants, whose value must be determined by ex-
periments,
M. Baumgarten and Capt. Boileau found that this relation was
best expressed by a slightly curved line.
TABLE I.
Telegraphic Meter, Lapointe Meter.
Velocity Co-efficient. Co-efficient.
in feet
per sec.
Observed. | Computed. | Difference. Observed. | Computed. | Difference.
0-3 | ......... 14-578 ......... . ......... sa e e a a e e A i is a tº e º 'º * * *
0.5 12 778 12-7()4 + 0-074 : ......... . ......... . .........
1-0 1 1 - 123 11-190 – 0.067 ......... . ......... .........
1 : 5 10.268 10-300 — 0.032 ......... 0-650 .........
2. () 9.722 9.662 + 0.060 0.571 0. 572 0.001
2.5 9-208 9-208 ......... 0.546 0. 540 -- 0:006
3 () 8-888 8.881 + 0.007 0-519 0.519 0.000
3 5 8.638 8-686 — 0-048 0-507 0-502 + 0.005
4-0 8-589 8'546 –– 0.043 0.49(5 O 493 -- 0:003
4.5 8.504 8-518 — 0-014 0.486 0-485 + 0-001
5-0 | ......... . ......... . ......... 0.477 O-480 — 0.003
9:5 | “....... . ......... . ......... 0 474 0 478 — 0-004
6-0 | ......... . ......... 0.466 ......... . .........
6-6 ......... . ......... . ......... 0.469 ........ . .........
7-0 | ......... . ....... . . ......... 0.473 ........ .........
80 l ......... . ........ . ...... .. 0.469 ......... . .........
Sums... X 0.015 ......... . ......... . ........ -- 0 007
Mean...] ......... × 0.0017 / ........, | ......... . ......... + 0.0009
→
In Table I. are given the observed and computed co-efficients for
the telegraphic meter, of the form shown in Fig. 3, and for La-
pointe's meter; the observed co-efficients of the latter being taken
from Morin's Hydraulique, page 100. The co-efficient of the latter
was obtained by placing the meter in the centre of a tube between
two reservoirs, and noting the number of revolutions made during
the time the lower reservoir was being filled. *
The observed co-efficient for the telegraphic meter was found by
hanging it beneath a boat drawn through still water at different ve-
locities, dividing the distance passed over by the number of revo-
'ſ
lutions, and grouping the results for every half a foot per second of
velocity.
These co-efficients decrease rapidly in the low velocities, and
more slowly in the higher. They plot in a curve which approxi-
mates closely to an ellipse whose axes are 828 and 3:44.
The vertex of the curve is taken at zero of the meter, or the
velocity of the current at which it stops revolving. The computed
co-efficients in the table are the ordinates of the ellipse, taken at
every half foot per second. The difference between these and the
observed co-efficients is quite small.
In the Lapointe meter the curve only goes to the velocity of 5-5
feet per second, while the observations were taken up 8-0 feet per
second.
The co-efficients beyond the curve differ so little from those at
5.5 feet that they seem to be best expressed by a straight line tan-
gent to the ellipse at that point.
As has been mentioned, the telegraphic meter used in these ex-
periments was made with hemispherical cups, and was, in fact, con-
structed from a “Robinson's Anemometer.
The ratio.of the resistance of a sphere to that of its great circle
when drawn through still water is given by Dubuat as 35 to 100;
and Beaufoy, with velocities from two to twelve feet per second,
made it 342 to 1000.
Robinson's Anemometer was based on these experiments, the
velocity of the cups being estimated at one-third that of the wind.
Taking the mean of the whole of the above observations, the velo-
city of the cups is 0:36 of the velocity of the water; but at 4:5 feet
per second it is only 0-189; therefore the velocities given by this
anemometer are too small when there is more than a light breeze
blowing.
The slight difference between the above mean ratio and that
found by Dubuat and Beaufoy, shows how little friction there is in
this meter, particularly when the resistance of the arms is taken
into account; and, in fact, there are no recorded observations of a
meter turning at so low a velocity as three-tenths of a foot per
second.
8
In Table II, are given the observed and computed cóefficients of another
Telegraphic Meter, much heavier than the last, of the form shown in figure
2; and of a Woltman's Meter which was tested with the Telegraphic
Meters,
*.
f
TABLE II.
Telegraphic Meter. Woltman Meter.
Veloci- Cöefficients. Cöefficients.
ty in feet
Persec. Observed. Computed. Difference Observed. Computed Difference
1.0 0.493
1.4 1.322
1.5 1.171 1.170 +0.001 0.420 0.412 +0.008
2.0 0.961 0.971 —0.010 || 0.388 0.387 +0.001
2.5 0.868 0.882 —0.014 || 0.375 0.374 +0.001
3.0 0.852 0.837 +0.015 0.369 0.369 000
3.5 0.840 0.825 +0.015 0.369 0.369 000
4.0 0.833 0.322 + 0.011 || 0.368 0.368 000
4.5 0.819 0.819 000 0.366 0.367 —0.001
5.0 0.813 0.816 —0.003 || 0.360 0.366 —0.006
5.5 0.808 0.813 —0.005 ;
| Sums. +0.010 —0.003 *
Means. +0.0011 +0.0004

These céefficients were obtained in a similar manner to the last, but
they decrease much more rapidly at first, so that the same curve cannot be
used. It was found that they would best agree with an ellipse having the
same minor axis as the other, 3.44, but a major axis only one-half as
great, 4.10. The computed cöefficients were obtained in the same manner
as those in Table I. After leaving the curve, the observed cóefficients fol-s
low a line making a small angle with the tangent.
Unfortunately, there were no observations at high velocities made with
the Telegraphic meters, therefore we cannot exactly determine the law of
change of the céefficients in fast currents, but they probably vary little
from the last ones given in the Tables.
Mr. Baumgarten” gives a series of observations to determine the cöeffi-
cient of the Woltman meters, constructed by him, and used in his velocity
measurements on the Garonne. These observations were made both by
drawing the meters through still water, and also by fixing them to an arm,
which could be revolved at different velocities around a vertical axis.
*Annales des Ponts et Chausses, Tome XX, Paris, 1847; page 327, et seq.
9
The cöefficients are given in Table III.
TABLE III.
CöEFFICIENTS,
Yº...] observed. computed. Diffe
feet per Sec. º Omp ge Ten Ce.
0.950 1.622
1.121 1,495 1.492 + 0.003
1.454 1,387 1.409 —0.022
2,116 I.329 1.329 000
2.57.1 1.313 I.305 + 0.008
2.842 1.313 1.298 +0.015
3.579 1.309 1.295 +0.014
3.732 1.2S5 1.295 —0.010
3.958 1.293 *1.294 —0.001
6.804 1.279
11.518 1.260
11,966 1.264
Sum. + 0.007
Mean. + 0.0009

The computed cóefficients in this table are taken from the second curve,
and as the observations were too few to group them for every half-foot per
second, they are given at the recorded velocities. These côefficients decrease
more beyond the curve than those of the other meters.
The cöefficients for the tachometer, hydrometric tube, etc., can be
obtained in a similar manner.
The Pitot's tube, constructed by M. Darcy was tested by all three
of the methods mentioned.
Compared with surface floats the coefficient was 1.006
Drawn through still water & 4 & 4 “ 1,034
In water of known velocity, “ & 4 “ ().993
The second is much too large, and this discrepancy was accounted for
Shy the fact that in these experiments, the tube was placed in front of a
boat which was drawn through a canal.”
The effect of the motion on the boat would be to raise the forward end,
and thus incline the tube to the direction of motion.
This would reduce the indications of the instrument a little, and thus
increase the côefficient. Other tubes however, have had a much smaller
cöefficient than this.
M. Baumgartent says: “Many persons have objected to the method of
obtaining the coefficient by drawing the meter through still water, * * *
and pretend that there would be a difference between this mode of obser-
vation, and that of noting the number of revolutions of the meter when
placed in a current of known velocity. I have undertaken a series of
experiments for ascertaining whether these doubts have any foundation”
-Recherches Hydrauliques. : Enterprises par M. H. Darcy et continués par M. H. Bazin, Paris,
1865, page 69.
tAnnales des Ponts et Chausses page 363.
10
He made two series of comparisons of the meter with surface floats, in
one of which the cöefficients were found to be a little larger than those
obtained by drawing through still water, and in the other a little smaller,
so that he concluded that these two methods gave practically the same
result.
The Telegraphic meters were also compared with surface floats. Table
TV gives the results of the comparison of meter No. 1, with floats in a
small canal at Ogdensburg.
The floats were run over a distance of 200 feet at the depth of three
feet, their time being taken by a chronometer, and the meter was placed at
the same depth, in the centre of the distance run, as nearly as possible on
the line of the passing floats.
TABLE IV.
Velocity of Current in Arithmetical
No. of Feet per Second. sum of Mean of Range of
observa- differences of differences. Differences.

single obser-
tions. - ti
Difference. VâûOIAS.
By Floats. By Meter.
24 I,992 1.980 |+ 0.012 2.314 0.096 + 0.219 to-0.162
6 1.876 1,916 || + 0.040 0.368 0.061 + 0.064 to—0.161.
6 1,476 1.434 — 0.042 | 0.443 0.074 +0.157 te—0.092
Here the differences between the mean velocities is small, while the
individual floats differ greatly from the meter velocities, the float only
giving the speed of the current at the moment of passing, while the meter
shows the mean velocity for the time the float was running over the whole
distance. Comparisons were also made between the Propeller meter and
floats, at the depth of one foot below the surface in St. Clair river.
The mean velocity given by 50 floats was - - * - 3.619
And that given by the meter was - - * - * - 3.655
Difference - - - †- - - - º- - 0.036
There was a light up stream wind at the time of observation, which
probably retarded the floats a little.
The method of observing these floats will be hereafter explained.
Navier gives a formula for the correction of the velocity obtained by
floats, assuming that the velocity of the water is uniform at the depth to
which the body is immersed:
2g MI #3
X=
m S
11
in which M-the displacement of the float,
S=the greatest immersed area,
g=the velocity due to gravity, *
m=a constant depending on the form of the float,
x=the correction required, which is always minus.
Mr. J. B. Francis found the constant m for floating rods to be 0.77,
nearly.”
The double float also requires correction, but its determination is much
more difficult than the others.
In 1867, I was ordered by the Superintendent of the U. S. Survey of
the North and Northwest Lakes, to make velocity measurements for the
determination of the outflow of the Lakes. The latest current measure-
ments were those made by Humphreys and Abbott on the Mississippi, and
moreover, as that was the largest river ever attempted to be gauged, I
adopted their methods of observation.
They used the double float, which had been previously tried by Mr.
Chas. Ellet, jr., on the same river.
Many kinds were tested, but the best was found to be an old paint keg
with the bottom knocked out for the lower float, and for the upper a tin
ellipsoid about 6 by 13, inches, bearing a small flag on a wire passing through
its center.
These floats were connected by a cord from one to two-tenths of an inch
in diameter, and of different lengths, according to the depths at which
the measurements were to be made.
It was assumed that the upper float would have no effect upon the lower,
but would merely serve as a surface guide to show its position.
Their method of observation was as follows: {
“Two parallel cross sections having been sounded out 200 feet apart, a
base line of the same length was laid off on the bank perpendicular to
both. An observer with a theodolite was stationed at each end of the line.
Two skiffs were stationed in the river, one considerably above the upper, and
the other below the lower section line, the former being provided with
several keg floats. At a signal from the Engineer at the upper station, a
float was placed in the river. The keg immediately sunk to the depth
allowed by the cord, and the whole float moved down towards the upper
line. The observer at the lower station followed its motion, keeping the
cross hairs of his telescope directed constantly upon the flag. At the word
‘mark,’ uttered by his companion when the float crossed the upper line, he
recorded the angle shown by his instrument, and then setting his telescope .
upon the lower line, watched for the arrival of the float. In the mean-
time the observer at the upper station, whose theodolite supported a watch
* Lowell Hydraulic experimments: By James B. Francis. New York, 1868; page 167.
# Report upon the Physics and Hydraulics of the Mississippi river, &c. Prepared by Capt. A. H.
Humphreys and Lt. H. L. Abbot, 1861.
f Report page 225.
12
with a large seconds hand, recorded the time of transit of the float across
the upper line, and followed the flag with his telescope. At the word
‘mark' given by his assistant, when the flag crossed the lower line, he
recorded the time and angular distance from the basé line. The float was
picked up by the lower boat.”
Where these observations were made, the river was from 2,000 to 4,000
feet wide, while the base was only 200 feet.
This base seems altogether too short ; for the exact location of a fiased
point is difficult when its distance is ten or more times the length of the
base, but when this point is moving from two to eight feet per second its
location must be very uncertain.
It would also seem to be impossible to read and record the angles and
time while a fast float was passing over so short a distance.
In the location of the soundings taken on these cross sections, a base of
from 400 to 1000 feet was used; and it seems strange that a shorter base
was accepted for the location of the little fast moving float, while the boat
could be held by its oars nearly stationary.
When this method was first tried on the Outflow of the Lakes, it was
found that in even this short distance there was a great difficulty in hearing
the call over the wash of the waves and the ordinary noises of a large river,
and also that the distance run by a fast float, between the time one observer
saw it passing the cross hairs of his telescope, and the other heard his call,
was quite appreciable.
The base line was therefore increased to one-third the width of the
rivers where measurements were made, and a wire from a battery run from
one station to the other, having a telegraphic relay or sounder and key in
the circuit, placed near each observer.
The assistant at the upper station was also provided with a chronometer.
Then when a float put out from the anchored boat approached the upper
section line, the observer commenced counting the beats of his chronometer
and called his assistant by a few rattles of his key. The latter turned his
theodolite upon the flag and followed it till a single touch of the key by his
companion signalled its passing the upper line, when he clamped his theod-
olite and read the angle; the observer at the upper station recording the
time of passing.
The float was signaled by the lower observer in the same manner when
it passed his station. -
Before the commencement of the second season's observation the Tele-
graphic meter was invented, and gave results which agreed well among
themselves, but quite different from the floats.
I3
A series of comparisons was made between the floats and the meter at
different depths, to obtain a correction for the work of the first season, or
the côefficient of the double floats. These comparisons were taken in the
St. Clair river where it was over fifty feet deep, the meter being anchored
opposite the middle of the base line, and as near as possible in the path of
the floats. The velocity of the current given by the floats and the meter
at different depths, with the resultant mean velocity and direction of the
wind parallel to the base line, are given in Table V.
TABLE V.
Wind parallel to Depth of Velocity of current, feet per sec.
No. of direction of the observation Corrected
Observations. river in miles below the difference,
per hour. | surface. By floats. By meter. Difference
50 3.26 up I ft. 3,619 | 3.655 — 0.036 0
62 1.92 “ 5 “ 3.759 3.782 –0.024 0.030
56 1.27 “ 10 “ -3.703 3,674 +0.029 0,070
50 0.48 “ | 15 “ 3.590 3,516 +0.074 0.120
54 0.29 down 20 “ 3.598 3.405 +0.193 0.170
31 0.53 up 25 “ 3.637 3.441 + 0.196 0.240
37 0.80 ** 30 “ 3.546 3.279 + 0.267 0.320
29 2.18 down 35 “ 3.556 3.166 +0.390 0.400
12 4.85 “ 40 “ 3.636 3.142 + 0.494 0.490
7 0.74 up | 45 “ 3.5 12 2,985 + 0.557 0.600

At the surface, and five feet below, the floats give a less velocity than
the meter ; a light wind was blowing up stream at the time, which would
probably retard the floats a little.
Below this point, the floats show a greater velocity than the meter, con-
stantly increasing towards the bottom. Making the difference at the
surface zero, a slightly curved line will pass through nearly all the
points of difference. The ordinates of this line are given in the last column
of the table headed “corrected difference;” the observed differences and
this curve being plotted in figure 6.
These comparisons, and other observations upon the practical working
of double floats, show the following to be the principal causes of error in
the measurement of sub-surface velocities by that method.
1ST. THE ERROR OF CROSS SECTION.
In order to ascertain the true mean area, it would be necessary to know
the exact depth of the river past the whole base line. Generally it is con-
sidered sufficient to sound out two or three lines across the river, and take
their mean depth. But when the bottom is not perfectly regular this
may differ considerably from the true mean depth. It is claimed that
one reason why so short a base was chosen in the Mississippi obser-
vations, was that the eddies and whirls are common, and irregula-
*
14
rities of the current, past a long base, vitiated the results; and Mr.
S. F. Abbert, who made some current observations with double floats
on the Arkansas River in 1869, shortened his base to 100 feet, be-
cause he found too much irregularity in 200. Then, if the base is
reduced to zero, all errors from this cause must disappear, which is
precisely what is done when meter observations are made.
However, on Gen. Abbert's suggestion, this matter was tested on
the St. Clair River. The base was divided into three nearly equal
parts, and floats were located and timed past each section.
It was found that, though individual floats varied greatly, the mean
velocity in each section was about the same as the mean velocity past
the whole base.
2D. THE PULSATION OF THE CURRENT.
This is, perhaps, the most curious phenomenon of flowing water.
All water in motion, from the jet of a toy fountain to the Gulf stream,
has an intermittent velocity, increasing and diminishing in accord-
ance with some yet undiscovered law. This fact has been known for
a long time; but without some such apparatus as the telegraphic me-
ter, the amount and duration of these pulsations, especially below the
surface, could not be measured. With this meter, however, by plac-
ing a Morse's paper register or, better, a chronograph in the circuit,
every revolution of the wheel can be recorded on the moving paper,
and thus every change in the velocity of the current noted.
Although these pulsations were found in every stream that was
tried, from the slow-moving St. Lawrence to the tail race of a mill,
no general law of variation has yet been deduced, and we can only
say that the lesser fluctuations are from half a minute to a minute in
duration, with larger ones every five or ten minutes.
They do not seem to be synchronous with the oscillations of the
surface level, which are also very irregular, and attain their maximum
fully as often during the increase as the decrease of the velocity.
Nor can M. Bazin's supposition be correct, that they are due to
eddies and whorls of the surface current, as they are much smaller
at the surface than toward the bottom. In fact, as near the bottom
as measurements could be made, the velocity was sometimes less than
half what it was at its maximum.
Capt. Boileau, speaking of the belief of Dubuat and other engi-
neers of the adherence of the fluid molecules in contact with the bot-
15
tom, says:* “But it would seem more rational, and more in accord-
ance with experiments, to consider the bottom and sides as forming
by their asperities eddies, which consume nearly all the work of the
motive forces, so as to leave the force of translation very weak. On
the other hand, we know that each eddy formed in a fluid current
causes, by the lateral communication of its movement, a second, of
greater size but with less force of rotation, and this gives birth to a
third, still more feeble, and so on until these gyratory movements be-
come insensible.”
The formation of ridges in the sand at the bottom of canals, as
shown by Dubuat's experiments, in precisely the same manner as the
sand dunes are formed by a pulsating wind, would seem to indicate
also the existence of such eddies at the bottom, which would of course
produce variations in the velocity of the current, decreasing towards
the surface, just as the observations show is the case with those pulsa-
tions.
But, whatever theory be accepted as to their cause, it is easy to
see that they must greatly affect the velocity obtained from floats,
particularly as we approach the bottom. j
Often, when two floats were put out from the upper boat, one min-
ute apart, the last one would gain upon and sometimes even pass the
first. As 150 floats is a good day's work, if we assume that each
float is one second in passing the centre foot of the base line where
the mean area is taken, the velocity will have been actually measured
for only 150 seconds during the day, and the mean would be greater
or less than the true mean velocity, according as the majority of the
floats had passed during the increase or decrease of the pulsations.
3D. THE UNCERTAINTY OF LOCATION.
Even when the base line was one-third the width of the river, and
the stations connected by telegraph, it was found almost impossible to
locate the exact point where the float crossed the section lines. As
the velocity is slower near the banks, the measured velocity would be
too small or too large, as the location was on one side or the other of
the true place of the float. Moreover, the floats rarely ran parallel
to each other, or in the same vertical plane.
Often, in still weather, when the upper boat was in the center of
* Traité de la mesure des eaux courantes. Par P. Boileau. Paris, 1854.
Page 339.
16
one of the 200 feet divisions (into which the rivers were divided by
imaginary lines for convenience in reduction), part of the floats would
run entirely out of the division, or more than 100 feet out of their
proper plane. The calculated path of the float was sometimes more
than half a foot in a hundred longer than the distance between the
section lines.
4TH. FLOATING BODIES Move FASTER THAN THE WATER IN WHICH
THEY ARE IMMERSED.
This error is very small compared with the others, and the formula
for its computation has already been given.
5TH. THE UPPER FLOAT DRAGS THE LOWER.
This error is also small, and depends upon the relative size of the
floats and the velocity of the current.
6TH. THE EFFECT OF THE CURRENT ON THE CONNECTING CORD.
The cord used in the Mississippi observations was from one to two-
tenths of an inch in diameter. It has a tendency to drag the lower
float; but, what is of more importance, it can never be perfectly
straight, but curves down stream, and therefore raises the lower float.
This brings the float into a faster current, and thus gives much too
large velocities, especially when the depth is considerable.
When starting the floats from the upper boat, both cannot be thrown
out together without danger of entangling the cord; but the upper
must be held until the lower has gone far enough to extend the cord.
The upper float must then move enough faster than the lower to as-
sume its proper position before they reach the upper section line,
otherwise it will pass the base with nearly the full velocity of the sur-
face current, instead of being retarded by the lower float. Whether
this actually takes place cannot be known; and the distance required
for the two floats to assume their proper relative positions will depend
on the length of the connecting cord and their velocity.
When the floats are in their proper relative position one cannot be
vertically over the other, but the cord must form a curve, whose chord
is the hypothenuse of a right-angled triangle, the perpendicular being
the actual depth of the lower float, and the base varying according to
the depth, velocity and relative size of the floats and the length and
size of the connecting cord.
Of these errors the first three are uncertain, and the last three are
17
always plus; but the sixth is the most important, as well as the most
difficult to eliminate. It is, however, evident that these errors must
increase with the depth, as the comparisons given in Table V show,
the lowest observation giving the error of floats over half a foot per
second. This correction would only apply to float observations in a
river of the same depth and velocity as the St. Clair. In the St.
Lawrence, with a current of only about one mile an hour, it is much
less. In the deep and swift-flowing Mississippi it must be much
greater. The discharge, computed from the meter observations, was
about ten per cent. less than that obtained from the floats; and we
may therefore call the co-efficient of double floats, when observed past
a long base, 0.9. When a shorter base is used, the errors of observa-
tion would probably somewhat decrease this co-efficient.
MAXIMUM VELOCITY.
About the year 1730 Pitot invented the tube which has since borne
his name, and made the first actual measurements of the subsurface
velocity of water in canals. Before that time many engineers held,
with Castelli, that the velocity of the molecules of water increased
directly as the depth, while others, following Guglielmini, believed
the increase was the square of the depth. None doubted, however,
that the maximum velocity was at the bottom, although Papin warned
his contemporaries against attempting to apply to fluids Galileo's laws
for the friction of solid bodies. .
Pitot's experiments proved that the velocity in canals increased
for a certain distance below the surface, and then decreased toward
the bottom. - - -
Since his time many observations have been made to determine the
law of variation in the velocity, with apparently very diverse results.
Most of these observations were made on small canals, and in them
the maximum velocity was found considerably below the surface;
while in the few European rivers and streams which have been gauged,
it was at or near the surface. The latest velocity observations on
canals are those made by MM. Darcy and Bazin; and, as they give
their observations in detail, we can readily find the locus of maximum
velocity. Table VI is compiled from their observations.” It shows
the kind of canal used, the nature of the bottom, the width and depth
of the water, the ratio of the width to the depth, and the distance
below the surface of the maximum velocity of the center vertical.
* Récherches Hydrauliques, page 187 et seq. ,
18
TABLE WI. º
Ratio Depth of
Form of Character Of Mean maximum
Series. ** Width. Depth. Dept Velocity
Canal. of Bottom. to Velocity. below
| Width. surface.
|Meters. Meters. Meters.
59—1 ||Rectangular Plank 1988 || 0-084| 0-04 1-20'ſ
62–1 ( & Strips 0.01 m. apart 1.988 || 0:199| 0-05 0-961
59–2 t {{ 1-994 0-134 || 0-07 1.573
65—l * { “ 0 05 m. apart 1.992 || 0-135 0-07 0-756
58—1 { { Plank 1.998 || 0-138|| 0-07 0-730
63—1 { { “ 0.01 m. apart “ 0° 144|| 0 07 1.454
62–2 ( ! { { t; 0- 1.58|| 0 08 1.336
61—1 { { {{ é à 0- 160|| 0-08 0.643
60–1 || “ Plank 1 994 0-180 0-09 2.297
59– 3 § { {{ t is 0-20} 0-10 2 501
65–2 { { “ 0-05 m. apart 1.992 || 0-207 0-10 1 - 000
58—2 ( : Plank I •998 || 0-2 15 0 - || || 0 95.3
63—2 ( ſ. {{ 0-01 m. apart * { 0-2 18; 0-11 1.925
61—2 {{ Plank {{ 0-244 0-12 0-854
62–3 i ( {{ 0-01 m. apart * { 0-248 0-12 1-702
68—1 Trapezoidal Plank 1-500 || 0 191 ()- 13 0-908
69—1 { { { { 1-048 0-133| 0-13 0.406
59–4 ||Rectangular {{ 1.984 0.265 (). 13 2-3 18
67—1 | 6 “ 0.05 m. apart| 0-80% 0-110 0-14 1 - 084
63—3 * { “ 0.01 m. apart 1988 0.286 0 14 2- 199
66–1 { { t{ 0-05 m. apart 1.994 || 0-288; 0-14 1 °464
55— 1 {{ Cement 1.812 || 0-269; 0-15 2-509
65–3 { { & 4 0.05 m. apart 1.992 || 0.322 0-16 1 - 295
62—4 { { “ 0.01 m. apart 1-998 || 0-320 0 16 1.979
58–3 { { Plank {{ 0-332 (). 17 1-248
61—3 { { “ 0.01 m. apart. “ 0.377 0-19 1 - 109
66–2 { { “ 0.05 m. apart 1994 0-380 0-19 I-675
68–2 |Trapezoidal Plank 1-580 0-300 0-19 1- 188
69–2 { { § { I - 148} 0-22 || 0-20 1-77 1
58–4 ||Rectangular {{ 1.988; 0.436|| 0 22 1-429 0-030
65—4 { { “ 0.05 m. apart 1992| 0:412. 0-22 I-511 0-130
69—3 Trapezoidal Plank 1 208; 0.275 0-23 l-992
68–3 { { { { 1-800 0.433 0-24 1-329
64–1 ||Rectangular “ 0 01 m. apart 1988 0.487| 0:24 0.856 0- 157
56–1 { { Gravel 1-832| 0-394 0-24 1-7 l 4
57–1 & Small stones 1-860; 0-452| 0-25 l-471 0-050
61—4 { { “ 0 01 m. apart 1988; 0.495 0-25 1-267 0- 155
73—l Semicircular Plank 1-100 0-270ſ 0-25 0-966
71—l {{ Cement 1-000 0-268|| 0-27 | 1.052
69—4 Trapezoidal Plank 1-28 li 0-342|| 0-27 2. 156
68–4 t{ {{ l'984 0-540|| 0-27 1 - 497 0-190
72–1 |Semicircular Fine Sand 1.000 || 0-292 0.29 0-954
69–5 |Trapezoidal Plank l:348i 0-393 || 0-30 2-28 l
73–2 |Semicircular {{ l-260 0-3.77|| 0-30 1-230
74—1 {{ Small stones 1-080 0-322 O-31 0.825
69–6 Trapezoidal Plank l-398 0-430 0-31 2-385 0- 130
64–2 ||Rectangular “ 0-01 m. apart 1.988; 0.660 0-33 0-948 0-230
71–2 Semicircular Cement 1 - 160|| 0-378; 0.33 l:300
72–2 {{ Fine sand 1 - 165 0-388 0-33 I •266
71–3 { { Cement l" 180|| 0:456) 0-39 l-534
74—2 {{ Gravel 1 - 170 0 458 0-39 1-028 0-120
73—3 ( & Plank 1 "300|| 0-554 O-41 1-450 0- 1 1()
72—3 {{ Fine sand l" 190|| 0-488| 0 - 41 1-392
71–4 {{ Cement 1-200 || 0-528 0-44 1.676
19
º
TABLE VI.-Continued.
Ratio Depth of
Form of Character Of Mean maximum
Series. Width. Depth. Depth Velocity
Canal. of Bottom. to Vecocity. below
Width. surface.
Meters Meters. Meters.
72—4 Semicircular Fine sand 1 - 2 l 0 || 0 °554 || 0-46 1.569
74–3 4 * Gravel 1-200 ( - 567 || 0:47 1 - || 62 0 - 160
70—l Triangular Plank O-800 0-380 || 0-47 | 1.406
71—5 Semicircular Cement 1 - 225 0.588 0-48 || 1.782
70–5 ||Rectangular Plank 1,400 0.686 || 0 49 2.218 (). 130
70—3 { { {{ J - 160 || 0-57 0 || 0-49 | -92.2 0-130
70–2 Triangular { { 0 980 || 0 - 480 || 0-49 1-719 0-130
70—6 ( [. { { 1-480 0.735 | 0:49 2 294 0-130
71–2 iSemicircular Cement 1-250 0.625 || 0 50 1-786 0 1 00
73—4 { { Plank 1-400 0-704 || 0-50 1 - 162 (). 030
74—4 { { Small stones 1 - 220 || 0-610 || 0 , 50 1 - 229 0-200
70—4 Triangular Plank 1 - 260 0-630 || 0-50 2 084 (). 130
71—8 |Semicircular Cement 1 - 250 0-632 || 0 51 1-8 l 0
71–7 {{ { { { { 0 662 0.53 1. 786
72–5 {{ Fine sand ( e. { { 0.53 | 679 0-050
67–3 ||Rectangular Plank 0-800 0.486 || 0-61 1-86() (). I 56
SMALL FEEDERS IN MASONRY.
7 5–1 |Trapezoidal Masonry 1 - 201| 0-150 0-13 I '824
7 5–2 { { {{ 1-203 0-234|| 0 - 19 2 352
7 5–3 { { { { l. 204 0.29 l 0.24 || 2:560
75—4 { { { { 2.750 0.464 () 29 2.707
}* Irregular { { - Pº Pº * ... O ſº fly
76—1 Fºia 2-750 0 464 0-1 6 0.266
76—2 { { { { 3-000; 0-623| 0:21 0.391 || 0.303
76–3 t ( {{ 3-200 0-746; 0.23 (J-461 0-453
76–4 { { { { 3. 400 0.835 | 0-25 (): 513 0.400
77–1 t t { { 1:996' 0" 663| 0-33 O - 253 0-203
77—2 ( ! ( { 2 : 060 0-853 || 0:42 0 - 4 13 0-500
77—3 {{ ( ſ. 2-080. 0.987 || 0:47 0-501 || 0-500
77–4 { { { { 2. 100 1-135 0-54 0-638 || 0-650
f t
* The descent of the maximum velocity in these observations does
not seem to depend on the velocity, but upon the ratio of the depth
to width, and somewhat upon the character of the bed. Thus, in
rectangular canals the maximum of the center vertical remains at the
surface until the depth is about one-fourth the width; but when the
depth is half the width it descends to about one-third the depth below
the surface.
In triangular canals, on the other hand, the surface velocity is the
greatest until the depth is about one-third the width ; and when this
ratio is one-half, the maximum is only about one-fifth the depth below
the surface.
In the trapezoidal and semicircular canals, the portion of the sides
20
enclosing the water is slanting until its depth is about one-half the
actual depth of the canal; then it becomes vertical, or nearly so; so
that below that point the effect is the same as in the triangular, and
above it, the same as in the rectangular canals.
The nature of the bed seems to have an influence upon the descent
of the maximum velocity, for when the lining of the canal was rough
the table shows it to be somewhat deeper than when the lining was
smooth.
This table only gives the locus of the maximum velocity of the
center vertical. In the rectangular canals it is often found at mid
depth near the sides, even when the ratio of depth to width is small
enough to carry it to the surface on the center vertical. Thus, lines
drawn through the points of equal velocity follow nearly parallel to
the bottom and sides of the retangular canals, until about mid depth,
when they bend inwards toward the center, as if forced off from the
upper portion of the vertical sides.
In the canals of other forms this effect is not so marked, and in the
triangular is scarcely noticeable.
M. Bazin remarks that the friction of the air has certainly much
less influence in retarding the upper layers of the water than this
curious and unexplained influence of the sides.
Capt. Boileau, in discussing the effect of the wind upon the surface
of the water in experimental canals, says:* “In the geometric lines
representing M. Hennocque's observations, when a strong down stream
wind was blowing, and also in two of those of M. De Fontaine, it is
very remarkable that those lines, after being curved to a certain point,
as if to give the maximum velocity below the surface, are inflected in
the contrary direction under the influence of the wind, so as to bring
the maximum to the surface.” Thus, this effect of the sides is appa-
rent even when a strong wind is blowing with the current, and car-
rying the maximum to the surface.
In other observations the data are seldom given with as much de-
tail as those of Darcy and Bazin. But such as are at present acces-
sible are as follows:
Dubuat. An experimental canal, about one foot deep, gave the
maximum velocity from # to § below the surface.
Boileau. In an experimental canal, about two feet wide and on
foot deep, it was from 4 to # the depth below the surface. 3.
Hennocque. On an arm of the Rhine, where the depth was about
* Mesure des eaux courantes, page 311.
21
eight feet, the maximum was about one-fifth the depth below the sur-
face.
Baumgarten, on the canal du Rhone au Rhin, where it was about
45 feet wide, found the maximum below the surface, except for about
three feet in the middle of the stream.
On the Garonne, Baumgarten observed the maximum to be gene-
rally at the surface; but in one section it was below for a portion of
the river about 325 feet wide near the middle.
De Fontaine found the maximum on the Rhine, where it was about
five feet deep, to be at the surface.
Racourt, in his velocity measurements on the Neva, found the max-
imum at the surface. The river at one point was 900 feet wide and
63 feet deep.
All these observations confirm the law deduced from the Darcy and
Bazin observations, that when water is flowing in a channel with ver-
tical sides, the maximum velocity will be found below the surface,
when the depth is greater than one-fourth the width; and when the
depth is less in proportion to the width, it will be at or near the sur-
face; also if the sides of the channel are sloping, the depth may be
one-third or one-half the width before the maximum leaves the Sur-
face.
In the Mississippi, the new theories of flowing water place the
maximum velocity three-tenths of the depth below the surface.
As these observations are the only ones which do not conform to
the general law, it will be well to examine them; particularly as the
authors say:* “The method of conducting the field work and of com-
puting the results is given in great detail, in order that it may be seen
what degree of confidence the conclusions hereafter to be drawn from
this material are entitled to.” - ---
On page 230 is the following: “As floats are compelled to pass
through nearly the same paths when starting from a fixed station, and
are consequently unaffected by the change in velocity due to the dif-
ference in distance from the banks, the principle was adopted of de-
pending entirely upon the elaborated sets of observations from
anchored boats. All the observations in each set being thus confined
to nearly the same vertical plane, one great cause of error was prac-
tically eliminated. From the position of the boat, found by triangu-
lation, the recorded gauge reading, and the known depths of the dif-
e
* Humphrey and Abbott's Report, page 221.
22
ferent parts of the river section, the depth of water in each vertical
place of passage was readily determined.”
It appears from this that the method of observing floats, so elabo-
rately described on page 225,” was not used at all, at least for the
determination of the velocities on which the formulae are based;
(although they say “that system was adopted for the observations
both of 1851 and 1858,”) but merely the time of passage of floats
between the two stations was observed.
This, of course, greatly simplified the work; but they were obliged
to assume that “floats starting from a fixed station are compelled to
pass through nearly the same plane.” How erroneous this assump-
tion is has already been shown, and, therefore, in place of eliminat-
ing any of the errors of float measurement, it merely adds another to
that list.
“It was evident that some combination of curves was necessary to
reconcile discrepancies of observations. The first method adopted
was to combine all curves where neither the depth of water nor velo-
city of river varied materially. * * The resulting mean curves
are exhibited on figure XI, the numbers being shown in the following
tables.”f Let us first examine these tables. Here are recorded 2170
floats, of which 233 are interpolations, 1937 floats being observed.
On page 246 other similar tables are given, which were used for
testing the formulae deduced from the first set. These contain 447
floats, of which 109 are interpolated, making in all 2275 floats used
in the computations.
In Appendix D, the velocities as determined at all of the stations
on the river are given, and the number of floats there recorded ex-
ceeds 36000.
During the first year at Carrolton there were nearly 6500 floats
passed, noted as at all depths; and though the remainder are given
as at or near the surface, yet, as in these tables observations at seve-
ral other places are recorded, we cannot be far wrong in assuming
that there were at least 10,000 floats passed at all depths; especially
as at Carrolton alone the work lasted over a year, and this amount
would give an average of only fifty floats a day for 200 working days.
Whether, if more than 2275 of these floats had been taken, the re-
sulting curves and formulae would have been changed, cannot be told;
* Humphrey and Abbot's Report, page 225.
f Ante, page 230. .
23
but probably these were considered to be the only reliable observa-
tions. e *
These floats give the velocity at 297 points on 69 verticals; and
the means of the nine tables or groups in which they are arranged
show the maximum velocity to be generally much below the surface,
the resulting curves having a decided flexure toward the surface, as
well as toward the bottom. *
In Table VII these verticals are taken separately, the depth of the
maximum velocity and the number of floats passed at that depth be-
ing given for each vertical.
The number of verticals in the first set of tables or series is 39,
and in 28 of them the maximum velocity is at the surface or one-
tenth of the depth below; in the second series, nearly half the ver-
ticals give the maximum near the bottom. •
TABLE VII.
First Series. | Second Series. Total.
Depth.
No. of No. of | No. of No. of | No. of No. of
Werticals|Floats |Verticals|Floats.|Verticals|Floats
0 12 63 8 17 20 80
‘l 1-10th 11 46 l 2 12 48
2-10ths 5 33 5 33
3-10ths 3. 20 l 3 5 23
4-10ths 4. 9 4 9
5-10ths 2 22 l 3 3 25
6-10ths 5 30 1 3 6 33
7-10ths 6 18 6 18
8–10ths 6 15 6 15
9-10ths 2 5 2 5
Bottom l 8 1 8
Sums 39 222 30 75 69 297


But, if we throw out those verticals where the maximum is at mid-
depth and below, which must certainly be erroneous, there remain in
the first series 31 verticals, of which 23, or (#) three-fourths have the
maximum at the surface or one-tenth the depth below ; and in the
second series 14 verticals, 9 of which—two-thirds—show the maximum
at or near the surface.
One of the verticals in the first series gives the maximum at the
bottom, or rather one foot below the bottom, as it is recorded.
Thus, nearly all of these selected observations follow the general
24
law deduced from the European observations; though the means of
the groups in which they are arranged do not.
In 1869, very careful observations were made on the St. Clair and
Niagara rivers, to determine, if possible, the law of change in the ve-
locities near the surface and bottom. In the measurement of the
velocities near the surface, the meter was suspended off the side of
the boat, on a pole long enough to remove it from the retarding influ-
ence of the boat. The means of the observations near the surface
are given in Table VIII. -
These velocities are plotted in Figures 7 and 8, being represented
by the circumscribed dots.
A free hand curve, shown by the full line, was drawn through as
many of the points as possible; and the ordinates of this curve are
given in the table.
TABLE VIII.
Depth below St. Clair River. Niagara River.
the Surface. Observed Ordinates Observed Ordinates
Velocities. of Curve. Velocities. of Curve.
Surface 0 0
l 3-607 0.025 2-924 0-0 15
2 3-651 0-040 2-943 0.035
3 3-651 0.055 2.965 0.045
4 3.662 0.065 2.956 0-057
5 3-689 0.075 2.985 . 0-070
6 3-676 0-070 ſ
T 3:672 0.065 : i
8 3-644 0.055 i
9 3-646 || 0-035 | i
10 3. 601 0-020 2.989 —0.005
No. of Verticals 47 45
i i | i
In these rivers, as the table shows, the maximum velocity is about
five feet below the surface, though the difference between it and the
surface velocity is quite small. The mean surface velocity in these
rivers was nearly three miles an hour; but in the St. Lawrence,
where the current is only about one mile an hour, the surface velocity
was often considerably greater than that five feet below; although,
unfortunately, too few observations of the surface velocity were made
to give it exactly. Thus, these great rivers connecting the lakes agree
with the general law heretofore given; the slight retardation at the
surface being probably due to the friction of the air. Therefore, in
25
large rivers, we may consider the surface velocity in a calm time to be
nearly the same as the maximum; while in narrow canals, especially
when their sides are vertical, it may be considerably less. -
One curious thing in regard to this descent of the maximum velocity
was discovered by MM. Darcy and Bazin in their experiments.” This
is shown in Table IX.
The covered canal was a tube, 0.80 m. wide and 0.50 m. high in
the first case, and 0.48 m. wide and 0.30 m. high in the second. In
both tubes the maximum velocity was found at the center. The
cover was then removed from the tubes, and the water allowed to run
through with a depth as nearly as possible equal to one-half the height
of the tube.
Of these experiments M. Bazin says: “The half of the discharge
of the first tube is 0.309 m. The open canal only discharges 0.307 m. ;
but it will be noticed that the surface of the current did not quite rise
to the center of the tube, being 0.042 m. below it. If we take this
into account, we shall find, all the proportions being the same, that the
discharge of the canal slightly exceeds that of the tube.”
TABLE IX.
Canal 0-80 m. wide! Canal 0-48 m. wide
Covered. Open. Covered. Open.
Meters. | Meters. | Meters. | Meters.
Height or depth of water 0.50 0-2458 || 0-30 0-1513
Inclination.................. 0-00427 | 0-00430 0.00627 | 0-00600
Discharge.................... 0-618 0.307 0.191 0.093
Maximum velocity......... 1-826 1.859 1 634 1.6 li;
Mean: do. ......... I-545 1-561 1.326 1.282
Maximum below surface
on center vertical...... 3 depth # depth 3 depth 0 -
Do. do. side vertical...... # depth # depth
And of the second set of experiments he says: “The half of the
discharge of the tube is 0.0955 m., or a little greater than that of the
canal. The surface of the water in the latter rises above the axis of
the tube 0-0018 m. Taking this into account, and also the difference
in the inclination, we shall find an excess in the discharge of the tube,
very small, it is true, but a result precisely the opposite of that before
obtained. * * The comparison of the preceding results appears to
demonstrate that in a calm time the resistance of the air has very
little influence on the discharge.”
* Récherches Hydrauliques, pages 176, 177.
26
But, notwithstanding the very small difference between the dis-
charge of the tube and the canal, the maximum velocity in the first
experiment descends from the axis of the tube to one-sixth the depth
below the surface on the center vertical, and to one-half the depth on
the verticals near the side.
M. Racourt measured the velocity of the Neva where it was cov-
ered with ice, thus forming a tube 900 feet wide and 63 feet deep.
The observations were made by cutting holes in the ice, and letting
down a ship's log.
The following were the measurements on the deepest vertical:
Velocity near the surface, . g . 191 feet per second.
& & a little below the center, . . 2:58 {{
& 4 near the bottom, . * . 1-67 £ 4
The maximum was found a little below the center of the deepest
vertical; while in summer—when the cover was removed—it rose to
the surface. These experiments confirm the conclusions drawn from
the observations on small canals, that the locus of the maximum velo-
city is due in part only to the resistance of the air, but mainly to the
form of the channel in which the water is flowing.
27
BOTTOM VELOCITY.
Capt. Boileau says:* “The observations should be continued to a
short distance from the bottom, for in that region the velocities de-
crease rapidly.” And again:f “Returning to the observations made
by Dubuat upon the movement by translation of grains of sand which
follow the bottom of the canal, let us compare these movements with
that of the liquid at the same depths. According to that engineer's
estimates, the velocity of translation was only about 0.003 of an inch
(0:00008 m.) per second in a current where the lower layer of water
moved at about one foot (0.325 m. per second, a result which at first
appears inexplicable; for, even admitting a notable error in the meas-
urement, these quantities still differ enormously. However, remark-
ing, in the first place, that Dubuat measured the bottom velocities in
his experimental canal, by means of small spheres 0.24 inch (0.006 m.)
or less in diameter, while the grains of sand were not over one-tenth
this size, it seems that the molecules of liquid very near the bottom
have a very slight movement, and that in the small inferior zone or
layer the velocities increase very rapidly to a certain height, where
this increase follows the laws discovered by means of the hydro-
metrical instruments;
“This circumstance seems to confirm the opinion of Dubuat and
the greater part of hydraulic engineers, who hold that the fluid mole-
cules in immediate contact with the bottom of the canals are made
immovable by their adherence.” Then follow his remarks (already
quoted;) on the formation of eddies.
To obtain actual measurements of the bottom velocity seems almost
impossible.
In small canals the distance between the lowest measurement and
the bottom affords ample space for great changes in the velocity,
as Dubuat's experiments show; and in deep rivers it is difficult to get
a meter close to the bottom, or rather to know exactly how mear the
bottom it is ; and, as the pulsations of the current are so great in
that region, the velocity will sometimes be so slow that the meter will
stop for a time, and thus make the mean too large. Floats, as we
have seen, are entirely unreliable for such observations, for the exact
depth of the lower float can never be known, and they invariably give
too high velocities.
Notwithstanding the experiments noted above, Dubuat gives a for-
* Mesure des eaux courantes, page 302. i Idem, page 339. † Ante.
28
mula for the bottom velocity which makes it a little more than half
that at the surface. He considered also that the bottom velocity was
the same for like surface velocities, whatever the depth of the stream,
and that it was not affected by the nature of the bed. *
Darcy and Bazin have shown that the nature of the bed materially
affects the velocity, a different constant being used in their formula
for beds of plank, masonry, cement, earth, &c., and the later observa-
tions seem to show that the bottom velocity changes with the depth
also.
The bottom velocities given by Dubuat's and other formulae, and
found in all hydraulic tables, can only be considered as the lowest
measured velocity. Dubuat, Blackwell and others have measured the
velocity of currents capable of moving different substances, which
may be condensed as follows:
16 feet a minute will start white clay.
40 “ & 4 “ move along coarse sand.
60 “ {{ { { § { fine gravel.
120 “ & 4 & & ( { rounded pebbles 1 inch in diameter.
200 “ & 4 § { “ flint stones size of hens' eggs.
Beardman's tables, which are computed by Dubuat's formula, give
a bottom velocity of 2.92 feet per second fore a surface velocity of
four feet per second, or about three miles an hour. This, according
to the above table, would move along pebbles over an inch in diam-
eter; while we know from experience that rivers of a much greater
surface velocity run over beds composed of soft sand or finely com-
minuted clay, with but little change in their cross section, through
quite long periods of time. In a small clear stream, I have watched
the gathering of small particles of vegetable or earthy matter, whose
specific gravity was so nearly that of the water that they would just
sink upon the small stones at the bottom, but if they were disturbed
so as to rise out of the lower layers of water, they would be imme-
diately carried off by the current.
At times, of course, when the conditions are favorable, the scouring
of rivers is very great, but ordinarily they appear to have but little
action on the bed.
In the report of Mr. Chas. Ellet, Jr., on the “Inundations of the
Mississippi,” is the following:* “To excavate a channel through soil
of a given texture, and to keep the same channel open when so exca-
wated, are two very distinct things, requiring very different applica-
* Ex. Doc., 20, 1852, page 67.
29
tions of force. We find, consequently, that it is no easy thing, even
with a great fall and a great volume, to open a new channel by the
mere action of the running water of the Mississippi. The first at-
tempt to make a cut off at Racourci, where the fall was at the rate of
six feet to the mile, were unsuccessful, although a considerable volume
of water was let through an artificial trench leading from the river
above to the river below the bend. Various other attempts to create
cut-offs across bends in the upper portions of the river have likewise
been unsuccessful, although sometimes aided by a descent of 7 or 8
feet per mile. $
“The Atchafalaya and the Plaquemine have probably been open
for ages—certainly from periods far beyond the reach of history or
tradition—the first having a fall more than twice as great, and the
other a fall ten times as great as the Mississippi itself; and yet, un-
aided by art, they have been found unequal to the task of increasing
the depth of their channels, or enlarging their respective waterways.
On the contrary, the Atchafalaya, in the view of the writer, seems to
have been contracting its original width for a great many years.
“The crevasse at Bonnet Carré discharged into Lake Ponchartrain
about the one-tenth part of the high-water burden of the Mississippi,
for many consecutive days, during the great flood of 1850, when the
water of overflow rushed down a plane descending about fifteen feet
in 4% miles; and yet the velocity and force of the torrent were not
sufficient to tear up the natural soil to any considerable extent. No
channel was excavated. The furrows left by the plow and the roots
of the crop remained on the field where it had been swept by the
water, after the flood had subsided.”
There were also two attempts to make artificial cut-offs during the
Rebellion, which were unsuccessful.
A curious story was told by the divers employed in the construc-
tion of the St. Louis bridge across the Mississippi. It must be taken
for what it is worth ; yet it seems hardly possible that it could have
been an invention of their own :
During the sinking of those piers, large scows were moored along.
side ; and the divers reported that the bottom of the river underneath
the scows was scooped out just about the amount they extended be.
neath the surface. This would seem to show that while the river was
in its normal condition—the discharge and cross section being in
equilibrium—the sand and silt of the bottom was undisturbed; but an
obstruction being placed in the stream which reduced the cross sec-
30
tion, the velocity at the bottom was increased till the equilibrium was
restored.
If an obstruction be placed in the bottom of a river, thus lessening
the cross-section, the surface level will be raised; and if an obstruc-
tion be placed near the surface, it would seem as if the bottom or
sides must yield.
These are, of course, only conjectures, but they all seem to indicate
that the real bottom velocity in rivers must be very small.
In the Mississippi observations, the bottom velocity exceeds that of
the surface in 28 out of the 69 selected verticals, in some cases being
more than one foot a second in excess, and in one vertical, as has
been already mentioned, the maximum velocity is recorded at one
foot below the bottom. ef
It was found impossible to obtain satisfactory results in the rivers
connecting the lakes below. about three feet from the bottom, both
from the uncertainty of position, and the intermittent velocity, as
before noticed.
The means of the measurements from that depth to ten feet from
the bottom are given in Table X, and are also plotted in figures 7 and
8, being represented by the circumscribed dots.
A free hand curve, shown by the full line, is drawn through as
many of these points as possible, and continued till it meets the bot-
tom; the ordinates of this curve being also given in the table :
TABLE X.
Distance St. Clair's River. Niagara River.
from
the Bottom. Observed Ordinates Observed Ordinates
Velocities of Curve. Velocities. of Curve.
10 3-053
9 2.988 0-060 0-040
8 2-935 0- 1 15 2- 124 0.095
7 2-870 0 - 180 2 034 0-175 ;
6 2.782 0-250 I-975 (). 260
| j 2.734 0 330 1-889 0 360
4 2-624 0-440 1-710 0-495
3 | 2.437 0-590 1-536 0.660
2 0.790 0-870
l | 1.050 1 - 120
|Bottom............ | l' 550 1-500
|No. of Verticals 33 20

These curves show a very rapid decrease in the velocity towards
the bottom, and thus agree with Dubuat's experiments on the trans-
31
lation of grains of sand; but the bottom velocities are still very large
compared with the slow motion of those particles, being nearly a foot
and a half a second in the St. Clair and seven-tenths of a foot in the
Niagara.
According to the table on page - the former velocity ought to
carry along coarse gravel; yet, while the bed of the St. Clair is com-
posed of clay so soft that a sounding lead sinks into it a foot or more,
the form of the bed has not changed materially in the past ten years.
The mean surface velocities corresponding to these bottom veloci-
ties are, in the St. Clair, 3-6 feet per second, and 2-9 in the Niagara.
The depth of the former river is over fifty feet, and of the latter
about 70 feet.
This shows, as far as a single case can, that the bottom velocity is
dependent on the depth as well as on the surface velocity, and the
character of the bed; which seems much more rational than that it
should be a function of the surface velocity alone.
In an article in “Nature,” by Mr. S. Login, upon the abrading and
transporting power of water, the following “Practical Conclusions”
are arrived at:
“1st. That all particles of water have an affinity to each other, as
to other bodies, and that force is required to separate them.
“2d. That friction sets these particles rotating in all directions, in
larger or smaller circles, and that the friction or force increases in
some proportion to the area of surface exposed.
“3d. That this rolling motion becomes rarer the larger the diameter
of the circles may be ; that is, the resistance decreases as the depth
and breadth of the stream increase; or, in other words, the velocity.
increases proportionally to the “hydraulic mean depth.’
“4th. Lastly, that any increase to the rapidity of this rotatory
motion must increase the abrading and transporting power of water,
by enabling it to remove from the channel of a stream grains of solid
matter, and hold them in suspension.”
And he quotes from a series of papers in “The Artisan,” the fol-
lowing “Supposed Law:”
“The abrading and transporting power of water increases in some
proportion as the velocity increases, but decreases as the depth in-
creases.”
This idea of the rolling motion of water caused by friction is the
same as that of Captain Boileau of the eddies, which he thought were
generated by the adhesion of the water to the bottom of the stream;
32
and if, as seems to be true, the resistance, and therefore the abrading
power of water decreases as the depth increases, the bottom velocity
must also decrease even should the surface velocity be the same.
ForM OF VELOCITY CURV Es.
Almost every writer on the subject of hydraulics adopts a new form
of velocity curve.
Woltmann found that a reversed parabola, with its vertex below the
bottom of the river, agreed best with his observations on the Rhine.
Defontaine, also on the Rhine, found the greatest velocity at the
surface; that it decreased slowly at first, and then more rapidly
towards the bottom, and he considered that this decrease approxi-
mated to two right lines of different inclinations, intersecting at a
point probably below mid-depth.
Racourt, in his current measurements on the Neva, found the de-
crease in the vertical velocity was best expressed by an ellipse whose
minor axis was a little below the surface. He also found that the
same curve would agree closely with the increase of the velocities in
a horizontal plane from the banks to the center of the stream. .
Funk adopted a logarithmic curve in his observations on the Weser.
Boileau found a parabola, with its axis at the surface, agreed best
with his observations on experimental canals.
Darcy and Bazin found the reversed parabola approximated most
closely to the velocities they obtained, but the parameter changed
with the character of the bed.
Humphreys and Abbot, in the Mississippi observations, adopted a
parabola with its axis three-tenths of the depth below the surface.
M. Baumgarten says: “In constructing the curve of different
velocities corresponding to each point of vertical height of a sound-
ing, we can assure ourselves that the engineers who have maintained
that the curves were ellipses, or right lines with a slight inclination
to each other, might all have been more or less nearly right (pouvaient
tous avoir plus ou moins raison), according to their observations; but
I believe that in reality no simple curve gives a rigorous expression
of the truth, and that they are all merely approximations, even as
ellipses are but approximations to the orbits really described by the
planets.” &
In the survey for the outflow of the lakes the method adopted for
obtaining the vertical curve of velocity was to combine all the obser-
vations or verticals whose depth did not differ more than three feet.
33.
Four of these combined verticals for each of the rivers of St. Clair,
Niagara and St. Lawrence are given in Table XI.
The velocities are a mean of all the observations at the places and
depths noted.
In the St. Clair and N iagara the surface and bottom velocities are
taken from the curves given in Tables IX and X.
TABLE XI.
ST. CLAIR RIWER.
Depth. 54.6 feet. 45.4 feet. 42-5 feet. 37-0 feet.
4034 3-86s
5 ft. | 4° 109 3-907
3-958 3 409
4°013 3. 484
10 4,043 3-821 3-900 3-379
15 3 958 3.709 3-798 3-223
20 3-887 3.608 3 663 3.050
25 3-778 i 3-496 3-50 l ; 2 784
;
|
30 3-624 || 3:309 3.228 2.529
35 3-503 || 3-100 2 904 || 2:379
40 ; 3.323 || 2:678 2-600 1-159
45 : 2.986 || 1:428 1:250 .
50 2.650 :
55 ; 1.472 º |
i :
No. of Werticals 1 l 32 18 14
|
TABLE XI—Continued.
NIAGARA RIVER,
! | t
Depth. 69.9 feet. 36.1 feet. 532 feet. 441 feet.
3 169 is 125 || 3:376 || 2:577
5 ft. 3 239 3.195 : 3-446 2-64'ſ
- 10 3-279 3.184 3-347 2-601
15 3-261 3 081 | 3, 17 1 2-499
20 3. 154 3 085 3-209 2-291
25 3.124 2.858 || 3-108 2-225
30 3.083 2.815 2-957 2-153
35 3.037 2.765 2-809 1:996
40 2.842 2,594 2.499 1-670
- 45 2.760 2-371 2-321 ()-652
. 50 2.65 1 l-877 •l-'766
55 2.333 1.252 0-931
60 2.014 0-807
($5 1-654
70 0.552
i t | 3. |
iNo. of Verticals|| 8 || 15 31 10
:
“Annales des Ponts et Chausses, tome xx, 1847, page 362;
34
Tannexi-Continued
ST. LAWRENCE RIVER,
Depth. 83-0 feet. 65.2 feet.|46. 1 feet.[42.5 feet.
5 ft. 1.638 1.533 1°332 1 - 176
10 1.609 1 -509 1.320 1 - 194
15 1-596 } -507 1-369 1-143
20 1 - 602 1-470 1-292 1-075
25 1-569 1 - 477 I 280 1 - 0.65
| 30 ° } • 549 1-4 17 l'214 0-969
- 35 1-576 1 -398 1 - 153 0-874
l 40 1 : 521 1.357 1.007 --
| 45 1-486 1-3 15
| 50 1 483 1-236
55 1-500 1. 167 |
60 1-504 || 1 - 092 |
65 1 385
70 1-289
75 1-180
i 80 1-202
85
No. of Werticals 4 66 14 5
| -
As these curves are not applicable to the St. Lawrence river, those
velocities are omitted. -
These velocities are plotted in figures 9, 10 and 11, being repre-
sented by the circumscribed dots.
In the deeper verticals, the velocities near the surface plot nearly
in a perpendicular line, while in those of less depth the velocities in-
cline more from the surface. It was found that an ellipse whose mi-
nor axis was at or near the surface, with the vertex a little below the
deepest vertical, would pass through nearly all the plotted points of
that vertical, and by raising the minor axis above the surface would
also pass through the plotted points of the other verticals at the same
station, except in those very near the banks, the velocities there de-
creasing too rapidly from the surface. This ellipse is shown by the
full line in the figures, being the same curve for the St. Clair and
Niagara, but more eccentric for the slow and deep current of the St.
Lawrence.
In figure 12 the velocities across the river St. Clair, at different
depths, are plotted. These were obtained by taking the mean of all
the observations every fifty feet across the river at the several depths.
As these measurements were made on different days, the means
35
would not probably show as regular a curve as the verticals, where
one or more series were taken during the same day.
They are given only to show that the same form of curve satisfies
the velocities in a horizontal as well as in a vertical plane, the full
lines representing two semi-ellipses meeting in the “Thalweg,” or
deepest part of the river.
As we have seen, the curve adopted in the Mississippi observations
was a parabola with its axis three-tenths the depth below the surface.
To compare the observations on these rivers with this curve, two
verticals were selected, one in the St. Clair and one in the St. Law-
rence. These verticals were chosen, partly because they were the
mean of the largest number of observations, and partly because, not
being the deepest verticals in the rivers, the velocities were not so
nearly equal near the surface, and therefore would compare more
favorably with the parabola. Also, instead of taking the axis at
three-tenths the depth, it was assumed at five feet from the surface,
or one-ninth the depth in the first vertical, and one-thirteenth in the
second.
In Table XII the observed velocities on these verticals and the
ordinates of the parabola are given. The ordinates of other curves
have been added for comparison with these observations.
These curves were calculated by the following formulae :
Parabola. x = W — (b v); (ºr, º *;
in which x = the ordinate at any depth, y,
v = the mean velocity of the river,
W = the velocity at the depth of the axis,
d = the depth of the axis,
b = a constant, the value of which for rivers is given as
0-1856.
Introducing the constants, we have for the
— 5
St. Clair, x = 3-580 – 0.771 (i.) *, and for the
St. Lawrence, x = 1.499 — 0.491 (...) 2.
«»
X º g tº
** ; in which x and y have the
Reversed Parabola. x* = C +
same value as the last ; and for the St. Clair, x = 2.5, y = 47-8, C
= 1.591. For the St. Lawrence, x = 0.624, y = 652, C = 0-950.
36
* 1. ey -
Logarithmie Curve. x = C + y – ; x and y being the same. C
3,
= 1.800, a = 0-77, for the St. Clair, and C = 0.790, a = 2:45, for
the St. Lawrence.
Ellipse. x = C –H º (2 A y + y”) ; ; for the St. Clair, A = 57,
B = 2-7, C = 1.300, and for the St. Lawrence, A = 83, B = 0.75,
These last curves are the same as those plotted in figures 11, 12
and 13. -
The straight lines were taken as intersecting at six-tenths the
depth.
Noting first the St. Lawrence observations, we see that in that slow
current the ordinates are so small in proportion to the depth that all
the curves approximate very nearly to a straight line, and therefore
there is but little difference in their agreement with the observations;
though even in this case we see that the parabola is the worst form of
curve to express the decrease in the velocities.
But in the St. Clair observations, where the velocity is more than
double, and the depth much less than that of the St. Lawrence, we
see at once how entirely it disagrees with the observations both at the
surface and bottom. Probably, were the true velocities of the Mis-
sissippi known, the differences between them and this parabola would
be much greater.
The reversed parabola also differs considerably from the measured
velocities; while the logarithmic curve, the straight lines intersecting
at six-tenths the depth, and the ellipse, agree very well with the ob-
servations. The straight lines cannot be correct near the bottom, on .
account of the more rapid decrease in the velocities there; to give a
close approximation to that part of the velocities, there should be a
third line, still more inclined, intersecting the second near the bot-
tom. In the selected vertical from the St. Clair the logarithmic curve
appears a little the best; but in the deeper verticals, where the velo-
city for a considerable depth differs but little from that near the sur-
face, the ellipse shows a much better agreement with the observations;
for near the extremity of the minor axis the curve approximates to
the tangent at that point.
The great discrepancy shown by the parabola naturally leads us to
examine more carefully the manner of obtaining it, especially as we
have seen that in full two-thirds of the selected verticals of the Mis.
TABLE XII.
ST. CLAIR RIVER—DEPTH FRom 44 to 47 feet.
~3 i Parabola. Reversed Logarithmic Straight lines. E}lipse.
; : . . . ; Parabola. Curve. intersecting at .
‘s.3 § 3 ; Humphreys and Q-6 depth. Racourt.
2: 3 := } Abbot. Woltmann. Funk. Defontaine.
º P. S} : , — — — — — r -- -- —----- –––. --
§§ : § Differ- {Differ- Differ- , Differ. . . . Differ-
P Š § - Ordi- ence Ordi- ence Ordi- ence Ordi- ence Ordi. en Gºe
O § nates from mates from nates from natesi from nates from
3- of Obser- of Obser. of Obser- of Obser- of Obser.
curve. Vations curve. vations. Curve vations. Lines. vations. Curve.; vations.
! t | ! ! ------ - --
- ! ; : i *
3-868 3-564 –0-304 4-091 4-0-223 i8-952 Ho 084 4.000 i–H 0.132 3.964 +0.096
5 3.907 || 3:573 –0-334 3 957 |-|-0.050 3-886 –G-021 3-892 —0 0.15 (3-916 --0-009
10 : 3 821 3-564 –0.257 (3.814 –0.007 3.812 –0 009.3 784 – 0 037 3-843 +9922
15 3,709 || 3:536 –0.173 3-662 –0-0473-725 +0.016 3.676 –0-033 3-744; +0.035
20 3:608 3,489 –0.129 i3:476 –0. 132 (3 623 ––0-015 3.568 i–0 040 3-618 —-0-01)
25 : 3-496 || 3-423 –0-073 33 is i-0118 3-501 |-|-0-005 3460 –0.036 (3:460 – () - 0.36
30 3.309 || 3:339|+0.0303-117 –0-1923.342 +6. 33 3.260 –0-0493-269-9.94%
35 3:100 3.237 --0-137 (2.885 –0.215 3-121 |-|-0-021 2-943 –0.057 3:005 –9.9%
40 2-678 3-115 |-|-0.437 |2-604 –0 077 (2.75l |-|-0 073 2,627 — () ()5 | 2.67. —0.007
42-4 || 2:388 3.050 |-|-0.662 2.431 +0 043.2-420 +0.032 2.475 H-0-087 2.447 +0.04]
45.4 1 +28 2.962 2:05. 1-800 2-200 2000 l
Sums 33 884 33-890 2-536 1164: 0.309 0.537 0-400
Means 0.254 0 - 1 16 0-03 l . () 054 0.04()
TABLE XII—Continued.
ST. LAWRENCE RIVER,--DEPTH FROM 64 to 67 FEET
! - ------ - - -
+3 Parabola. Reversed Logarithmic | Straight lines, Ellipse.
g . 2: Parabola. Curve. intersecting at -
3.3 § 3 || Humphreys and 0-6 depth. Racourt.
a 3 # 3. Abbot. Woltmann. Funk. Defontaine.
--> P. QX - - - º 1 -- - - - - --- - 1 -- - -
; ; .9 §. Differ- iDiffer. ; Differ- ! Differ- Differ-
2.3 § 3. Ordi- ence Ordi- ence;Ordi- ence Ordi- ence Ordi- € In Ce
O $ nates from mates from nates from nates, from i nates; from
ſº of Obser- of Obser- of Obser- of Obser- of on-ser-
curve. vations.{Curve. Vations. Curve.' vations. Lines. vations. Curve. Vations.
1 5 42 l 496 | - 0.046 1-574 |-|-0-032 |l:537 –0-005 |1-560 --0-018 il 548 --0.006
5 I -533 1499 –0-034 1-550 |-|-0-0) 7 |1-521 |–0-0 12 1-537 |-|-0-004 ii.536 +0.003
10 ! •509 1.496 – 0-013 |1:524 |-|-0-015 1.506 1–0-003 il-5 14 +0.005 521 +9:012
1 5 |-507 1.487 0 020 .49? —0-0 || 0 | 1.490 —0 017 1 - 491 – 0-0 16 || 495 —0-0 12
20 } 1 - 47 0 l:473 ––0-003 || 469 |–0-001 |i-470 1468 ; –-0-002 || 483 +0-013
25 | 477 l'453 –0.024 | 1.440 ||—0-037 | 1.450 —0.027 il. 445 –0-032 || 458 –0-019
30 1 - 4 1 7 1 427 |-|-0-010 || 408 –0-004 il:426 -–0 009 || 422 —0-005 || 429 +0-012
35 1.398 1.395 – 0-003 |1-374 –0-034 |I-399 |-|-0-001 |l-399 |-|-0 001 I 394 —0.004
40 1.357 l 258 |-|-0-001 |1-327 –0-030 || 367 |-|-0 010 1-373 +0 016 |1364 –0 003
45 1 - 3 15 1-314 –0.001 || 1:296 |-|-0-019 11.328 ––0-013 (1.296 –0-019 1307 –0-008
50 1.236 1.265 |-|-0-029 |1 250 +0.014 1-277 --0-041 1-219 –0.017 || 249 +0.013
55 1 - 167 1-2 10 |+0.043 |1-195 |-|-0 028 1-207 |-|-0 º: –0.024 || 178 --0-011
60 1 - 092 1-150 |+0.058 [1-127 |+0.035 | 1.087 —0.005 |1-066 –0.026 (1.078 —0-014
62.5 0 800 1-080 0-950 Q-790 0.985 ! 0-8 ; 8
|
Sums | 18-020 i 18:025 0.285 0-2'ſ l 0- 183 0-185 0-13 ()
i
i ł
Means 0-022 0.02 I 1:014 # 0 014 0-0 l ()
t : t



37
sissippi observations the maximum velocity is at or near the sur-
face.
As has already been stated, Capt. Boileau found that in his obser-
vations on the experimental canals at Metz, the decrease in velocities
was best expressed by a parabola with its axis at the surface. As these
observations are rather curious they are exhibited in figure 13. The
velocity below the surface first increases rapidly, then follows closely
a vertical line to about one-third the depth, then decreases towards
the bottom in nearly a straight line; thus agreeing with M. Defon-
taine's two straight lines, except that their intersection is at one-third
the depth, which is much higher than is given by any other observa-
tions.
In the Mississippi Report (page 251) these observations are given,
reduced to English feet; and the authors found that a parabola with
the axis about two-tenths the depth below the surface would agree
nearly as well with the observations as one with its axis at the sur-
face, and besides express the decrease above the point of maximum,
which Capt. Boileau thought followed in law. This parabola is shown
by the full lines in figure 13.
In December, 1859, about a year after the close of the Mississippi
observations, Lieut. Abbot made a series of float observations, on a
feeder of the Chesapeake and Ohio canal, at the Little Falls of the
Potomac.” -
They were very nicely and carefully executed, and are as good as
any double float observations can be, and would probably, in such a
small stream, give a very near approximation to the true velocity
of the current. These observations are plotted in figure 14, being
represented by the circumscribed dots.
The canal was rectangular, about 23 feet wide and 7 feet deep.
According to the law of the descent of the maximum velocity, it
should be found at about two-tenths the depth below the surface,
which is a little less than is given by the observation.
By rejecting the surface and lowest measurement, a parabola simi-
lar to the one found to agree with Capt. Boileau's observations would
pass very near to the other points. This parabola is shown by the
dotted line.
An ellipse, represented by the full line in the figure, passes through
the lowest observation, and agrees nearly as well as the parabola with
the remainder, while it cuts the bottom at an acute instead of an ob-
tuse angle.
* Mississippi Report, page 253.
38
It is to be regretted that this curve was not chosen rather than the
parabola, for it certainly better expresses the decrease of the sub-
surface velocities in flowing water.
MEAN VELOCITY.
Many engineers have attempted to find the ratio between the mean
velocity of rivers and canals and the maximum velocity at the sur-
face. Some have been contented with a simple expression of the
ratio, and the equation v = 0.8 W is generally adopted, in which v =
the mean velocity of the current, and W = the maximum surface ve-
wº sº - ſ —- W º
locity. Dubuat gives the formula v = vºw , W being the bottom
velocity. Young makes v = V + 05 1/W H 0.25, V being given
in inches per second.
V (V -- 7-78188)
W–EIO-34.508 but this gives generally too
Prony found v =
large a value for v.
Baumgarten multiplied it by 0:8, to make it agree with his observa-
tions on the Garonne. The formula then becomes
W – 0.8 V (V -- 7-78188)
r W –– 10-34508
the velocities being expressed in feet per second. In this form it
will probably give a very close approximation to the mean velocity of
FlverS.
But the great desideratum has been a formula by which the mean
velocity could be obtained from the cross section and inclination of the
bed. Almost every writer on this subject has given a new formula,
and proved the incorrectness of all those which were established be-
fore him. r
The oldest and simplest of these formulae is that of Chezy, v = C VRI,
R being the hydraulic mean radius, or the cross section a divided
by the perimeter p. In large rivers this quantity, R, differs but
little from the mean depth. I = the inclination of the bed of the
stream per running foot, or the fall, h, divided by the distance in
feet, l.
C = a constant to be determined by experiment.
The value of this constant has been differently estimated by differ-
ent writers, varying from 0-68 to 1:00.
Dubuat, from his own observations on small canals and a few
others, deduced his celebrated formula,
39
*------- 307 - 0.3
v= VR = Oi (, II, LuII.3) T")
1, being the hyperbolic logarithm, or the common logarithm, multiplied
by 2-3025851.
When R and I are both very great, the formula may be written,
-- 307
v = V R (w/º L. L.I
Dr. Young, in his experiments upon the circulation of the blogd,
found that the resistances could not be expressed by the simple square
of the velocity, and he modified Dubuat’s formula as follows:
- 02) nearly.
W = VRI + * c in which b = 0000001 (418 —- º *
—- 3 1
b b 3 Tb
1440 180 900 d? 1
-— “T” — —- = .0000001 – ‘’’ ” + . . .
d + 12-8 d -- ...sº), and #i85 " vºi
13:21 , 1.0563
(1085 + ~d- + **) d = four times the hydraulic mean depth.
For large rivers v = V20000 JI nearly.
He gives a table of the values of b and c for all ordinary streams
and canals. In this and in Dubuat’s formula the quantities are ex-
pressed in inches, and in all others, in feet.
Eytelwin's well-known formula, the one most commonly used in
practice, especially by German engineers, partly on account of its
extreme simplicity is, v = }} (10560 R. I.) #. -
Darcy and Bazin's formula for streams with natural beds, deduced
from their own and preceding observations is, v VR I
A being = 000085 ( 1+º
the value of A is different.
Humphrey's and Abbot's formula, obtained from measurements in
the small rectangular feeder of the Chesapeake and Ohio Canal and
from the selected Mississippi observations, is as follows:
) For beds of cement, plank, &c.,
W = / 225 a V/T # 0. — ”
Y 00081 b + ( #) i"9° V b) , in which
B = the width of the stream and
40
In 1868, after discussing the observation of MM. Darcy and Bazin,
Gen. Abbott added a second term to this formula 34 1/ v'
- I-EP
v' = the value of the first term in the expression for v. He has
also prepared tables to facilitate the computation. Dr. Hazen, col-
lating all the observations from which the above formulae were de-
duced, and computing the constants by means of the method of least
squares, gives the following : v = 4.39 y R x "VI.
The above are a few of the formulae which have been deduced by
different engineers; each one first showing that none of the previous
formulae would conform to their theories and experiments.
At the commencement of this century, Dr. Robinson, of Edinburg,
speaking of the state of the science of hydraulics in his day, says:
“As to the uniform course of the streams which water the face of the
earth, and the maxims which will certainly regulate this agreeably
to our wishes, we are in a manner totally ignorant.
Who can pretend to say what is the velocity of a river of which
you tell him the breadth, depth and declivity ? Who can say what
swell will be produced in different parts of its course, if a dam or
wier of given dimensions be made in it, or a bridge be thrown across
it; or how much its waters will be raised by turning another stream
into it, or sunk by taking off a branch to turn a mill 2 >k >}:
>k >k >{< >}< >k * Yet these are
most important questions. The causes of our ignorance are the want
or uncertainty of our principles; the falsity of our theory which is
belied by experience; and the small number of proper observations
or experiments, and the difficulty of making such as shall be service-
able.” *
Dupuit says:* “The comparison of velocities, observed in large
bodies of running water is far from confirming the preceding form-
ula;” and he also says that M. Menard states in his “Cours de Con-
struction,” that he found the actual outflow at a certain point in the
river Meuse to be 33.4 m., while Prony's formula, applied to different
sections, gave 53 m., 56 m., 74 m., 69 m., 34 m. and 29 m. At
another point the computation gave 28 m., 11 m., 30 m., 39 m. and
4m., while the actual discharge was 25 m., 5.”
Such being the state of the science, it is not strange that the new
* Etudes théoriques et pratiques sur le mouvement des eaux, par I. Dupuit.
Paris, 1863, page 55.
41
theories of the flow of water in rivers, and the new formulae deduced
by Humphreys and Abbot were at once accepted by the scientific
world. Dr. Hagen, of Berlin, the oldest and ablest of German
hydrauliciens, after speaking of the many commendatory notices of
the Report on the Mississippi, says:* “Thus the new formulae were
so praised, that the former unshaken belief in Eytelwin's formula
suddenly vanished, and those of Humphreys and Abbot seemed about
to take its place. I was therefore even then induced to warn my co-
workers against such an unconditional acceptation of them, by show-
ing their very weak foundation.” Again, after describing his own
very simple formula, given above (which he compared with the Missis-
sippi observations and found the error only one-half as great as with
their own very complicated one), he says (page 20): “If they only
wished to obtain an analytical formula which would best agree with
their observations, there was no need of taking into account the
assumed resistance of the air, nor the other members which they have
introduced.” Humphreys and Abbot give a table of their own, and
other observations computed by the different formulae.
The difference between the observed and the computed velocity is
in some cases as much as three feet per second.
On the other hand, these very formulae, when used for the calcula-
tion of the discharge of water through pipes, give results which agree
very well with the observations.
Dr. Brewster, in his Encyclopedia, gives an example of a pipe, 4
inches in diameter and about three miles long, with a fall of 51 feet,
of which the average discharge for five years had been obtained. He
computed the discharge by several formulae, and gives the results as
follows:
C. F.
Measured discharge, 5 years' average, . tº . 11-333 per m.
Calculated by Eytelwin's formula, te & tº 11-355 “
* * * Girard's 4 ſº e * & . 11'265 “
§ { Dubuat's 4 & tº & te 11.257 “
§ { Prony's 4 & e & * . 11.502 “
§ { Young's $$. e º * 11.45% “
The agreement between the observed and calculated discharge is
very close, and the simpler formulae are rather better than the more
complicated. The hydraulic mean depth of a river can be measured
with nearly as great accuracy as the diameter of a pipe, and there-
* Bewegung des Wassers in Strömen, von G. Hagen. Berlin, 1865. Page 5
42
fore the error which is found when we attempt to apply these formulae
to rivers must arise from inaccuracy in the measurement of the incli-
nation. In a pipe, the fall can be obtained with great accuracy, and
probably in a canal, where the cross section is uniform, the measured
would not differ much from the true slope; but in a river, where the
depth and width are constantly varying, and the fastest current is
changing from side to side, the water cannot be a plane or a simple
curved surface, but must form a warped surface, inclining towards one
side or the other; for the fall, measured on opposite sides of a given
reach of river, is rarely the same. Generally in swift rivers there is
a back flow near the banks. Mr. Ellet states, in regard to the slope
of the Mississippi, that “*it not unfrequently happens that while the
mass of the water which its channel bears is sweeping to the south at
at a speed of four or five miles an hour, the water next the shore is
running to the north at a speed of one or two miles an hour. It is
no unusual thing to find a swift current and a corresponding fall on
one shore, towards the south, and on the opposite shore a visible cur-
rent and an appreciable slope towards the north.”
The surface level is also constantly changing, often sufficient to en-
tirely mask the fall in a short distance.
In a length of a thousand feet, near the mouth of the Niagara River,
where the current and inclination were ordinarily down stream, this
change in the water level did not take place simultaneously at the
ends of the selected line, and thus made the apparent inclination as
often up stream as down.
The fall, therefore, must be measured for long distances not less
than one or two miles, and great care must also be taken to have the
water-level stations so situated that the velocity of the current near
them is the same, for the water rises with an increase of velocity. M.
Baumgarten found that in the swiftest current in the Garonne the
water was over 0-4 foot higher than near the banks.
But in this long distance there may be bends, which by increasing
the friction will decrease the velocity, and there will certainly be
changes in the cross section and distance of the maximum velocity
from the banks. These and other causes render the accurate deter-
mination of the inclination a very delicate matter, and in some cases
an impossibility. -
Dr. Hagen, in speaking of the observations of Brunnings, upon
* Report, page 27.
43
which Eytelwin based his formula, says:* “These observations are
divided into certain groups, and a single inclination given for each
group. • . #
“It is noticeable that this agrees for one, and generally the first.
observation in each group, with that computed by Eytelwin's formula.
This coincidence is so striking that we can hardly consider it acciden.
tal. * * * In fact, the falls were not measured, but computed by
Dubuat's formula, and then compared with the velocity observations.
Naturally, they should agree; indeed, they only disagreed because
Funk computed the inclinations for each group, and not for each ob-
servation. That Eytelwin did not refer directly to Wiebeking's notes,
but used Funk's computations, is proved by the fact that he copied a
typographical error in Funk's work.” And, after collating all the
known observations when the velocity and slope were both obtained,
he says of the few observations selected by Humphreys and Abbot,
for comparison with their formula:f “Of these observations of Du-
buat they only used two which best agree with their theory. These
are the last-mentioned, and of one of these Dubuat says that, on ac-
count of the opening of the sluice, the velocity was too great. The
other eight observations, which would certainly give a very different
result, they rejected altogether. Besides these, nine other observa-
tions are noted in the American work, made in the Netherlands, Italy
and Russia, of which detailed notes are missing.”
In a supplement to the Mississippi Report, written by Gen. Abbot
for the “Essayons Club” of Engineer Officers, the observations of
MM. Darcy and Bazin are examined, the new formula applied to
them, and the Mississippi observations computed by their formula.
The arithmetical sum of the differences between the observed veloci-
ties and those computed by both formulae are as follows:
TABLE XIV.
- | SUMS OF DIFFERENCEs.
OBSERVATIONS.
H. and A. | D. and B.
Formula. | Formula.
Darcy and Bazin canals............................ I 4° 085 7. 158
Bayous and selected European observations. 3-792 8: 653
Mississippi River...................................... 2-584 13-284
* Bewegung des Wassers, page 4.
i Bewegung des Wassers, page 24.
44 º
º
Thus the error of the new formula applied to Darcy and Bazin's
observations is double that of their own formula, while in the Missis-
sippi the sum of the differences of the observed and computed veloci-
ties is nearly six times as much by the latter as by the former.
In Table XV the observed and computed velocities of the Missis.
sippi are given, that we may examine them in detail.
TABLE XV.
. Cross Section. Mean Velocity. Discrepancy.
to l —
3-4 ſº
3.3 |Area in 3|Inclination D. and B. H. and A. D. and B. H. and A.
3 = gº. -- Observed
--> cº rºs
É 2. sq. feet.: 3 Formula. Formula. Formula. Formula.
Ft.
1 | 193-968 || 72-0 0-0000205 5'929 4.047 5-891 || --1-882 || --0-038
2 195-349 || 72-4|| 0-0000171 5'887 3.710 5-644 +2-177 | +0.243
3 180-968 || 73.6|| 0-0000034 || 4-034 1-671 3-775 +2-363 || --0-259
4 183-663 74-4|| 0-0000038 3.977 1 - 781 3-911 +2-196 || --0-066
5 | 1.48-042 || 65-9| 0-0000680 || 6-957 7-044 7-766 —0.087 | –0-809
6 || 178. 137 64. 1 || 0-0000638 6-949 6-7 10 7.409 | +0.239 —0,460
7 || || 79-502 || 64-5|| 0-0000486 || 6-825 5.570 6-754 | +1-255 | +0.071
8 78-828 31.2| 0-0000223 || 3:523 2- 681 3.920 | +0.842 —0.397
9 : 134.942 52° 1' 0-0000303 || 3. 55.8 4' l 40 5:515 | +1-418 +0.043
10 | 150-354 57.4|| 0-0000481 || 6-319 5-495 6-517 | +0.824 —0.198
Noticing, first, the relation of the inclination to the velocity in Nos.
1 and 2, the slope is 205 and 171—omitting the cyphers, with a
velocity of nearly six feet per second; while in 3 and 4 it is only 34
and 38, or one-fifth or sixth of the former, the velocity being about
four feet per second, or two-thirds of the former, with nearly the same
cross section. Again, in No. 6 the inclination is 638, with a velocity
of about seven feet per second; while in 7 the velocity is nearly the
same, but the inclination is only about two-thirds that of No. 6, the
cross sections being nearly equal.
We also see that the D. and B. formula gives the velocity of that
great river correctly when it is 7 feet per second, while at from 4 to 6
feet per second it gives less than half the observed velocity. There
must be some error in these measurements, or the velocity of the
water in the Mississippi is dependent upon something besides the fall.
The H. and A. formula agrees, of course, with these observations,
as it was based on them. These inclinations were measured by the
civil assistants, one of them being re-run five times, and their disagree-
ment is another proof of the small dependence we can place upon the
measurement of the fall of a river.
45
In fact, with the exception of the experimental canals and small
streams, there seem to be few if any inclinations recorded which are
really trustworthy. &
The fall of nine miles of the St. Clair River was very carefully
measured, two lines of levels being run on each side of the river, and
the surface level obtained at five places on each side. These points
were chosen where the velocity was nearly the same, and stakes were
simultaneously driven to the water surface, and afterwards connected
with the marks on the level lines. There was but little difference
between the two determinations, and the mean inclinations are given
in Table XVI.
TABLE XVI.
t i
we & * º *
* * * * . anadian Side. |
Stations. American Side | C 3. |
- |
i
Distance. Fall. Inclination Distance. Fall. Inclination
F.TTF.T. Ft. Ft.
99.93.5 0.1410 0-000074l 4; 10226-5 0-7253 0-00007092
1251 1-7 0-98.05} 0-00007839; 11867 - 2 ; 0-6725 0.00005667
96.95.5 0-87 l ; 0.00008990 l 1534-0 || 0-8402 0-00007283
17530.7 0-8805 || 0-00005023; 1407.2.0 0-1775 0.00008369
i §
:
:
:
i
The stations are lettered the same on both sides of the river, A
being below the city of St. Clair, and E near Marysville, and were
as nearly opposite each other as suitable places for them could be
found.
Between the extreme stations the river runs almost in a straight
line, and there are no obstructions except an island between D and E
and a shoal opposite St. Clair. Yet the table shows great differences
in the slopes, not only between the stations, but also between the op-
posite sides of the river. For about a mile above the station A, the
river remains at nearly the same width and depth; then it widens
until it is nearly twice as broad, near the town of St. Clair. There-
fore it would seem as if the slope for this whole distance ought to be
rather less than that near station A, where current measurements were
made.
But the measured inclination introduced into the mean velocity for-
mulae gives much too large a result, as will be seen in Table XVII,
where the mean velocity is calculated by the different formulae hereto-
fore given, the smallest inclination, or that on the Canadian side,
being used. - -
46
TABLE XVII. *
Mean Velocity of the St. Clair River, feet per second.
A = 66147 ft. R = 38.06 ft. I = 0.00007092.
Calculated by Formulae.
observed. - | i .
| - Chezy Darcy Humphrey
i Dubuat. Young. Eytelwin and Hagen and
Co-eff. 80. Bazin. Abbot.
|
3-272 4' I 56 4-272 4'490 4'ſ 23 5-359 5-5 10 5-858.
Although there is considerable difference in the velocities as com-
puted by the several formulae, the least value is nearly a foot in a sec-
ond greater than the observed velocity; therefore the measured incli-
nation must be much larger than its true value at station A, instead
of being smaller, as the increase in the cross section near station B.
ought to make it.
The slope for the whole distance is just about the same as that be-
tween stations A and B, except that the larger inclination is on the
Canadian side, being 0-000071 there and 0.000070 on the American.
As there is scarcely any change in direction in the river, the correc-
tion for bends must be very small; and the only apparent cause for
a decrease in the velocity below that due to the slope, is the diversion
of the current by the island between D and E and the shoal opposite
the town of St. Clair.
If the true fall in such a reach of river cannot be measured, every
possible care being taken to prevent error, it must certainly be impos-
sible to obtain the inclination of ordinary rivers whose cross section
and direction is constantly changing. Therefore, until Some better
method for the dermination of the fall is discovered, we shall appa-
rently have to be content with the measured velocity, and not attempt
to calculate it from the inclination. Surely the velocity determined
from a few surface floats could not possibly differ so much from the
true mean velocity as that computed by the best of the preceding for-
mulae. But, as will hereafter be shown, by means of the telegraphic
meter, a very near approximation to the mean velocity can be readily
obtained by a few observations.
Among the formulae given by Humphreys and Abbot is what they
call the “mid-depth formula,” which, if correct, would be very useful.
47
It is as follows: v. = W1 — ſº, (b v), in which
v, - the mean velocity in any division,
Wi = the mid-depth velocity,
v = the mean velocity of the whole river,
b => *-
(D -- 1:5),
D = the depth of the river in each division,
a = the area of each division.
The mid-depth velocities were chosen because the ratio of the velo-
city at that depth to the mean velocity “is independent of the width
and depth of the stream—except for their almost inappreciable effect
upon bi-absolutely independent of the depth of the axis, and from
the small numerical value of Tº bº nearly independent of the mean
velocity.”
In order to test this formula, the mid-depth velocity in each division
of the several rivers where not directly observed was taken from ver-
tical velocity curves, and the mean depth of each division from the
cross section profiles. These quantities for the St. Clair River are
given in -
TABLE XVIII.
! ſ w t i
S4– - l i
© tº | - i :
# #| \,P V b a . Lui l-a < a.
#3 bº Mid Depth I-69 partial V × “ is". is "X."
#5|Division. Velocities. (D + 1-5); Areas. |
Ft. |
13, 15-68 1-420 0-40'ſ? 1568 2226.5 0.053 84.67
} b 29-04 3.050 0 3058 2904 8857.2 0.046 133-58
2 39-65 3-820 0-2636 7930 || 30355-5 0.043 340-99
3 43 55 3-975 0-2506 87 10 34622-3 ; 0.042 405-82
4 50-03 3-800 0-2348 10060 38228-0 0-040 402-40
5 : 52-25 3-750 0-2305 10450 38383-8 ; 0-040 418.00
6 : 47-65 3.41 7 0-2409 9510 || 32495-7 || 0-041 389-91
7 40-77 2-880 0-2509 8.155 23.486-4 || 0-042 350-67
8 28-20 2-3 16 0-3 l 55 5640 | 13062-3 || 6-047 265.08
9 : 15-25 1 - 1 18 0-4129 1220 1864' 0 || 0-054 65:00
66147 |223081.7 2856-12

The value of b was found to be 0.1856 for the Mississippi, and it
was considered that this value would generally be applicable to all
streams over twelve feet in depth; but this table shows its value cal-
culated by the formula is always greater.
In fact, in one of the divisions in another river it was over 0-7.
48
Multiplying the sum of the partial areas by v, we have 661470 +
2856-12 V3 = 223081.7 ; therefore v = 3-294.
In all the observations on these rivers, the mean velocity was found
at about six-tenths the depth, a fact which has been noticed by many
engineers, and Dupuit theoretically placed it at 0:58 depth.
The velocity at six-tenths the depth and the partial areas for the
St. Clair are given in -
TABLE XIX.
Number vº Area Velocity at
Of at 0:6 of each | 0:6 Depth
Division. Depth. Division. into Area.
la 1.390 1568 2179
1b 2.950 2904 8567
2 3.700 7930 2934.1
3 3-820 8710 33272
4 3.645 10060 36669
5 3-620 10450 37.829
6 3.250 951 0 30908
7 2-700 8.155 220 18
8 2: 240 56 40 12633
9 1 : 110 1220 1354
| t
Sums..... 66- 147 214770

Dividing the sum of the last column by that of the last but one,
we have v = 3-247.
The mean observed velocity was 3.272; obtained by the mid-depth
formula 3-294. -
Thus the velocity at six-tenths the depth is a little nearer the ob-
served velocity than that calculated by the mid-depth formula.
Probably had the velocities been taken at 0:58 of the depth it
would have been still nearer. This, of course, greatly simplifies the
calculation, the correction iſ, (b v)} only serving to reduce the mid-
depth velocities to about six-tenths the depth.
Therefore this is a still easier method for measuring the discharge
of a stream, as it is only necessary to obtain the velocities at several
points at six-tenths the depth, and multiply the mean by the area of
cross section.
We have seen, by Table XII, that the two straight lines intersect-
ing at six-tenths the depth approximate closely to the observed velo-
cities; if, then, the velocities are measured at three points on the
swiftest vertical, namely, at the maximum, at six-tenths the depth,
49
and as near as possible to the bottom, and plotted on a large scale,
joining these points by straight lines, we can find the velocity at any
desired position on this vertical—very nearly,
MM. Darcy and Bazin and Capt. Boileau have shown that lines
drawn through equal velocities assume very nearly the form of the
beds, and this was also found to be the case in the observations on the
rivers connecting the Great Lakes; therefore, if we draw lines parallel
to the bottom and sides through any chosen velocities on the measured
vertical, their intersections with any other assumed verticals will give
the velocities at the points of crossing, and thus a close approximation
to the true velocity of the river can be obtained by only three meas-
urementS.
Of course such observations require some such apparatus as the
telegraphic meter, for by this the mean velocity for at least half an
hour could be obtained at each of the required points.
A PE2 E N DIX.
After the foregoing article was written for the Journal, and before it was
published, there appeared in the annual report of the Chief Engineer U. S.
A. to the Secretary of War, for the year 1870, reviews of my progress re-
ports submitted to the Superintendent of the Lake Survey, Gen. W. F.
Raynolds, and annexed as appendices to his reports to the Bureau.
These reports mainly gave the data which where to be embodied in a gen-
eral report at the close of the work.
Before the work of gauging the rivers connecting the lakes was sanc-
tioned, Gen. Raynolds being officially asked what method he expected to
adopt, replied, “that proposed in the Mississippi report.” This method was
followed the first season, but the second season the Telegraphic meter was
used, and much better results were obtained.
During the fall and winter following I had considerable correspondence
with Gen. Abbot in regard to the use of the floats and meter, he of course
urging the superiority of the floats, but saying nothing to discourage me
from pursuing the investigation; much less intimating that I had better
study the rudiments of hydraulics before I essayed the use of new methods
of measurement. The reports for 1868 and 1869 were forwarded to the
Department by Gen. Raynolds, and were published without an intimation
being given him that the course of investigation was disapproved.
Gen. Raynolds was removed from the charge of the Survey, Feb. 9th,
1870. One month afterwards Gen. A. A. Humphreys, Chief Engineer,
went through the form of submitting his reports to Gen. H. L. Abbot for
examination. In one week's time Gen. A. has his ‘review ready.
Gen. Raynolds' final report was forwarded to the Department about the
middle of July, 1870. On the 22d it was referred to Gen. Abbot, probably
reaching Willet's Point on the 24th. On the 25th the second ‘review' was
ready.
Neither of Gen. Abbot's papers were furnished to Gen. Raynolds, whose
reputation as well as my own was attacked, nor had he the least informa-
tion that the investigations were to be stopped. But as soon as he had left
Detroit, at least one of them was sent to his successor.
52
In his last report Gen. R. stated especially, that owing to the extent of
the work under his charge, (he had been doing double duty for years, and
two officers were sent to relieve him,) the details of the river gauging had
been left to me and that he expected me to bear the responsibility. When
that report was printed that paragraph was cut out, but Gen. Abbot's stric-
tures were published, and then for the first time had Gen. R. any intima-
tion of the mortal offence he had given to the Chief.
These reviews would have been noticed long before this, had I not known
that Gen. R. felt called upon to reply to the attack upon himself and to de-
fend the general method of conducting the work which had been approved
by him.
Naturally he desired to have the same publicity given to his answer as
had been allowed to Gen. A.'s reviews, and his paper was therefore forwarded
to the Chief Engineer with the request that it might be published in the forth-
coming annual report, but no notice was taken of his wishes. After the
report (for 1871) was issued, Gen. R. asked permission to publish his views
at his own expense.
This request was not only refused, but his manuscript was returned,
probably to prevent the possibility of its ever seeing the light through
official channels; and he was informed that one officer could not review the
work of another without being called upon to do so by the proper authority.
Thus it is evident that so far as the authors of the report on the Mississippi
have power to prevent, no investigation of the merits of the system of
double floats adopted on that survey will be permitted; they using the military
power of the Government—even to the removing of an officer and mutila-
ting his reports—to sustain the “new theories and formulae of flowing
water,” fearful that their otherwise extremely weak foundation might be
made manifest.
As, therefore, no official answer will be allowed these reviews, I have con-
sidered it due to Gen. Raynolds as well as to myself to make as public a
reply as possible.
Gen. A. A. Humphreys who had charge of the Mississippi Survey, had
been appointed Chief of the Corps of Engineers, and it would seem that he
and General Abbot found that the result of the work upon the Lakes was
likely to conflict with the new theories of flowing water so elaborately dis-
cussed in their report, for in the report of the Chief Engineer for 1870 we
find the following letter: -
53
OFFICE of THE CHIEF OF ENGINEERs,
WASHINGTON, D. C., March 19, 1870.
GENERAL:—With the annual reports of 1868 and 1869 upon the Survey
of thc N. and N. W., Lakes, transmitted to this office by the officer in charge,
there are sub-reports from one of the assistants, Mr. D. F. Henry, upon the
gauging of the lake rivers, which should be carefully examined. As the
subject is one with which you are familiar, the reports are herewith trans-
mitted to you for examination and report.
A. A. HUMPHREYS,
- BRIGADIER GENERAL AND CHIEF ENGINEER.
BRT. BRIGADIER GENERAI, H. ſ. ABBOT. U. S. A.
Major of Engineers, Willet's Point, N. Y.
As only a week elapsed between the dates of the order and of the review,
it would look a little as if the review was written £rst and the order issued
afterwards, so as to bring it officially before the Bureau.
“WILLET’s PoinT, N Y. March 17th, 1870.
GENERAL:—In accordance with your instructions dated the 9th inst., I
have attentively examined the river gauging reports of Assistant D. F.
Henry, constituting appendices to the last two annual reports of the officer
in charge of the N. and N. W. Lakes, and have the honor to submit the
following report thereon.”
Then follows a rambling discussion of what General Raynolds designed to
to do and what he should have done, which merely shows General Abbot's
ignorance of the lake region; and then he says:
“Assistant Henry seems to have misapprehended the character of the
operations approved and ordered by the Department, for * * I assume
that if any funds had been available for the advancement of our theoretical
knowledge of the laws of flowing water, they would have been expended
as recommended in the report of the Mississippi River on streams of a very
different character from any included in these operations, and upon quite
different subjects of investigation from those selected by Assistant Henry.”
That is we settled all those matters in our report, for all time, and don’t
choose to have them disturbed.
“His operations were analagous to those of an astronomer who wishing
to run a boundary line, should begin by making a star catalogue of the
heavens, and should not make it correctly.”
But suppose that in place of a star catalogue which had grown by the
patient labor of hundreds of observers, through many long years, and which
those who had contributed the most would acknowledge was still far from
perfect, the science was yet in its infancy, and that the few who had made
observations had each constructed a star catalogue of his own, and no one
|had yet been found who could reconcile them, when at last one wise in his
own conceit, discarding all the instruments which the men of science who
preceded him had been slowly bringing to perfection, takes up an old wooden
astrolabe, and by its aid makes a new catalogue, differing more from all its
54
predecessors than they did among themselves, and proclaims to the world that
he has at last found the true positions of the stars, and that his is the only
genuine catalogue which ever has been or ever can be made,-would it not
be well for our friend who is about to run the boundary line, to make a few
observations on the stars himself, before he accepts this last catalogue as
being all that its author claims, or in fact, as being any better than the others?
Next follows a criticism on the reports, mainly upon verbal inaccuracies;
apparently considering those two reports of progress, final reports upon the
subject; and of which the fairness is well shown in the following extract:
“In the operations of 1868 a new set of values for these mean annual
discharges is determined, and it is concluded that the first were in error
about 10 per cent, a discrepancy which Mr. Henry attributes to the errors.
inseparable from the use of floats.”
The truth was that after the comparison of the floats and meter, I ascer-
tained that the floats ran 10 per cent faster on an average than the velocity
recorded by the meter, as shown in the table given on page 13 (ante; ) and
then finding that the total discharge measured by floats in 1867 was about
10 per cent greater than that obtained from the meter observations in 1868,
I naturally concluded that the floats were in error by at least that amount.
He then quotes the table where the mean velocity as measured by the
meter is given and compared with that calculated from the same data by his
mid-depth formula, and with the mean of the velocities at six-tenths the
depth, similar to the table on page 48 (ante; ) and says:
“The discrepancy of 14 per cent is only about one-tenth of the discrep-
ancy existing between Assistant Henry's own deductions in different years,
and so small a discrepancy as 1} per cent between his latest revised result.
and that obtained with incomparably less labor by taking advantage of the
general laws which govern flowing water, might naturally be supposed to
be sufficiently close to satisfy one as to the truth of these laws, but it is not
}}
SO.
This is criticism Many observers have found that the mean velocity in
any vertical is at about six-tenths the depth. Gen. A. not apparently hav-
ing heard of this, discovered that by applying a small correction to the mid-
depth velocities, he would approach very near to the mean. In the table
in question I merely show that there is no need of such a correction, as the
velocities at six-tenths of the depth come nearer to the mean than those
calculated by his cumbersome mid-depth formula. While he compares this
discrepancy with that between the floats and meter to show the great
superiority of the formula and of float measurements! And there are
whole pages devoted to such criticism as this.
But let us pass on to the arraingment of the meters. After quoting
the six reasons (ante p. 13 et seq.) against the use of the double float, he
says: g. &
“These are the imaginary difficulties of a theorist, well known as
such by any hydraulic engineer of experience who has practically used the
double float of the pattern and in the manner recommended in the Missis-
sippi report. Assistant Henry felt himself called upon to improve on these
recommendations, and hence arose most of his difficulties. Thus, he
restricted his soundings to the ends instead of properly distributing them
along the whole front of his base (whence arose his first difficulty)”—here
he is mistaken,_*and used an enormous base from 700 to 1,100 feet in
length instead of 200 feet as recommended, (whence arose his second dif-
ficulty)”—this will be noticed hereafter, “and he attempted to observe his
floats at all depths at each station (whence is derived all practical value of
his fifth and sixth difficulties).”
That is, if I had confined myself to the observations of surface floats,
and had had unbounded faith in the Mississippi report, I would never have
wanted any meter nor caused Gen. Abbot any trouble.
“His third difficulty can only be serious to one lacking manual dexterity
in the use of a theodolite. As well might it be claimed that no one can
estimate time to the tenth of a second because a tyro in practical astronomy
cannot do so.”
The boomerang effect of this shot will be fully appreciated when we
know that the Mississippi Survey was commenced three years before Gen. A.
finished his military education, and when he was ordered on that work he
was a callow fledgling of West Point, an Institution which, whatever else
it may do for its graduates, eertainly sends them forth with less practical
knowledge of Instruments than any other scientific school in the country;
while the gentlemen who were assigned to me to take charge of the guaging
parties, were graduates of the scientific department of the University of
Michigan, had been several years connected with the Lake Survey, and
were remarkably skillful in the handling of Instruments.
“A length of 200 hundred feet is amply sufficient for a width of 2000
feet, wider than this both banks can be occupied, or the base may be a little
lengthened. Of course the angles for the more distant floats will be small,
but on the other hand nothing like absolute accuracy is required. In so
wide channels the change of horizontal velocity in a space of several feet is
hardly appreciable, and, since both ends of the path are located, any bad
angle will be at once detected on the plot by want of parallelism of the path
to those of others in the vicinity.”
This has been discussed before, (ante p. 16) and I will only here men-
tion thefact that the uncertainty of location has probably led Gen. A. into this
error, as in practice the floats rarely run parallel, often being out more than
56
100 feet in 700 feet, as has been shown and can at any time be further
demonstrated by the original plats in the Lake Survey office.
“Moreover, I think he is disposed to be a little unfair to them. He says:
‘One hundred and fifty floats a day is as many as a single party can put
out and locate, the mean of the day's work will give the velocity for say
one hundred and fifty seconds; while from the meter we can obtain it for
the whole time or any part thereof.” I confess myself unable to see why
his floats, ‘often ten minutes in passing,’ are not entitled to have credit for
six hundred times an hundred and fifty seconds in comparing them with
the meter.”
Another instance of close reasoning. The float only gives the velocity
of the current in which it is carried past the center foot of the base line, to
which all the velocities are reduced, while the meter not only records that
current but all the variations between the times the floats are passing.
“Let us now consider Assistant Henry's meter. Never having seen the
instrument used, I cannot form an opinion respecting its relative merit as
compared with older types, but from his description it appears to be equal
if not superior to the best of them. He has certainly obtained by its aid
some interesting and novel data respecting pulsations in flowing water.
* †. In my opinion, founded on a somewhat close study of the sub-
ject, instruments of this class are pretty toys which have contributed more
to retard the progress of discovery in the science of river hydraulics than
any other cause. This is due to the fact that they register their results in
a kind of cypher to which we can by no means be sure that we possess the
key. To translate a given number of revolutions of a submerged wheel
into velocity per second, and by this means to detect laws whose existence is
denoted only by a difference of a few tenths of feet in this velocity, is so
delicate an operation that errors in the céefficent have generally masked
the laws. * * * Now to assume that a cóefficient thus obtained will
translate exactly the recorded number of revolutions into velocity per
second, while the meter is subjected to the current of the river by being
held fast at different depths from the surface to the bottom, is too wide a
generalization. That it may give a close approximation when the meter is
suspended three feet under the center of a small boat anchored in the stream,
is probable.”
Suspended thus it would not even give an approximation to the true
velocity, for, as has already been shown, at even five and ten feet from the
surface the velocity beneath the boat was much retarded by the friction on
the bottom of the boat.
“That by its use in a proper manner a tolerably correct idea of the dis-
charge of a river may be obtained is not improbable. But that by its aid it
is possible to determine in a scientific manner so delicate a change in the
velocity as that existing between the surface and the bottom is doubted.
The errors of the double float may be cancelled by multiplying sufficiently
the observations, but errors in the cóefficient of the meter always act in the
same direction under similar circumstances and cannot therefore thus be
annulled.”
57
To this special pleading in favor of the double float it might sufficient
to mention such names as Woltmann, Defontaine, Baumgarten, Racourt
and others, who have used various meters with success; but there is incon-
testable proof of the accuracy of the meters used on the Lake Survey.
1st. In table W. (ante page 13) which gives the comparison between the
floats and meter, it will be seen that there is little difference in the velocity
recorded by either, at or near the surface, and that little is accounted for
by the light wind at the time of observation. So that if the surface float
gives the true velocity of the current, the coefficient of the meter is correct
for that locality. *
2d. In April, 1870, that distinguished engineer, Mr. E. S. Chesbrough—
in a note to me—asked whether it would be possible to send over a meter
to Chicago to determine the velocity of the water in the lake tunnel, so as
to obtain its discharge otherwise than by the pumps. The matter being re-
ferred to Gen. Raynolds, he gave his consent, and subsequently Assistant
L. Foote, who had been in charge of one of the gauging parties from the
first, was given a leave of absence without pay, by Gen. Comstock, (Gen.
R.'s successor), and went over to make the observations.
This tunnel was excavated under the bed of the lake, to provide the city
with pure water. It is two miles long and five feet in diameter. The cen-
ter is about sixty-five feet at the outer, and sixty-seven feet at the shore
end, below the lake surface. At the shore end it empties into a well about
fifteen feet in diameter, from which, above the main tunnel, another tunnel
of the same size leads off to the pump wells. Both tunnels are somewhat
increased in size as they approach the main well.
Assistant Foote, under Mr. Chesbrough's directions, made four series of
measurements, three in the upper tunnel leading to the pump wells, at forty
feet distant from the main well, and one in the main tunnel forty feet from
the shore end, (they are numbered 1, 2, 3 and 4, respectively, in Table No.
Mr. Chesbrough has kindly allowed me the privilege of examining his
notes, and I copy his tables of final results. The coefficient used in the re-
duction of the meter revolutions, was the same that was used in the measure-
ment for the out-flow of the lakes. Table No. XX gives the velocity in feet
per second, at different points in the tunnel, and Table No. XXI, the means
for the several points. In Table No. XXII, the velocity, as computed from
the discharge of the pumps at each observation, with the corresponding
head in the tunnel is given and compared with the velocity measured by the
meter, and also with that computed from the head, by various formulae.
*
58
TA BLE NO. XX.
RESULTS DEDUCED FROM OBSERVATIONs witH HENRY's METER IN CHICAGO
TUNNEL AND FROM STROKES OF ENGINES.
(Corrected velocities correspond to discharge of 16,000 Gall, per min.)
LAKE

º
32.
s
*f;
Date Time Location
Of of Of
Observation. Day. Meter.
- 1870. h. m. h. m.
May 27th, 1.38 to 2.20 ! At Center.
& & & K 2.43 to 3.44 || 15 in. from N. side,
& C & & 5.38 to 6.08 || 12 “ & { {
& C & C 5.12 to 5.30 || 9 “ & K & 4
& & « C 4.44 to 5.04 || 6 “ & C & K
& & & & 4.02 to 4.29 || 4.25° C & & C
May 28th, 2.28 to 3.00 At Center.
& “ « , 1.45 to 2.14 15. in...frem N. side,
C & & & 12.00 to 12.33 12 “ “ & K
& C & & 11.19 to 11.50 9 ** * * & Cº.
C & < * 9.10 to 9.40 || 9 ‘‘ ‘‘ g
& & & K 9.53 to 10.26 6 ‘‘ ‘‘ & K
“ . “ 10.35 to 11,07 || 4 “ ‘‘ ‘‘
May 28th, 5.08 to 5.21 15 in. from bottom.
& C & C. 5.32 to 6.0S 15 ‘‘ ‘‘ & &
& K & 4 4.46 to 5.01 | 12 ‘‘ ‘‘ & C
& C & & 4.14 to 4.36 9 ‘‘ “ i &
& & C 6.15 to 6.36 9 & 4 & K & &
& c. g & 3.26 to 3.59 5 & C & & & C
May 30th, 3.19 to 3.29 || Af, Center.
& & & & 3.29 to 4.12 || “. & 4
& & & 4 4.12 to 4.37 || “ & C
C & & & 4.43 to 5.03 || “ « &
& & & & 1.33 to 2,06 || “ ‘,
& C & & 12.21 to 12.31 15 in. from S. side.
& K & & 12.41 to 1.01 || 15 “ g & & &
& C & 4 12.00 to 12.18 12 ‘‘ ‘‘ ( &
& 4 & & 5.27 to 5.48 || 9 “ “ & C
& K & & 11.34 to 11.56 || 9 ‘‘ ‘‘ & K
& C & & 2.56 to 3.12 || 9 “ ‘‘ N. Side.
& C C & 2.83 to 2.51 6 “ “ & Cº.
& & & & 10.45 to 11.15 || 6 “ ‘‘ S. Side,
& C & & 5.08 to 5.24 || 6 ‘‘, ‘- & C
& & & & 10.09 to 10.26 4 “ “ & c.
Rate
of dis-
charge
per
min.
per
En-
gines.
gal-
lons.
15766
15655
15660
16000
15980
15666
TT|T6375
16322
16150
16340
15875
16144
16262
11625
16140
16228
11666
16272
I4125
13380
10200
16000
15975
15S20
16025
11433
16116
16050
16140
16060
10.133
15940
;
;
Means.
I6535 | 1.8
15400 || IsøIS
i
:
:
:
i
1.7450
1.6270
1,4836
1.8068
1.6827
1.6699
1,6763
1.6887
1.7094
1.6837
1.7794
1.8020
1.8180
1.8109
1.7292
1.7047
1.9655
1.9892
1.9865
1.9747
1.9427
1.9193
1.8672
1.8942
1.7209
1.8205
1.7848
1.7416
1.7469
1.5632
1.5565
I
9
3
3
:
:
i
:
‘59
TABLE No. XXI.
SHOWING THE MEAN WELooITY AS DEDUCED IN TABLE No. XX, THE NUMBER of GALLons
PER MINUTE, CORRESPONDING TO THE MEAN WELocITY, AND TO THE MEAN op EACH SET of
OBSERVATIONS, &c.

Location Observa- |Observa- Observa- Observa- || Grand
of Meter. Ition No. 1.jtion No. 2.]tion No. 3.}tion No. 4. Mean.
At Center. 1.8140 1.8068 1.8068 1.97.05 I-8495
15 in.from side 1.8050 1.6827 1 7907 1.8932 1.7930
12 in. : : 1.7450 1.6699 1.8180 1.8942 I.7968
9in. . . 1,7450 1.6825 1.7700 1.7754 1.7432
6in. “ “ 1,6270 1.7094 1.7047 1.6839 1.6812
4 in. “ “ 1.4836 1.6837 1.5565 1.5746
O { % ºf 1,3000 1.3000 1.3000 1.3000 1.3000
*** * * *
ºm tº
viding the diseharge by the area
of the tunnel.
Average velocities obtained by di- Hy
ag y } 1,653 1.652 1.704 1.712 1,677
Gallons per minute in section of
# 20,468 sq. ft. 15,186 15,179 15,464 157,74 15,415
. Per cent. of 16,564 Gals. } |
quantity shown by pumps—plus 91.7 91.6 | 93.1 95.2 93.08
quantity for condensing.
\60
TABLE NO. XXII.
TABULAR COMPARISON OF CHICAGO LARE TUNNEL ExPERIMENTS.
For formulae, see Brooklyn Water Works report, 1858. Diameter of Tunnel 5.1 feet. Length of
Tunnel, 10,867 feet.
Values of velocity in feet per second.

Mean
head un-Deduced ||Deduced By By By By By By By By
der which from dis-|from ob- Hawkes- | Black- Prony's Prony's Eytel- D’Au- | D'Au- | Weis-
discharge charge served ley’s well’s formula, formula | win’s buison’s buison's bach’s
occurred shown by velocities|formula formula, 1) (2) formula formula formula formu-
in feet. pumps. Table (A.) (B.) § (D) (E.) (1) 2) la.
No. XX. (F.) G) (H.)
1.933 1.716 1.621 1.448 1,463 1.403 1,378 1.444 1.396 1.408 || 1.433
1.991 1.704 1.652 1.469 1.485 1.425 | 1.400 | 1.465 | 1.418 | 1.430 | 1.458
2,016 1.706 1.729 1.479 1.495 1.434 1.410 1.474 1.427 1.439 | 1.468
2.081 1.742 1.679 1.502 1.5.19 1,458 | 1.436 | 1.498 || 1.452 I.463 | 1.495
1.983 1.740 1,621 1,466 1.482 1.421 1.397 1.462 | 1.415 | 1.426 | 1.454
1.966 1,706 1.547 1.460 1,476 1.41 1.391 | 1.456 | 1.408 || 1.420 | 1.464
2.061 1.784 1.677 1.495 1.511 1.451 1.427 | 1.491 | 1.444 1.456 | 1.486
2,056 1.777 1,606 1.493 1.509 1.449 | 1.429 | 1.489 | 1.442 1.454 | 1.481
2,079 1.759 1.575 1,502 1.518 1.457 1.434 | 1.497 1.451 | 1.464 | 1.494
2.064 1.779 1.647 1.496 1.512 1.452 1.428 1.492 1.445 1.457 | 1.488
1,976 1.728 1.612 1.464 1.480 1.4.19 1.394 1.460 1.414 1.424 | 1.450
1.989 1.75S 1.721 1,469 1.485 1.424 1.399 1.464 1.4.17 I.429 | 1.457
2.016 1.771 1.722 1.479 1.495 1.434 1.409 1.474 1.427 1.438 | 1.468
2,091 .805 1.766 1.506 1.522 1.462 | 1.438 | 1.501 | 1.455 | 1.467 | 1.516
I fe
0,994 1.266 1.224 1.038 1.049 0.986 0.950 1.036 0.978 || 0.986 0.986
1.997 i.758 1.709 1,464 1.480 1.446 1.395 1.460 1.4.13 1.424 | 1.452
2.008 1.766 1.776 1.475 1.491 1.431 1.406 | 1.471 1.425 1.426 | 1.464
0.997 1.264 1.213 1.039 1,051 0,986 0.951 1.037 0.980 O.987 || 0.988
2,047 1,772 1.730 1.490 1.506 1.446 1.422 1.485 1.439 1.451 | 1.486
1.823 1.677 1.716 1.406 1.421 1.360 1.334 1.402 1.354 | 1.364 | 1.389
1.548 1.538 1.593 1,296 1.310 1.247 1.219 1.292 1.241 1.250 | 1.265
1.367 1.467 1,506 I.218 1.231 1,168 | 1.137 | 1.214 | 1.161 | 1.170 | 1.179
0.730 1.111 1.142 0.890 0.899 0.833 0.795 0.888 0.827 0.833 || 0.832
1.985 1.742 1.762 1.467 1.433 1.426 1.398 1.463 1.416 | 1.427 | 1.455
1.978 1,739 1793 1.464 1.480 1.420 1.396 1.460 | 1.413 | 1.425 | 1.453
1.920 1.723 1.726 | 1.443 1.458 1.398 || 1.372 1.438 | 1.392 | 1.402 | 1.428
1.982 1.745 1.771 1.466 1.482 1.421 1.397 1.461 1.414 | 1.426 | 1.454
1.049 1.245 1.185 1.066 1.078 1.013 0.979 1.063 1.007 1.015 | 1.011
2.061 1.754 1.765 1.495 | 1.511 1.450 1.427 1.491 1.444 1,454 | 1.486
1.916 1.748 1.722 1.441 1.457 1.396 I.371 1.437 1.390 1.401 | 1.426
2.056 1.758 1.747 1.493 1.509 1.449 | 1.425 1.489 1.442 1.454 | 1.484
2.002 1.749 1.750 1.473 1.489 1.429 1.405 1,469 1.422 1.434 | 1.463
0.699 1.104 0.987 0.871 0.880 0.814 || 0.776 0.868 0.808 0.813 || 0.807
1,939 1.736 1.651 1.450 1.466 1.402 ' | 1.380 1.446 | 1.398 1.410 | 1.436
Table No. XXIII gives the percentage of difference between the meas-
ured discharge; that obtained from the meter observations and that computed
by the several formulae mentioned in Table XXII.
TABLE XXIII.
PERCENTAGE OF VELOCITY SHOWN BY METER AND DIFFERENT FORMULE.


Mean Velocity in feet per second computed from
Formula |Formula |Formula |Formula |Formula |Formula |Formula | Formula
A. B. C. D. E. F tº
Discharge. Meter.
1.335 1.345 | 1.369
1.651. | 1.610 1.386 | 1.406 1.213 1.315 1. 383
— |
we a * | * | * | *
* | **-m-s-- ****
Per cen t
81 81 83.
61
The meter gives 97.5 per cent. of the discharge; and Mr. Chesbrough
estimates that the ordinary leakage of the pump valves would make the
actual discharge at least two per cent. less than that given by the pumps,
while the discharge calculated by the formulae is only a little more than
80 per cent. The same or greater difference has been found in the Brooklyn,
Croton, Cochituate and other water-works, where the large mains discharge
fully 20 per cent, more than the estimates. Thus we have another test of
the meter, at depths of nearly 40 and 60 feet below the surface, and still
the cóefficient is found correct.
Therefore Gen. Abbot's fears that the translation of the revolutions into
velocity is impossible, are shown to be groundless, and he can rest assured
that the “key” to this “pretty toy” has been discovered.
Of course, from so few observations, we can hardly expect to obtain the
curve of velocities in the tunnel, but they are sufficient to show that it
approaches the elliptical form.
In Table No. XXIV, the mean observed velocities are compared with
an ellipse computed by the formula given on page 36.
x = C + *{ 2Ay—y”) # in which C =1.215 A=32 and B–8.
TABLE XXIV.

Distance of points of observation Velocity in feet per second.
from the side of the tunnel
Observed Computed | Difference.
0 - 1.215
: inches 1.575 1.608 +0.033
9 s & I-68, 1,660 —0.021
12 “ 1,743 | 1.7:1 —0.022
15 g & 1.797 I,765 —0.032
30 & & I,793 1.809 +0.016
——| 1.850 | 1.885 - +0.035
Sum of differences. 0.159
Mean. 0.026
The differences are quite small considering the paucity of observations.
“The uncertainty as to the cóefficient is my first objection to the use of
a meter, and my confidence in its validity is strengthened by the following
discussion of Assistant Henry's table of comparison between floats and a
meter in the St. Clair river, the only data he furnishes where the depth
varies. The first point is the considerable absolute difference between
some of the velocities indicated by the two methods. This possesses no
significance, however, owing to the great length of the base, which was
62 g
f
about 700 feet. In traversing so long a distance, the floats must have had
many different velocities, and there is no reason to suppose that the mean
of the whole of them (which was measured), was identical with that with
which it passed the meter.”
As Gen. Abbot, in his letters to me, was continually speaking of the
errors due to the length of the base, apparently mistaking what I call “irreg-
ularities of flow' for a want of permanent motion in the current, I divided
the base into three nearly equal scetions, and observed the time of passage
and location of the floats past the four points. The results of these obser-
vations are given in Table No. XXV, where we have the mean velocity
past each section, and also past the whole base. The difference in the
means is very small, although the individual floats, as might have been
expected, differed greatly.
TABLE XXV.
SECTIONAL OBSERVATION OF VELOCITY BY FLOATs, ST. CLAIR RIYER. BASE NEARLY
FEET.
Velocity in feet per second past equal sections of Base.
Depth be-| No. of
low sur- | Floats
face. passed 1st. 2d. 3d. Whole Base.
*-* -º- a -m-, -º-º: * * |
5 33 3.980 4.074 4.005 4.000
10 32 4.003 4.014 , 3.944 3.996
15 20 3.959 4.120 3.965 4.015
20 22 3.934 3.922 3.751 3.831
25 14 3.869 3.927 3.827 3.835
30 15 3.799 3.787 3.625 3.717
35 9 3.616 3.704 3.518 3.598
40 4 3.824 3.835 3,481 3.081
Mean. 149 3.873 3.934 3.765 3.833
Thus we see that as far as the mean velocity is concerned, there is little.
difference between floats observed past a short or a long base—which is
merely equivalent to saying that the current had a “uniform motion” where
these observations were made.
Gen. A. then goes on to discuss the differences between the floats and
meters at various depths; but as on page 13, the observations of 1868 are
combined with those of 1869, and discussed in a much clearer and more
scientific manner, it will be useless to copy Gen. A.'s review. There is,
however, one very characteristic sentence which must be quoted.
“d, (the depth of the axis of the parabola) was taken so as to cause the
best accordance between the observations and the parabola, viz: at # D.
(depth of river) for the float curve and # D for the meter curve. * * * *
This fact proves that the float curve accords best with the general laws of
63
flowing water in natural channels so far as the position of maximum.
velocity with reference to the surface is concerned.”
Both the float and meter observations that he was discussing give the
maximum velocity at the surface, but that fact was nothing compared with
the other “fact” that a parabola could be found which did n't disagree
much with the observations, only it made the maximum velocity # the
depth below the surface; and therefore the float observations agree best
with the laws of flowing water discovered and modestly announced by
Gen. Abbot. But to proceed: &
“This discussion seems to me to show 1st, that the two curves (by floats
and meters) are parts of one and the same parabola predicted by the above
general formula derived from the observations on the Mississippi.
“2. The divergence between the floats and meter curves is that due to
raising the axis of the parabola ͺs D above its usual position for this class
of streams. - º
“3d. This divergence indicates a radical error in one of the methods of
measuring velocity, which in these curves seems to have its minimum value
about mid-depth.”
(Although the observations, as will be seen in Table V (ante p. 13)
have their minimum divergence at the surface )
“In order to discover which instrument gives erroneous results varying
with the depth, let us compute how much of this discrepancy can be attrib-
uted to errors of the double float—a matter which admits of close calcula-
tion. * * * The surface of his upper and lower floats exposed to the
current, appear to have been about 2 square inches and 110 square inches
respectively; the connecting cord was tº of an inch in diameter. * * *
Let us compute the numerical value attributable to his fifth and sixth
objections.
Supposing that the water offered no resistance whatever to the dragging
of the lower float by the upper, we should have for the rate of motion of
110 × 2.875 + 2 × 3.720
110 + 2
Hence 2.890—2.875=0.015 foot, would be the acceleration under that
supposition. But in reality the effect of this mutual dragging will be
divided between the upper and lower floats in the ratio of their areas of
resistance to the water; i.e.; the lower float will be accelerated 0.015 x *,
= 0.0003. This represents the extreme dragging effect of the upper float
by the lower.”
=2.890
the combination
I have already stated that this dragging effect was of but little conse-
quence, and therefore it is hardly worth while to comment on the inaccu-
racies in the above calculation. . .
“Let us now consider the connecting cord. Its dragging effect computed
in the same manner as the above is 0.0193 foot. Its raising the lower float,
and hence accelerating the velocity by transferring it into a swifter current.
is due to two causes: First, its bending into a curved form from the differ-
st
64
ence of velocity in the current acting at different parts of its length ; and
second, the shortening of the perpendicular distance between the lower float
and the water surface due to the upper float advancing a little to the front.
If the lower float"was perfectly ballasted, neither of these causes would sen-
sibly raise it ; but nevertheless, let us assume that the cord took the form
of the curve of sub-surface change in velocity, and that the upper float
resisted the influence of the 0.845 foot retarding effect of the lower, suffi-
ciently to move forward in front of the vertical five feet, giving a slope of
about one-seventh to the cord, both extreme estimates. The first would
raise the lower float 0.13 foot and the latter 0.36 foot, total 0.49 foot, which
according to the meter curve would give an increased velocity of 0.0246
foot. Hence the maximum possible acceleration of the deepest float due to
all disturbing causes was .
Dragging influence of the upper float * > * 0.0003
do do do cord g- * 0.0193
Effect of raising the lower float * *- - 0.0246
Total. gº - - *& º - 0.0442
* * * * * Hence, we are forced to conclude that the meter is in error
at this depth 0.330–0.044 = 0.286 foot. This error is chiefly attributable
to the meter cóefficient being inaccurate.”
Now it has already been shown that the cóefficient cannot be inaccurate,
and it would seem as if Gen. Abbot willfully ignored the fact that the
double float is not a rigid system, but is liable to be swept about by every
varying current. For the best form must be that in which the difference in
the size of the floats is the greatest, and the upper presents the least resist-
ance to the movements of the lower. -
As we have seen, the form recommended in the Mississippi report was a
tin spheroid 6 x 1% inches for the uppper float, and an old paint keg 12 x 8
inches for the lower. The total displacement of the upper float is thirteen
ounces, but it was ordinarily sunk only a little below mid-depth. This had
to sustain over fifty square inches of tin, twelve inches of No. 9 wire, with
the solder weighing nearly six ounces, a flag six inches by two to four
inches—generally wet—at least one-half ounce, the connecting cord from
one to two-tenths of an inch in diameter and sometimes over one hundred
feet in length, certainly half an ounce; and we have remaining only one
and a half to two ounces for the weight of the keg, giving its specific
gravity probably less than 1.02, which, taking into account its size would
make it almost like a feather or a thistle down in the air. If the system
were rigid, the above calculation might be taken as correct, but unstable as
it is, it practically seems almost impossible to compute the actual position
of the lower float in the ever changing sub-surface velocities. But Gen.
Abbot assumes that the floats were improperly ballasted to give even the
65
small error his calculation shows, and in the Mississippi report he affirml,s
with as much confidence as if he had been able to see the whole course of
both floats through its turbid waters, that when placed in the stream, the
lower float “immediately sunk to the depth indicated by the connecting
cord.”
“A second objection to the meter applies when used in most natural
channels. The irregularities of the bottom always give rise to local upward
currents. They often may be seen breaking at the surface where they form
whirls sometimes a few inches and sometimes feet, in diameter. They are
evidently small internal currents of water which being swept along by the
surrounding element, pierce the mass in an upward direction with much less
velocity, viz: that with which they were originally deflected from the
bottom, diminished by subsequent friction on the surrounding water. * *
Now what would assistant Henry's meter do if placed in a current affected
by an internal whirl! Since the revolving part is hung in a yoke, and back
of it are vanes to keep the cups in the direction of the current, it would
evidently tip upward and register a velocity which is oblique to the vertical
plane of cross section. But it is from the perpendicular component alone that
the discharge can properly be estimated. Hence, the bottom velocities,
(where these irregularities of flow are most frequent and powerful), may
not be correctly registered by the meter. Floats always indicate this com-
ponent correctly, and by soon escaping from a whirl will usually be little
affected by it.”
This graphic description of the whirls is no doubt correct in the main,
except the omission of the fact that these whirls are swept along with nearly
the velocity of the current. Therefore, while the meter would be momenta-
rily tilted to record the velocity of the passing whirl, and then immediately
resume its normal position, thus affecting the mean of the revolutions
through five or ten minutes but little ; the lower float when caught in one
of these whirls would be borne upward by the ascending current—like a
feather floating in the air—into the swifter flowing water near the surface,
and would scarcely be able to free itself from the whirl in the short distance
of 200 feet; and even past the longer bases used on the Lake Survey would
hardly have time to sink again to its former depth. I confess that before
I read Gen. Abbot's description, I had not given weight enough to this
cause of error; but it certainly must be nearly equal to the sixth error in
raising the lower float, and thus giving too great a velocity. These whirls
or eddies are also noticed by Capt. Boileau (see ante p. 15).
“My third objection applies to the meter when used in natural channels
where the water contains earthy matter in suspension. There is usually
more of this matter near the bottom, how can we assume that it does not
increase the fricvion on the axle, and thus cause the instrument to register
too slow a relative velocity in that vicinity!”
66
This objection has a little weight, though with meters constructed as
those used on the Lake Survey, and in the clear rivers connecting the lakes,
its effect would be very small—practically of no account. Possibly, were
the meter to be used in the turbid waters of the Mississippi, this friction
would need to be guarded against. w
“My fourth objection, Assistant Henry himself discovered to be a valid
one. It is, that the boat over the meter retards the flow of the water. How
can he or any one else tell how far below the boat its influence extends. It
may materially modify the vertical curve which his meter is measuring.”
Suppose it did have that effect, it would certainly be greatest near the
surface, and there the records of both meter and floats agree, while they
diverge as they near the bottom.
“Moreover it is not improbable that the two anchor lines, the meter
weight frames, &c., all held rigidly in the current, may modify its delicate
adjustments. -
“A fifth objection, to which Assistant Henry apparently attaches con-
siderable importance is, that if the line supporting the meter be not drawn
perfectly taut, the depth of the latter below the surface will be over esti-
mated.” -
This is no doubt true, and as I have stated, I do not consider observa-
tions less than three feet from the bottom reliable in deep water, but it has
no weight in this discussion, as the meter showed considerably less velocity
than the floats, and if the cord was not taut, the recorded would be greater
than the actual velocities.
Thus, of the five objections of Gen. Abbot to the use of the meter, only
one has any weight; that is that the solid matter held in suspension may
increase the friction on the axle, of which the effect must in any case be
quite small. -
“Since Assistant Henry observed the velocity from surface to bottom at
nearly every station occupied by his boats, it would naturally be supposed
that he had accumulated a mass of interesting sub-surface data. Unfortu-
nately, his float observations possess little or no value, because, with so long
bases they cannot in general correspond to any single vertical plane; and
because as printed, he has grouped them in divisions of 200 feet in width.”
Gen. Abbot has a great deal to say about the wonderful accuracy of float
observations when such a base as he recommended is used. Dr. Hagen
whose review of the Mississippi report has already been alluded to (Bewe-
gung des Wassers in stromen, p. 7), remarks: “The distance over which
the time of the passing float was recorded, was so short that its velocity
could not be determined with any accuracy nearer than tenths of a foot.
To reduce these observations to ten thousandths of a foot is therefore an
67
exaggeration.” This will no doubt be found to be correct by any reliable
observer.
“The curve discussed above includes all his data except those obtained
with his meter, which, as has already been explained, cannot be accepted
as free from the influence of instrumental errors, that may mask the true
form of the curve. Believing there is little left to be learned respecting the
form of the curve, which after a thorough investigation upon the Mississippi
survey was experimentally determined in so complete and satisfactory a man-
ner as both to bear the test of subsequent experimental investigation and to
satisfy the theoretical conditions of the problem in the minds of such men as
Prof. Pierce and Prof. Barnard in this country, and M. M. Dupuit, Grebenau,
Messadaglio and others in Europe, it is with some surprise that I find Assist-
ant Henry quoting meager experiments made in the earlier part of this
century, as among the most trustworthy of existing data; and finally (ignor-
ing the logical indication of his own instrument), even falling back upon
the old a priori method of reasoning in vogue two centuries ago, and telling
us that the curve must be one whose vertex is at or near the bottom. Such
reasoning does not require an elaborate refutation; and I shall only refer
to two grave errors which might mislead a reader who was not critically
examining Assistant Henry's arguments. The first of these errors falls
within the scope of elementary school books, but nevertheless pervades
Assistant Henry's reports. I refer to the matter of interpolation. His
views are thus expressed : ‘If there are but few observations missing in a
large number taken, it will make but little difference whether we interpo-
late or not, for it will make but a slight variation in the mean, but if the
missing observations are numerous, then (as they must be interpolated
either in a straight line drawn between the known points or in a curve),
the mean will approximate more or less to the line or curve by which the
interpolations were made.”
Hence, in general, he rejects interpolation in reducing his means. I
cannot better illustrate the absurdity of these views than by an example.
Let us take two parabolas, and computing a number of corresponding points
of each, find a true mean ; then conceiving certain points of one parabola
to be unknown, let us find the corresponding points of the mean curve,
first by the usual methods of interpolation, and second by Assistant
Henry's method. The following table exhibits the result: the points con-
ceived to be missing being those at #, D and Pol).

TABLE XXVI.
Bepth Velocity. Mean |& 2.É Mean curve deduced by
Para- || Para- curve |E|3: ~3 to
below * from T. & 5 Usual § 53
bola | bola | Nos. 1|T.3, Error.]: #5 | Error.
axis. and 2.3 3.5 met’od #:
No. 1 | No. 2. 3 3R a
0 4.180 | 1.480 2-830
0.1 D 4.171 | 1.495 i 2.832
0.2 “ | 4.144 | 1.459 || 2.802
0.3 ** | 4,100 | 1.433 2,767
0.4 “ 4,035 | 1.395 || 2.715 4.027 2.711 || 0.004 || 1.395 || 1,320
0.5 ‘‘ 3.954 | 1.349 || 2.651
0.6 ‘' | 3.854 1.289 2.572
0.7 “ 3.736 | 1.220 | 2.478 || 3.754 2,487 || 0.009 || 1,220 1,258
68
Hence, to avoid an error in the third place of decimals, Assistant Henry
would introduce one in the unit's place. It is however, due to him to
remark that his curves do not differ quite so largely in absolute velocity as
these two ; and that the number of his curves is usually more than two,
both of which facts would tend to reduce the numerical value of his errors.”
General Abbot's sense of justice is unique, and cannot sufficiently be
praised. Let us look into this matter of interpolation a little.
While trying to work up the notes of the first season's work with floats,
it was found impossible to plot the observations in any regular curve ;
neither did they agree among themselves in the same or in different rivers.
I still had great faith in the Mississippi Report, and thought the failure
must be due to my methods of reduction. I therefore more carefully
examined that portion of the report describing those methods. I do not
think there can be found, at least in any scientific work, a more mystify
ing description; and yet it is so skilfully written as to seem to the casual-
reader perfectly straightforward and simple.
In the first place, on pages 222 and 223, we have an elaborate descrip-
tion of the method of putting out floats and making the observations
which we are told was used in all the work on the river; but just beyond,
on page 231, it is stated that for the purpose of investigating the form of
curve &c., the observations from anchored boats were used.
Again, we find that the daily observations were plotted on a large scale,
and missing ones interpolated ; then different days were combined, and
again missing observations interpolated, and the mean of all these observa-
tions and interpolations were plotted in the curves in Plate XI. If from
these observations the final velocity curves could be plotted, why seek
further, and why not give us at least the means from which these curves
were derived? But no, we are referred to Appendix D for the observa-
tions, and there we find only the velocity at the surface or five feet below.
Passing on to page 231, we find a series of observations embracing the
result of about 2,000 floats. This is probably the most incongruous mass
of data that was ever attempted to be used for the determination of any
scientific question. Any ordinary observer would have discarded at least
half of them. Dr. Hagen says, page 6: “Extraordinary anomalies are
shown when these series are graphically represented. In some of them the
curves are quite regular, but in others the velocity remains nearly the same
from the surface to the bottom; but frequently with increasing depth it
increases and decreases considerably. Generally, the velocity decreases
with the depth, but a few series have an opposite tendency, that is, a sur-
: ; r *#.
, jº
69
prising increase near the bottom of the river. * * * * If we com-
pare the several verticals in each series, we find that only the first one in
the series forms a tolerably regular curve; the rest, and all the verticals
in the second and third series are so very irregular, that hardly any
decrease of velocity near the bottom can be noticed. There is also an-
other peculiarity; some of the observations are one foot below the bottom
of the river, In the fourth series, the verticals form more regular curves,
and still more so with one exception, in the two latter series. These
series follow in the order of the dates of observation; and it seems that
the observers gained accuracy by practice, and obtained more correct results
by the middle of the year 1851.”
As we have heretofore seen, 23 out of the 39 verticals give the max-
imum velocity below mid-depth, and should have been rejected at any
rate. These series, we are again told are plotted in the curves in Plate XI.
But why use these in preference to the more numerous observations of the
floats? Let us look further. On page 251 are given some observations
made on a small canal near Washington, after the completion of the Mississipi
Survey. It was found that these velocities would plot very near to a par-
abola similar to one used by Capt. Boileau in reducing his observations
on a small experimental canal. Now, have we not the probable reason for
not giving the means of the regular velocity observations, and using in their
stead those few selected ones from the anchored boat observations just
noticed! Finding it impossible to work out any results from the regular
observations, these measurements in the little &anal were made, and the
observations from anchored boats chosen to compare with them. And as
we have before seen, the maximum velocity having been correctly found
below the surface in the little canal, those observatičns which almost any
one would have rejected were chosen, in order to make the means of the
series show the same And now comes the real interpolation in this work.
As these series were at different' depths, they could not be combined
directly, and therefore they were plotted on a large scale, and horizontal
lines drawn dividing them in tenths of depth, and the points where these
lines intersected the curves taken But what curves? Why the curves on
Plate XI, or in other words the parabola which was found to fit the obser-
Vations in the little canal. Therefore the real data from which the new
theories were obtained, were not even these anomalous observations, but
were the plotted points of these parabolas. We are afterwards told with
great exultation, that the combined series differed but little from a calcu-
lated parabola. Probably had it not been for unavoidable errors, it would
70
not have differed at all. Would it not be well hereafter, to change the
phrase “arguing in a circle,” to arguing in a parabola After Gen. Abbot
had arranged these data, his assistants worked out the formulae, and appar-
ently no one except Dr. Hagen has attempted to examine their foundation
—the superstructure seeming perfect.
This is what has misled the scientific men who have examined the work,
and it is an evidence of the infantile condition of the science of Hydrau-
lics that such work could be allowed to pass unchallenged by such scientists.
Thus, interpolation was used in the Mississippi survey in four ways:
1st. Each day's work was corrected by interpolating the missing obser-
vations.
2d. The several day's means were combined and missing days inter-
polated.
3d. The observations at about the same depth of river and velocity of
current were collated, and any missing ones interpolated.
4th. To combine these latter means, they were plotted on a large scale,
divided by horizontal lines passing at tenths of depth, and the points noted
where these lines crossed parabolas drawn through as many as possible of
the plotted points. And therefore it was only by accident that the finally
accepted points were identical with the observations.
Such are the data upon which the new theories of flowing water were
based. And if ten days' work (in which time these observations from
anchored boats might easily have been made), is sufficient for the establish-
ment of these theories and formulae, why spend months and years in daily
observations on the river!
In the meter observations of 1868 there were so few observations
missed, that the change in the velocities at the different depths in the
several verticals would not have been affected beyond the thousandth's
place had the missing observations been interpolated; (except at the sur-
face—for reasons given in the report, and which were corrected the next
year.) The missing observations were not probably more than one in
twenty. In the first year with the floats the difference might have been a
little more, but much less than the error of observation.
[NotE.—As all hydraulic formulae which have been proposed by different engineers will give good
results when used for the same classes of streams as those on which the observations were made from
which they were deduced (though they generally fail under other conditions), the fact that Mr. James
B. Francis, in the Lowell Hydraulic experiments, found that the H. & A. formulae would give a near
approximation to the measured discharge of his experimental canal, which was very similar to the one
on which Gen. Abbot made these observations, while on the rivers connecting the Lakes, their formulae
were found to be totally unreliable, goes to prove the correctness of the above supposition that their
formulae were founded on those canal observations.] -
71
This shows what a merely superficial knowledge Gen. Abbot has of the
subject, when he says:
“This unfortunate method of deducing his means, renders it necessary
to re-compute the sub-surface velocity tables entirely before they can be
used.” *
We can also see how just is Gen. A.'s criticism, when he compares only
two verticals, and those in different rivers, one with a velocity of over three
miles and the other about one mile an hour, and calls it; “Henry's method.”
“He has selected two of them in their uncorrected state by which to
test the several curves proposed by different engineers. In applying this test
he committed the second error referred to above. * * * * He has
assumed so far as I can see a depth for the axis for which no authority can
be found in the Mississippi Report, where the proper manner of determining
this depth is clearly indicated, * * * The true axis of the St. Clair
curve is thirteen feet above, and that of the St. Lawrence three feet below
the surface.”
This is too absurd. Placing the axis of the parabola, or the locus of
maximum velocity, thirteen feet above the surface is nearly as bad as giving
the maximum one foot below the bottom. The formula for the depth of the
axis as given in the Mississippi report is
d=(0.317 + 0.06 f) r,
in which d, is the depth of axis, f, the force of the wind and r, the mean
radius of the river.
Now as the wind effect at the time the above observations were made, was
slightly down stream, the depth of the axis should have been taken at about
one-third the depth of the river. *
Instead of this I took it at the locus of maximum velocity, or five feet
below the surface, thus making the comparison more favorable to the para-
bola. Gen. Abbot on the other hand discards his own formula, and places
the axis by trial where the curve will best fit the observations. Knowing
this, the next paragraph reads as if Gen. A. had got a little beyond his own
depth of axis.
“Considering that the Humphreys and Abbot curve is not simply a para-
bola, but a parabola whose parameter varying with the velocity and depth
is exactly fixed by an equation, while Assistant Henry's ellipses follow no
general law, but were computed to represent these particular observations,
it seems to me patent from these tables, that his new data confirm the con-
clusions of the Mississippi. Report, instead of, as he imagines, contradicting
them.” -
After this speak of “a priori deductions!”
The review closes with a general recapitulation of the errors into which
I have fallen.
72
The following letter submitted the report for 1870, to the same person
whose just criticism, logical deductions and full knowledge of the subject
were made so plainly manifest in the former review.
ſ “OFFICE OF THE CHIEF ENGINEER,
\ WASHINGTON, D. C. July 22, 1870.
GENERAL:—
You will please examine at your leisure the accompanying
paper upon the subject of the outflow of the lakes, which forms one of the
appendices to General Raynold's annual reports upon the survey of the lakes
this year, and give your opinion as to its merits, and whether it, or any part
of it, should necessarily be printed in the next annual report from this office,
as a sequel to the same subject in the last.
By command of Bvt. Maj. Gen. HUMPHREYs.
Very Respectfully º
J. B. WHEELER,
Maj. Corps Eng. Bwt. Col.”
§
IBvt. Brig. Gen, H. L. ABBOT,
Maj. Corps Eng. Willets Point, N. Y.
Gen. Abbot only took a couple of days to reply to this order; his answer
being dated July 25th, and it only contains two or three points which are
worth considering.
“The first point last season in Assistant Henry's programme of general
hydraulic investigation, is thus stated by him: ‘Whether floats passed with
the same velocity through all parts of a given distance.’ The ambiguity of
the question is preserved in the answer, which is contained in three tables.
* * * A perusal of any standard discussion of permanent motion would
probably have prevented so barren an investigation, the result of which
is necessarily strictly local,” -
The third of these tables has already been given—Table XXV-and, as
we have seen, the work was undertaken at Gen. Abbot's suggestion, and I
thought all the time he was talking so much about long bases making the
work unreliable, that “the perusal of any standard work on permanent
motion” would have shown him his error.
“Upon the fourth point—‘The form of the vertical velocity curves at
surface and bottom,’ Assistant Henry reports: “We found it impossible to
make the meter run continuously below two or three feet from the bottom.”
Instead of distrusting the meter in consequence, as would seem to be most
rational, since floats will move at this depth, Assistant Henry concludes the
bottom velocities are exceedingly small. To support this view, he refers to
a bed of finely comminuted clay found by exploring forty feet below the
bottom of the Detroit river, which he argues would not resist the present,
73
current as measured. This might be granted without affecting the question
under consideration.”
Why? This seems to need a little explanation.
“He next proceeds to quote from Mr. Chas. Ellet, jr., to refer to fail-
ures in making cut-offs, &c., all to confirm his peculiar theory of a very
small bottom velocity. It is not often that an engineer resorts to argu-
ments like these to explain away accurate observations noted from the days
of Dubuat to the present time.”
If Gen. Abbot has ever read any works on hydraulics he must have a
very short memory to make such a statement as the above. For it was
Dubuat who first, discovered the wonderfully small velocity in the lower
layers of water; and even Capt. Boileau, from whom Gen. Abbot obtained
his vertical velocity curve, says that below the points observed, the veloci-
ties probably decrease rapidly toward the bottom. Gen. Abbot also forgets
that I have merely attempted to reconcile the observations on small canals
and large rivers; that I have shown that up to a certain proportion of the
depth and width, the maximum velocity has always been found below the
surface, above that, at or near the surface; and also that almost all the
European observations point to the fact that the velocity in immediate con-
tact, with the bottom must be very small. He alone, of all observers, finds
the locus of maximum velocity to vary from thirteen feet above the surface
to a foot or more below the bottom.
IHe too it is who has founded upon, say ten days' observations—and
such observations—on the Mississippi, and one day on the little canal, a
set of new theories and formulae of flowing water; and then has the effron-
tery to say of those scientists who have spent years in careful experiments:
“The fallacy of the prevailing idea that the maximum velocity is neces-
sarily at or very near the surface, is apparent from these diagrams.” M.
Report p. 255.
One other extract showing Gen. Abbot's ignorance of the Lake region:
“Tables XXVII to XXX would appear more properly to belong to the
metorological report.” - *
The tables referred to are on evaporation, showing the difference
between evaporation on land and on the water, the temperature Of
the air, water &c., being noted. The amount of evaporation from the
enormous lake surface is a quantity which cannot be omitted from any
comparisons of discharge and downfall; but Gen. A. did not need it on the
Mississippi, and therefore sees no use for it here. -
But to leave this unpleasant part of the subject. As we have seen, the
science of hydraulics needs more observers, and more reliable observations
i
j
74
to put it on the same basis as other sciences. Heretofore these observa-
tions have been so expensive and laborious that we have had no amateur
observers such as in the sister sciences have helped to extend the area of
facts which form their foundation. Thus, almost all our reasoning is em-
pirical; which is shown by the fact that more than a century after Pitot
with his “tube” had disproved the assertions of the older authors that the
maximum velolocity must be at the bottom, the scientific world with
hardly an exception accepts as correct, and without question, a set of
observations in which the maximum velocity is given not only at the bot-
tom, but a foot below.
But by the aid of the telegraphic meter, any one who lives near a stream
or river can sit in his own parlor and observe its velocity. For if a small
punt or large float be anchored in the stream, and the meter attached as
heretofore described, insulated wires can be run from it to any convenient,
place on shore where one can at his leisure record the observations and
study the laws of flow ; and all this with a small expenditure of money
and of such fragments of time as may be spared from other pursuits. A
few years of such observations would do more toward lifting the science
from its present darkness toward the light of the perfect day than all
those which have gone before ; and we might then be able to put our tables
and formulae into the hands of the engineer who wished to know the dis-
charge of certain streams with as much confidence as the astronomer now
feels in referring to his almanacs and catalogues, those who wish to use the
stars as guide-boards across the trackless waters of the ocean.
[The ill-success attending the use of the double float and the Humphrey's and
Abbot formulae, and their course in quashing all further investigation, lest it should
disturb their published theories, may remind the reader of an adventure of the
immortal Gil Blas of Santillane, which was something in this wise: It will be remem-
bered that Dr. Sangrado took this worthy into partnership, instructed him in the
two principles of medical science—bleeding and warm water—and sent him forth to
practice. After sometime Gil Blas meets his preceptor and complains to him that
although he has followed most rigidly his master's instructions, he cannot save a
single patient. It seemed as if they took a malicious pleasure in dying just to
cast discredit on his system. My friend, says the worthy Doctor, I own that the
same thing happens to me. Why then, says the younger practitioner, would it
not be worth while to take up a different system for a time, and should it fail it
would be easy to return to the lancet and syringe. There is reason in what
you say, my poor Gil Blas, replies the chief, but the fact is I have written a LARGE
Book in defence of my theory, and what, I ask you, would beeome of that if I now
acknowledge myself mistaken 2 Ah well ! says the youthful aspirant, that indeed
alters the case. Let the poor devils of sick folks perish. Long live science, and
long live Dr. Sangrado's big book ||
75
I have been permitted by that distinguished mathematician, PROF.
S. W. Robinson, of the Illinois Industrial Institute, to use the following:
MATHEMATICAL INVESTIGATION OF THE USE OF FLOATS IN GAUGING
RIVERS AND STREAMs.
Three questions arise in the use of the double float for the determination
of the velocity of a current.
1st. Will the upper float be vertically over the lower?
2d. Will the connecting cord be straight !
3d. Will the observed velocity of the upper float give the true velocity
of the current at the lower'ſ
The answers to these questions may seem self-evident, but they certainly
demand some investigation, from the fact that extensive gaugings have
heretofore been made upon the assumption, that the error of floats, if any,
was so small as to be safely neglected, and it may therefore be necessary to
apply a correction to all such observations to bring them to that standard
of accuracy, which the care and precision with which the work has been con-
ducted warrants.
Fig. 15 shows a system of floats as acted upon by the current. The
lower float being much larger than the upper, will tend to move with the
velocity of the strata of current in which it is immersed. But as the cur-
rent is usually faster near the surface, its pressure upon the upper float and
connecting cord will affect to a greater or less extent the common velocity
of the system.
Let v. = The surface velocity.
m V, - Velocity of current at any depth.
u v' = Velocity of current at lower float.
tº v, = Observed velocity of floats.
ſ
P = Pressure of current against the cord.
m P = Pressure of current on upper float.
iſ P2 = Tension of Cord.
11 Pa = Pressure of current on lower float.
n P. = Horizontal component of tension of cord.
tº ai = Sectional area of upper float exposed to current.
it as = Sectional area of lower float.
it r = Radius of the connecting cord.
1 c, - Coëficient, found by experiment, giving pressure of current upon
units section of upper float at units velocity.
it c, - Coëficient similarly found for lower float.
76
Let c = Similar coèfieient for connecting cord.
it t = Tension of cord at any point.
n i = Angle of inclination of cord with horizon.
m W = Immersed weight of lower float.
iſ x and y = Coördinates of the curve of cord, origin at upper float. **.
The general expression for the pressure of a current of water upon a
2 -
body resting in it, is p = 6 k *::: in which 6, is the weight of a unit volume
O
5
of water = 62.5 lbs; a = the sectional area; g, component of gravity = 32.16,
and k = a coèficient found by experiment, which is -0.77 for a cylinder
moving sideways, and = 0.51 for a sphere. By substitution and reduction
we find :
p = k a vº–0.748 a v" for cylinder,
and = 0.495 a vº for sphere.
The effective velocity past the upper float is
Vo — VI, . . . P =c, a (vo–V.)”
For lower float, P. = c, as (v-v')*= P, + = & P.
Dividing the cord into elementary lengths, an elementary area will be
2r dy. Hence .
p=c 2r dy (v.-vi.)”,
is one of the elementary pressures, and
> . P = 2 cr (vº-vi)*dy,
is the total pressure from surface to depth y,
Considering the weight of the cord when wet as without disturbing
effect, (unless the inclination of the cord to the current gives rise to an ob-
lique pressure, which, however, would be so small that it may be neglected),
the vertical component of tension will be the same at all depths; and ob-
serving that the action of the current upon the cord is partly a horizontal
pressure and partly friction against its sides ; we have
* P,-- W =t sin. i --. P. = P, + s : P =t cos. i.
The above expressions are based on the principles of the equation of
forces ; the equation of moments enables us to establish two more equations.
Taking into account all the forces from surface to depth y, and the origin
of moments at the upper float, we have -
WX=P, y—y: ; P;
in which y = the depth of the centre of pressure of the current on the
cord. Letting x, and y, be the coördinates of the position of the float for
any paritcular length of cord; we have
WX = P, y. - 7, > § P.
77
Of course y, will not be equal to the length of the connecting cord,
unless it is straight, and the upper float vertically over the lower.
In finding the value of s : P, we notice that v, is a variable, and must
be expressed in terms y, before integration is possible. This expression
depends upon the law of change of the current velocity in a vertical. Al-
though this may be one thing sought, it will probably be sufficiently
accurate so far as the effect of current upon the connecting cord is con-
cerned, to assume some law which has been found to be nearly correct.
Different observers have, however, given different laws, though gener-
ally varying as the ordinates of an ellipse or parabola. The curve, which
probably approximates the nearest in all cases, is an ellipse, with its axis
horizontal and slightly below the surface.
The equations are, however, much simplified by placing the axis at the
surface. The equation of an ellipse, with the axis at the surface and
origin at centre, is --
x* d” + v.” y” = v,” d”, whence
2
x*=v,”=v,” (-; )
In the equation of the parabola, axis vertical, we have
x*=2|P (nd—y), or x*=v,”=v,” (1-#),
in which n = depth of vertex below surface divided by the depth of the river d.
Taking the axis of the parabola horizontal, and at a depth h, below the
surface, with the origin a distance = b, horizontally from the vertex, we
have
h?
—w
(y-by-r
We have then the following fundamental equations for the solution of
-(b—x), or x = v. = b – ‘Fºlº (y –h)*.
0
the problem.
P = e, a (vo–Vi)*. (1)
Ps = ca as (vi-v')* = P + s § P. (2)
> . P = 2 cr ſ: (v.–V)* dy. (3)
P, - W = t sin. i. (4)
P. = P + = . P = t cos. i. (5)
WX = Py—ys; P (6)
WX = Psy-y, E § P. (7)
78
v: = v. (-; , for elliptical law. # (8)
v: = v. (-i). for parabolic law. (9)
Il
b—v
h?
The value of > 3 P – the pressure of the current on the cord—might
at first thought be deemed inappreciable ; but in deep rivers with a
cord of ordinary size, say one-twelfth of an inch in diameter, it is too great
to neglect, and in fact it becomes the chief disturbing cause when the upper
float is small, with a total displacement of say twelve cubic inches. For if
the cord be 60 feet in length, the area presented to the current will be 60
square inches, while that of the float, if half immersed, will be only six
square inches.
° (y – h)”. (9a)
or V. = b —
To find the value of > y P, substitute in Eq. (3) the value of Ve from
Eqs. (8), (9), or (9a), and integrate. -
Thus substituting from Eq. (8);
y —;
4: = p = 2 ſ. (V, V-1, -way
2 -3 am=mmº-ºººº-
= 2 c r 2 2 _ Yo Y f 2 . . . . )
or (ºy 3 d * : V → V, air #)
for the elliptical law, or when the current velocity is taken as varying
according to the ordinates of an ellipse tangent to the bottom of the stream,
with the horizontal axis at the surface. In the same manner we can find
the integral when the velocity curve is taken as varying according to the
parabolic law. These expressions give the pressure upon the cord for any
depth, y. To find the pressure upon the whole cord make y = y, in the
above. The difference between the pressure computed by any two of the
above, for a river 50 feet deep, with the lower float anchored at the bottom,
is about ten per cent. of the total pressure on the whole cord.
To find the equation of the curve assumed by the cord when the system
of floats and connecting cord are set free, we have, by combining Eqs. (4)
and (5),
d x_ = . P + c, a (v, -vi)*.
t sin. i dy W
79
§
Substituting the value of s : P, found above, we have for the elliptical
law :
* f ( (v. vº v; y” ——
X = —— Vo" + V. — — — —- — V a V. V y
# / (6. ') y – i. ows VII-
... — 1 y
Vo VI d sin d + C1 all (Vo * v.)” ). V,
and by integration : • y
2cr. ſſ., , , , y” väy’ vow, dº [. T 2.
sºm *** * Vn + W. ). - – -: *-- - - - - --—- *— 9 | y
x = w (vi '); 12d? 3 | 2 |V_j.
2 *
3. Vo v1 d” – vov, y d sin º, + =w=(V, - v.) y.
In a similar manner the integrals for the curve, according to the para-
bolic laws, can be obtained.
In using these equations to obtain the ordinates of a curve, to locate
the float, we observe, first, that all the quantities entering into the second
members of the equations are constant except y, which changes according
to the desired frequency of the points to be plotted in tracing the curve.
Several points should therefore be obtained for each length of cord
used, or for each depth of lower float. This will give a series of curves
equal in number to the depths of floats ; but it should be particularly
observed that in computing the ordinates for the curve for any depth of
float, that the proper value of vi, the observed velocity at that depth should
be used. -
These equations enable us to answer the first two questions stated.
First, the cord connecting the floats is not a straight line, because the
expressions are equations of the third to the sixth degree. If, however,
the effect of the cord could be rendered inappreciable by decreasing its
size, and increasing W, the second member of these equations will be
reduced to the last term, and all the equations will then be the first
degree, or of straight lines.
Second, as all the quantities have appreciable values, the last term can
not become zero unless vi = Vo, which is only possible at small depths. At
such depths the effect of the current upon the cord is slight, and the floats
will then be nearly in the same vertical, but it is evident that this cannot
be true at great depths, and the lower float, as it falls back, will rise
above the depth indicated by the length of the connecting "ord.
This rise of the lower float, and consequently its passing into a swifter
current, was surmised before the above analysis was attempted, by Mr. D.
80
Farrand Henry, Chief Engineer of the Detroit Water Works, while
engaged in extensive river gauging operations on the rivers connecting the
Great Lakes. He suspected that the neglect of this effect, or even that of
the swifter current upon the cord and upper float was not only unscientific,
but led to gross inaccuracies, causing the results to be in no respect com-
mensurate with the great care and efforts for accuracy, bestowed mpon
the observations. *
An inspection of the above analysis will show that it can easily be made
applicable to an ellipse when its origin is shifted to the right or left, by a
simple substitution. But when the horizontal axis is taken above or
below the surface, a constant is introduced, so that the equation must be
reintegrated; which though possible (having been effected by the writer),
leads to such a complication of the formulae as to overweigh the advantage
gained. The errors arising from the use of the simpler formulae will
increase with the size of the connecting cord, and in practice the cord
has been taken so large as to make the most accurate location of the ellipse
necessary; but it may be taken so small as to render the above formulae
sufficient. k
The first modification proposed, however, will be found useful, as the law
can be more completely expressed, and the formulae are practically correct
in most cases. This modification suggested itself while examining the
equation of an ellipse which Mr. Henry found would agree very closely
with a set of velocity observations, obtained by floats at different depths in
the Mississippi, as given in Humphrey's and Abbot's Report.
The equation of the ellipse with its origin and horizontal axis at the
surface is,
2 -
x-v.–B/ K. : C ;
in which A and B are the vertical and horizontal axes respectively, and C
the distance of the origin from the centre reckoned, in a negative direction.
Substituting this value of v, in Eq. (3), it becomes,
= . P = 2a ſ. (e Wºº-º-o) dy ;
As
an equation which is identical with that heretofore obtained by substitut-
ſº
ing the value of v, from Eq. (8), and making B = v., A = d, and vi – c = Wi.
Therefore it is not necessary to perform the integrations, but merely to
substitute in the former integrals or formulae these values. Hence we
have,
81
& ** Bºy” { —,
> . P = 2 c r (*16-o), -º-be-ory VI:
B (v) –C) A sin - X) ;
which is an expression for the total pressure of the current against the
cord between the surface and the depth y.
In the same manner for the equation of the curve we have,
2Cr * L (vr y? Bºy" A* y” tº tr?
x=# (ºr 6-cy); -º-be-o) (; )/- X."
;B & O A-Bº, -o) A y in º º(B-6-0)'s,
and for the velocity of the current at the lower float we have,
- --- P
v' = v, - We a (B-6-0) y=% *
3.
C; a 3
> § P being found from the first modified equation above by making y=y.
In the first term of the second member of the equation, v, does not become
v, - C, because the first term in Eq. (6) from which the above is derived
has not entered into the integrations; besides it is evident that the quan-
tity under the radical gives the difference between v' and vi, so that v'
must remain the observed velocity of the floats, to make the v' the true
velocity of the current.
In the calculations for the value of > § P, the actual depth of the
lower float may differ considerably from y, the assumed depth or length
of the connecting cord used. If so, it must be obtained by plotting the
computed coördinates of the curve of the cord for each depth of float, and
laying off on the curve the length of cord used. The point thus found
will show the true position of the lower float, and its actual depth below
the surface can be measured.
An inspection of the formulae will show that x decreases with the in-
crease of W and the decrease of r ; also that as W becomes greater, a will
be increased—that is the float will sink deeper—its form remaining the
same. If the cord is thick, the influence of the current upon it will be
greater than upon the upper float. It is therefore advisable to increase W
to a certain limit, which, however, depends upon so many conditions that
it is not easily determined.
82
The form of the upper float also affects the value of x, for this value
decreases with the decrease of c1.
It can be shown by analysis that a broad, thin upper float presents less
area of resistance for a given displacement than one of a more compact
form, both a and x being decreased. Therefore a flat ellipsoid is one of
the best forms for the upper float, as bodies of that shape are among those
offering the least resistance.
In future observations with double floats, the endeavor should be to -
keep the deviation of the cord from a vertical and straight line as small
as possible, and thus save much labor in the computations. This can
be done by making the upper float of the best form, increasing W to its
limit, and using a small connecting cord, or perhaps a fine wire.
But the errors in past work must be eliminated before float observa-
tions can be compared with velocities measured by a Tachometer or other
accurate instrument.
S. W. ROBINSON
School of Mechanical Engineering,
Illinois Industrial University.
To test these formulae, and also the correctness of my views in regard
to the Mississippi observations, the following example has been fully
worked out by Professor Robinson and myself.
On page 230 et seq., of the Mississippi report are given, as we have
before noticed, certain series of float observations on which the “new the-
ories and formulae " are based. The means of the first of these series are
given in the second column of the following table.
TABLE I.
OBSERVED VELOCITIES COMPARED WITH ORDINATES OF ELLIPSE AND PARA-
BOLA.
Depth of Observed -— PARABOLA. - ELLIrs;
Float Velocity of Ordinates | Difference Ordinates | Difference
in Feet. Float. of from Of from
y Curve. observed vel. Curve. - observed vel.
0 4.230 4.245T TO.005 TT.I.314 +0.084
1S 4.298 4.285 – 0.013 4.306 +0.00S,
36 4.346 4.300 –0.046 4.281 –0.065
54 4.274 4.270 +0.004 4.236 —0.038
72 4.158 4.190 + 0.032 4.16S + 0.010
90 4.053 4.062 +0.009 4.063 + 0.010
102 3.948 3.965 + 0.017 3.954 + 0.006
110 bottom 3. S75 3.840
114 == A 3.664__
'TVean. T 0.018 0.031
1. 2 —s 4 5 6


83
*
The ordinates of the parabola are taken from the trace of the curve in
Plate XI., as they are nowhere given in figures. In the 5th column are the
ordinates of an ellipse calculated by the equation given in the foregoing mod-
ified formulae in which A = 114, B = 0.65, and C = 3.664, and as this curve
agrees nearly as well with the observed velocities as the parabola it was
used as the law of change for the vertical velocities, in the computations for
the velocity of current. On page 224 of the report, we are told that dur-
ing the year 1851 the upper floats used were of cork, 8" square by 3" deep,
and one-half immersed. The connecting cord was 0.2 inch in diameter,
and the lower float a keg 10" by 15". From these data, ai = 0.083, c = 1.3
for a prism, r = 0.083. W may be taken at about 1 pound. The lat-
ter quantity is not given in the report, but is deduced by deducting from
the displacement of the upper float the probable weight of that float, its
wire and flag, and of the immersed cord. With the upper float a tin ellip-
soid, such as was used in the latter work on the Mississippi, and recom-
mended in the report, the weight W would be about 0.1 pound as hereto-
fore shown. -
Taking c = 0.748 and c, = 0.495 we have,
9 a nº c. .
# * * = 0.0125 . . . * * = 0.108 . . . c, a = 0.78.
W W
In this example it was thought advisable to compute the ordinates x,
for all values of y, even though it might prove to be unnecessary labor.
Taking the values of vi, from the column of observed velocities in
Table I., and the constants as above given, the coördinates of the different
curves of the cord for the several depths were obtained from the modified
formula. These values are given in the following table :
TABLE II.
COMPUTED COóRDINATES OF THE CURVE Assu MED BY THE CORD witH FLOATs
AT WARIOUS DEPTHS.
Length of the Connecting Cord in feet, or supposed depth of Lower Float.
}
: *— 38 54 == 72 90 || 102
i y X y | x |--|--|-ºn-- X | y N y X.
! -------- ! ------- || --- ~- || “. - - - -- " - - - - - - -
i is |OTO is 3.00 is Toº 18 0.10 1S 0.27 | 18 0.6S
} i 36 || 0 oil 36 º | }} | * | *ś | 3 || ||
| n 54 0.03 | 3 || 3:3: | # || 3 || 3 || 3.49
t i | 72 0.79 || 72 2.39 || 72 5.47
! | | 90 3 90 7.64
\ l
, 31 90 7.
__ (102) (3.96) | 102 9.12
N. B. —The figures in brackets give a point in the curve below the float.
84
Plotting the last curve, which deviates most from a vertical, the real
depth of the float is found to be only about nine inches less than the
length of the cord, and as the decrease is less for the other curves, the
determination of the coördinates xi and y, is not necessary, nor was even
the computation of the ordinates in the above table. -
From the last of the above modified formulae the true velocities of the
current, v', at the several depths were computed, and the results given
in Table III.,
TABLE III.
CoMPARISON of TRUE VELOCITY OF CURRENT AT THE SEVERAL DEPTHS WITH
THE OBSERVED veloCITY OF FLOATs.”

Computed -
Observed e
Deuth. velocity by º Difference.
- Floats. of CuTrent
- at Float.
y Vi V’ v'—v,
0 4.230 4, 230 0.000
18 4.298 4.29S 0.000
36 4.346 4.346 0.000
54 4.274 4.250 —0.024
72 4.158 4.015 —0.143
90 4.053 3. 785 -—0.268
102 3.94S 3.275 —0.673
110 2.670.
114 (1.800)
\ e
As before stated the values of a, as, c, ci, cs, and W, should be very
carefully determined. In this example they are only approximately
known, but they were taken so as to favor the floats as much as possible.
Were they accurately known, the computed velocities would be undoubt-
edly nearer those given by the meter, or other accurate instrument. Be-
sides the ellipse chosen was made to agree as near as possible with the
observed velocities v. As calculations show that that ellipse does not give
the law of change in the true velocities, the quantities A, B, and C should
be redetermined, which might be done from the above corrected velocities
v'. This would give a second approximation, which would agree still
better with the meter observations.
However, these computations remarkably confirm the correctness of the
comparative observations of the floats and meter given in Table V., page 13,
and prove that observations on the Mississippi made with accurate instru-
ments, would show the same decrease of velocity towards the bottom, as
has been found in other rivers.
* Compare with Table V., page 13, where the differences between the float and meter observations
are given. *
85
To illustrate the effect of decrease in the weight W, the following ex-
ample has been worked out :
A, B, C, and c were taken as before ; 2 r = Tº inch, c, for a flat ellip-
soid = 0.5, ai = 0.053 ft., ca =0.75, as about 1 foot, and W assumed = 0.1
pound.
The following table shows the computed ordinates of the several curves,
which are also graphically represented in Figure 15:
TABLE IV.
CoMPUTED CoöRDINATES OF THE CURVE ASSUMED BY THE CORD WITH Lower
FLOAT AT v ARIOUS DEPTHs. W, BEING VERY SMALL.
Length of the Connecting Cord in feet, or supposed depth of Float.
ls i 30 54 —º- 90 102
––––. * - Y | * *— — —
18 0.01 18 || 0.02 18 0.02 18 0.36
| 36 0.05 || 36 0.06 || 36 | 1.14
j 54 0.11 || 54 2.23
} | 72 3.46
{
| | |
The great increase of the ordinates x, is
entirely due to the small value of W, because
the form of the upper float and the size of the
cord are both better than in the first example.
At the depth y = 90, that part of x which
is due to the pressure of the current against
the cord is 13.25, while that due to the action
of the upper float is only 1.59, showing that
the cord, small as it is, gives rise to about
nine-tenths of the horizontal displacement of
the lower float.
The velocities of currents at different
depths, computed for the last example do not
differ from those of the former at like
depths. -
It will be noticed that the data in the last example, except the depth of
the river and velocities of current, are the same as used by Gen. Abbot in
his proof of the infallibility of float observations, quoted on page 63.
As we have previously seen, these series of float observations were in
all probability selected so as to make the mean agree with the parabola


86 $º
found from the little canal velocities, and therefore many of the verticals
taken, were such as almost any one would have rejected; giving as they do,
the maximum velocity at or below the bottom, but as their influence on the
mean was to carry the locus of maximum velocity down to one-third of the
depth, they were retained. In the series given above, the proportion of
this class of verticals is rathem too large; for the velocity at 36 ft. is much
greater than is shown by even the parabolic curve at the same depth.
Were the corrections computed for each vertical of which this series is
composed, we should probably have a result which would agree still closer
with the meter observations, and would show errors in the velocities by
floats above the depth of fifty feet.
*
+
small Canal at Metz
Journal Franklin Inst. Vol. LXII. * - On the Flows of Waterim Rivers 8 Canals. --- —i. Plate II.
Fig. 6. y n
Z2.
sº * ta. 70. |
Fig 6. Surface & bottom Velocity Fig * v. & º |
º 7 t º H ntal Velo city Curves 73
* M. Curve, NIAGARA RIV. y Vertical Velocity Curves orizo S. º ir Ri y A&
Difference of Floats& Meter TNLAGARA RIV. . ins clair aver captbOILEAftobservations
Width inal |
|
.
.
Fig. 7.
Surface & bottom. 4.
Velocity Curves
St Clair River
36 3.7 ft Second
Fig. 9.
Vertical Velocity Curves in St. Clair Riv.
Feet -pr: Second
4. 1. 2
Płg. //
Vertical Velocity Curves.
i
St. Lawrence
Second
1.5
Aºg 4.
Vertical Velp city Curve
O
Feeder of O & C. Canal
obtained by Floats
HUMPHREY & ABBOT
2-0
2.sft.
Second 30
3.5
Jasimº Guigan.Latlu Philada.