Chapter XI: On Streams and Rivers

§ 123. _Catchment Basin._--A stream or river is the channel for the
discharge of the available rainfall of a district, termed its
catchment basin. The catchment basin is surrounded by a ridge or
watershed line, continuous except at the point where the river finds
an outlet. The area of the catchment basin may be determined from a
suitable contoured map on a scale of at least 1 in 100,000. Of the
whole rainfall on the catchment basin, a part only finds its way to
the stream. Part is directly re-evaporated, part is absorbed by
vegetation, part may escape by percolation into neighbouring
districts. The following table gives the relation of the average
stream discharge to the average rainfall on the catchment basin
(Tiefenbacher).

+-----------------------------+-----------------+--------------------+
| |Ratio of average |Loss by Evaporation,|
| | Discharge to | &c., in per cent of|
| |average Rainfall.| total Rainfall. |
+-----------------------------+-----------------+--------------------+
| Cultivated land and spring- | | |
| forming declivities. | .3 to .33 | 67 to 70 |
| Wooded hilly slopes. | .35 to .45 | 55 to 65 |
| Naked unfissured mountains | .55 to .60 | 40 to 45 |
+-----------------------------+-----------------+--------------------+

§ 124. _Flood Discharge._--The flood discharge can generally only be
determined by examining the greatest height to which floods have been
known to rise. To produce a flood the rainfall must be heavy and
widely distributed, and to produce a flood of exceptional height the
duration of the rainfall must be so great that the flood waters of the
most distant affluents reach the point considered, simultaneously with
those from nearer points. The larger the catchment basin the less
probable is it that all the conditions tending to produce a maximum
discharge should simultaneously occur. Further, lakes and the river
bed itself act as storage reservoirs during the rise of water level
and diminish the rate of discharge, or serve as flood moderators. The
influence of these is often important, because very heavy rain storms
are in most countries of comparatively short duration. Tiefenbacher
gives the following estimate of the flood discharge of streams in
Europe:--

Flood discharge of Streams
per Second per Square Mile
of Catchment Basin.

In flat country 8.7 to 12.5 cub. ft.
In hilly districts 17.5 to 22.5 "
In moderately mountainous districts 36.2 to 45.0 "
In very mountainous districts 50.0 to 75.0 "

It has been attempted to express the decrease of the rate of flood
discharge with the increase of extent of the catchment basin by
empirical formulae. Thus Colonel P. P. L. O'Connell proposed the
formula y = M [root]x, where M is a constant called the modulus of the
river, the value of which depends on the amount of rainfall, the
physical characters of the basin, and the extent to which the floods
are moderated by storage of the water. If M is small for any given
river, it shows that the rainfall is small, or that the permeability
or slope of the sides of the valley is such that the water does not
drain rapidly to the river, or that lakes and river bed moderate the
rise of the floods. If values of M are known for a number of rivers,
they may be used in inferring the probable discharge of other similar
rivers. For British rivers M varies from 0.43 for a small stream
draining meadow land to 37 for the Tyne. Generally it is about 15 or
20. For large European rivers M varies from 16 for the Seine to 67.5
for the Danube. For the Nile M = 11, a low value which results from
the immense length of the Nile throughout which it receives no
affluent, and probably also from the influence of lakes. For different
tributaries of the Mississippi M varies from 13 to 56. For various
Indian rivers it varies from 40 to 303, this variation being due to
the great variations of rainfall, slope and character of Indian
rivers.

In some of the tank projects in India, the flood discharge has been
calculated from the formula D = C[3root]n², where D is the discharge
in cubic yards per hour from n square miles of basin. The constant C
was taken = 61,523 in the designs for the Ekrooka tank, = 75,000 on
Ganges and Godavery works, and = 10,000 on Madras works.

§ 125. _Action of a Stream on its Bed._--If the velocity of a stream
exceeds a certain limit, depending on its size, and on the size,
heaviness, form and coherence of the material of which its bed is
composed, it scours its bed and carries forward the materials. The
quantity of material which a given stream can carry in suspension
depends on the size and density of the particles in suspension, and is
greater as the velocity of the stream is greater. If in one part of
its course the velocity of a stream is great enough to scour the bed
and the water becomes loaded with silt, and in a subsequent part of
the river's course the velocity is diminished, then part of the
transported material must be deposited. Probably deposit and scour go
on simultaneously over the whole river bed, but in some parts the rate
of scour is in excess of the rate of deposit, and in other parts the
rate of deposit is in excess of the rate of scour. Deep streams appear
to have the greatest scouring power at any given velocity. It is
possible that the difference is strictly a difference of transporting,
not of scouring action. Let fig. 129 represent a section of a stream.
The material lifted at a will be diffused through the mass of the
stream and deposited at different distances down stream. The average
path of a particle lifted at a will be some such curve as abc, and the
average distance of transport each time a particle is lifted will be
represented by ac. In a deeper stream such as that in fig. 130, the
average height to which particles are lifted, and, since the rate of
vertical fall through the water may be assumed the same as before, the
average distance a´c´ of transport will be greater. Consequently,
although the scouring action may be identical in the two streams, the
velocity of transport of material down stream is greater as the depth
of the stream is greater. The effect is that the deep stream excavates
its bed more rapidly than the shallow stream.

§ 126. _Bottom Velocity at which Scour commences._--The following
bottom velocities were determined by P. L. G. Dubuat to be the maximum
velocities consistent with stability of the stream bed for different
materials.

Darcy and Bazin give, for the relation of the mean velocity v_m and
bottom velocity v_b.

v_m = v_b + 10.87 [root](mi).

But

[root]mi = v_m [root]([zeta]/2g);

.: v_m = v_b/(1 - 10.87 [root]([zeta]/2g)).

Taking a mean value for [zeta], we get

v_m = 1.312 v_b,

and from this the following values of the mean velocity are
obtained:--

+-----------------------+---------------+-------------+
| |Bottom Velocity|Mean Velocity|
| | = v_b. | = v_m. |
+-----------------------+---------------+-------------+
| 1. Soft earth | 0.25 | .33 |
| 2. Loam | 0.50 | .65 |
| 3. Sand | 1.00 | 1.30 |
| 4. Gravel | 2.00 | 2.62 |
| 5. Pebbles | 3.40 | 4.46 |
| 6. Broken stone, flint| 4.00 | 5.25 |
| 7. Chalk, soft shale | 5.00 | 6.56 |
| 8. Rock in beds | 6.00 | 7.87 |
| 9. Hard rock. | 10.00 | 13.12 |
+-----------------------+---------------+-------------+

The following table of velocities which should not be exceeded in
channels is given in the _Ingenieurs Taschenbuch_ of the Verein
"Hütte":--

+--------------------------------+---------+---------+---------+
| | Surface | Mean | Bottom |
| |Velocity.|Velocity.|Velocity.|
+--------------------------------+---------+---------+---------+
| Slimy earth or brown clay | .49 | .36 | .26 |
| Clay | .98 | .75 | .52 |
| Firm sand | 1.97 | 1.51 | 1.02 |
| Pebbly bed | 4.00 | 3.15 | 2.30 |
| Boulder bed | 5.00 | 4.03 | 3.08 |
| Conglomerate of slaty fragments| 7.28 | 6.10 | 4.90 |
| Stratified rocks | 8.00 | 7.45 | 6.00 |
| Hard rocks | 14.00 | 12.15 | 10.36 |
+--------------------------------+---------+---------+---------+

§ 127. _Regime of a River Channel._--A river channel is said to be in
a state of regime, or stability, when it changes little in draught or
form in a series of years. In some rivers the deepest part of the
channel changes its position perpetually, and is seldom found in the
same place in two successive years. The sinuousness of the river also
changes by the erosion of the banks, so that in time the position of
the river is completely altered. In other rivers the change from year
to year is very small, but probably the regime is never perfectly
stable except where the rivers flow over a rocky bed.

If a river had a constant discharge it would gradually modify its bed
till a permanent regime was established. But as the volume discharged
is constantly changing, and therefore the velocity, silt is deposited
when the velocity decreases, and scour goes on when the velocity
increases in the same place. When the scouring and silting are
considerable, a perfect balance between the two is rarely established,
and hence continual variations occur in the form of the river and the
direction of its currents. In other cases, where the action is less
violent, a tolerable balance may be established, and the deepening of
the bed by scour at one time is compensated by the silting at another.
In that case the general regime is permanent, though alteration is
constantly going on. This is more likely to happen if by artificial
means the erosion of the banks is prevented. If a river flows in soil
incapable of resisting its tendency to scour it is necessarily sinuous
(§ 107), for the slightest deflection of the current to either side
begins an erosion which increases progressively till a considerable
bend is formed. If such a river is straightened it becomes sinuous
again unless its banks are protected from scour.

§ 128. _Longitudinal Section of River Bed._--The declivity of rivers
decreases from source to mouth. In their higher parts rapid and
torrential, flowing over beds of gravel or boulders, they enlarge in
volume by receiving affluent streams, their slope diminishes, their
bed consists of smaller materials, and finally they reach the sea.
Fig. 131 shows the length in miles, and the surface fall in feet per
mile, of the Tyne and its tributaries.

The decrease of the slope is due to two causes. (1) The action of the
transporting power of the water, carrying the smallest debris the
greatest distance, causes the bed to be less stable near the mouth
than in the higher parts of the river; and, as the river adjusts its
slope to the stability of the bed by scouring or increasing its
sinuousness when the slope is too great, and by silting or
straightening its course if the slope is too small, the decreasing
stability of the bed would coincide with a decreasing slope. (2) The
increase of volume and section of the river leads to a decrease of
slope; for the larger the section the less slope is necessary to
ensure a given velocity.

The following investigation, though it relates to a purely arbitrary
case, is not without interest. Let it be assumed, to make the
conditions definite--(1) that a river flows over a bed of uniform
resistance to scour, and let it be further assumed that to maintain
stability the velocity of the river in these circumstances is constant
from source to mouth; (2) suppose the sections of the river at all
points are similar, so that, b being the breadth of the river at any
point, its hydraulic mean depth is ab and its section is cb², where a
and c are constants applicable to all parts of the river; (3) let us
further assume that the discharge increases uniformly in consequence
of the supply from affluents, so that, if l is the length of the river
from its source to any given point, the discharge there will be kl,
where k is another constant applicable to all points in the course of
the river.

Let AB (fig. 132) be the longitudinal section of the river, whose
source is at A; and take A for the origin of vertical and horizontal
coordinates. Let C be a point whose ordinates are x and y, and let the
river at C have the breadth b, the slope i, and the velocity v. Since
velocity × area of section = discharge, vcb² = kl, or b =
[root](kl/cv).

Hydraulic mean depth = ab = a [root](kl/cv).

But, by the ordinary formula for the flow of rivers, mi = [zeta]v²;

.: i = [zeta]v²/m = ([zeta]v^(5/2)/a) [root](c/kl).

But i is the tangent of the angle which the curve at C makes with the
axis of X, and is therefore = dy/dx. Also, as the slope is small, l =
AC = AD = x nearly.

.: dy/dx = ([zeta]v^(5/2)/a) [root](c/kx);

and, remembering that v is constant,

y = (2[zeta]v^(5/2)/a) [root](cx/k);

or

y² = constant × x;

so that the curve is a common parabola, of which the axis is
horizontal and the vertex at the source. This may be considered an
ideal longitudinal section, to which actual rivers approximate more or
less, with exceptions due to the varying hardness of their beds, and
the irregular manner in which their volume increases.

§ 129. _Surface Level of River._--The surface level of a river is a
plane changing constantly in position from changes in the volume of
water discharged, and more slowly from changes in the river bed, and
the circumstances affecting the drainage into the river.

For the purposes of the engineer, it is important to determine (1) the
extreme low water level, (2) the extreme high water or flood level,
and (3) the highest navigable level.

1. _Low Water Level_ cannot be absolutely known, because a river
reaches its lowest level only at rare intervals, and because
alterations in the cultivation of the land, the drainage, the removal
of forests, the removal or erection of obstructions in the river bed,
&c., gradually alter the conditions of discharge. The lowest level of
which records can be found is taken as the conventional or approximate
low water level, and allowance is made for possible changes.

2. _High Water or Flood Level._--The engineer assumes as the highest
flood level the highest level of which records can be obtained. In
forming a judgment of the data available, it must be remembered that
the highest level at one point of a river is not always simultaneous
with the attainment of the highest level at other points, and that
the rise of a river in flood is very different in different parts of
its course. In temperate regions, the floods of rivers seldom rise
more than 20 ft. above low-water level, but in the tropics the rise of
floods is greater.

3. _Highest Navigable Level._--When the river rises above a certain
level, navigation becomes difficult from the increase of the velocity
of the current, or from submersion of the tow paths, or from the
headway under bridges becoming insufficient. Ordinarily the highest
navigable level may be taken to be that at which the river begins to
overflow its banks.

§ 130. _Relative Value of Different Materials for Submerged
Works._--That the power of water to remove and transport different
materials depends on their density has an important bearing on the
selection of materials for submerged works. In many cases, as in the
aprons or floorings beneath bridges, or in front of locks or falls,
and in the formation of training walls and breakwaters by _pierres
perdus_, which have to resist a violent current, the materials of
which the structures are composed should be of such a size and weight
as to be able individually to resist the scouring action of the water.
The heaviest materials will therefore be the best; and the different
value of materials in this respect will appear much more striking, if
it is remembered that all materials lose part of their weight in
water. A block whose volume is V cubic feet, and whose density in air
is w lb. per cubic foot, weighs in air wV lb., but in water only
(w--62.4) V lb.

+----------------------+-----------------------------+
| | Weight of a Cub. Ft. in lb. |
| +--------------+--------------+
| | In Air. | In Water. |
+----------------------+--------------+--------------+
| Basalt | 187.3 | 124.9 |
| Brick | 130.0 | 67.6 |
| Brickwork | 112.0 | 49.6 |
| Granite and limestone| 170.0 | 107.6 |
| Sandstone | 144.0 | 81.6 |
| Masonry | 116-144 | 53.6-81.6 |
+----------------------+--------------+--------------+

§ 131. _Inundation Deposits from a River._--When a river carrying silt
periodically overflows its banks, it deposits silt over the area
flooded, and gradually raises the surface of the country. The silt is
deposited in greatest abundance where the water first leaves the
river. It hence results that the section of the country assumes a
peculiar form, the river flowing in a trough along the crest of a
ridge, from which the land slopes downwards on both sides. The silt
deposited from the water forms two wedges, having their thick ends
towards the river (fig. 133).

This is strikingly the case with the Mississippi, and that river is
now kept from flooding immense areas by artificial embankments or
levees. In India, the term _deltaic segment_ is sometimes applied to
that portion of a river running through deposits formed by inundation,
and having this characteristic section. The irrigation of the country
in this case is very easy; a comparatively slight raising of the river
surface by a weir or annicut gives a command of level which permits
the water to be conveyed to any part of the district.

§ 132. _Deltas._--The name delta was originally given to the [Greek:
Delta]-shaped portion of Lower Egypt, included between seven branches
of the Nile. It is now given to the whole of the alluvial tracts round
river mouths formed by deposition of sediment from the river, where
its velocity is checked on its entrance to the sea. The characteristic
feature of these alluvial deltas is that the river traverses them, not
in a single channel, but in two or many bifurcating branches. Each
branch has a tract of the delta under its influence, and gradually
raises the surface of that tract, and extends it seaward. As the delta
extends itself seaward, the conditions of discharge through the
different branches change. The water finds the passage through one of
the branches less obstructed than through the others; the velocity and
scouring action in that branch are increased; in the others they
diminish. The one channel gradually absorbs the whole of the water
supply, while the other branches silt up. But as the mouth of the new
main channel extends seaward the resistance increases both from the
greater length of the channel and the formation of shoals at its
mouth, and the river tends to form new bifurcations AC or AD (fig.
134), and one of these may in time become the main channel of the
river.

§ 133. _Field Operations preliminary to a Study of River
Improvement._--There are required (1) a plan of the river, on which
the positions of lines of levelling and cross sections are marked; (2)
a longitudinal section and numerous cross sections of the river; (3) a
series of gaugings of the discharge at different points and in
different conditions of the river.

_Longitudinal Section._--This requires to be carried out with great
accuracy. A line of stakes is planted, following the sinuosities of
the river, and chained and levelled. The cross sections are referred
to the line of stakes, both as to position and direction. The
determination of the surface slope is very difficult, partly from its
extreme smallness, partly from oscillation of the water. Cunningham
recommends that the slope be taken in a length of 2000 ft. by four
simultaneous observations, two on each side of the river.

§ 134. _Cross Sections_--A stake is planted flush with the water, and
its level relatively to some point on the line of levels is
determined. Then the depth of the water is determined at a series of
points (if possible at uniform distances) in a line starting from the
stake and perpendicular to the thread of the stream. To obtain these,
a wire may be stretched across with equal distances marked on it by
hanging tags. The depth at each of these tags may be obtained by a
light wooden staff, with a disk-shaped shoe 4 to 6 in. in diameter. If
the depth is great, soundings may be taken by a chain and weight. To
ensure the wire being perpendicular to the thread of the stream, it is
desirable to stretch two other wires similarly graduated, one above
and the other below, at a distance of 20 to 40 yds. A number of floats
being then thrown in, it is observed whether they pass the same
graduation on each wire.

For large and rapid rivers the cross section is obtained by sounding
in the following way. Let AC (fig. 135) be the line on which soundings
are required. A base line AB is measured out at right angles to AC,
and ranging staves are set up at AB and at D in line with AC. A boat
is allowed to drop down stream, and, at the moment it comes in line
with AD, the lead is dropped, and an observer in the boat takes, with
a box sextant, the angle AEB subtended by AB. The sounding line may
have a weight of 14 lb. of lead, and, if the boat drops down stream
slowly, it may hang near the bottom, so that the observation is made
instantly. In extensive surveys of the Mississippi observers with
theodolites were stationed at A and B. The theodolite at A was
directed towards C, that at B was kept on the boat. When the boat came
on the line AC, the observer at A signalled, the sounding line was
dropped, and the observer at B read off the angle ABE. By repeating
observations a number of soundings are obtained, which can be plotted
in their proper position, and the form of the river bed drawn by
connecting the extremities of the lines. From the section can be
measured the sectional area of the stream [Omega] and its wetted
perimeter [chi]; and from these the hydraulic mean depth m can be
calculated.

§ 135. _Measurement of the Discharge of Rivers._--The area of cross
section multiplied by the mean velocity gives the discharge of the
stream. The height of the river with reference to some fixed mark
should be noted whenever the velocity is observed, as the velocity and
area of cross section are different in different states of the river.
To determine the mean velocity various methods may be adopted; and,
since no method is free from liability to error, either from the
difficulty of the observations or from uncertainty as to the ratio of
the mean velocity to the velocity observed, it is desirable that more
than one method should be used.

INSTRUMENTS FOR MEASURING THE VELOCITY OF WATER

§ 136. _Surface Floats_ are convenient for determining the surface
velocities of a stream, though their use is difficult near the banks.
The floats may be small balls of wood, of wax or of hollow metal, so
loaded as to float nearly flush with the water surface. To render
them visible they may have a vertical painted stem. In experiments on
the Seine, cork balls 1(3/4) in. diameter were used, loaded to float
flush with the water, and provided with a stem. In A. J. C.
Cunningham's observations at Roorkee, the floats were thin circular
disks of English deal, 3 in. diameter and ¼ in. thick. For
observations near the banks, floats 1 in. diameter and 1/8 in. thick
were used. To render them visible a tuft of cotton wool was used
loosely fixed in a hole at the centre.

The velocity is obtained by allowing the float to be carried down, and
noting the time of passage over a measured length of the stream. If v
is the velocity of any float, t the time of passing over a length l,
then v = l/t. To mark out distinctly the length of stream over which
the floats pass, two ropes may be stretched across the stream at a
distance apart, which varies usually from 50 to 250 ft., according to
the size and rapidity of the river. In the Roorkee experiments a
length of run of 50 ft. was found best for the central two-fifths of
the width, and 25 ft. for the remainder, except very close to the
banks, where the run was made 12½ ft. only. The longer the run the
less is the proportionate error of the time observations, but on the
other hand the greater the deviation of the floats from a straight
course parallel to the axis of the stream. To mark the precise
position at which the floats cross the ropes, Cunningham used short
white rope pendants, hanging so as nearly to touch the surface of the
water. In this case the streams were 80 to 180 ft. in width. In wider
streams the use of ropes to mark the length of run is impossible, and
recourse must be had to box sextants or theodolites to mark the path
of the floats.

Let AB (fig. 136) be a measured base line strictly parallel to the
thread of the stream, and AA1, BB1 lines at right angles to AB marked
out by ranging rods at A1 and B1. Suppose observers stationed at A and
B with sextants or theodolites, and let CD be the path of any float
down stream. As the float approaches AA1, the observer at B keeps it
on the cross wire of his instrument. The observer at A observes the
instant of the float reaching the line AA1, and signals to B who then
reads off the angle ABC. Similarly, as the float approaches BB1, the
observer at A keeps it in sight, and when signalled to by B reads the
angle BAD. The data so obtained are sufficient for plotting the path
of the float and determining the distances AC, BD.

The time taken by the float in passing over the measured distance may
be observed by a chronograph, started as the float passes the upper
rope or line, and stopped when it passes the lower. In Cunningham's
observations two chronometers were sometimes used, the time of passing
one end of the run being noted on one, and that of passing the other
end of the run being noted on the other. The chronometers were
compared immediately before the observations. In other cases a single
chronometer was used placed midway of the run. The moment of the
floats passing the ends of the run was signalled to a time-keeper at
the chronometer by shouting. It was found quite possible to count the
chronometer beats to the nearest half second, and in some cases to the
nearest quarter second.

§ 137. _Sub-surface Floats._--The velocity at different depths below
the surface of a stream may be obtained by sub-surface floats, used
precisely in the same way as surface floats. The most usual
arrangement is to have a large float, of slightly greater density than
water, connected with a small and very light surface float. The motion
of the combined arrangement is not sensibly different from that of the
large float, and the small surface float enables an observer to note
the path and velocity of the sub-surface float. The instrument is,
however, not free from objection. If the large submerged float is made
of very nearly the same density as water, then it is liable to be
thrown upwards by very slight eddies in the water, and it does not
maintain its position at the depth at which it is intended to float.
On the other hand, if the large float is made sensibly heavier than
water, the indicating or surface float must be made rather large, and
then it to some extent influences the motion of the submerged float.
Fig. 137 shows one form of sub-surface float. It consists of a couple
of tin plates bent at a right angle and soldered together at the
angle. This is connected with a wooden ball at the surface by a very
thin wire or cord. As the tin alone makes a heavy submerged float, it
is better to attach to the tin float some pieces of wood to diminish
its weight in water. Fig. 138 shows the form of submerged float used
by Cunningham. It consists of a hollow metal ball connected to a
slice of cork, which serves as the surface float.

§ 138. _Twin Floats._--Suppose two equal and similar floats (fig. 139)
connected by a wire. Let one float be a little lighter and the other a
little heavier than water. Then the velocity of the combined floats
will be the mean of the surface velocity and the velocity at the depth
at which the heavier float swims, which is determined by the length of
the connecting wire. Thus if v_s is the surface velocity and v_d the
velocity at the depth to which the lower float is sunk, the velocity
of the combined floats will be

v = ½(v_s + v_d).

Consequently, if v is observed, and v_s determined by an experiment
with a single float,

v_d = 2v - v_s

According to Cunningham, the twin float gives better results than the
sub-surface float.

§ 139. _Velocity Rods._--Another form of float is shown in fig. 140.
This consists of a cylindrical rod loaded at the lower end so as to
float nearly vertical in water. A wooden rod, with a metal cap at the
bottom in which shot can be placed, answers better than anything else,
and sometimes the wooden rod is made in lengths, which can be screwed
together so as to suit streams of different depths. A tuft of cotton
wool at the top serves to make the float more easily visible. Such a
rod, so adjusted in length that it sinks nearly to the bed of the
stream, gives directly the mean velocity of the whole vertical section
in which it floats.

§ 140. _Revy's Current Meter._--No instrument has been so much used in
directly determining the velocity of a stream at a given point as the
screw current meter. Of this there are a dozen varieties at least. As
an example of the instrument in its simplest form, Revy's meter may be
selected. This is an ordinary screw meter of a larger size than usual,
more carefully made, and with its details carefully studied (figs.
141, 142). It was designed after experience in gauging the great South
American rivers. The screw, which is actuated by the water, is 6 in.
in diameter, and is of the type of the Griffiths screw used in ships.
The hollow spherical boss serves to make the weight of the screw
sensibly equal to its displacement, so that friction is much reduced.
On the axis aa of the screw is a worm which drives the counter. This
consists of two worm wheels g and h fixed on a common axis. The worm
wheels are carried on a frame attached to the pin l. By means of a
string attached to l they can be pulled into gear with the worm, or
dropped out of gear and stopped at any instant. A nut m can be screwed
up, if necessary, to keep the counter permanently in gear. The worm is
two-threaded, and the worm wheel g has 200 teeth. Consequently it
makes one rotation for 100 rotations of the screw, and the number of
rotations up to 100 is marked by the passage of the graduations on its
edge in front of a fixed index. The second worm wheel has 196 teeth,
and its edge is divided into 49 divisions. Hence it falls behind the
first wheel one division for a complete rotation of the latter. The
number of hundreds of rotations of the screw are therefore shown by
the number of divisions on h passed over by an index fixed to g. One
difficulty in the use of the ordinary screw meter is that particles of
grit, getting into the working parts, very sensibly alter the
friction, and therefore the speed of the meter. Revy obviates this by
enclosing the counter in a brass box with a glass face. This box is
filled with pure water, which ensures a constant coefficient of
friction for the rubbing parts, and prevents any mud or grit finding
its way in. In order that the meter may place itself with the axis
parallel to the current, it is pivoted on a vertical axis and directed
by a large vane shown in fig. 142. To give the vane more
directing power the vertical axis is nearer the screw than in ordinary
meters, and the vane is larger. A second horizontal vane is attached
by the screws x, x, the object of which is to allow the meter to rest
on the ground without the motion of the screw being interfered with.
The string or wire for starting and stopping the meter is carried
through the centre of the vertical axis, so that the strain on it may
not tend to pull the meter oblique to the current. The pitch of the
screw is about 9 in. The screws at x serve for filling the meter with
water. The whole apparatus is fixed to a rod (fig. 142), of a length
proportionate to the depth, or for very great depths it is fixed to a
weighted bar lowered by ropes, a plan invented by Revy. The instrument
is generally used thus. The reading of the counter is noted, and it is
put out of gear. The meter is then lowered into the water to the
required position from a platform between two boats, or better from a
temporary bridge. Then the counter is put into gear for one, two or
five minutes. Lastly, the instrument is raised and the counter again
read. The velocity is deduced from the number of rotations in unit
time by the formulae given below. For surface velocities the counter
may be kept permanently in gear, the screw being started and stopped
by hand.

§ 141. _The Harlacher Current Meter._--In this the ordinary counting
apparatus is abandoned. A worm drives a worm wheel, which makes an
electrical contact once for each 100 rotations of the worm. This
contact gives a signal above water. With this arrangement, a series of
velocity observations can be made, without removing the instrument
from the water, and a number of practical difficulties attending the
accurate starting and stopping of the ordinary counter are entirely
got rid of. Fig. 143 shows the meter. The worm wheel z makes one
rotation for 100 of the screw. A pin moving the lever x makes the
electrical contact. The wires b, c are led through a gas pipe B; this
also serves to adjust the meter to any required position on the wooden
rod dd. The rudder or vane is shown at WH. The galvanic current acts
on the electromagnet m, which is fixed in a small metal box containing
also the battery. The magnet exposes and withdraws a coloured disk at
an opening in the cover of the box.

§ 142. _Amsler Laffon Current Meter._--A very convenient and accurate
current meter is constructed by Amsler Laffon of Schaffhausen. This
can be used on a rod, and put into and out of gear by a ratchet. The
peculiarity in this case is that there is a double ratchet, so that
one pull on the string puts the counter into gear and a second puts it
out of gear. The string may be slack during the action of the meter,
and there is less uncertainty than when the counter has to be held in
gear. For deep streams the meter A is suspended by a wire with a heavy
lenticular weight below (fig. 144). The wire is payed out from a small
winch D, with an index showing the depth of the meter, and passes over
a pulley B. The meter is in gimbals and is directed by a conical
rudder which keeps it facing the stream with its axis horizontal.
There is an electric circuit from a battery C through the meter, and a
contact is made closing the circuit every 100 revolutions. The moment
the circuit closes a bell rings. By a subsidiary arrangement, when the
foot of the instrument, 0.3 metres below the axis of the meter,
touches the ground the circuit is also closed and the bell rings. It
is easy to distinguish the continuous ring when the ground is reached
from the short ring when the counter signals. A convenient winch for
the wire is so graduated that if set when the axis of the meter is at
the water surface it indicates at any moment the depth of the meter
below the surface. Fig. 144 shows the meter as used on a boat. It is a
very convenient instrument for obtaining the velocity at different
depths and can also be used as a sounding instrument.

§ 143. _Determination of the Coefficients of the Current
Meter._--Suppose a series of observations has been made by towing the
meter in still water at different speeds, and that it is required to
ascertain from these the constants of the meter. If v is the velocity
of the water and n the observed number of rotations per second, let

v = [alpha] + [beta]n (1)

where [alpha] and [beta] are constants. Now let the meter be towed
over a measured distance L, and let N be the revolutions of the meter
and t the time of transit. Then the speed of the meter relatively to
the water is L/t = v feet per second, and the number of revolutions
per second is N/t = n. Suppose m observations have been made in this
way, furnishing corresponding values of v and n, the speed in each
trial being as uniform as possible,

[Sigma]n = n1 + n2 + ...

[Sigma]v = v1 + v2 + ...

[Sigma]nv = n1v1 + n2v2 + ...

[Sigma]n² = n1² + n2² + ...

[[Sigma]n]² = [n1 + n2 + ...]²

Then for the determination of the constants [alpha] and [beta] in (1),
by the method of least squares--

[Sigma]n²[Sigma]v - [Sigma]n[Sigma]nv
[alpha] = -------------------------------------,
m[Sigma]n² - [[Sigma]n]²

m[Sigma]nv - [Sigma]v[Sigma]n
[beta] = -----------------------------.
m[Sigma]n² - [[Sigma]n]²

In a few cases the constants for screw current meters have been
determined by towing them in R. E. Froude's experimental tank in which
the resistance of ship models is ascertained. In that case the data
are found with exceptional accuracy.

§ 144. Darcy Gauge or modified Pitot Tube.--A very old instrument for
measuring velocities, invented by Henri Pitot in 1730 (_Histoire de
l'Académie des Sciences_, 1732, p. 376), consisted simply of a
vertical glass tube with a right-angled bend, placed so that its mouth
was normal to the direction of flow (fig. 145).

The impact of the stream on the mouth of the tube balances a column in
the tube, the height of which is approximately h = v²/2g, where v is
the velocity at the depth x. Placed with its mouth parallel to the
stream the water inside the tube is nearly at the same level as the
surface of the stream, and turned with the mouth down stream, the
fluid sinks a depth h´ = v²/2g nearly, though the tube in that case
interferes with the free flow of the liquid and somewhat modifies the
result. Pitot expanded the mouth of the tube so as to form a funnel or
bell mouth. In that case he found by experiment

h = 1.5v²/2g.

But there is more disturbance of the stream. Darcy preferred to make
the mouth of the tube very small to avoid interference with the
stream and to check oscillations of the water column. Let the
difference of level of a pair of tubes A and B (fig. 145) be taken to
be h = kv²/2g, then k may be taken to be a corrective coefficient
whose value in well-shaped instruments is very nearly unity. By
placing his instrument in front of a boat towed through water Darcy
found k = 1.034; by placing the instrument in a stream the velocity of
which had been ascertained by floats, he found k = 1.006; by readings
taken in different parts of the section of a canal in which a known
volume of water was flowing, he found k = 0.993. He believed the first
value to be too high in consequence of the disturbance caused by the
boat. The mean of the other two values is almost exactly unity
(_Recherches hydrauliques_, Darcy and Bazin, 1865, p. 63). W. B.
Gregory used somewhat differently formed Pitot tubes for which the k =
1 (_Am. Soc. Mech. Eng._, 1903, 25). T. E. Stanton used a Pitot tube
in determining the velocity of an air current, and for his instrument
he found k = 1.030 to k = 1.032 ("On the Resistance of Plane Surfaces
in a Current of Air," _Proc. Inst. Civ. Eng._, 1904, 156).

One objection to the Pitot tube in its original form was the great
difficulty and inconvenience of reading the height h in the immediate
neighbourhood of the stream surface. This is obviated in the Darcy
gauge, which can be removed from the stream to be read.

Fig. 146 shows a Darcy gauge. It consists of two Pitot tubes having
their mouths at right angles. In the instrument shown, the two tubes,
formed of copper in the lower part, are united into one for strength,
and the mouths of the tubes open vertically and horizontally. The
upper part of the tubes is of glass, and they are provided with a
brass scale and two verniers b, b. The whole instrument is supported
on a vertical rod or small pile AA, the fixing at B permitting the
instrument to be adjusted to any height on the rod, and at the same
time allowing free rotation, so that it can be held parallel to the
current. At c is a two-way cock, which can be opened or closed by
cords. If this is shut, the instrument can be lifted out of the stream
for reading. The glass tubes are connected at top by a brass fixing,
with a stop cock a, and a flexible tube and mouthpiece m. The use of
this is as follows. If the velocity is required at a point near the
surface of the stream, one at least of the water columns would be
below the level at which it could be read. It would be in the copper
part of the instrument. Suppose then a little air is sucked out by the
tube m, and the cock a closed, the two columns will be forced up an
amount corresponding to the difference between atmospheric pressure
and that in the tubes. But the difference of level will remain
unaltered.

When the velocities to be measured are not very small, this instrument
is an admirable one. It requires observation only of a single linear
quantity, and does not require any time observation. The law
connecting the velocity and the observed height is a rational one, and
it is not absolutely necessary to make any experiments on the
coefficient of the instrument. If we take v = k[root](2gh), then it
appears from Darcy's experiments that for a well-formed instrument k
does not sensibly differ from unity. It gives the velocity at a
definite point in the stream. The chief difficulty arises from the
fact that at any given point in a stream the velocity is not
absolutely constant, but varies a little from moment to moment. Darcy
in some of his experiments took several readings, and deduced the
velocity from the mean of the highest and lowest.

§ 145. _Perrodil Hydrodynamometer._--This consists of a frame abcd
(fig. 147) placed vertically in the stream, and of a height not less
than the stream's depth. The two vertical members of this frame are
connected by cross bars, and united above water by a circular bar,
situated in the vertical plane and carrying a horizontal graduated
circle ef. This whole system is movable round its axis, being
suspended on a pivot at g connected with the fixed support mn. Other
horizontal arms serve as guides. The central vertical rod gr forms a
torsion rod, being fixed at r to the frame abcd, and, passing freely
upwards through the guides, it carries a horizontal needle moving
over the graduated circle ef. The support g, which carries the
apparatus, also receives in a tubular guide the end of the torsion rod
gr and a set screw for fixing the upper end of the torsion rod when
necessary. The impulse of the stream of water is received on a
circular disk x, in the plane of the torsion rod and the frame abcd.
To raise and lower the apparatus easily, it is not fixed directly to
the rod mn, but to a tube kl sliding on mn.

Suppose the apparatus arranged so that the disk x is at that level in
the stream where the velocity is to be determined. The plane abcd is
placed parallel to the direction of motion of the water. Then the disk
x (acting as a rudder) will place itself parallel to the stream on the
down stream side of the frame. The torsion rod will be unstrained, and
the needle will be at zero on the graduated circle. If, then, the
instrument is turned by pressing the needle, till the plane abcd of
the disk and the zero of the graduated circle is at right angles to
the stream, the torsion rod will be twisted through an angle which
measures the normal impulse of the stream on the disk x. That angle
will be given by the distance of the needle from zero. Observation
shows that the velocity of the water at a given point is not constant.
It varies between limits more or less wide. When the apparatus is
nearly in its right position, the set screw at g is made to clamp the
torsion spring. Then the needle is fixed, and the apparatus carrying
the graduated circle oscillates. It is not, then, difficult to note
the mean angle marked by the needle.

Let r be the radius of the torsion rod, l its length from the needle
over ef to r, and [alpha] the observed torsion angle. Then the moment
of the couple due to the molecular forces in the torsion rod is

M = E_t I[alpha]/l;

where E_t is the modulus of elasticity for torsion, and I the polar
moment of inertia of the section of the rod. If the rod is of circular
section, I = ½[pi]r^4. Let R be the radius of the disk, and b its
leverage, or the distance of its centre from the axis of the torsion
rod. The moment of the pressure of the water on the disk is

Fb = kb(G/2g)[pi]R²v²,

where G is the heaviness of water and k an experimental coefficient.
Then

E_t I[alpha]/l = kb(G/2g)[pi]R²v².

For any given instrument,

v = c [root][alpha],

where c is a constant coefficient for the instrument.

The instrument as constructed had three disks which could be used at
will. Their radii and leverages were in feet

R = b =

1st disk 0.052 0.16
2nd " 0.105 0.32
3rd " 0.210 0.66

For a thin circular plate, the coefficient k = 1.12. In the actual
instrument the torsion rod was a brass wire 0.06 in. diameter and 6½
ft. long. Supposing [alpha] measured in degrees, we get by calculation

v = 0.335 [root][alpha]; 0.115 [root][alpha]; 0.042 [root][alpha].

Very careful experiments were made with the instrument. It was fixed
to a wooden turning bridge, revolving over a circular channel of 2 ft.
width, and about 76 ft. circumferential length. An allowance was made
for the slight current produced in the channel. These experiments gave
for the coefficient c, in the formula v = c [root][alpha],

1st disk, c = 0.3126 for velocities of 3 to 16 ft.
2nd " 0.1177 " " 1¼ to 3¼ "
3rd " 0.0349 " " less than 1¼ "

The instrument is preferable to the current meter in giving the
velocity in terms of a single observed quantity, the angle of torsion,
while the current meter involves the observation of two quantities,
the number of rotations and the time. The current meter, except in
some improved forms, must be withdrawn from the water to read the
result of each experiment, and the law connecting the velocity and
number of rotations of a current meter is less well-determined than
that connecting the pressure on a disk and the torsion of the wire of
a hydrodynamometer.

The Pitot tube, like the hydrodynamometer, does not require a time
observation. But, where the velocity is a varying one, and
consequently the columns of water in the Pitot tube are oscillating,
there is room for doubt as to whether, at any given moment of closing
the cock, the difference of level exactly measures the impulse of the
stream at the moment. The Pitot tube also fails to give measurable
indications of very low velocities.

PROCESSES FOR GAUGING STREAMS

§ 146. _Gauging by Observation of the Maximum Surface Velocity._--The
method of gauging which involves the least trouble is to determine the
surface velocity at the thread of the stream, and to deduce from it
the mean velocity of the whole cross section. The maximum surface
velocity may be determined by floats or by a current meter.
Unfortunately the ratio of the maximum surface to the mean velocity is
extremely variable. Thus putting v_o for the surface velocity at the
thread of the stream, and v_m for the mean velocity of the whole cross
section, v_m/v_o has been found to have the following values:--

v_m/v_o

De Prony, experiments on small wooden channels 0.8164
Experiments on the Seine 0.62
Destrem and De Prony, experiments on the Neva 0.78
Boileau, experiments on canals 0.82
Baumgartner, experiments on the Garonne 0.80
Brünings (mean) 0.85
Cunningham, Solani aqueduct 0.823

Various formulae, either empirical or based on some theory of the
vertical and horizontal velocity curves, have been proposed for
determining the ratio v_m/v_o. Bazin found from his experiments the
empirical expression

v_m = v_o - 25.4 [root](mi);

where m is the hydraulic mean depth and i the slope of the stream.

In the case of irrigation canals and rivers, it is often important to
determine the discharge either daily or at other intervals of time,
while the depth and consequently the mean velocity is varying.
Cunningham (_Roorkee Prof. Papers_, iv. 47), has shown that, for a
given part of such a stream, where the bed is regular and of permanent
section, a simple formula may be found for the variation of the
central surface velocity with the depth. When once the constants of
this formula have been determined by measuring the central surface
velocity and depth, in different conditions of the stream, the surface
velocity can be obtained by simply observing the depth of the stream,
and from this the mean velocity and discharge can be calculated. Let z
be the depth of the stream, and v_o the surface velocity, both measured
at the thread of the stream. Then v_o² = cz; where c is a constant
which for the Solani aqueduct had the values 1.9 to 2, the depths
being 6 to 10 ft., and the velocities 3½ to 4½ ft. Without any
assumption of a formula, however, the surface velocities, or still
better the mean velocities, for different conditions of the stream may
be plotted on a diagram in which the abscissae are depths and the
ordinates velocities. The continuous curve through points so found
would then always give the velocity for any observed depth of the
stream, without the need of making any new float or current meter
observations.

§ 147. _Mean Velocity determined by observing a Series of Surface
Velocities._--The ratio of the mean velocity to the surface velocity
in one longitudinal section is better ascertained than the ratio of
the central surface velocity to the mean velocity of the whole cross
section. Suppose the river divided into a number of compartments by
equidistant longitudinal planes, and the surface velocity observed in
each compartment. From this the mean velocity in each compartment and
the discharge can be calculated. The sum of the partial discharges
will be the total discharge of the stream. When wires or ropes can be
stretched across the stream, the compartments can be marked out by
tags attached to them. Suppose two such ropes stretched across the
stream, and floats dropped in above the upper rope. By observing
within which compartment the path of the float lies, and noting the
time of transit between the ropes, the surface velocity in each
compartment can be ascertained. The mean velocity in each compartment
is 0.85 to 0.91 of the surface velocity in that compartment. Putting k
for this ratio, and v1, v2 ... for the observed velocities, in
compartments of area [Omega]1, [Omega]2 ... then the total discharge
is

Q = k([Omega]1v1 + [Omega]2v2 + ... ).

If several floats are allowed to pass over each compartment, the mean
of all those corresponding to one compartment is to be taken as the
surface velocity of that compartment.

This method is very applicable in the case of large streams or rivers
too wide to stretch a rope across. The paths of the floats are then
ascertained in this way. Let fig. 148 represent a portion of the
river, which should be straight and free from obstructions. Suppose a
base line AB measured parallel to the thread of the stream, and let
the mean cross section of the stream be ascertained either by sounding
the terminal cross sections AE, BF, or by sounding a series of
equidistant cross sections. The cross sections are taken at right
angles to the base line. Observers are placed at A and B with
theodolites or box sextants. The floats are dropped in from a boat
above AE, and picked up by another boat below BF. An observer with a
chronograph or watch notes the time in which each float passes from AE
to BF. The method of proceeding is this. The observer A sets his
theodolite in the direction AE, and gives a signal to drop a float. B
keeps his instrument on the float as it comes down. At the moment the
float arrives at C in the line AE, the observer at A calls out. B
clamps his instrument and reads off the angle ABC, and the time
observer begins to note the time of transit. B now points his
instrument in the direction BF, and A keeps the float on the cross
wire of his instrument. At the moment the float arrives at D in the
line BF, the observer B calls out, A clamps his instrument and reads
off the angle BAD, and the time observer notes the time of transit
from C to D. Thus all the data are determined for plotting the path CD
of the float and determining its velocity. By dropping in a series of
floats, a number of surface velocities can be determined. When all
these have been plotted, the river can be divided into convenient
compartments. The observations belonging to each compartment are then
averaged, and the mean velocity and discharge calculated. It is
obvious that, as the surface velocity is greatly altered by wind,
experiments of this kind should be made in very calm weather.

The ratio of the surface velocity to the mean velocity in the same
vertical can be ascertained from the formulae for the vertical
velocity curve already given (§ 101). Exner, in _Erbkam's Zeitschrift_
for 1875, gave the following convenient formula. Let v be the mean and
V the surface velocity in any given vertical longitudinal section, the
depth of which is h

v/V = (1 + 0.1478 [root]h)/(1 + 0.2216 [root]h).

If vertical velocity rods are used instead of common floats, the mean
velocity is directly determined for the vertical section in which the
rod floats. No formula of reduction is then necessary. The observed
velocity has simply to be multiplied by the area of the compartment to
which it belongs.

§ 148. _Mean Velocity of the Stream from a Series of Mid Depth
Velocities._--In the gaugings of the Mississippi it was found that the
mid depth velocity differed by only a very small quantity from the
mean velocity in the vertical section, and it was uninfluenced by
wind. If therefore a series of mid depth velocities are determined by
double floats or by a current meter, they may be taken to be the mean
velocities of the compartments in which they occur, and no formula of
reduction is necessary. If floats are used, the method is precisely
the same as that described in the last paragraph for surface floats.
The paths of the double floats are observed and plotted, and the mean
taken of those corresponding to each of the compartments into which
the river is divided. The discharge is the sum of the products of the
observed mean mid depth velocities and the areas of the compartments.

§ 149. _P. P. Boileau's Process for Gauging Streams._--Let U be the
mean velocity at a given section of a stream, V the maximum velocity,
or that of the principal filament, which is generally a little below
the surface, W and w the greatest and least velocities at the surface.
The distance of the principal filament from the surface is generally
less than one-fourth of the depth of the stream; W is a little less
than V; and U lies between W and w. As the surface velocities change
continuously from the centre towards the sides there are at the
surface two filaments having a velocity equal to U. The determination
of the position of these filaments, which Boileau terms the gauging
filaments, cannot be effected entirely by theory. But, for sections of
a stream in which there are no abrupt changes of depth, their position
can be very approximately assigned. Let [Delta] and l be the
horizontal distances of the surface filament, having the velocity W,
from the gauging filament, which has the velocity U, and from the bank
on one side. Then

[Delta]/l = c^4 [root]{(W + 2w)/7(W - w)},

c being a numerical constant. From gaugings by Humphreys and Abbot,
Bazin and Baumgarten, the values c = 0.919, 0.922 and 0.925 are
obtained. Boileau adopts as a mean value 0.922. Hence, if W and w are
determined by float gauging or otherwise, [Delta] can be found, and
then a single velocity observation at [Delta] ft. from the filament of
maximum velocity gives, without need of any reduction, the mean
velocity of the stream. More conveniently W, w, and U can be measured
from a horizontal surface velocity curve, obtained from a series of
float observations.

§ 150. _Direct Determination of the Mean Velocity by a Current Meter
or Darcy Gauge._--The only method of determining the mean velocity at
a cross section of a stream which involves no assumption of the ratio
of the mean velocity to other quantities is this--a plank bridge is
fixed across the stream near its surface. From this, velocities are
observed at a sufficient number of points in the cross section of the
stream, evenly distributed over its area. The mean of these is the
true mean velocity of the stream. In Darcy and Bazin's experiments on
small streams, the velocity was thus observed at 36 points in the
cross section.

When the stream is too large to fix a bridge across it, the
observations may be taken from a boat, or from a couple of boats with
a gangway between them, anchored successively at a series of points
across the width of the stream. The position of the boat for each
series of observations is fixed by angular observations to a base line
on shore.

§ 151. _A. R. Harlacher's Graphic Method of determining the Discharge
from a Series of Current Meter Observations._--Let ABC (fig. 149) be
the cross section of a river at which a complete series of current
meter observations have been taken. Let I., II., III ... be the
verticals at different points of which the velocities were measured.
Suppose the depths at I., II., III., ... (fig. 149), set off as
vertical ordinates in fig. 150, and on these vertical ordinates
suppose the velocities set off horizontally at their proper depths.
Thus, if v is the measured velocity at the depth h from the surface in
fig. 149, on vertical marked III., then at III. in fig. 150 take cd =
h and ac = v. Then d is a point in the vertical velocity curve for the
vertical III., and, all the velocities for that ordinate being
similarly set off, the curve can be drawn. Suppose all the vertical
velocity curves I.... V. (fig. 150), thus drawn. On each of these
figures draw verticals corresponding to velocities of x, 2x, 3x ...
ft. per second. Then for instance cd at III. (fig. 150) is the depth
at which a velocity of 2x ft. per second existed on the vertical III.
in fig. 149 and if cd is set off at III. in fig. 149 it gives a point
in a curve passing through points of the section where the velocity
was 2x ft. per second. Set off on each of the verticals in fig. 149
all the depths thus found in the corresponding diagram in fig. 150.
Curves drawn through the corresponding points on the verticals are
curves of equal velocity.

The discharge of the stream per second may be regarded as a solid
having the cross section of the river (fig. 149) as a base, and cross
sections normal to the plane of fig. 149 given by the diagrams in fig.
150. The curves of equal velocity may therefore be considered as
contour lines of the solid whose volume is the discharge of the stream
per second. Let [Omega]0 be the area of the cross section of the
river, [Omega]1, [Omega]2 ... the areas contained by the successive
curves of equal velocity, or, if these cut the surface of the stream,
by the curves and that surface. Let x be the difference of velocity
for which the successive curves are drawn, assumed above for
simplicity at 1 ft. per second. Then the volume of the successive
layers of the solid body whose volume represents the discharge,
limited by successive planes passing through the contour curves, will
be

½x([Omega]0 + [Omega]1), ½x([Omega]1 + [Omega]2), and so on.

Consequently the discharge is

Q = x{½([Omega]0 + [Omega]_n) + [Omega]1 = [Omega]2 + ... + [Omega](n-1)}.

The areas [Omega]0, [Omega]1 ... are easily ascertained by means of
the polar planimeter. A slight difficulty arises in the part of the
solid lying above the last contour curve. This will have generally a
height which is not exactly x, and a form more rounded than the other
layers and less like a conical frustum. The volume of this may be
estimated separately, and taken to be the area of its base (the area
[Omega]_n) multiplied by 1/3 to ½ its height.

Fig. 151 shows the results of one of Harlacher's gaugings worked out
in this way. The upper figure shows the section of the river and the
positions of the verticals at which the soundings and gaugings were
taken. The lower gives the curves of equal velocity, worked out from
the current meter observations, by the aid of vertical velocity
curves. The vertical scale in this figure is ten times as great as in
the other. The discharge calculated from the contour curves is 14.1087
cubic metres per second. In the lower figure some other interesting
curves are drawn. Thus, the uppermost dotted curve is the curve
through points at which the maximum velocity was found; it shows that
the maximum velocity was always a little below the surface, and at a
greater depth at the centre than at the sides. The next curve shows
the depth at which the mean velocity for each vertical was found. The
next is the curve of equal velocity corresponding to the mean velocity
of the stream; that is, it passes through points in the cross section
where the velocity was identical with the mean velocity of the stream.

HYDRAULIC MACHINES

§ 152. Hydraulic machines may be broadly divided into two classes: (1) _Motors_, in which water descending from a higher to a lower level, or from a higher to a lower pressure, gives up energy which is available for mechanical operations; (2) _Pumps_, in which the energy of a steam engine or other motor is expended in raising water from a lower to a higher level. A few machines such as the ram and jet pump combine the functions of motor and pump. It may be noted that constructively pumps are essentially reversed motors. The reciprocating pump is a reversed pressure engine, and the centrifugal pump a reversed turbine. Hydraulic machine tools are in principle motors combined with tools, and they now form an important special class.

Water under pressure conveyed in pipes is a convenient and economical means of transmitting energy and distributing it to many scattered working points. Hence large and important hydraulic systems are adopted in which at a central station water is pumped at high pressure into distributing mains, which convey it to various points where it actuates hydraulic motors operating cranes, lifts, dock gates, and in some cases riveting and shearing machines. In this case the head driving the hydraulic machinery is artificially created, and it is the convenience of distributing power in an easily applied form to distant points which makes the system advantageous. As there is some unavoidable loss in creating an artificial head this system is most suitable for driving machines which work intermittently (see POWER TRANSMISSION). The development of electrical methods of transmitting and distributing energy has led to the utilization of many natural waterfalls so situated as to be useless without such a means of transferring the power to points where it can be conveniently applied. In some cases, as at Niagara, the hydraulic power can only be economically developed in very large units, and it can be most conveniently subdivided and distributed by transformation into electrical energy. Partly from the development of new industries such as paper-making from wood pulp and electro-metallurgical processes, which require large amounts of cheap power, partly from the facility with which energy can now be transmitted to great distances electrically, there has been a great increase in the utilization of water-power in countries having natural waterfalls. According to the twelfth census of the United States the total amount of water-power reported as used in manufacturing establishments in that country was 1,130,431 h.p. in 1870; 1,263,343 h.p. in 1890; and 1,727,258 h.p. in 1900. The increase was 8.4% in the decade 1870-1880, 3.1% in 1880-1890, and no less than 36.7% in 1890-1900. The increase is the more striking because in this census the large amounts of hydraulic power which are transmitted electrically are not included.