ANALYSIS OF FLOW IN NETWORKS OF
CONDUITS OR CONDUCTORS
I. INTRODUCTION
1. Type of Problem.-As indicated later, various types of problems
occur in connection with flow in networks. In what is perhaps its
simplest form, the problem is usually as follows:
The quantity of fluid or of energy flowing into the system at one
point is known and the point of delivery is also known. The sizes
and lengths of the conductors or pipes in the system are given or
assumed, and also the law controlling the relation between quantity
of flow in the conductor and the loss of head or voltage in a given
length.
It is usually desired to determine the total loss of head or voltage
between inlet and outlet. If a single conductor connected these two
points, the loss of head for given flow could be computed directly
from the relation between flow and head loss. In a network, however,
this loss depends on the distribution of the flow in the system. If
such distribution is known, the drop of potential in each conductor
can be determined directly, and the total drop found as the sum of
the drops along any path connecting inlet and outlet, the total drop
being of course the same whatever path is chosen.
The difficulty arises in determining the distribution of flow in the
network. This is controlled by two sets of conditions, both simple
and obvious:
(a) The total flow reaching any junction equals the total flow
leaving it (continuity of flow)
(b) The total change in potential along any closed path is zero
(continuity of potential).
These sets of conditions, together with the relation between flow
and potential drop, lead to sets of equations in which either the flows
in the individual conductors or the potentials at the junction points
are taken as the unknowns.
If the flows are taken as the unknowns, the equations will be
those for continuity of potential; if the potentials are the unknowns,
the equations will be those for continuity of flow.* In either case,
*The two methods here presented represent general methods applicable to many engineering
problems. The physical conditions controlling engineering relations often consist of two groups of laws
which are quite independent of each other. In such cases either set of relations may be first expressed
in terms of the other, and then a formal or an approximate solution may be obtained to satisfy the second
condition. Compare the analysis of continuous frames by methods such as the theorem of three
moments, where the equations to be solved represent the geometrical relations, the unknowns having
been previously interrelated by statics, with the method of slope-deflection where the equations are
those of statics, the unknowns having been previously related from geometrical considerations.
ILLINOIS ENGINEERING EXPERIMENT STATION
the order of the equations will be that of the relation between flow
and loss of potential. If this relation is linear, the equations will be
linear. If, however, the relation is not linear, serious difficulties
arise in solving the equations.
In those cases where the relation between flow and change of head
is linear, the methods to be presented may be thought of as a book-
keeping procedure for solving linear equations; where the relation is
not linear, the method changes the problem into that of a succession
of linear relations by use of the fundamental principle of the dif-
ferential calculus.
2. Flow of Water in a Network of Pipes.-In any network of pipes
such as is shown in the problems discussed in the following pages,
it is known that in each closed circuit the sum of all changes in head
is zero, and that at each junction the quantity flowing into the
junction equals the quantity flowing away from the junction.
It is assumed further that we know the law determining the loss
of head in any length of pipe for a given flow. This law usually
takes the form
h = CVn
where h is the change in head accompanying flow in any length of
pipe, C is the loss in the pipe for unit velocity of flow, and V is the
velocity.
Since the quantity of water flowing in the pipe is A V, this relation
may be rewritten
h = rQn
where r is the loss of head in the pipe for unit quantity of flow. The
quantity r depends on the length and diameter of pipe and on its
roughness.
The problem is to find the amount of water flowing in each pipe.
When the distribution of flow is known, the losses of pressure
throughout the system are readily computed.*
It is important to note that, except as noted in the footnote,
only relative values of r are needed to determine distribution of flow.
II. METHODS OF ANALYSIS
3. Methods of Analysis Proposed.-Two methods of analysis are
proposed. In one of these the flows in the pipes or conductors of
*In ordinary cases (on the assumption h = rQ» with n the same for all pipes) the distribution
of total flow is independent of the quantity flowing. On certain assumptions as to the relation between
head and flow, this will not be true. The matter does not seem of immediate practical importance,
though it may be of scientific interest in some cases.
ANALYSIS OF FLOW IN NETWORKS
the network always satisfy the condition that the total flow into and
out of each junction is zero, and these flows are successively corrected
to satisfy the condition of zero total change of head around each
circuit. In the other method the total change of head around each
circuit always equals zero, and the flows in the pipes of the circuit
are successively adjusted so that the total flow into and out of each
junction finally approaches or becomes zero.
The former method is, for convenience of reference, here desig-
nated as the "Method of Balancing Heads," the latter as the
"Method of Balancing Flows." The method of balancing heads is
directly applicable where the quantities flowing at inlets and outlets
are known. The method of balancing flows is directly applicable
when the heads at inlets and outlets are known; in this case it will
probably be found more convenient than that of balancing heads.
In some problems it may be desirable to combine the two methods.
Both methods depend on the principle that the resistance to
change in flow in any pipe equals approximately nrQ(" - ) where
h = rQ".
III. METHOD OF BALANCING HEADS
4. Statement of Method.-The method of solution is as follows:
(a) Assume any distribution of flow.
(b) Compute in each pipe the loss of head h = rQn. With due
attention to sign (direction of potential drop), compute the total
head loss around each elementary closed circuit 1h = IrQ".
(c) Compute also in each such closed circuit the sum of the
quantities R = nrQ(n - 1) without reference to sign.
(d) Set up in each circuit a counterbalancing flow to balance the
head in that circuit (to make IrQ" = 0) equal to
SrQ" (with due attention to direction of flow)
ZnrQ(" 1) (without reference to direction of flow)
(e) Compute the revised flows and repeat the procedure.
Continue to any desired precision.
In applying the method it is recommended that successive com-
putations of the circuits be put on identical diagrams of the system.
In office practice such diagrams will usually be white prints. Write
in each elementary circuit the value IR, and outside the circuit write
first (above) the value 2h for flow in a clockwise direction around the
circuit and second (below) the value Zh for flow in a counterclock-
wise direction around the circuit. On the right of these figures put
an arrow pointing ) or ) to the larger figure. This arrow will show
10 ILLINOIS ENGINEERING EXPERIMENT STATION
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ANALYSIS OF FLOW IN NETWORKS
correctly the direction of counterflow in the circuit. This technique
is illustrated in the problems following.
5. Proof of Method.-If the distribution of flow assumed in the
first place were correct, the change of head around any single closed
circuit would be zero. This change of head is IrQ". Considering
for the present a single circuit, write for each pipe
Q = Qo0 + A
Then, rQn = r (Qo + A)" = r (Qn + nQ (-1) A + )
If A is small compared with Qo the remaining terms in the expansion
may be neglected.
Then, for
SrQ" = 0
SrQ' = -Ao2nrQ"n - 1)
STh yrQn
A = -- =- ---
IR InrQgn - 1)
Of course, if A is relatively large compared with Qo and n is greater
than unity, the approximation is not very good, but this is less
important than it might at first seem, because in any case we must
correct for the unbalanced head produced in one circuit by corrections
in the adjacent circuits, which in general requires a recomputation of
all circuits. The convergence is, for practical purposes, sufficiently
rapid.
6. Illustrative Problems.-
Problem 1.-Single Closed Circuit
(Figure 1)
This problem shows the elementary procedure for a single circuit
in each of three cases: (a) where h varies as Q (streamline flow, or
electrical resistance, E varies as I); (b) where h varies as Q1-5 merely
as an illustration of a fractional exponent; and (c) where h varies as
Q2, a common approximate value for water circuits.
In all of these cases it is required to distribute a flow of 100
between two pipes, one of which is four times as long as the other,
but which are otherwise alike. In each case, also, to show the con-
vergence, the first assumption is the worst possible, namely that the
total flow follows the longer path.
ILLINOIS ENGINEERING EXPERIMENT STATION
Of course each of these cases is readily solved directly. Thus, in
all cases let Qi be the flow in the shorter, and Q2 that in the longer
pipe. Then
(a) Q1 4 4
( - a QQ. = - 100 = 80
Q2 1 5
Q1 (4)% 2.52
(b) - Q, = - - 100 = 71.5
Q2 (1)% 3.52
Qi (4)M 2
(c) - Q1 = -. 100 = 66.7
Q2 (1) 3
The computations have in these cases been arranged in the order
explained already, which has been found very convenient, namely,
(1) Write the divisor ZnrQ(n - 1) within the circuit.
(2) Write on one side of the circuit first the sum ZrQ" for clock-
wise flow, and below this the sum SrQ" for counterclockwise flow.
If, then, the arrow indicating the direction of flow is written on the
right of these figures and pointing to the larger flow, it will correctly
indicate the direction of counterflow needed to balance the circuit.
In complicated problems observance of some such system is necessary.
Problem 2.-Simple Network-h Varies as Q2
(Figure 2)
This shows a very simple network with one inlet and one outlet.
It is here assumed that h varies as Q2. All pipes are assumed to be
alike. The purpose of the problem is to illustrate the method of
arrangement in a case slightly more difficult than that of a single
circuit.
Here, as before, a very bad first trial value was intentionally
chosen to illustrate the procedure. Of course, the exact solution is
at once known by inspection to be as shown, and almost any reasonable
trial converges rapidly.
The arrangement of computations is that previously recom-
mended. Note the relatively small change in 2,nQ( 1).
Procedure
Some distribution of flow (without excesses or deficiencies at the
junctions) is first assumed.
(1) Compute the unbalanced heads around each circuit. (In
this case h = Q2 in each pipe.) As previously noted, these are
written to one side of the circuit, first (above) the heads for clock-
wise flow within that circuit, next (below) those for counter clock-
ANALYSIS OF FLOW IN NETWORKS
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ILLINOIS ENGINEERING EXPERIMENT STATION
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METHOD OF BALANCING HEADS
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ANALYSIS OF FLOW IN NETWORKS
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FIG. 3. DISTRIBUTION OF FLOW IN NETWORK-TWO INLETS;
METHOD OF BALANCING HEADS
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ILLINOIS ENGINEERING EXPERIMENT STATION
wise flow. The difference is the unbalanced head. The arrow, )
or ), written to the right of these figures, points to the larger head
loss, and indicates the direction of counterflow.
(2) Compute the divisors 2,nrQ(n - 1). In this case nrQ(" - 1) = 2Q
in each pipe, since r = 1. These are written within the circuits.
(3) Revise the flows from these counterflows, and repeat the
process as often as required.
In the foregoing solution the method is intentionally applied
blindly, without any judgment at all. It is at once evident that all
circuits in the first trial are badly unbalanced, all flows in each circuit
being in one direction. Usually the circuits which are badly un-
balanced will at once be thrown more nearly into balance by guess.
Problem 3.-Complex Network-Two Inlets-h Varies as Q
(Figure 3)
This is a more complicated network, with two inlets and one
outlet. The solution involves no new principles. Here it is assumed
that h varies as Q. Of course, in this case the divisors for each circuit
are constant throughout.
No attempt has been made to compute fractional values. It is,
however, clearly possible to go to any reasonable degree of precision,
but with considerable increase in the computation required if results
correct to three figures are wanted.
The arrangement of computations follows that previously ex-
plained.
Problems 4 and 5.-Systems of Pipes in Different Planes Interconnected
(Figures 4 and 5)
In general, systems for distributing water in cities may, for pur-
poses of analysis, be considered as in a single plane. In other cases,
as, for example, in distributing steam or hot water to a heating system,
the distribution may take place in several planes, with interconnection
between the planar systems of distribution.
This type of problem presents no especially new features except
that successive distribution must be made in circuits closed by the
risers as well as in the circuits which lie in a plane. The pipes
chosen on each floor to close the circuits containing the risers are
selected arbitrarily. It will be noted that in such problems any pipe
may lie in only one circuit (an outside pipe in a floor) or in two
circuits, three circuits, or even in four circuits (two floor circuits and
two riser circuits). The total change in flow in the pipe is the sum
of the changes in all the circuits of which it is a member.
ANALYSIS OF FLOW IN NETWORKS
Problem 4 shows a rather impractical layout selected for simplicity
of illustration. The distribution is carried through only two steps to
show the procedure.
Problem 5 differs from Problem 4 only in having more risers.
Clearly the technique used in recording the flows is a matter of
individual choice. Some may prefer to use isometric diagrams
throughout the analysis.
It is believed that the diagrams are self-explanatory.
7. Characteristics of Procedure.-Certain characteristics of the
procedure will be noted. When the flow is adjusted in any circuit
the flow is increased in some pipes and decreased in others, so that
the quantity IR = InrQ(n - 1) is not very much changed, and need
not usually be recomputed for each change of flow.
Since the first adjustments are in a sense preliminary, it is useless
to attempt precision in making them.
The answer, when finally obtained, is inevitably correct, since it
satisfies the conditions that the quantities balance at each junction,
and that the heads balance around each circuit. Moreover, errors in
the procedure are not cumulative, and, if made, are ultimately
eliminated.
IV. METHOD OF BALANCING FLOWS
8. Statement of Method.-In the method of analysis by balancing
heads just presented, the flow at any junction is balanced throughout
the analysis, but the head around any circuit is balanced by successive
correction.
Another method is to keep the head balanced around any circuit
throughout the analysis, in which case the flow at the junctions is
balanced by successive correction.
We may, then, assume a series of heads throughout the system
and compute the flow in each pipe corresponding to the differences
of head. From these find the total flow to each junction except inlets
and outlets. Distribute this flow to the pipes connecting at the
junction in inverse proportion to their resistances. (R = nrQ(n - 1)).
This, of course, causes an excess (or deficiency) of flow at the next
junction, but by successive distribution the excess flow will ulti-
mately be squeezed out at the inlets and outlets of the system.
Note, in the first place, that if the flow is distributed as in the
ILLINOIS ENGINEERING EXPERIMENT STATION
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ANALYSIS OF FLOW IN NETWORKS
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ILLINOIS ENGINEERING EXPERIMENT STATION
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ANALYSIS OF FLOW IN NETWORKS
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ILLINOIS ENGINEERING EXPERIMENT STATION
foregoing the head difference between any inlet and outlet is approxi-
mately unchanged, the change of head in one direction along one
part of any path connecting two openings being equal and opposite
to the change of head along another part. Of course, the values
R = nrQC" - 1) are not exact when n is greater than unity, because the
values of Q are not exact. Hence revision of the values of R is
necessary.
Note also that the total flow into and out of the system is in
general not balanced in the beginning, but that any excess of inflow
will untimately be squeezed out by successive distributions. Similarly,
if there is a deficiency, we may imagine that it will inflow at the inlets
and outlets.
9. Illustrative Problems.-
Problem 6.-Simple Network-h Varies as Q2
(Figure 6)
The problem analyzed in Fig. 6 is the same as that previously
analyzed by balancing heads in Fig. 2. The elevation at the inlet
is assumed to be 100. The outflow takes place at one point as shown.
It is assumed that in each pipe-run of the system the loss of head
equals the square of the quantity flowing. The procedure is as follows:
Procedure
(1) Assume heads arbitrarily for inlet and outlet. In this case
assume inlet at El. 100 and outlet at El. 0. Guess at heads at other
junctions as closely as possible. These values are shown in circles
near the junctions.
(2) Compute flow (or relative flow) in each pipe for heads as-
sumed. In this case it is assumed that h = rQ2, and therefore that
(3) Compute excess (+) or deficiency (-) of flow at each junc-
tion except inlet and outlet. Write these in the squares shown for
each junction.
(4) Distribute these excesses or deficiencies to the pipes con-
necting at each junction in inverse proportion to R = nrQ(n - 1). In
this problem all values of r are assumed equal to unity, except in
the corner pipes which are twice as long, and hence have r = 2.
1
Also n = 2. Hence the flows are distributed in proportion to -,
rQ
where.Q is the flow already computed.
(5) Carry over in the pipes connecting at each junction (except
inlet and outlet) the flows distributed from adjacent junctions to
ANALYSIS OF FLOW IN NETWORKS
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FIG. 6. DISTRIBUTION OF FLOW IN SIMPLE NETWORK;
METHOD OF BALANCING FLOWS
THREE
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ILLINOIS ENGINEERING EXPERIMENT STATION
find new values of excesses and deficiencies.
(6) Distribute the new unbalanced flows and repeat to conver-
gence, thus determining the relative amount of the total inflow
flowing in each pipe.
Problem 7.-Multiple Inlets and Outlets
(Figure 7)
In this problem it is assumed that the elevations are known at
several inlets-namely, El. 40 at inlet A, and El. 30 at inlet B-and
at several outlets-namely, El. 10 at outlets B and D, and El. 0 at
outlets A and C. It is desired to determine the distribution of total
flow at inlets and at outlets, and, in general, the distribution of flow
in the system. For simplicity of presentation it is assumed that the
resistances are the same in all pipe-runs. It is also assumed that
Q2
h = -- in any pipe.
1000
First assume heads at intermediate junctions as shown by figures
in the circles at each intermediate junction.
Compute flows-or relative flows-in each pipe. Here the flow
is computed as Q = V 1000h.
Compute excess or deficiency of flow at each junction, and write
this in the rectangle near that junction. Distribute these unbalanced
1
flows in proportion to - at each junction. The relative values
Q
for distribution at each junction are written on the pipes at that
junction.
Compute the excesses and deficiencies produced by the new flows
and distribute these.
It may be assumed that in a problem of this type what is really
wanted is the distribution of outflow and of inflow. Table 1 shows
these flows after successive distributions in this case. The con-
vergence here is slow and not very satisfactory, partly because the
first guess as to intermediate heads is not very good. Nevertheless,
the computations indicate quickly the approximate distribution of
flow at inlets and outlets.
Some circuits will be found to be unbalanced as regards head.
These may now be adjusted, if desired, starting with the distribution
found for total flow.
10. Remarks on Method.-In such relatively complex problems as
that just shown, it seems clear that the method of distributing flows
has advantages. In general, however, it is thought to be less simple,
obvious, and expeditious than that of balancing heads.
ANALYSIS OF FLOW IN NETWORKS
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FIGa. 7. DISTRIBUTION OF FLOW IN NETWORK-SEVERAL INLETS AND OUTLETS;
METHOD OF BALANCING FLOWS
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ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 1
PROBLEM 7.-SucCESSIVE VALUES OF INFLOW AND OUTFLOW
Values Values Values Values Estimated True
Assumed After After After After Values
Values One Dis- Two Three Four
tribution Distri- Distri- Distri-
butions butions butions Value Per Cent
Inlet A................ 488 552 525 543 535 537 59
Inlet B................ 342 388 348 370 356 361 41
Total Inflow. ........ 830 940 873 913 891 898 100
Outlet A............... 244 279 259 270 263 268 30
Outlet B............... 200 158 170 162 166 166 18
Outlet C............... 244 279 259 270 263 268 30
Outlet D .............. 244 182 200 192 196 196 22
Total Outflow ........ 932 898 888 894 888 898 100
V. TYPES OF PROBLEMS ENCOUNTERED IN NETWORKS
11. Typical Problems.-In systems carrying water, and in other
networks of circuits, the problems may take various forms.
(a) Figure 8a. Given the inflow at one point and the point of
discharge, to determine the distribution of flow and variation of head
within the network. This problem represents the most elementary
application of the method of balancing heads. (Problems 1, 2, 4, 5).
It may also be solved by balancing flows. (Problem 6).
(b) Figure 8b. Given the distribution of inflow at several inlets
and the distribution of outflow at several outlets, to determine the
distribution of flow and the variation of head. (Problem 3). The
problem is practically the same as that just discussed. It is more
conveniently solved by balancing heads than by balancing flows.
(c) Figure 8c. Given the heads at various points of inflow or
outflow, to determine the total flow and its distribution. Such a
problem may occur in studying flow between reservoir systems. This
may be solved directly by the method of balancing flows. (Problems
6, 7).
(d) Figure 8d. Given the heads at various points of inflow and
the relation of head to flow at various points of outflow. This prob-
lem may be reduced to the problem just stated by imagining that at
each outflow point an additional pipe or conductor is connected.
This imaginary discharge pipe is to have such a resistance that for
unit flow the loss of head would be the same as for unit flow through
the outlet.
(e) Figure 8e. Given the heads at certain points of inflow or
outflow and also the flow, independent of head, at other points, to
ANALYSIS OF FLOW IN NETWORKS
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FIG. 8. TYPES OF PROBLEMS ENCOUNTERED IN DISTRIBUTING
FLOW IN NETWORKS
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ILLINOIS ENGINEERING EXPERIMENT STATION
determine the distribution of flow. This problem is essentially the
same as that of fixed heads in (c). At the points of fixed flow allow-
ance for the flow is to be made in computing excess or deficiency
of flow. Moreover, in applying the method of distributing flows,
the points of fixed flow are not to be considered as openings at which
water may be squeezed out or sucked into the system.
Evidently various combinations of these problems may occur.
All the problems just stated deal with constant conditions of flow.
The heads in the reservoirs, for example, are assumed to remain
constant in spite of the flow. A common problem, however, is that
in which the flow affects the heads as the flow continues, and it is
desired to study the changes in flow conditions with time. The
present paper makes no direct contribution to the solution of this
problem, except as it indicates methods of solution for fixed condi-
tions. However, such problems may be conveniently studied by the
method of increments, thus:
(1) Compute the flow conditions for the initial heads.
(2) Compute from inflows and outflows the changes in head in
any convenient short interval of time.
(3) Compute the changes of flow for these changes of head. This
procedure, while approximate, is quite direct.
(4) Repeat the procedure.
VI. CONCLUDING REMARKS-OTHER APPLICATIONS OF METHODS
As previously stated, this monograph deals primarily with flow of
water in networks of pipes. The problems are restricted to cases in
which the relation between loss of head and quantity of flow is of
the relatively simple form h varies as Qn. It is sometimes proposed
that the flow of liquids may be controlled by a relation of the form
h = aQ + bQ2. In this case we can deduce - = (a + 2bQ), and the
AQ
counterflow required to balance the head in any circuit becomes
I (aQ + bQ2) with due attention to direction of flow
AQ = -
S (a + 2bQ) without reference to direction of flow
Ah
In more general terms, if h = f(Q), then - = f'(Q) and the
AQ
counterflow
Sf(Q) with due attention to direction of flow
AQf'(Q) without reference to direction of flow
y2f'(Q) without reference to direction of flow
ANALYSIS OF FLOW IN NETWORKS
This suggests that such arithmetical application of the principle
of the differential calculus may have application in the solution of
certain problems in simultaneous equations, but no effort has been
made to explore this field.
Applications of the methods in the study of electrical circuits
are evident.
This paper deals directly only with networks of definite conductors.
Clearly, however, the methods may also be applied to study distri-
bution of flow in those cases where no pipes really exist, but where
the flow is diffused, as in cases of percolation through earthen dams
or through soil strata where the operation of wells is to be investi-
gated. In such cases flow in a series of imaginary pipes is substituted
for the diffused flow. This leads, of course to the "flow net" picture.
The principal value of the method in this connection is in checking
and revising an assumed flow net, the pipes being assumed to be
elements of the flow net and of uniform resistance per unit of length.
This suggests extension of the method to a general method of
studying flow nets in moving water.
The methods apparently have important applications in the study
of any field of potential.
Clearly many variations of technique are possible in applying
the methods. Thus, a circuit is unbalanced only by counterflow in
adjacent circuits, and so it is possible to merely "carry over" a cer-
tain fraction of such counterflows. Also it is possible to use the
results from balancing other circuits in computing each successive
circuit. Considerable experimenting with various procedures leads
the writer to believe that the technique recommended will be found
most satisfactory because of its simplicity.
It will bear repetition that the first approximations need not be
made very formally. With some experience it is possible to nearly
adjust a network at once by a little judicious guessing.

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