The Polarographic Analysis of Nickel Plating Solutions
J.V. Petrocelli and G. Tatoian, Director of Research and Chemist, respectively,
The Patent Button Company, Waterbury, CT
ABSTRACT A brief review is given of polarographic methods
and their potentialities in the study and control of plating solutions. The
general principles
and techniques
are outlined and a new manual polarograph is described which has been specifically
designed and developed for the application to electroplating solution analysis.
Experimental data are presented showing how the polarographic method may
be applied to the determination of the principal constituents of the
Watts’ type
nickel plating solution. Standard curves and detailed procedures are given
for the determination
of nickel sulfate, nickel chloride and boric acid. The effect of pH and brighteners
is briefly discussed.
INTRODUCTION
Polarographic methods of analysis have been used successfully and with
great advantage in many fields of applied analytical chemistry. They
generally
require less chemical manipulation and expenditure of time than the regular
chemical
methods and are especially useful for the determination of small quantities
of substances.
A consideration of the basic principles
and techniques involved in polarography indicates that it should lend itself
readily and with great
advantage to
the analysis and the control of plating solutions and electrodeposits.
It also should
be a very useful tool in plating research. Among the many possible uses
of the techniques in this latter field, the following may be mentioned:
the
mechanism
of electrode reactions, adsorption, the study of the potentials at which
metals may be reduced from various solutions, the formation of complex
ions and their
effect on electroplating, and the reaction of brighteners on electrode
potentials.
Since the polarographic method is
based upon electrochemical phenomena such as electrolysis, polarization,
and the measurement of current-voltage
curves,
it
should appeal to the electrochemist and the plater as an elegant and
interesting tool for the control and study of his processes.
In light
of these attractive features, it is not surprising that interest is increasing
in the practical application of polarography to plating
control.
Gordon and Roberts1 have
successfully applied the polarograph to the analysis of brass plating solutions
and deposits.
Sazanova
and Korshunov2 have
developed
polarographic methods for the analysis of the acid-zinc solution.
Vyakhiirev3 used the polarographic method in developing
a procedure for the amperometric
titration of sulfates in plating solutions. Recently, Diaz and
Lindemann4,5 have reported methods for the polarographic determination
of tin, lead and zinc
in their respective plating solutions.
The work reported in this
communication is a part of a general study undertaken in the authors’ laboratory
to develop polarographic methods for the control of some of the more common
plating solutions and to design and develop a simplified
polarographic instrument which can be used readily by the control
chemist. These procedures have been used in the control laboratory of the
writers’ company
for several years with great success. It has been found that
the polarographic methods save a considerable amount of time; not only
are they
much more rapid
but the time consumed in making and maintaining the standard
solutions usually necessary for the more conventional chemical methods
is reduced considerably.
Procedures have been developed
for determining the principal constituents
of nickel plating solutions, silver, copper, zinc, brass and
cadmium-cyanide plating
solutions, and chromium plating solutions, and for the detection
and determination of some of the impurities usually found in
these solutions.
The data presented
here are concerned with the determination of the principal
constituents of a Watts’ type and a cold nickel plating solution; it
is planned to present data on other plating solutions in subsequent
papers.
GENERAL PRINCIPLES
As previously indicated, the principles of polarographic analysis
are based upon subjects already familiar to the electroplater,
namely, electrolysis and polarization.
Since these fundamentals are treated in great detail in Kolthoff
and
Lingane’s
excellent monograph6 only a very brief outline of the fundamentals
will be presented here.
The method is based on the polarization
characteristics of
a micro-electrode when a solution containing an electro-oxidizable
or an electro-reducible
substance is electrolyzed. The system consists of a micro-electrode
and a relatively large
nonpolarizable electrode in a solution containing a small
concentration (about 10-3 M/l) of the substance to be determined,
a relatively
large concentration
(0.1 M/l) of a supporting electrolyte and a small amount
of a
maximum suppressor such as gelatin.
The most versatile micro-electrode
is the dropping mercury electrode (D.M.E.) which consists of a capillary
glass
tube connected to
a leveling bulb filled
with mercury. The diameter of the capillary and the height
of the leveling bulb are adjusted so that the mercury
drops from
the capillary
in small
uniform drops.
Usually, each drop has a lifetime of about 3 seconds.
The nonpolarizable electrode may consist of a large pool of
mercury or a calomel
half-cell; the saturated
calomel half-cell makes a very satisfactory electrode
for this purpose. A schematic cell arrangement is shown in
Fig. 1.
Fig. 1—Schematic
polarographic circuit and electrode assembly.
The substance which is to be determined
should be in solution form and a suitable portion added to the supporting
electrolyte,
gelatin
is added
and
the resulting
solution is transferred to the polarographic cell.
Since
oxygen reacts at the D.M.E. it is desirable to remove
dissolved air
from the solution
by passing
nitrogen
through it before electrolysis. After shutting off
the nitrogen the solution is electrolyzed by slowly applying
an increasing
voltage. The current
is recorded at various values of the applied voltage
and the results plotted.
This gives
the current-voltage curve known as a polarogram. It
is then possible to determine the nature and concentration
of the
reacting substance
from the
polarogram.
The following typical example is given in order to
show the technique involved.
As a typical polarographic cell
reaction, one may consider the electrolysis of a deaerated solution containing
0.001 M NiCl2,
0.10 N KNO3 and
a small amount
of gelatin. The cell and the basic polarizing circuit
are shown schematically in Fig. 1.
The basic circuit
consists of a uniform slide wire R2 through which a steady current passes.
The total
potential
drop,
V, across the
slide wire is adjusted
by means of the resistance, R1, usually at 1.00
v or 2.00 v.
The slide wire contact is moved
so as to vary the voltage, E, applied to the cell. Generally, the
slide wire has
some indicating
mechanism
such
as a graduated
dial so that the applied voltage, E, may be read
directly. The switch, S, is used for setting
the polarity of
the dropping mercury
electrode.
A sensitive
galvanometer, G, with its appropriate shunt.
R3, is generally calibrated as
a micro-ammeter.
If the solution is electrolyzed
by making the dropping mercury electrode the cathode and the
applied voltage
is increased
slowly from –0.3 v to –1.5
v at first only a very small current flows,
which increases approximately linearly with the voltage
and is known as the residual current. When
the voltage is reached
that is high enough for the discharge of nickel
ions at the dropping electrode, the current
rises very rapidly and finally reaches a
limiting diffusion
current.
If the values of the current are plotted
versus the corresponding values of the applied voltage,
the curve, C, in Fig. 2 will be obtained.
This curve is the
polarization curve for the deposition of
nickel on mercury and is known as a ”polarogram” or
polarographic wave. Curve D represents the
polarogram for a greater concentration of
nickel while curve
B represents the wave of lesser concentration.
Fig. 2—Schematic
polarographic curves. A—residual; B, C & D—various concentrations
of reacting ion.
If
the solution did not contain nickel, but
only 0.1 M KNO3, the curve A, in Fig. 2, would
be
obtained. This consists
only of the
residual
current and the
difference between curves C and A, at any
voltage along
the
top of the plateau
of the wave, is the current increase due
to the nickel ion and is called the diffusion
current,
Id.
The residual current is due predominantly
to the flow of current required to maintain
the
surface
charge
on the
continually renewed surface of
the mercury
drop.
As each mercury drop grows
and falls, the are changes from practically zero to some
maximin
value. The
current’ therefore, oscillates
between some minimum and maximum value
so that
the galvanometer spotlight performs oscillations
at
each current value. For purposes of measurement,
maximin values are satisfactory.
The gelatin
at a concentration of about, 0.01 per
cent is necessary in order to
suppress the usual
tendency
of the
current-voltage
curve to
develop an
abnormal rise in current before the
limiting current is reached.
The diffusion current is
due to the extreme state of concentration polarization
which
is reached
at the
dropping electrode
with respect to the deposition
of nickel. Generally there is a linear
relationship between the concentration
of
nickel in the solution and the diffusion
current. Ilkovic7 derived the following
equation for
the diffusion current
at the dropping
mercury electrode at constant
temperature:
Id =
607 n D1/2 m2/3 t1/6 C
Id is
the limiting diffusion current expressed in microamperes, n is the number
of electrons
involved in the electrode
reaction, D
is the
diffusion
coefficient
of the electro-active material in the
units cm2 sec-1,
C is the concentration of the electro-active
material expressed in millimoles
per liter,
m is the mass of mercury per drop expressed
in mg sec-1 and
t is the
lifetime
of a
drop in
seconds.
This linear relationship
is the basis for the application of polarography
to quantitative
analysis. The qualitative
aspects,
the identification
of the ion
is based on the measurement of the
half-wave potential, E. This is the
voltage at
which the current has
reached one-half
the
value of
the diffusion
current.
The half-wave potential is characteristic
of
each ion in ally given supporting
electrolyte.
If several substances,
which undergo reaction at the dropping mercury
electrode, are
present in
the solution7 each one
will produce its
own characteristic
polarographic wave. It is thus
possible to obtain qualitative and quantitative
analysis
for
several substances in one solution.
It is necessary, however, that
the half-wave
potentials
of the
substances are not
too close together.
Fig. 3—A schematic calibration curve.
Calibration
Curves
As shown above, the diffusion current
of a substance reacting at the
dropping mercury
electrode is
proportional to the
concentration of
the substance
in any given supporting electrolyte,
and given
capillary characteristics and
temperature.
There may be at times some deviations
from a linear relationship; it
is therefore highly desirable
to prepare calibration
curves.
A calibration curve should
be prepared by running polarograms
on a series
of test solutions
of
varying and known
concentrations of the
substance
to be determined.
It is absolutely necessary
that all other factors remain constant
and
exactly as
those encountered
when the
unknown is run. These
factors are: drop time
at a given voltage, supporting
electrolyte, concentration
of maximum suppressor and temperature.
A typical
calibration curve is shown
in Fig. 3.
INSTRUMENTATION
The essential parts of an instrument
for obtaining polarographic
curves are a polarizing
unit
by means of which a known
and variable voltage
may be applied
to the cell and a sensitive
current measuring device
such as a galvanometer.
It soon became apparent during
the course of the authors’ investigations
that a relatively simple
manual instrument is all
that would be required for
the applications to plating
solution work. Since most
of the commercial instruments
were more elaborate and hence,
much more costly than required,
a division of the authors’ company
developed (Electropolarizer
Model C1, manufactured by
Patwin
Instrument
Division,
The Patent Button
Company,
Waterbury, Conn.) a unit
to overcome such objections.
It
has been especially designed
to meet the requirements
at hand; it is simple
in its construction and by
the elimination of features
which are not important to
this application its cost
was held down
to a vicinity that would
be attractive to the control
chemist and plater. The
result was achieved by simplification
and not by sacrificing quality
of parts; precision components
have been used
throughout.
The polarizing
unit of the instrument
contains a 10
turn helical potentiometer
with a
linearity tolerance
of 0.1
per cent and
is operated by an integrating
dial graduated in one-thousand
units. The span voltage
may be varied from
1 to 3
volts. With
the 3-volt
span
the accuracy
obtained
is
more than sufficient
for
polarographic work of this
nature. The current is
provided by dry
cells and
regulated by
a variable resistor.
A voltmeter of
1 per
cent accuracy
designates
the span
voltage. A sensitive galvanometer
(about 0.005 µa/
mm) with a low internal
resistance
and a long period of swing
is used as a micro-ammeter.
This is shunted
by an 11 point multiplying
shunt providing suitable
current ranges. A photograph
of the instrument is shown
in Fig. 4.
Fig. 4—The Patwin Electro-Polarizer Model C1.
EXPERIMENT Apparatus and Materials
The exploratory work and
the determination of
the calibration
curves were
performed on a precision
manual instrument
assembled in the
authors’ laboratory.(Currently
manufactured as Model
R1 by Patwin
Instruments Division,
The
Patent Button Company,
Waterbury, Conn.)
The experiments and actual
analyses of plating solutions
were carried out with
the simplified model
which as previously
stated had been designed
especially for this application.
A polarographic cell,
also made by the company,
in combination
with a saturated calomel
half-cell was used. The
side arm of the calomel
cell
which serves as a liquid
junction was filled with
agar gel containing a
solution
of saturated KCI. The
agar plug together with
the sintered glass disc
in the
polarographic cell prevent
contamination of the
solution by chloride.
The cell
was immersed in a constant
temperature water bath
when calibration curves
were
obtained.
The capillary was made
from glass barometer
tubing.(Supplied by The
Corning Glass Works,
Corning,
N. Y.) It was about
8.0 cm in length
with a bore diameter
of about
0.06 cm.
The constant
m2/3 t1/6 was
1.830
mg2/3sec-1/2. The drop
time,
t, was about 4.5 sec
at
-1.375 v.
Standard solutions
were made up with distilled
water and
chemically pure
salts. The concentrations
were
determined by standard
chemical methods.
The gelatin solution
was made by dissolving
0.200
gram of
powdered gelatin in
100 ml of distilled
water which
had been
boiled
previously for about
10 minutes,
and then cooled to
60° C. A few drops
of toluene were added
and the flask firmly
fitted with a glass
stopper. All the operations
were performed in the
flask in which the
solution was stored.
A solution usually
kept for several days.
Bacterial action produces
a cloudiness
which is readily visible
and whenever bacterial
action was suspected
a fresh
solution was prepared.
Fig. 5—Polarographic
waves for chloride, nickel and boric acid as obtained on a nickel plating
solution.
Tank
nitrogen was used
without further purification;
this
was bubbled through
a wash bottle containing
distilled water before
entering
the solution to
be polarographed
in order to saturate the
gas with
water.
Supporting
Electrolyte
A study of the half-wave
potentials of nickel,
chloride ion and
boric acid indicated
that
it might be feasible
to use
one supporting
electrolyte
for
all three constituents.
Exploratory experiments
showed that 0.1
N KNO3 (10 g/l)
containing 150
g/l of
mannitol would
be a suitable
electrolyte.
In this
medium a polarogram
extending from
+0.40 v to –1.90 v yields three waves in succession, the chloride,
the nickel and
the boric acid wave. The chloride wave is anodic with E +0.26 v while the
nickel and boric acid waves are cathodic with E of about –1.07
v and –1.65
v, respectively.
A typical polarogram
is shown in Fig.
5.
It was found
that
boric acid does
not show
a wave in KNO3 without
the presence
of
mannitol. This
is probably
due
to the fact
that boric acid
is a very
weak acid, and
as is well known8
mannitol
forms a
relatively strong
acid
with
boric acid.
The
concentration of KNO3 has a
definite effect
upon
the
boric acid
wave. If the concentration
of KNO3 is
greater than about
0.25 N the
wave is
obscured by
the beginning
of the
much larger
potassium wave.
The effect
is shown in Fig. 6. As
a result
of these
experiments
a solution
of
0.1 N KNO3 containing
150 g/l of
mannitol was chosen
as a supporting
electrolyte.
Fig. 6—Effect of KNO3 concentration
on the boric acid wave. A—1.0 N, B—0.50 N, C—0.25 N and D—0.10 N KNO3.
Deaerator It is essential
to deaerate
the solution
when nickel
and boric
acid are being
determined.
The second wave for
the
reduction
of oxygen
occurs in the
same region
as the nickel
wave so that
the diffusion
current
for nickel
will
be about 1.5
to 2.0 pa too
high. The effect
on the
boric
acid
is of even
greater importance.
The
hydroxide ion
produced by
the reduction
of oxygen
combines with
an equivalent
amount
of hydrogen
ion at the
electrode surface
and low results
will be obtained
for boric acid.6
Since the calomel
half-cell contains
a high concentration
of chloride
ion, there
is the
possibility
that some chloride
may diffuse
into
the polarographic
cell
if sufficient
time
elapses. It
is, therefore,
desirable
to determine
the chloride
ion as soon
as possible
after the
half-cell is
connected to
the
polarographic
cell. This
determination
may be performed
before the
solution
is deaerated.
The solution
may
then be deaerated
and the
nickel and
boric acid
determined.
Procedures
and Calibration
Curves
The methods
developed
in this study
are based
upon
the use
of a 0.25-ml
sample
of plating
solution
diluted to
a final
volume
of
50 ml for
the polarographic
solution.
This dilution
yields
a satisfactory
concentration
(about 1.0
to
10.0 millimoles/l)
of reacting
substance
for polarographic
analysis.
The
concentration
of supporting
electrolyte
is high enough
so that either
of the two
following
procedures
may
he
used. A 0.25-ml
sample
of plating
solution
may be used
directly
or a
5.0 ml sample
may
be diluted
to 100 ml
with distilled
water
and a 5.0-ml portion
of
the resulting
solution
used.
Calibration
curves
were obtained
as follows:
a series
of standard
solutions
were made
containing
known and
varying
concentrations of NiSO4 · 7H2O,
NiCl2· 6H2O
and
H3BO3.
A 5.0-ml sample
was
diluted
to 100
ml
with distilled
water and
a 5.0-ml
sample
of the
resulting
solution
was pipetted
into
a 50-ml
volumetric
flask,
1.0 ml of 0.2
per cent
gelatin
solution
was
added and
the whole
diluted
to the
mark with supporting
electrolyte.
About 25
ml of this
final
solution
was transferred
to the
polarographic cell and
nitrogen
gas was
slowly bubbled
through
the solution
for about
15 minutes
and
then disconnected.
The D.M.E.
was
then polarized
anodically
from +0.10
to +0.50
v
in
order to
obtain
the chloride
wave;
and cathodically
from –0.70
v to –1.90
v to obtain
the nickel
and boric
acid waves.
Fig. 7—Calibration curves for NiSO4 · 7H2O
Since
the waves
are
well
defined and
reproducible,
the diffusion
currents
were
obtained
by pointer
readings.
The currents
were
read at
+0.40
v for chloride, –1.38
v for
nickel and –1.82
v for
boric acid. A
correction
was not
made
for the residual
current,
therefore
this
current appears
as the
intercept
of the
calibration
curves
on the
current
axis.
The calibration
curves
were
obtained at three
temperatures,
68°,
77° and
86°F.
All
concentrations
are
reported as those
in
the sample,
e.g.,
the
plating solution;
where
the
total
nickel
content
as
obtained from
the
polarogram is expressed
as
NiSO4 · 7H2O, the
total
chloride
content
as
NiCl2· 6H2O and
the
boric acid
as
H3BO3. It should
be
noted that
in
order to obtain
the
actual concentrations
of
NiSO4 · 7H2O a
correction
must
be
applied tor that
portion
of
the nickel
present
as
NiCl2· 6H2O. This
is
the
usual correction
which
is
applied when
standard
chemical
methods
are
used.
Calibration
curves
are
shown
in
Figs.
7,
8
and 9.
The
numerical
values
are
given
in
Table
I;
where
the
diffusion
current
has
been
corrected
for
the
residual
current.
It
will
be
seen
that
a
good
linear
relationship
between
current
and
concentration
is
shown
for
all
three
constituents.
Fig. 8—Calibration curves for NiCl2· 6H2O
Fig. 9—Calibration curves for H3BO3
Effect
of pH
It
was found
that if
the pH
of the
plating solution
is lower
than about
3.0, the
results for
boric acid
are too
high. Polarograms
ran on
solutions which
did Tot
contain boric
acid showed
that a ”hydrogen” wave is obtained
when the pH is lower than 3.0.
Although
it is
possible to
make a
correction for
the ”hydrogen” wave,
it appeared more desirable to neutralize the strong acid prior to running
the
boric acid polarogram. The following modification was therefore introduced
into the procedure when the pH of the
plating solution is below 3.0. After the sample
of plating solution is pipetted into the 50-ml volumetric flask, the flask
is only partially filled with the supporting
electrolyte; a few drops of bromphenol blue are added and then a solution
of
NaOH (about 01 N) is slowly added dropwise until the indicator just changes
color
(from yellow to blue). Then 1 ml of 0.2 per cent gelatin is added and
the
solution
is diluted to the mark with more supporting electrolyte. The polarograms
then are run as usual.
TABLE
I. DIFFUSION CURRENT CONSTANTS FOR PRINCIPAL CONSTITUENTS
OF A NICKEL
PLATING SOLUTION*
Constituent
Conc. in Plating Solution, oz/gal
Diffusion Current Id Microamperes
Constant Microamperes per oz/gal
86°F
77°F
68°F
86°F
77°F
68°F
NiSO4 · 7H2O
18.3
12.98
11.9
10.7
0.710
0.650
0.585
25.8
18.2
16.3
14.9
0.705
0.632
0.578
36.9
26.3
24.2
21.9
0.713
0.656
0.594
55.3
39.0
35.8
32.2
0.705
0.647
0.582
Average
0.708
0.646
0.585
NiCl2· 6H2O
3.16
4.30
4.14
3.97
1.36
1.31
1.26
4.41
6.11
5.75
5.36
1.39
1.30
1.22
6.30
8.80
8.18
7.60
1.40
1.30
1.21
12.67
17.6
16.4
15.4
1.39
1.29
1.22
Average
1.39
1.30
1.23
H3BO3
2.19
4.20
3.85
3.68
1.92
1.76
1.68
3.07
5.80
5.60
5.24
1.89
1.82
1.71
4.40
8.20
7.90
7.40
1.86
1.80
1.68
6.58
12.4
—
—
1.89
—
—
Average
1.89
1.79
1.69
*m2/3t1/6
= 1.83 mg2/3sec–1/2.
Values are for 0.25
ml of plating solution diluted to final volume of 50 ml.
TABLE
II. COMPARISON OF ANALYTICAL RESULTS AS OBTAINED
BY POLAROGRAPHIC AND
STANDARD CHEMICAL METHODS*
NiSO4 · 7H2O,
oz/gal
NiCl2· 6H2O,
oz/gal
H3BO3,
oz/gal
Chemical Method
Polarographic
Method
Chemical Method
Polarographic
Method
Chemical Method
Polarographic
Method
29.6
29.4
4.8
5.0
3.9
3.6
30.6
30.8
4.8
4.8
3.7
3.5
29.0
29.2
7.3
7.6
3.8
3.7
30.4
30.4
6.6
6.7
3.9
3.8
*Representative
values obtained with a production nickel
plating solution.
Effect of Brightening and
Other Addition Agents Although it was not possible
to examine the
effect of
all the
commercial brighteners,
the effect
of several
of the
more common
ones was
determined. It
was found
that only
one of
those examined
had any
appreciable effect:
it caused
high results for the
boric acid.
The result, however, is also too high when conventional chemical methods are
used and is due to the presence of boric acid in the brightener.
The
procedure was used on a bright cobalt-nickel bath9
and yielded satisfactory results.
No
significant effects on the results were
found when
wetting agents,
hydrogen peroxide
or both
were added
to the
bath.
DISCUSSION As
previously indicated,
the above
methods have
been used
successfully in
the control
laboratory of
the authors
for several
years to
control both
a Watts’ type
solution and a plain cold nickel solution containing ammonium chloride.
Generally,
it has been found that the precision for the nickel and chloride determination
is about 2 per cent. Since the diffusion
current for boric acid is about one-third that for nickel and occurs
after
the
nickel wave, the precision in this case is somewhat lower. A comparison
of the
polarographic results with those obtained by the conventional chemical
methods
shows that they generally agree to within about 0.25 oz/gal. The values
shown in
Table II were chosen at random and are representative.
It
should be
pointed out
that it
is not
necessary to
make special
solutions in
order to
obtain calibration
cures. If
one wishes
to set
up the
polarographic methods,
a very
convenient and
satisfactory procedure
is to
use the
plating bath
with its
known composition
as a
primary standard.
Suitable samples
of the
plating bath
may be
diluted in
such a
manner that
the resulting
solutions will
represent various
concentrations of
the constituents.
Polarograms may
be run
on each
dilution and
a calibration curve plotted.
ACKNOWLEDGMENT The
authors wish
to express
their appreciation
to Miss
E. Lawler
and Mr.
D. Lake
who assisted
performing the
experimental work
reported in
this study.
BIBLIOGRAPHY
1.
H. E.
Zentler Gordon
and Eric
R. Roberts,
Trans. Electrochem.
Soc. 90,
27 (1946).
2.
I. A.
Korshunov and
L. N.
Sazanova, C.
A. 42,
No. 22,
8701 (1948).
3.
D. A.
Vyakhirev, C.
A. 40,
No. 10,
2759 (1946).
4.
R. Diaz,
PLATING 40,
45, 261
(1953).
5.
R. Diaz
and E.
H. Lindemann,
PLATING
40,
762 (1953).
6
I. M.
Kolthoff
and
J. J.
Lingane, ”Polarography” 2nd
Edition, Interscience
Publishers, N. Y.
(1952).
7.
D. Ilkovic,
C. A.
29, No.
9, 2858
(1935).
8.
M. Ho]lander
and W.
Rieman,
III,
Ind. Eng.
Chem.
(Anal.
Ed.) 17,
602 (1945).
9.
W. Blum
and G.
B. Hogaboom, ”Principles of Electroplating and Electroforming” 3rd
Edition, McGraw-Hill
Book Co., Inc., New
York (1949).
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