Correction of Data for Errors of Measurement—W. A. =
Theory of the Howling Telephone with Experimental ‘a
Confirmation—Harvey Fletcher ...... 27
Electric Circuit Theory and the Operational Calculus S
Model, Part K. Darrow . .... 96

J. J. Carty Bancroft Gherardi FP. B. Jewett
E. B. Craft _ L. F. Morehouse O. B. Blackwell
E. H. Colpitts H. P. Charlesworth H. D. Arnold
R. W. King, Editor J. O. Perrine, Asst. Editor

Company, the Bell Telephone Laboratories, Inc., and the Associated
Companies of the Bell System

Address all correspondence to the Editor
Information Department
AMERICAN TELEPHONE AND TELEGRAPH COMPANY
195 BROADWAY, NEW YORK, N. Y.
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Previous to 1907 there were maintained in the Bell System
three laboratories and departments of development, research and
experiment,—one by the American Telephone and Telegraph
Company at Boston, one by the Western Electric Co at
Chicago and one by the Western Electric Company at New York.

In 1907, in the interest of economy and efficiency, these were
consolidated so far as laboratory and experimental work were
concerned and the Bell System Laboratory at Bethune and West
Streets, New York, was established. This was incorporated as
the Bell Telephone Laboratories, Inc., January first, 1925. The
expense of operation is divided between the American Telephone
and Telegraph Company and the Western Electric Company ac-
cording to the nature of the work done.

In the Bell System the American Telephone and Telegraph
Company undertakes, through constant association with the oper-
ating organizations, to formulate the requirements, present and
future, of the Bell System. Out of these requirements come the

roblems of the American a and Telegraph Company’s
tment of Development Research and the System lab-
oratory. After the problems have been eatisfactority ee
the Department of Development and Research adopts as s
the systems, equipment and apparatus thus produced which are
then specified for their proper uses in the Associated Companies
the Engineering Department of the American Telephone and
elegraph Company. When different departments and companies
are mentioned in this publication, so far as they are es of the
Bell System, they are parts of one working organization.

N the brillant galaxy of investigators to whom we owe out
knowledge of electrical science, Joseph Henry stands out as ot
the first magnitude; and for those who are associated with the Bell
System, the present is a most appropriate time to review his researches
which had an important guiding influence on the development ot
electrical communication. The present vear marks the fiftieth since
the invention of the telephone by Alexander Graham Bell, and among
the scientists with whom Bell conferred at that time, he gave a place
of honor to Henry. Ina letter to his parents written in March, IS75,
while he was busy in an effort to perfect the harmonic telegraph,
and before he had turned his attention to the telephone, Bell wrote:

“Now to resume telegraphy. When | was in Washington, | had
a letter of introduction to Professor Henry, who is the Pyndall ot
America. [| had found on inquiry at the Institute of Technology,
that some of the points I had discovered in relation to the applica-
tion of acoustics to telegraphy had been previously discovered by
him. I thought I would, therefore, explain all the experiments, and
ascertain what was new and what was old. He listened with an
unmoved countenance, but with evident interest to all, but when |
related an experiment that at first sight seems unimportant, [Twas
startled at the sudden interest manifested.

“T told him that on passing an intermittent current of electricity
through an empty helix of insulated copper wire, a noise could be
heard proceeding from the coil, similar to that heard from the tele
phone. He started up, said, ‘Is that so? Will you allow me, Mr.
Bell, to repeat your experiments, and publish them to the world
through the Smithsonian Institute, of course, giving vou the credit
of the discoveries?’

“T said it would give me extreme pleasure, and added that T had
apparatus in Washington, and could show him the experiments mysell
at any time.

strument working and he sat at a table for a long time with the empty
coil of wire against his ear listening to the sound. IT felt so much
encouraged by his interest that I determined to ask his advice about
the apparatus I have designed for the transmission of the human
voice by telegraph. | explained the idea and said, ‘What would
vou advise me to do, publish it and let others work it out, or attempt
to solve the problem myself?’ He said he thought it was ‘the germ
of a great invention,’ and advised me to work at it myself instead of
publishing. IT said that I recognized the fact that there were mechan-
ical ditficulties in the way that rendered the plan impracticable at
the present time. IT added that IT felt that I had not the electrical
knowledge necessary to overcome the difficulties. His laconic answer
was, “GET IT.’

“T cannot tell you how much these two words have encouraged
me. Such a chimerical idea as telegraphing vocal sounds would
indeed to most minds seem scarcely feasible enough to spend time in
working over. I believe, however, that it is feasible, and that I
have got the cue to the solution of the problem.

“Professor Henry seemed to be much interested in what IT told
him, and cross-questioned me about my past life, and specially wanted

Joseph Henry was born in Albany, New York, in 1799, and coming
to full maturity of mind at the beginning of a century which will
probably never be surpassed for fruitful research in the field of elec-
tricity, he demonstrated, at the very outset of his career, his right
to stand for all time with the foremost investigators in this depart-
ment of natural science. Henry was, moreover, a many-sided man.
His distinguished career leads into many fields and before reviewing
his researches on clectro-magnetism we may note briefly the very
diversified and yet important character of his other work.

During the latter half of his life, official duties as the director of
the Smithsonian Institution consumed an ever increasing portion of
his time, but he still found opportunity to prosecute many original
inquiries,—for example, into the application of acoustics to building,
into the best construction and arrangement of lecture rooms, and into
the strength of various building materials. As one of his first ad-
ministrative acts, he organized a widespread corps of observers for
simultaneous weather and meteorological reports by means of the
telegraph which was vet in its infancy. He was the first to have the
daily atmospheric conditions indicated upon a map of the country
and to utilize this information in making weather forecasts.

He was an active and long-standing member of the Lighthouse
Board of this country and his diligent investigations into the efficiency
of various illuminants and the best conditions for their uss

improved the beacons which dotted our coasts. During the dark
days of the Civil War, Henry clearly saw the tremendous advantage
to be derived from a mobilization of the nation’s scientific men for
cooperative service. His vision, backed by his tremendous energy
and ability, resulted in the formation of the National Academy of
Sciences, under a Congressional charter signed by Abraham Lincoln

ismi inspired by Henry, there re-occurred, in 1916, under presidential
proclamation, a mobilization of the nation’s scientific and technical
men.

known, his researches are of the most enduring character, and for all
time must enter intimately into the lives of all civilized mankind.
He was without peer among the American physicists of his time, and
itis well attested by every record that he was a man of varied culture,
of Jarge breadth and liberality of views, of generous impulses, of
great gentleness and courtesy of manner, combined with equal firm-
ness of purpose and energy of action.

Let us now turn to Henry's investigations of clectro-magnetism,
which were among his earliest scientific undertakings. He began
his career in 1826 in New York State at the Albany Academy, where
he had only the apparatus he could construct with his own hands
and, out of each vear, but a single month uninterrupted by other
duties to devote to his researches. It was there-~independently of
Faraday and on some fundamental points prior to him—that Henry
discovered the laws of current induction. At the same time he under-
took a study of the electromagnet which prepared the way for not
only the telephone and telegraph, but also for all types of dynamos
and motors.

The electromagnet was discovered by Sturgeon in’ England, but
Henry's contributions to. our knowledge of it were so great that after
his work, a powerful instrument suitable for many uses replaced
what had been a feeble toy. When he started his work on the electro-
magnet its design was not understood; when he had completed his
work he had developed a magnet, the design of which was understood
and which could be adapted, according to the rules which he laid
down, to a multitude of purposes.

With reference to the making of electromagnets, Henry pointed
out the improvements which resulted from insulating the conducting
wire itself, instead of the rod to be magnetized, and by covering the
whole surface of the iron with a series of coils in close contact. This
was effected by insulating a long wire with silk thread, and winding
this around the rod of iron in close coils from one end to the other.
The same principle was extended by employing a still longer insulated
wire, and winding several strata of this over the first, care being taken
to insure the insulation between each stratum by a covering of silk
ribbon. By this arrangement the rod was surrounded by a compound
helix formed of a long wire of many turns instead of a single helix of a
lew turns.

Thus Henry laid down the rules, which, in general, are followed
today in the construction of commercial electromagnets; namely,
that the wire should be insulated, that it should be wound in layers,
and that there should be several lavers, one above the other. He

also did another thing in his actual construction: he adopted what
may be called the spool construction, the placing of the windings on
spools, and then the sliding of the spools on the core. Phat is a
standard method of building clectromagnets today.

Soon after doing this work Henry built a magnet to be used at
Yale University, which was in its time a wonder and would even
today be considered very powerful. He also built a series of magnets
in which the emphasis was placed upon the lifting power in relation
to the weight of the magnet and succeeded in designing one which, when
energized by a single small cell, could support 420 times its own weight.

The improvements which Henry made in magnets suggested to him
applications of magnetic attraction to the production of mechanical
motion. He realized that electromagnets such as he built were easy
to control, and believed that he could design a machine by which he
could get power from an electric current and this at a time when the
only source of current were primary batteries as the dynamo did not
vet eXIst,

His electric motor was the first ever built to use electromagnets:!
it Was extremely simple consisting of an electromagnet supported at
its center by a pivot so that it could rock back and forth under the
alternating pulls of two permanent magnets. To effect the reversal of
magnetization of the electromagnet and hence the alternation ot pulls,
mercury cups were arranged so that wires would dip in them as the
suspended magnet rocked to and fro. These contacts were the proto-
type of the commutator which is found in every direct current motor
and dynamo today. It is interesting to note the words in which
Henry deseribed this invention. In Silliman’s American Journal of
Science for 1831 he wrote, “T have lately succeeded in) producing
motion in a little machine by a power which [T believe has never before
been applied mechanics -by magnetic attraction and repulsion.
Not much importance, however, is attached to the invention since
the article in its present state can only be considered a philosophical
toy; although in the progress of discovery and invention it is not
impossible that the principle or some modification of it on a more
extended scale may hereatter be applied to some useful purpose.”

ossible applications of the principle there shown cannot fail to com-
PI

' Faraday has some years before shown that a wire carrying a current could be

caused to revolve continuously around the pole of a permanent magnet. Henry's
advance over this was considerable in that he materially increased the force causing
motion by employing the attraction between two magnets, one permanent and one
generated by current. The motor using electromagnets throughout did not come

mand our admiration. Of course, until the dynamo was invented
at a later date, and a substantial electric current became available,
the motor could not be much more than he characterized it, “a
philosophical toy.”

Henry also became interested in a determining whether an electro-
magnet could be operated from a distance so that the doing of some
work—for example the ringing of a bell—could be controlled from a
distant station. From his investigations directed to this end, Henry
was the first to appreciate that the effect of the resistance of long
lengths of wire to the passage of electric current could be minimized
by properly proportioning the battery and the magnet windings to the
length and resistance of the line wires.

Efforts had been made by others prior to Henry's time to devise
successful electric telegraphs. They had failed, however, because
they did not know how to proportion their magnets and their batteries
so as to operate over any substantial length of line. The literature of
that time contains a number of demonstrations of the impossibility
of operating an electric telegraph, because scientists could arrange
instruments which would operate successfully when separated by
a few feet, or even one hundred feet, but they would not work at a
distance of thousands of feet because of the resistance of the long
line wire.

What Henry did was to determine the proportioning of the various
parts of the system so as to secure operation. He found, when his
magnet was connected by a short wire to the battery, that the greatest
magnetizing effect was obtained by joining the cells of the battery in
parallel, but that a series arrangement of the battery would give the
greatest pull if a long wire (a length of a mile or more was used in
some of his experiments) carried the current. He also obtained the
best operation over a short line when the magnet winding consisted
of several distinct coils, all connected in multiple; and for operation
over a long line he found it best either to connect these coils in series
or to apply to the magnet a single long winding. Henry was there-
fore the first to produce an electric telegraph, and more than that,
the transmission of electrical energy to a distance. That first tele-
graph paved the way for all the telegraph systems, all the ocean
cable systems, and contained the principle of all telephone call bells.

One of Henry’s greatest discoveries from the standpoint of electrical
science, but a discovery in which he must vield the first place to
Faraday, is that of mutual induction—the fact that a wire when
moving with respect to a magnetic field has an electromotive force
generated in it. Although Henry made his discovery independently

of Faraday, the latter was the first to make known his observations
to the world, and it is no trifling index of Henry's character that he
never in any wavy intimated that he was entitled to share with Faraday
credit for the discovery.

Because Henry was anticipated in the publication of his observation
of mutual induction, he does not appear to have left a verbal record
of the steps of reasoning by which he was led to the discovery. How-
ever, he does tell us what the arrangement of apparatus was and if
we bear in mind that he Was seeking a method ot generating an electri
current from a magnet--this magnet, in turn, being itself the product

Writing of his original observations, Henry savs he ‘succeeded in
producing electrical effects in the following manner, which differs

from that emploved by Mr. Faraday and which appears to me to
develop some new and Interesting Lacts;, A piece of copper wire,
about thirty feel long and covered with elastic varnish, Was closely
coiled around the middle of the soft iron armature of a galvanic
magnet... which, when excited will readily sustain between six
and seven hundred pounds. The armature thus furnished with wire
was placed in its proper position across the ends of the magnet and
fastened so that no motion could take place. The two projecting
ends of the helix were connected with a distant galvanometer by
means of two copper wires each about forty feet long. This arrange-
ment being completed, TP stationed myself near the galvanometer and
directed an assistant at a given word to suddenly immerse the galvanic
battery attached to the magnet. At the instant of immersion. the
north end of the needle was detlected 30° to the west, indicating a
current of electricity from the helix surrounding the armature. The
effect, however, appeared only as a single impulse, for the needle
after a few oscillations, resumed its former undisturbed position,
although the action of the battery was still continued. [T was, how-
ever, much surprised to see the needle suddenly detlected) from a
state of rest to about 20° to the east, when the battery was suddenly
withdrawn from the acid, and again deflected to the west when it
was re-immersed. This operation was repeated many times ino suc-
cession, and uniformly with the same result.”

It was in this same paper that Henry announced his observation
of the phenomenon of self-induction, a most important discovers
and one for which he holds full credit for having first made it known
to the world. He writes, “I may, however, mention one fact which
[ have not seen noticed in any work, and which appears to me. to
belong to the same class of phenomena as those before described; it
is this: when a small battery is moderately excited by diluted acid.
and its poles, which should be terminated by cups of mercury, are
connected by a copper wire not more than a foot in length, no spark
is perceived when the connection is either formed or broken; but
if a wire of thirty or forty feet long be used instead of the short wire,
though no spark will be perceptible when the connection is made,
vet when it is broken by drawing one end of the wire from its cup of
mercury, a vivid spark is produced... The effect appears
somewhat increased by coiling the wire into a helix.” In a some-
what later paper we find the following statement. “A ribbon of
sheet copper nearly an inch wide, and twenty-eight and a half feet
long, Was covered with silk, and rolled into a flat spiral similar to the
form in which woolen binding is found in commerce. With this a

vivid spark was produced, accompanied by a loud snap. ‘The same
ribbon uncoiled gave a feeble spark.”

Henry tried many modifications of this experiment and in the end
drew the conclusion that the after-current he was observing was due
to the inductive effect of the current in the wire upon itself, and that
this became particularly apparent when the wire was so coiled that
its various turns lay close together. The discovery of mutual in-
duction by Faraday and the discovery of self-induction by Henry
constitute two halves of a whole, and it is appropriate that to these
men should go equal recognition in the matter of having electrical units
names after them. Of the three units by which the properties of every
electric circuit are measured, the unit of capacity was named after
Faraday, and unit of inductance after Henry; the third unit, that of
resistance, recognizes the fundamental researches of Ohm.

A few vears later, after having accepted the chair of physics at
Princeton University, Henry returned to the subject of induced cur-
rents. In his earlier work he, like Faraday, had used the continuous
currents which a voltaic battery generates. He now chose the cur-
rents which flow when a Levden jar is discharged. To register the
inductive effects of the fleeting currents of discharge Henry adopted
a device consisting of an unmagnetized needle placed in a small coil
of wire. Through this coil the induced current had to flow. The
use of the needle as an indicator led Henry to an important observa-
tion. He noticed that following a discharge, the direction of magnetiza-
tion of the needle depended upon the distance across which the induc-
tive effect had occurred. To account for this curious result, he ad-
vanced the hypothesis later shown to be correct that the discharge
is oscillatory.

Here was the germ of a great discovery. The oscillatory character
of the discharge is one of the fundamental and important properties
of certain types of electric circuit. Henry did not have the facilities,
however, for carrving his investigations in this field far enough to
attract the attention of the scientific world. It was not until IN75.
some thirteen vears later, when Lord Kelvin was led independently
by mathematical considerations to believe that the discharge 1s
oscillatory, that the significance of the phenomenon began to be
understood.

Henry's work contained the germ of vet another important dis-
covery. Some of his experiments on induction by Leyden jar dis-
charges involved the transmission of electric force without wires
through distances as great as two hundred feet, and through the
floors and walls of buildings. And in similar experiments in which he

observed the effects of lightning flashes in place of sparks from a
Levden jar, he found that he could get the lightning to magnetize
needles up to a distance as great as eight miles. This was about
IS42. Here we have the earliest evidence of ether waves of the type
that the radio engineer emplovs. But again the significance of
Henry's work was not recognized. This could only have come after
much fuller investigation. However, it is instructive to reflect for a
moment on what might have been had Henry possessed the time and
facilities for carrying his work further. Needless to sav, there is a
wide gulf between the wireless telegraph of today and its earliest
precursor with which Henry received an electromagnetic signal from
a lightning flash eight miles away, but it is wholly possible that,
had Henry not been called to other work, the world might have
possessed a wireless telegraph capable of sending messages over sub-
stantial distances many vears before it did.

Writing of Henry, Simon Newcomb, the celebrated astronomer
said? “His scientific work is marked by acuteness in cross-examining
nature, a clear appreciation of the logic of science, and an enthusiasm
for truth without respect to its utilitarian results.” A man of the
highest scientific ability, Henry spent the better part of his life as the
head of an institution dedicated to ‘tthe increase and diffusion of
knowledge among men.”’

“The mantle of Franklin has fallen upon the shoulders of Henry,”
wrote Sir David Brewster,’ the eminent English scientist, and it is
reported that Abraham Lincoln declared, when he became acquainted
with Henry after assuming the Presidency, ‘The Smithsonian In-
stitution must be a grand school if it produces such thinkers as Henry.”
He was, in every way and in the best that the word implies, a scientist,
and the interest in scientific questions which dominated his life,
remained with him to the very end,—-almost the last words to pass his
lips were whether the transit of the planet mercury had been suc-
cessfully observed. If we use the word ‘Dean’’—so rich in academic
association—to stand at once for the greatest usefulness to one’s
fellowmen as well as for the highest achievements in the field of
scholarship and research, for lifelong devotion to public service, for
breadth of view and tolerance regarding all questions, whether arising
in science or directly out of human relations, and as epitomizing all
that is best and highest in man’s intellectual life, we may well call
Joseph Henry the Dean of American scientists.

° Biographical Memoir; prepared by Prof. Asa Gray in behalf of the Board of
Regents of the Smithsonian Institution.

4 accepted truth is the result of every-day experience. From
the simplest type of measurement, such as determining the length
of a board with an ordinary tape measure, to the most refined type
of measurement, such as determining the charge on an electron,
errors are bound to creep in.

Now, a manufacturer must constantly make measurements of one
kind or another in an effort to control his production processes and
to measure the quality of his finished product in terms of certain of
its characteristics, but, before he can safely determine the significance
of observed differences in his production processes or in the quality
of his product as given by these measurements, he must make allow-
ance for his errors of measurement; i.e., for the fact that the observed
differences may be larger or smaller than the true differences. To
make such allowances for the errors of measurement of any character-
istic, to find out what the true magnitude of the characteristic most
probably is, to find out, as it were, what a thing most probably is
from what it appears to be, presents an endless chain of interesting
problems to be solved.

Three important types of problems arising in engineering practice
are discussed in this paper. They are:

2. Error correction of data taken periodically to detect significant
changes in quality of product.

3. Error correction of data taken to relate observed deviations in
quality of product to some particular cause.

The solution of the first one is presented here for the first time.
The solution of the second has been generalized to include cases not
previously solvable. All three types of problems are illustrated.

Tyer 1 ERROR CORRECTION OF DATA TAKEN TO SHOW THE QUALITY
OF A PARTICULAR Lor

Let us take a specific problem first. Assume that we have a lot
consisting of 15,000 transmitters ' and a machine with which to measure
the efficiency of cach instrument. Suppose we make one observation
on cach transmitter-—a total of 15,000 measurements. Suppose we
find, as in the distribution illustrated in Fig. 1, that one measure-
ment is in the efficiency range — 1.75 to —1.50, 17 within the range

Kig. 1-—Typical frequency distribution. Chart showing observed number of trans-
mitters versus efficiency

1.50 to —1.25 units, and so on. The vertical height of a point
represents the number or frequency of occurrence of observations
falling within the corresponding interval laid off on the horizontal
axis of the chart.

So far so good, but suppose a customer wants to buy these trans-
mitters. We know that some transmitter which appeared to have an
efficiency within the range of 1.25 to 1.50 units say, may actually have
had an etheiency within some other interval. We know too that,
because of the errors of measurement, the transmitters appear to
differ more among themselves than they really do. We therefore

‘Of course, the efficiency of a transmitter does not remain constant during a

series of tests but these inherent variations in the transmitter may be considered,
for our purpose, as forming a component part of the resultant error of measurement

desire to find the most probable numbers of transmitters within tl
different intervals indicated in Fig. 1.

the interval of efficiency from Y to VY+dN is It is this
function f7(XY) that we want to find. Similarly let us assume that
there is some function such that vives the observed
number of transmitters appearing to have ethiciencies within” the
interval Y to Y+dY where the measurements are made by a method
wherein the probability of making an error within the interval x to a
+dx is fi vide. [tis reasonable to expect that, if two of these func

tions are known, the third can be easily determined. We shall pro
ceed to show that this is the case. Let us first tind the law of errot
experimentally.

given magnitude in measuring the efficiency of any transmitter
Naturally, the only way of doing this is to make a series of measure
ments on a single transmitter from which we can determine the
observed frequency of occurrence of measurements which differ from
the average by some fixed amount, and thus tind what percentage of
the total number of measurements may be expected to fall within
anv given range on either side of the average. Common sense and
intuition may tell us that we may expect to find a large percentage
of the measurements within a narrow range on either side of the
average, that there will be just as many measurements greater than
the average by a certain amount as there are less than the average
by the same amount, and that large deviations from the average may
be expected to occur with less frequency than small deviations
Suppose we make 500 observations of the ethciency of a single trans
mitter and find the distribution given in Fig. 2. Just as we might
have expected, the observed values of the efficiency of the transmitter
are grouped symmetrically about the average of all the observed
values. We see that the maximum deviation between observations
on a single transmitter is quite large (33°, ) compared with the actual
maximum differences observed between the efficiencies of the trans
mitters.

The results reproduced in Fig. y sugvest that the deviations for the
case in hand are distributed in a manner closely approximating the

bell-shaped distribution so familiar in the theory of errors. We often
find, as we do in this case, that the observed distribution can be closely
approximated by a function fx(x) of the form

where fr(x)dx is the probability that an error x will lie within the
interval x tox+dyx, ¢ is the root mean square or standard deviation,
X is the arithmetic mean value and (Y —Y) is the deviation x. The

Fig. 2—Typical form of distribution of errors of measurement. Chart showing
number of measurements on a single transmitter versus efficiency

function f(x) is referred to in the literature as the normal law of
error. If we try to fit such a curve to the deviations® given in Fig. 2,
we obtain the results shown in Fig. 3. This figure is the same as
Fig. 2 except for the addition of the smooth normal curve of error
calculated for the observed data. Without further consideration, we
shall assume the law of error to be normal and hence of the form

We have next to consider the choice of the function to represent
the true distribution f7-Y). Often we have reason to believe that this
* If the average of the observed values of the 500 observations of efficiency given

in Fig. 3 is assumed to be the true value of the efficiency of the transmitter, then the

deviation of an observed value from this mean is also the error of this observed value.
We shall use the terms “error” and ‘‘deviation” interchangeably in this sense.

is also approximately normal, and hence we shall consider first the
method for finding the observed distribution f,(Y) for the special case
when both the true distribution f7(.Y) and the law of error fr(X) are
normal; t.e., when they are both of the form given by Equation (1

We shall first obtain an experimental answer to this problem.
Suppose we take, say, 1,000 instruments of some kind which are

Fig. 3—Chart showing the observed distribution of errors fitted by a tvpical smooth
curve. Data of Fig. 2 fitted by normal law of error, Eq. 1

known to be distributed in normal fashion, in respect to some char-
acteristic, with a standard deviation op. Let us measure each of
these instruments by a method subject to the normal law of error
whose standard deviation oy, is } or. The results of one such experi-

ment are given in Fig. 4. The observed frequencies of occurrence
are represented by the circles. It was found that this observed
distribution could be closely approximated by a normal law f.(X)

in a succeeding paragraph. The theorem is: When the true distri-
bution f7(¥) and the law of error fe(x) are both normal (hence ex-
pressible in form indicated by Equation (1)) with root mean square
or standard deviations % and ¢,% respectively, the most probable ob-
served distribution will be normal in form with a standard deviation
V 03.

normal as it should be if fr(X¥) and fr(x) were both normal. We
must therefore, try some other function for f7CY).

Of course, experiments might be performed for other types of true
and error distributions, but in all such cases the results, as in the
ilustration just considered, would be subject to errors of sampling.

Fig. 4 -Experimental results shpwing effects of errors of measurement. Normal curve
fitted to observed points, when the true distribution and the law of error are both
normal

Hence we shall proceed at once to the analytical treatment of the
problem.

Assuming the law of error to be normal, we see that the fraction
fe(xjdx of the number of objects having magnitudes between Y +x
and Y+x+dsx will be measured with an error between —x and —x—dx

lor the particular case treated in a previous paragraph where both
the true distribution f7(Y) and the law of error fe(x) are normal,
we may write Equation (2) in the form

the true and error distributions respectively. Integration of Equation

I-quations (4) and (5) are the analy tical eX pression for the rule stated
previously, for finding the observed distribution (..Y) when both
the true and error distributions are normal, because Equation (4

shows it to be normal and Equation (5) expresses the standard devi-
ation o, of the observed values in terms of those of the true values
and of the errors.

In practice, however, we often tind that the true distribution is
non-symmetrical or skew and can be more nearly approximated
by the function

where ky is a measure of the asymmetry or skewness, the modal o1
key
most probable value of X being at a distance —° 97 from the average
See Appendix 1 where another method of solution is given
‘ This is often referred to in the literature of statistics as the secon
tion. It is in fact the first two terms of the Gram-Charlier series

18 BELL SYSTEM TECHNICAL JOURNAI
value of YY. Substitution of this expression and a normal error

function in Equation (2), vields upon integration ® the following
distribution f.(.V) of the observed values

We see that the distribution f,CY), Equation (7), of the observed
values is of the same form as that f7(.Y), Equation (6), of the true
values. The standard deviation of the errors of measurement ¢p,
as in the previous case, has equal weight with the standard deviation
7, in influencing the standard deviation o, of the observed values.
The degree of asymmetry of the observed distribution as measured
by the skewness k, is, however, less (Equation (8) ) than that of the
true distribution as measured by the skewness ky of the true dis-
tribution.

Now we can correct the observed distribution, Fig. 1, for the errors
of measurement, because we find that the observed frequencies, Fig. 1,
can be closely approximated by a function of the twpe defined by
Equation (7). Knowing that the law of error, Mig. 3, is normal we
conclude that the true distribution f7CY) must be a function of the
same type as f,(Y) was found to be except that the true standard
deviation 77 will be, from Equation (5), ¥/¢2+ 3, and the true skewness

lated from the observed distribution, Fig 1, and 7%, can be determined
by the data given in Fig. 3.

Thus finding the values of ¢; and kr and substituting them in
Equation (6), we have the function f7(.Y) representing the true dis-
tribution which we started out to find. Krom this knowledge of
fr(X) we can now get the most probable frequencies of occurrence
of the different efficiencies. Subtracting these frequencies from those
observed and shown in Fig. 1, we get the corrections plotted in Fig. 6,
expressed as percentages of the observed frequencies.

We are now in a position to summarize the practical routine to be
followed in finding the most probable distribution f;(.Y) of quality
when the observed distribution is given.

To tind f7CV), we must first Know the law of error fe(v). We
must show this to be normal and tind the standard deviation ¢%

Fig. 6—Correction which must be applied to the observed distribution of transmitters
Fig. 1, because of the existence of errors of measurement

cision we attain in finding f7(Y) depends upon the number of obser-
vations made in finding 7p.

Having found ¢% to the required degree of precision, we must next
discover whether or not the observed distribution f;(X) is either
normal or the second approximation. Standard statistical methods
can be used for this purpose.

and, if f (XY) is second approximation, we know that f7(.X) is given
by Equation (6), where op and ky can be found with the aid of Equa-

tions (5) and (S) in terms of the observed values of op, o) and k..
In other words we have

CORRECTION OF DATA TAKEN PERIODICALLY TO DETECT SIGNIFICANT
CHANGES IN QUALITY OF PRODUCT

Irrespective of the care taken in defining and controlling the manu-
facturing processes, the units of a product will differ among them-
selves in respect to any measurable characteristic. Random fluctua-
tions in such factors as humidity, temperature, grade of raw material,
and wear and tear on machinery may produce such differences be-
tween units of a product. Such random variations in the factors
underlying the manufacturing process usually yield a product in
which the units differ in random fashion according to some law of
probability.

Customarily, product is inspected periodically, and the data are
analyzed to determine if the observed difference in two samples is
greater than can be accounted for as a random variation. If it is,
we may assume that the manufacturing processes have changed
significantly for some reason which further investigation should dis-
close. Now, the presence of errors of measurement effectively in-
creases the magnitude of the random differences to be expected from
one sample to another and hence makes it harder for us to detect
trends or fluctuations in product. Let us investigate this effect of

Symbolically we may assume that the probability of production of
a unit of product having a characteristic Y within any range Y to
+dN is fp( N)dX, where the characteristic Y is measured by a method
subject to a law of error fe(x), so that fe(x)dx represents the proba-
bility of occurrence of an error x within the range x to x+dx. The
problem is to find the corresponding distribution f,CY) for the ob-
served magnitudes.

Obviously the observed magnitude Y, is the algebraic sum of the
true value Y and the error x. Assuming that there is no correlation
between these two quantities, the probability of a unit having a value
of Y within the range Y to VY + dX being measured with an error

within the range x to x + dy is fy(.N)dN fyp(xidx. Assuming that
Y,=N+.x we may write the probability

because f,U.Y,) is obtained by taking into account that all possible
values of x between + % and — * may be combined with a given Y,.
This integral is of the same form as that given in Equation (2). Inte-

where as before ¢,=\/o}+o;. This result is well known as the law
of propagation of error.

When f(x) is normal and fy XY) ts given by the first two terms of
the Gram-Charlier series, Equation (6), with skewness ky; and stand-
and deviation o7, the observed distribution f,CY,) 1s of the same func-
tional form as the true distribution f7(.Y) and has values of standard
deviation o, and skewness k&, given by Equations (5) and (8) in Part I.
This result appears to be new.

Now for the case where the true distribution f7CY) and the law of
error fe(x) are both second approximation type, the integration ts
somewhat tedious, but we can approach a special case of this problem
easily from a slightly different angle as indicated in Appendix 2.
Under certain special conditions therein set forth, the resultant dis
tribution is also second approximation form with a skewness which

Exam ple of Applications to Determine Most Economical Way of \easur-
ing Quality

Let us next consider a very simple method of using the above
results to indicate the most economical method for determining the

What is the most economical way of determining the quality of
product within some predetermined range Y+AY with a known
probability P, where XY is the average quality? Let us assume that:

dy =cost of making each measurement,
=number of units selected,
ts =number of measurements made on each unit,
o, =standard deviation of the errors of observation.
o,=o0,=standard deviation of the true distribution

cent. of the observations, and hence AY =3cy.
The average of 7, measurements made on one unit is the observed

Over 99 per cent. of the averages of samples of size .V drawn from
a product whose law of distribution is f7(.X) where fy(Y) is either
normal or second approximation may be expected to lie within the

falls outside these limits, this fact is taken as probably indicating the
existence of a trend or cyclic fluctuation in product, the cause of
which should be sought. The presence of errors of measurement
increases the separation of these limits to 6o, trom 6e;. Our pre-
cision of detecting trend or cyclic fluctuation is thereby decreased.

25 per cent. greater than oy. In some instances ¢@ has been found
to be nearly 50 per cent. greater than o;.

ERROR CORRECTION OF DATA TAKEN TO RELATE OBSERVED
DEVIATIONS IN QUALITY OF PRODUCT TO
SoME PARTICULAR CAUSE

In many practical cases it is not possible to write down an equation
to show how the quality of a finished product depends upon the
factors controlled by different manufacturing steps. To cite one such
case, we may know that the quality of the finished article depends
upon the control of the temperature to which some of the piece parts
are heated in the process of manufacture. Thus the microphonic
properties of carbon depend upon the temperature to which the
carbon is heated. In cases where the relationship between quality
and some factor (such as temperature in the above illustration) can
only be determined through a study of the correlation existing between
the quality and the particular factor, use must be made of the correl-
ation coefficient r which is defined as

where x and y represent respectively deviations from the average
quality Y and the average magnitude Y of some factor which is

to be controlled by the manufacturing process, and .V is the number of
observations. Now, if errors of observation are made in determining
x and y, the observed correlation coefficient ry,y,, is known to be given
by the expression

gx, and oy, being the root mean square errors of observation of x» and
y respectively.

Attention is directed to Equation (10) which shows that the observed
correlation coefficient 7, is always less than the true correlation co-
efficient: 7, irrespective of the number of observations made. Ob-
viously, this point is of considerable commercial importance as we
shall now see.

If the observed correlation is small, we customarily assume that
there is little need of trying to control the quality Y by controlling the
manufacturing factor Y, whereas this conclusion cannot be justified
unless it can be shown that the true correlation has not been masked
by the errors of measurement.

This point has had to be taken into account in the development
of machine methods for testing transmitters and receivers, because
the calibration curves of the machines in terms of ear-voice tests
depend upon the correlation coefficient.

It may be of some interest to certain readers to note that the results
given in Equations (4) and (7) can also be obtained in the following
way by the method of moments so often used in statistical investi-
vations.

If we substitute a normal form for fr(Y) in Equation (11) and
solve for the moments of {,CY), we find that the odd moments are zero

and the ratio of the 4th moment to the square of the 2nd is numerically
3 which indicates that f,CY) is normal in form.

A similar substitution of the 2nd approximation form for fy(Y) in
Equation (11) yields a distribution f,.Y) from whose moments we

deduce Equation (7). Use is made in this proof of the easily demon-
strated theorem that
/ =O
x
if i where is the jth derivative of the normal law function,

It is well known that the normal law of distribution may result from
a system of » (nv being large) causes each of which produces an incre-
ment AY measured from some fixed origin with a probability p=! and
no increment with a probability g= 3. Furthermore the second ap-
proximation may result from a similar system in which p+g and »
is large. Under such systems of causes, the probabilities of the oc-
currences of m, u—1,...3, 2, 1, 0 increments are given by the suc-
cessive terms of the point binomial (p+4q)”.

Let us assume that the symbols pr, gp, v7, AN and pp, de, Ne, AX
refer to the systems of causes controlling the product and_ errors
respectively. The probabilities of observed combinations my; +
neAx, (np—WAN + (ne—1)Ax,... are given by the successive terms
of the expansion (pr+qr)"’ (petgqr)"®. Now for the special case
pr=pr=p and AX =Ax we have the resultant probability distribu-
tion with skewness
q—P

Now if p=q, the skewness kj is zero and the observed distribution
is more nearly normal than either component, and its standard devi-
ation ¢o is the square root of the sum of the squares of %p and 7p.
This result is similar to that given by Equation (4) of this paper.

We may also consider by this method a case not treated in this
paper. When the skewness ky of the true values is equal to that

kr of the law of ertor, or, more particularly, when mp=ne_=n, pr=
Pe=p, dr=qi=q, p= q, we see that the observed distribution is given

or
times the standard deviation of either of the true or error distributions.

The Theory of the Operation of the Howling
Telephone with Experimental Confirmation

sufficient numerical data are given to enable one to calculate the intensity
and frequency of howling for various types of systems. Detailed con-
sideration is given to the following three systems, namelv, one where the
transmitter and receiver disphragms are coupled together mechanically
by a lever system, one where they are coupled by a small box of air, and
one where they are coupled by a long tube of air. The ty pe ot electrical
circuit to use with each of these systems depends upon the type of perform-

front of the mouthpiece of the transmitter, a shrill note is
emitted. A sustained oscillation is set up in the electro-mechanical
system which is frequently called “howling” or “singing” or “hum-
ming.”

This phenomenon was first observed by A. S. Hibbard ot
the United States in IS90.) Frank Gill was the first to publish an a
count of the phenomenon. He first noted that the pitch of the howling
note was changed by reversing the telephone receiver connection. — In
summarizing further his experimental results, he states “that the pitch
of the note appears to be determined by the length of the column of air
between the two diaphragms and the conditions of the circuit. As the
periodic time of the circuit is increased, the time of the note rises. To
some extent, the pitch is governed by the rate of the diaphragm, but I
do not think this is so important a factor as the others. The main
factors appear to be the angle of lag and the length of the column of
air between. the diaphragms. Although the vibration is a forced
one, we could almost see that its rate is largely dependent on the
free period of the circuit.” !

In 1908 Kennelly and Upson extended Gill's work and made ex
tensive experimental investigations of the case in which the trans
mitter and receiver are coupled together acoustically by means of a

'Taken from a paper on ‘Notes on the Humming Telephone” by F. Gill, read
at a meeting of the Dublin Loeal Section of the Society of Telephone Engineers

and published in the Journal of the Institution of Electrical Engineers, Vol. NXNXI,
1901,

hollow circular tube of varying lengths and electrically by means of
an induction coil, The summary of the conclusions is as follows: *

(1) The mean frequency of the humming-telephone note is de-
termined solely by the receiver diaphragm, and its natural free rate
of vibration. (2) The ascending intersections of the frequency
zig-zag with the mean frequency line will be formed approximately
at tube lengths of (8. 4+) v mo cm. for one connection, and of (144
m) von, em. for the other connection, of the receiver; where v is the
velocity of sound in air, 7 is the mean frequency in cycles per second,
and m is any positive integer, within the working range of the tube.
The constants 3,4 and 1/4 may be modified by the presence of con-
densers, and other circumstances. (3) The range of pitch variation.
and the breaking positions, are determined by the transmitter, and
by the reinforcing capability of the system. For systems that are
weak, either electrically or acoustically, the range of pitch, above or
below the mean, will be small. (4) The primary current, as measured
by a DC instrument, is ordinarily a minimum at the mean frequency,
and a maximum at a break. (5) Transmitters may be tested for
effectiveness, by measuring their hum-extinguishing resistances in
the primary or secondary circuit. The tube length should be such
as to produce mean frequency if one connection of receiver only is
used, but should favor both connections equally, if both connections
of receiver are used.”

They also give a first approximation theory to account for the
changes in frequency as the length of the coupling tube is changed.

In 1917, H. W. Nichols gave the general equations for the
special case where the two diaphragms act as pistons closing the
ends of a tube of air. This case was given as an illustrative example
of the “Theory of Variable Dynamical Electrical Systems.”

This paper gives a theoretical treatment of the behavior of a system
containing a transmitter and a receiver coupled together acoustically
and electrically, and with a source of electrical energy feeding the
transmitter. Formulae are deduced which give the frequency and
intensity of howling in terms of the physical constants of the system.
Numerical calculations are given sufficiently detailed) solution
of some special cases are given to enable one, who is interested in
using the howling telephone as a source of alternating current. or
for other experimental work, to design the set for his particular
purpose.

?“Humming Telephone” by 4. E. Kennelly and Walter L. Upson, American
Philosophical Society, July 20, 1908.

The elements of a telephone system which is howling are the trans-
mitter, the receiver, the mechanical coupler and the electrical coupler
as indicated in Fig. 1. Hf there is a source of electrical power in the
electrical coupler, which is released by movements of the transmitter
diaphragm in the form of clee trical Vibrations, and also, if there is a

howling will result. In other words, if the gain in the transmitter
due to its amplifving action is just equal to the losses in the electrical
and mechanical circuits, then a steady oscillatory state will be
maintained. The problem is to determine the aature of these
relationships.

Assume that the conditions are such that a steady oscillatory
state has been set up. Under such conditions let 7) be the electrical
impedance of the transmitter, R the impedance looking away from
the transmitter terminals into the electrical coupler, and Zr the
impedance of the receiver. [tis well known that the impedance Zp
is dependent upon the velocity of motion of the receiver diaphragm.
Also, T is dependent upon the amplitude of motion of the transmitter
diaphragm as well as upon the direct current supplied to it. Conse-
quently, the impedances detined above are not only dependent upon
frequency but also upon the mechanical coupling and magnitude of
the current supplied to the transmitter.

current flowing through it Loth expressed in root mean s;uare values‘,
then
e=(I+R)i (1)
It is cousenent to define a quantity JZ which shall call the uni-
lateral mutual impedance by the qeuation

where ¢; is the electromotive force created in the transmitter when
a current 7; flows in the recetver circui.. [It is a quantity which is
closely related to the etfectiveness of the mechanical coupling and
the efficiencies of the transn: iter and receiver.

If the electrical coupler be considered part of the receiver, and
the transmitter and rec iver circuits are connected together as in
Vig. J, then e=e,, and f=7,. Consequently

the eoadition .or sustained oscillation. This condition is in effect
a pair of condtiiors, as the two sides of the equation must be equal
both in amplitude and in phase. These two conditions are sufficient
to determine the irequency and intensity of howling.

In order to express Jé and Rin more fundamental physical constants,
it is necessary to examine more closely the mechanical and electrical
connections. Before doing this for some important special cases,
it will be necessary to discuss some of the electro-dynamical properties

For the sake of clarity the discussion will be confined to permanent
magnet receivers and carbon transmitters. The modifications neces-
sary for other types of instruments will, | think, be evident from the
discussion. Representing by Fr and Fr the forces acting on the
diaphragms of the receiver and transmitter respectively, and by y¥
and s their displacements, we have the following equations defining
the ‘stiffness factors’’ Spr and

‘In what follows all quantities involving periodic variations will be expressed
as root mean square values unless otherwise specified, and the vector notation
will be used for denoting phases.

These factors are usually complicated functions of the frequency
while Sp likewise depends on the kind and amount of agitation. In
the case of a system of a single degree of freedom which mav_ be
regarded as a first approximation to this case

where w is 27 times the frequency. When referring to the movements
of a diaphragm, the quantity m represents the mass, r the mechanical
resistance, and s the elastic constant. The stiffness factor S divided
by jw is usually called the mechanical impedance.

<x
>
VALUES OF THE FORCE FACTOR
OF THE RECEIVER
z . FORCE FACTOR Z
< }
|
30
0 200 400 600 800 1000 (200 1400
FREQUENCY
Fig. 2

An important constant which enters into the determination of the
unilateral mutual impedance \/ is the force factor of the receiver
which will be designated by Z. It is defined as the force in dynes
acting upon the diaphragm per unit of current. For the receivers
used in this investigation, its values in magnitude and phase are shown
for various frequencies in Fig. 2. These were determined by the
method outlined by Wegel2 In the region of the resonant frequency
its value in absolute units can be approximately represented by

The impedance Zp of the receiver varies with frequency and de-
pends upon the load on the diaphragm. If S is the loaded stiffness
of the diaphragm, that is, its resistance to force under actual working
conditions, and Z; is the impedance of the receiver when the diaphragm
is prevented from moving, then it is well-known that

It was found that Zy expressed in ohms could be represented in the
frequency region near resonance by the formula

The electromotive force e created in the transmitter, the direct
current J flowing through it, and the displacement of the diaphragm
are related in a rather complicated way. For describing this rela-
tionship it is convenient to define a modulation factor h# by the
equation

which shows that the modulation factor is also an important one in
determining the unilateral mutual impedance. For sustained
oscilation the factor Ji does not enter into the periodic variation
and may be thought of as an electro-mechanical impedance between
the electromotive force created in the button and the displacement
of the diaphragm of the transmitter. However, for a different condi-
tion of sustained oscillation which results in giving 2 a different mag-
nitude the value of 2 changes. In other words / is dependent upon
the agitation of the carbon as represented by s, and also upon the
direct current supplied to the transmitter. [It is mainly this variable
character of A that makes it possible to fulfill the conditions for sus-
tained howling.

Simultaneous measurements of e, J and s were made upon several
transmitters of the type used in this investigation. From the results
obtained and from the defining equation (10) for /, it was found that

the following empirical equation would represent approximately the
relation between /, J and s, namely

where 2 is expressed in) microns and J in amperes ¢ in volts and h
in ohms per micron. To facilitate solving for z when k and TJ are
given, a set of curves showing this relation is given in Fig. 3.) It
is this modulation factor # which measures the efficiency of the trans-
mitter button.

- Relation Between the Modulation
Factor and the Displacement
“2” of the Transmitter Diaphragm

It is also necessary to know the dependence of 7 upon 2 and J.
To obtain this relation corresponding values of e and Vy the DC
drop across the transmitter as measured by direct current measuring
instruments, were obtained for various degrees of agitation and
amounts of direct current. Four transmitters were used in establish-
ing the relation, the results being shown in Fig. 4. Then, for any
value of the supply current J a value of 7 can be obtained from V.
From the corresponding e a value of and s can be obtained from

and 6 were obtained. It is thus seen that for a given type of trans-
mitter if the direct current and any one of the four quantities e, /, ¢,
or T are known, the others are determined and may be obtained from
suitable curves.

THE RELATION BETWEEN THE MODULATION FACTOR fh
AND THE TRANSMITTER RESISTANCE “T"

Commercial receivers and transmitters have constants which vary
largely from those given above. These values represent the general
behavior of such instruments and are useful in understanding their
operation in a howling circuit. Inasmuch as the performance of such
instruments particularly the transmitter depends very largely upon
the condition of operation the constants given cannot be applied
with confidence to conditions greatly different from those mentioned
in the paper. With these facts concerning telephone instruments
in mind we are now in a position to treat some special cases.

CASE 1—DIAPHRAGMS CONNECTED MECHANICALLY BY A RIGID
AND WEIGHTLESS LEVER

To illustrate the method of solution this special case will be solved
in some detail. A diagrammatic sketch illustrating the connections
is shown in Fig. 7. Neglecting the reaction of the air, the vibration of
the receiver diaphragm is controlled by the force Zi exerted by the

receiver winding and the opposing force Y exerted by the connecting
rod.

13)
= (
Sr
If the lever is rigid and weightless and has an arm ratio c, then
Fr=cFr (14)
and due to the restraint
cx
Y= cs = (10)
5;
Using these equations together with equation (11) it is seen that
IhZ
(16)
CSR

where Rpc is the direct current resistance of the receiver winding
and k is the line resistance. The condition (3) for howling then
becomes

Phis is equivalent to two scalar equations and taken together with
19) and the curves of Fig. 6 gives the necessary four equations to

The solution, however, is not) straightforward since the rela
tion between 7. and J is only given empirically by a set of
curves. By “cut and trv” methods the solution for anv numerical
case can be obtained. The last term of (20) is usually nevgligibl
or at least it is of second order of magnitude.  ¢ onsequently, th
sum ot the phase angles of the other factors must be APpronimiate hy
equal to the phase of Z. This completes the formal solution for

The solution of a numerical case throws considerable light) upon
the physical phenomenon taking place, and also upon the method of
calculation. Let the arm ratio be unity, a case corresponding to that
when the diaphragms are connected directly together, and assum
that the supphy current is furnished by a battery of 24 volts throug!
a line having a resistance of 300 ohms. Using the constants for the
receivers and transmitters given above and expressing f in kiloewcles
Tin ohms, J in amperes and # in ohms per micron, equations (14

If J is positive there is no solution for f/, since the angle of the first
factor is in the first quadrant, and that of the second factor either
in the first or second; consequently, the phases cannot match at any
frequency. If the supply current is reversed, then J is negative or
180° is added to the phase of the left hand member making it a positive

It a value of ¢ equal to 2.7, which is approximately equal to the
square root of the ratio of mechanical impedances of the two dia-
phragms, then the solution for reversed DC supply becomes

It is thus seen that changing the ratio arm has increased the howling
intensity, but the increase for the various clements is greatly different.
The frequency is slightly lowered, the values of # and J have been

reduced by 26°, and 14° respectively, while the values of y, 2, 7,
t and have been increased 40000, 105°C, 5706, 2206 and respect-
ively.

If the circuit of Fig. 7 is modified as shown in Fig. 8, the induct-
ance L being very large, then the condition for howling becomes

The solution for values of A= 1 mi, K=1/2 mf, and K= 1/5 nit
are given in Table T. When A= 1 mf and the supply current is direct
the solution which satisfies the phase equality is f= 506. This corre
sponds to h= 220 which is an impossible value. Therefore, no howling
will be sustained for this condition. For A= 1 2 mf the svstem will
howl for both direct and reversed supply current, the frequency
changing suddenly from 839 to L119 eveles as the current is reversed
while the other variables change only slightly.

It is interesting to note the change in the howling frequency as
the value of AK increases. When the supply current is negative, and
for values larger than 1 mf, the frequency of howling is always close
to 1000, as K goes from 1 to 1,2 the frequency increases to above
1100. For smaller values of A the frequency continues to slowly
increase until, for values smaller than 1 3, the system ceases to
sustain oscillations. — For positive values of supply current no howling
will result until A becomes smaller than 2.3 where the frequency is
around 800.) The frequency then increases reaching a howling fre-
quency around 1000 for A= 17.) For smaller values of K no howling
will be sustained.

It will be assumed that the air chamber is so small that the phase
of the pressure variation is the same on both diaphragms. Let V
be the volume of air between the diaphragms. Then

where Vis the volume of air in the undisturbed state and Or and Q;
are the effective areas of the receiver and transmitter diaphragms
respectively.

The pressure variation in the chamber (changes considered adiabatic
is given by

When the steady state is set up this may be considered a vector
equation and the variables expressed in rms values.

In this case the ratio between zs and y is not fixed, but depends upon
Sr which is a function of the frequency.

For the transmitter and receiver used
Qr = 6.5,
10.5.
Let the volume of entrapped air be taken as 10 cc., then

where J is expressed in amperes, 7 in ohms, f in kiloeveles and # in
ohms per micron.

Comparing this to the case where the diaphragms are coupled by a
lever having an arm ratio 2.7 it is seen that the air coupling produces
a greater e.m.f. in the transmitter and only a slightly increased AC
current. The receiver diaphragm in this case, however, has a smaller
amplitude than the transmitter diaphragm. At this” particular
howling frequency the transmitter diaphragm stiffness is only about
1/4 that of the receiver diaphragm stiffness which explains this
anomalous result. Also, it will be seen that the diaphragms vibrate
almost oppositely in phase.

These cases are sutticient to illustrate the method of calculation, but
there is one other important case for which I desire to give the results
as this is the case handled experimentally by Kennelly and Upson.

Case III—DIAPHRAGMS CONNECTED ACOUSTICALLY BY A TUBE OF
AIR OF UNIFORM CROSS-SECTION WITH AN
AIR CHAMBER AT ENpDs

In this case the two diaphragms are connected acoustically by the
air, but since the tube has considerable length phase differences exist

at different points along it. The connections are shown schematically
in Fig. 9.
The equation of motion for the receiver diaphragm is

where dpr and dpr are the pressure variations in the air chambers
at the receiver and transmitter ends of the tube respectively.

The equations of motion for a gas in which the movements are
small and in only one direction and in which the fluid friction is
neglected are as follows: °

where ¢ is the velocity potential, ¢ the time, a the velocity of sound in
the air, x the distance along the tube, p the pressure and p the density
of the air.

For the case in which we are interested, a sinusoidal oscillation is
sustained, so that the special solution

is suitable for our problem. Quantities A and B are arbitrary con-
stants which are determined by the end conditions. Substituting
this value of ¢ in equation (35), there results

dp = —pjwe’ (A cos®* +B sin“) (38)
a a
It remains then to determine the arbitrary constants A and B.

At the receiver end of the tube, the displacement, ¢x of the air
diaphragm across the end of the tube is related to the displacement
y of the receiver diaphragm. This relationship is established by the
following consideration. If g is the cross-section of the tube, the
increase in volume in the air chamber is given by

Assuming that the air chamber is so small that the pressure change
at any instant is the same throughout, and that it takes place adia-
batically, we have:

dpr=—} dVr (40)
Vr
Combining equations (35), (39), and (40), we obtain:
QSrtr = OrZi— ( Sr + Ox® )d pe (41)
p
Similarly,
gSrér = (= S7 +O *\dp (42
YP

Then the following conditions must be fulfilled at the two ends of a
tube of length 7.

yp/ Or yp Or
Or ~ On , « o
+S) + Sr Spr 32)
Or V1 vp
The unilateral mutual impedance J is given by
ThZ
M=
N sin— +P cos
ad ad
The condition for sustained howling becomes
IZ wl wl
h=Nsi +P cos—. 54)

If the two diaphragms work directly into the connecting tube as
pistons, then Qr=Qr=q=Q and Vr=Vr=O0 and the expressions
for MW and S become

The method of solution is the same as that given for the sim ler
cases, although it is evident that the actual work of calculation is
more involved.

It is seen that in such a system the intensity and frequency depend
upon a large number of quantities, namely: S; and Spr, the diaphragm
stiffness factors; Og and (Ur the effective areas of the two diaphragms;
Vr and Vy the volumes of air entrapped between the diaphragm
and the opening into connection tube; the length / and the cross
section q of the connecting tube; the pressure a, the density s, and the
velocity of sound_a for the*gas in the connecting tube; the resistance
T, direct current_J and modulation factor / of the transmitter; and

7 These two equations were given by H.W. Nichols in essentially this form in
the Physical Review, Vol. 10, p. 171; 1917.

the force factor and impedance of the receiving circuit. Modification
of any of these may produce marked changes in the resulting howling.

The way the length / enters the formula (54) for sustained howling
indicates that the curves representing the possible frequencies of
howling, that is, frequencies which produce equality of phase on both
sides of the equation, vary periodically with the length.

The intersection of the branches of these curves on any given
frequency line will be separated by distances corresponding to rs
that is, corresponding to a wave length at the pitch corresponding
to f. Also, if the supply current is reversed, that is, the sign of J

changed, and the length of the tube varied until the frequency of
howling is brought back to the original value, the change in length
a
must be equal to —-
2f

cosine factors. Adding a half wave length is equivalent to adding 7
to the angle which makes the left hand member the negative of its
first value, and consequently, restores the phase equality.

Using the circuit shown in Fig. 10 for the electrical coupling, the
frequency of howling was computed for various tube lengths, the
results being given in Fig. 11.

The instrument constants were those used before, the other values
being Vr=1.6ce., Vr=6.4 ce., and g=.97 cm", a=3.43 X 104 cm/sec.
o =.001203 gm/cm*. Using these values the formulae for N and P
become
N= (—1.31f°+7.5f3 — 9.68f 13.26 )X 105+ j(.141f' — .63/°+.36) X 108,

The points on the calculated curves of Fig. 11 were obtained by
direct experimental observation with the circuit shown, and with
various lengths of brass tube coupling the transmitter and receiver
together. The agreement between the calculated and observed

values is well within the experimental error involved in determining
the constants used in the calculation.

In Fig. 12 are shown similar calculated curves for a transmitter
called “hollow,” that is, for one having a lower natural period of
vibration. It is coupled to the same receiver as used before. The
dotted curves in each case represent the behavior for reversed current.

In Figs. 13 and 14 are shown the probable frequencies of howling
for these two transmitters as the tube length of the coupler Is In-
creased The shaded areas are the so-called breaking points where
the howling may be at either of the frequencies shown.

With these facts in mind let us review the conclusions reached by
Kennelly and Upson given in the beginning of this paper. It is seen
that conclusion

s not warranted. The transmitter and. circuit
conditions as well as the receiver diaphragm influence the mean

of the curves representing the relation between frequency and tube
length is correct and the explanation has just been given. This
periodic relation is not only true of the mean frequency line but for
every constant frequency line.

factors including the circuit and end conditions. Conclusion (3) is
partially correct, the range of the howling frequencies depending
upon the efficiencies of the transmitter, receiver, and circuit is evident
from equation (54). Calculations show that conclusion (4) is generally

When the transmitter and receiver are coupled by the air in an
open room the behavior is somewhat similar to the case yust solved
The size and shape of the room as well as the disposition of articles
of furniture will all influence the intensity and frequency of howling
In general when the two instruments are moved apart the trequency

R Impedance looking away from Transmitter Ti
Zr Impedance e of Receiver.

V, Volume of Air in Front ot Receiver Diaphrag
V, Volume of Air in Front of Fransmitter Diaphrag

On Effective Area of Receiver Diaphragm
Or Effective Area of Transmitter Diaphrag:
pb Air Pressure.

ce Displacement of Au Particle at Receiver End of 1
Displacement of Air Particle at Trans nitter End of 1

electrotechnics are to the theory of the propagation of current
and voltage along transmission systems. Of such transmission sys-
tems the simplest is the non-inductive cable. The theory of the
non-inductive cable is not only of great historic interest, relating as it
does to Kelvin’s early work on the possibility of transatlantic teleg-
raphy, but is also of very considerable practical importance today,
and serves as a basis for the theory of submarine telegraphy over long
distances. We shall therefore consider the propagation phenomena
in the non-inductive cable in some detail.

The propagation phenomena in any type of transmission system
are isolated and exhibited in the clearest possible manner when we
confine attention to the infinitely long line, with voltage applied
directly to the line terminals. Furthermore, as we shall see later,
the solution for the infinitely long line is fundamental and can be
extended to the more practical case of the finite line with terminal
impedances. We therefore, in this chapter, shall confine our atten-
tion to the case of the infinitely long cable with voltage applied directly
to the cable terminals.

Consider a cable of distributed resistance R and capacity C per
unit length, extending from x=0 along the positive x axis. From a
previous chapter (see equations (64) and (65) ), we are in possession
of the operational equations of voltage and current; they are, for the
infinitely long line,

(162)
ap ¢ var Vo= \ We
where a=x°RC, and V. is the terminal cable voltage at x=0. Let
“unit then
V=e7 Var, (164)

The solution of (164) for V was considered in some detail in the
preceding chapter; it is, by (129)

where t=4t/a=4t/x°RC. Series expansions of this solution were
also given. Another equivalent form is, by (131)

This last form, recognizable also from inspection of the series expan-
sion (132), is useful because the integral term is what is called the error
function and has been completely computed and tabulated.

Before discussing these formulas and the light they throw on propa-
gation phenomena in the non-inductive cable, we shall derive the
solution for the current. A very simple way of doing this is to make
use of the differential equation (57)

It is worthwhile verifying the formula by direct solution from the
operational equation (165). From formula (g) of the table of in-
tegrals, we have

Comparison with the operational equation shows that they are iden-
tical, within a constant factor provided we put =a 4. Conse-
quently the solution of (165) is
Cc !
[= e~ =
N N cRt

example of the utility of the table of integrals in solving operational
equations.

This formula is easily calculated for large values of ¢ by expanding
the exponential function; it ts

The propagation phenomena of the non-inductive cable are there-
fore determined by the pair of equations

spt; that is, of 4¢ divided by the total
resistance and capacity of the cable from x=0 to x=x. The same
statement holds for the form of the current wave: its magnitude,

however, is inversely proportional to xR, or the total resistance of
the cable up to point x. Consequently a single curve, with proper
time scale serves to give the voltage wave at any point on the cable.
Similarly a single curve, with proper time and amplitude scales,
serves to depict the current wave at any distance from the cable
terminals. These curves are given in Figs. 3 and 4.

Referring to the curve depicting the current wave, we observe that
it is finite for all values of £>0; consequently, in the ideal cable, the
velocity of propagation is infinite. This is a consequence, of course,
of the fact that the distributed inductance of the cable is neglected.
Actually, of course, the velocity of propagation cannot exceed the

point will be discussed and explained more fully in connection with the

Its subsidence to its final zero value is very slow; for example, when
7 = 100 its value is still
(0.10).
Turning to the voltage curve, Fig. 4, we see that it is negligibly
small until 7 reaches the value 0.25, at which point it begins to build
up. Its maximum rate of building up occurs when 7+=2/3, after

which it builds up more and more slowly. Its approach to its final
steady value is in accordance with the formula

Even, therefore, when 7 is as great as 100, V differs sensibly from its
ultimate value, unity, its value being 0.8876.

The power curve VI is given in Fig. 5. V.I is the rate at which
energy is being transmitted past the point x of the cable.

The fact that the form of the current and voltage waves depends
only on 4t/x°RC is at the basis of Kelvin’s famous “KR” law, long
applied to cable telegraphy and sometimes incorrectly applied to
telephony. When the first transatlantic telegraph cable was under
consideration, Kelvin attacked the problem of propagation along the
non-inductive cable and arrived at formulas equivalent to (169) and

(170). From these formulas he announced the law that the ‘‘speed”’
of the cable, i.e., the number of signals transmissible per unit time,
is inversely proportional to the product of the total capacity and total
resistance of the cable (KR in the English notation). To see just
what this means requires a little digression into the elementary
theory of telegraph transmission.

Telegraph signals are transmitted in code by means of “dots”
and ‘‘dashes.””. The “dot” is the signal which results when a battery
is impressed on the cable for a definite interval of time, after which
the cable is short circuited. A “dash” is the same except that the
time interval during which the battery is connected to the cable is
increased. The “dots” and ‘dashes’ are separated by intervals,
called “spaces”, during which the cable is short circuited. Now when
the cable is short-circuited we may imagine a negative battery im-
pressed on the cable in series with the original battery. Conse-
quently the current in the cable, corresponding to a signal composed
of a series of dots, dashes and spaces, will be represented by a series
of the form

where, in the cable under consideration, J(‘) is given by (168). fy is
the duration of the first impulse, fg—f; of the first space, fs— tz of the
second impulse, etc.

xCR
Then the signal can be written as
2
o(7) — +O(7—T2)— (172)
xRV x '
Now if the relative time intervals 7;, 72... are kept constant (as the
length of the cable is varied), the actual time intervals f;, fo... are

proportional to x°CR or to KR, and the wave form of the total signal
is independent of KR, when referred to the relative time scale r.

Hence, if 7 is the total time of the signal, 7 is proportional to x°7CR
or to KR). Phat is to say, if the duration of the component dots,
dashes, spaces of the signal are proportional to the “KR” of the
cable, the wave form of the received signal, referred to the 7 time
scale, is invariable, and the total time required to transmit the signal
is proportional to the “KR” of the cable. Now the maximum theo-
retical speed of transmission on the cable is limited by the require-
ment that the received signal shall bear a recognizable likeness to
the original svstem of dots and dashes: in other words there is a

maximum allowable departure in wave form between received and
transmitted signals. If, therefore, the actual speeds of two cables
are inversely proportional to their “KRs,” the wave form will be the
same. This establishes Kelvin’s “KR” law. As a corollary, if the
length of the cable is doubled the speed of signaling is reduced to one-
quarter, assuming the same definition of signals.

The foregoing will be somewhat clearer, perhaps, if we refer to
curves, 4, 5, 6, 7 of Fig. 6 which illustrate the distortion suffered by
elementary dot signals in cable transmission. Curve 4 shows the
dot signal produced by a unit battery applied to the cable terminals
for a time interval /=2 a while curves 5, 6 and 7 are the cor-
responding dot signals when the battery is applied for the time in-

affect the shape of the transmitted dot, whi
speed has reached its theoretical maximum. These curves, it should
be observed, can be interpreted in two wavs. First, we can regard the
length x of the cable as fixed and the duration of the impressed dot
as varied. On the other hand, we can regard the actual duration of
the impressed dot as constant and the length of the cable as varied
From the latter standpoint the curves illustrate the progressive dis
tortion of the signal as it is transmitted alone the cable

The dot signal of relative duration 7 can be written as
D=TI(r),
=I(r)—I(7-7),
and the second expression can be expanded in a Tavlor’s series, giving

Hence when the dot signal is of sufficiently short relative duration
I’, the wave shape of the received signal is constant, 7’(7), and its
amplitude is proportional to the relative duration of the dot.

This can be generalized for any type of transmission svstem:
Let the dot signal be produced by an e.m.f. f(t) of actual duration 7’.
Then the received dot signal, by formula (31), is

Hence for a sufficiently short duration of the impressed e.m.f. the
received dot signal is of constant wave form, independent of the shape
of the impressed e.m.f., and its amplitude is proportional to the time

integral of the impressed e.m.f. These principles are of considerable
practical importance in telegraphy.

The leaky cable, that is, a cable with distributed leakage conductance
G in addition to resistance R and capacity C, is of some interest. The
differential equations of the problem are given in equations (70); the
operational formulas for the case of voltage directly impressed on the
terminals of the intinitely long line are

Writing CRx? =a and RGx? =8, G/C=X, and assuming a “unit e.m.f.”
impressed on the cable, this becomes

These equations are readily solved by means of the table and formulas
given in a preceding chapter.

But first let us attempt to solve the operational equation (175) for
the voltage by Heaviside methods, guided by the solution of the
operational equation

of the preceding chapter. Expand the exponential function in (175)
in the usual power series; it is

Now identify Vp with 1 \/ xt in accordance with the Heaviside rule,
and 1/pwith jdt. We get

Now in the terms of the expansion (178) identify p” with d"/dt"
and substitute (180); we get

This series is hopelessly complicated to either interpret or compute.
It is, in fact, an excellent illustration of the grave disadvantages under
which many of Heaviside’s series solutions labor. We shall there-
fore attack the solution by aid of the theorems and formulas of a
preceding section. The simplicity of the solutions which result is
remarkable.

Now the operational formula for the voltage in the non-leaky cable
is (see equation (164) )

In order to distinguish between the two cases, let us denote the voltage
in the latter case by V’; thus

the superiority of the definite integral to the series expansion- -compare

with the series expansions —and secondly as exhibiting
clearly the etfect of leakage on the propagated waves of current and
voltage. We see that in both the current and voltage the effect. of
leakage is two-fold: first it attenuates the wave by the factor e ™
(X=G C), and secondly it adds a component consisting of the pro-
gressive integral of the attenuated wave. This, it may be remarked,
is the general effect of leakage in all types of transmission svstems.
Its effect is, therefore, easily computed and interpreted.

Formulas (185) and (186) are very easy to compute with the aid
of a planimeter or integraph; or, failing these devices, by numerical
integration. However, for large values of f, the character of the waves
is more clearly exhibited if we make use of the identity

The first two terms of these formulas are clearly the ultimate stead,
state values of the voltage and current waves, and can be deter
more

mined by evaluating the infinite integrals. A far simpler at
direct wav, however, is to make use of the fact that the ultimate
steady values of Vand J are gotten from the operational formulas
by setting p=0. That this statement is true is easily seen if we
reflect that the steady voltage and current are gotten from. ti
original differential equations of the problem by assuming a steady
state and setting d dt=0.
From the operational formulas we get, therefore

Using the values of WV’ and J’, as given by (169) and (170), it is ex
tremely easy to compute V and I, for large values of ¢, from (193
and (194),

So far we have considered the current and voltage waves in re-
sponse to a “unit e.m.f.,”” impressed on the cable at «=O. It is of
interest and importance to examine the waves due to sinusoidal
e.m.fs., suddenly impressed on the cable, particularly in view of

(194-a)
— cos ut | sin wt.h'(t)dt.
Jo
Similarly, if the impressed e.m.f. is cos at,
x-(t) =cos wt | cos wt.h’(t)dt
eJ0
(194-b)

The investigation of the building-up of alternating currents and
voltages, therefore, depends on the progressive integrals

For the case of the voltage waves on the non-inductive, non-leaky
cable these integrals, by aid of equations (169), become, if we write

a ELECTRIC Cll
For small values of 7 and ’ these integrals can be numerically
evaluated without great labor. Mechanical devices, such as the

the Coradi Analyzer gives these progressive integrals automaticalls

It may be said, therefore, that a complete mathematical investiga
tion of the building-up of alternating current: and voltage waves
on the non-inductive cable presents no serious ditticulties, although
the labor of computation is necessarily considerable. One tact makes

posed. This is, if the foregoing integrals are calculated for a given
value of w’, the results apply to all lengths of cable and all actual
frequencies w

such that aw is a constant. Then if we double the
length of the cable and quarter the frequency, the integrals are un-
attected.

The solid curve of Fig. 7 shows the building-up of the cable voltage
in response to an e.m.f. cos wf, impressed at time f=). The fre-

culated from equations (194-b) and (194-e). The dotted curve shows
the corresponding steady-state voltage on the cable; that is, the voltage
which would exist if the e.m.f. cos wt had been applied at a long time
preceding ‘= °. We observe that, for this frequency, the building-up
is effectually accomplished in about one cvcle, and that the transient

The case is very much different when a higher frequency is ap-
plied. Fig. 8 shows the building-up of the alternating current in the
cable when an e.m.f. sin wt is applied at time ‘=0. The frequency
is so chosen that w’=aw/4=107. The outstanding features of this
curve are that the initial current surge is very large compared with
the final steady-state, and that the transient distortion is relatively
very large. It is evident that the frequency here shown could not be

employed for signaling purposes. This curve has been computed
from the steady-state formulas, and equations (160) and (161) for
the transient distortion.

If the applied frequency w, 27 is very high, the steady-state becomes
negligibly small, and the complete current is obtained to a good
approximation by taking the leading terms of (160) and (161). Thus
if the applied e.m.f. is sin wf, and w is sufficiently large, the cable
current is

We now take up the more important and difficult problem of
investigating the propagation phenomena in the transmission line.
The transmission line has distributed series resistance R and _ in-
ductance L, and distributed shunt capacity C and leakage conductance
G. It is the addition of the series inductance L which makes our
problem more difhcult and at the same time introduces the phenomena
of true propagation with finite velocity, as distinguished from the
diffusion phenomena of the cable problem. The cable theory serves
very well for the problems of trans-oceanic telegraphy * but is quite
inadequate in the problems of telephonic transmission.

If J denotes the current and V the voltage at point x on the line, the
well known differential equations of the problem are:—

d }
L +R) I=—y,
( dt x0
(195)
(c +G)V=-
d x
Replacing d/dt by p, these become
(Lp+R) I= — —V,
(196)
(Cp+G)V= — —I.
x
From the second of these equations
oV
— =
ox
and substitution in the first gives
: (Lp+ R)(Cp+G)I=—.I. (197)
Similarly if we eliminate J, we get
(Lp+R)(Cp+G) V=—\V. (198)
3 § With the installation of the new submarine cable, continuously loaded with per
; malloy, this statement must be modified. In this cable, the inductance plavs
3 important part, and is responsible for the greatly increased speed of signaling

Be
where «lf and B are arbitrary constants, substitution shows that the
solution satisties the differential equation for V7 provided

Now restricting attention to the infinitely long line extending along
the positive x axis, with voltage |, impressed directly on the line at
v=0, the retlected wave vanishes and we get

= ,[(b+p)?— (202)
where
v=1/V LC,
RG
&
Then setting V,=1, the operational equations of the problem become
(203)
G e Vip+p 0
I=0( C+ \p (204)
p p)* =o"

operational equation (205). This equation can be solved by aid of
the operational rules and formulas already given. The process is
rather complicated, and there is less chance of error if we deal instead

Now let us search through our table of definite integrals. We do
not find this integral equation as it stands, but we do observe that
formula (m) resembles it, and this resemblance suggests that formula
(m) can be suitably transformed to give the solution of (208). We
therefore start with the formula

7) for being the Bessel
function of order zero. We now transform (m) as follows:

| eH J, )at= = (m.3
where \;=-—-A and w,=—- wu. (They are, of course, as yet, arbitrary
parameters, except that they are restricted to positive values).
(4) Now if we compare (m.3) with the integral equation (208
for F, we see that they are identical provided we get
Mi>Ppy
—1,
which is possible, since p>o.
Introducing these relations, we have

Here J, denotes the Bessel function of imaginary argument; thus
It follows from (m.4) and the integral equation (208) that

Having now solved for F= F(t), the current and voltage are gotten
from equations (206) and (207). Thus

Here I)(¢ VY 72?—x°/v") is the Bessel function of order 1: thus —iJ,(iz) =
I,(z). The function is entirely real. The derivation of formula
(211) is a little troublesome, owing to the discontinuous character of
the function F: the detailed steps are given in an appendix.

The preceding solution depends for its outstanding directness
and simplicity on the recognition of the infinite integral identity (m
into which the integral equation of the problem can be transformed.
When such identities are known their value in connection with the
solution of operational equations requires no emphasis. On the
other hand, we cannot always expect to find such an identity in the
case of every operational equation; and, particularly in the case of
such an important case as the transmission equation it would be
unfortunate to have no alternative mode of solution. Fortunately
a quite direct series expansion solution is obtainable from the oper-
ational equation, and this will now be derived. As a matter of con-
venience we shall restrict the derivation to the voltage formula

As a further matter of mere convenience we shall assume that G=4,
so that ¢o=p and (203) becomes
V =e (203-a)
where
The method holds equally well for the current equation (204) and
for the general case op.
Write (203-a) as

If the coethcients 3), bs... are evaluated, a simple matter of elemen-
tary algebra, the foregoing expansion in the retarded time t—7 will
he found to agree with the solution (211) when @ is put equal to p.

We shall now discuss the outstanding features of the propagation
phenomena in the light of equations (210) and (211) for the current
and voltage. We observe, first, that we have a true finite velocity of
propagation 7=1 y LC. No matter what the form of impressed
e.m.f. at the beginning of the line (x=), its effect does not reach the
point x of the line until a time f=x v has elapsed. Consequently
v=x fis the velocity with which the wave is propagated. This is a
strict consequence of the distributed inductance and capacity of the
line and depends only on them, since v=1 LC. It will be recalled
that in the case of the cable, where the inductance is ignored, no
finite velocity of propagation exists.

The question of velocity of propagation of the wave has been the
subject of considerable confusion and misinterpretation when dealing
with the steady-state phenomena. It seems worth while to briefly
touch on this in passing.

As has been pointed out in preceding chapters, the symbolic or com-
plex steady-state formula is gotten from the operational equation by
replacing the symbol p by where 7= — andw, 27 is the frequency.
If this is done in the operational equation (203) for the voltage, the
symbolic formula is

V=e “VN
If the expression Is separated into its real and im-
aginary parts we get an expression of the form

along the line with velocity dx dt=v, 8, the phase of the wave will
remain constant. This is interpreted often as meaning that the

velocity of propagation of the wave is @ 3. Now since 3 is greater
than unity and only approaches unity as the frequency becomes
indefinitely great, the inference is frequently made that the velocity
of propagation depends upon and increases to a limiting value
with the frequency. This velocity, however, is not the true velocity
of pr ypagation, which is always but is the veloc ity of phrase propagation
in the steady-state. This distinction is quite important and failure
to bear it in mind has led to serious mistakes.

Returning to equation (211) and (210) we see that after a time
interval /=x v has elapsed since the unit e.m.f. was impressed on the
cable, the voltage at point x suddenly jumps from = zero to the
value e ’*° while the current correspondingly jumps to the value

which will be recognized as the steady-state attenuation factor for high
frequencies. Similarly y CL is the steady-state admittance of the
line for high frequencies. The sudden jumps in the current and
voltage at time f=y v are called the heads of the current and voltage
waves. If, instead of a unit e.m-f., a voltage f(¢) is impressed on the
line at time ¢=0, the corresponding heads of the waves are fioje *
and VC. L floje for voltage and current respectively. These
expressions follow at once from the integral formula

x(t) = f(t—r)h(r)dr
dt, 0
=f(o)h(t)+ | f'(t—r)h(r)dr.
The tails of the waves, that is, the parts of the waves subsequent
to the time ‘=x v, are more complicated and will depend on the
distance x along the line and on the line parameters p and o. The two
simplest cases are the non-dissipative line, and the distortionless line
The ideal non-dissipative line, quite unrealizable in practice, is one
in which both R and G are zero. — In this case p=o=0, and formulas
(210) and (211) become
t<x/2,

Both current and voltage jump, at time ¢=.x/v, to their steady values.
If an e.m.f. f(t) is impressed on the line at time ¢=0, the corresponding
current and voltage waves are

Consequently the ideal non-dissipative line transmits the waves
with finite velocity v, without attenuation or distortion. Such a line
is, of course, the ideal transmission system.

The non-dissipative line is, of course, purely theoretical and un-
realizable in practice; the disfortionless line is, however, approxi-
mately realizable, and as the name implies, transmits without distortion
of wave form. The distortionless line is one in which the line con-
stants are so related that

=e for x/v.
Furthermore, if the impressed e.m.f. is f (4), the corresponding current
and voltage waves are :—
I=0 for t<x/v,
axf(t—x/v) for
V=0 for i<x/y,
=e *f(t—x/v) for t2=x/v.
The distortionless line, therefore, transmits the waves without dis-
tortion of wave form, but attenuates the waves by the factor e~**.
Such a line is an ideal transmission system as regards preservation
of wave form, but introduces serious attenuation losses. For example,
if a line has normally negligible leakage, and leakage is introduced

to secure the condition R/L=G/C, the line is thereby rendered dis-
tortionless but the attenuation is doubled.

One of Heaviside’s most important contributions to wire trans-
mission theory was to point out the properties of the distortionless
line, its approximately realizable character, and to base on it a correct
theory of telephonic transmission.

The character of the wave propagation when the parameters
p and o are not restricted to special values, can only be roughly in-
ferred from inspection of the formulas, and then only when the prop-
erties of the Bessel function 7, and J; have been studied. Fortunately
these functions have been computed and tabulated for small values
of the argument, and have simple asymptotic expansions for large
values. It is therefore a simple matter to compute and graph a
representative set of curves which show the current and voltage
waves for various values of p, ¢ and x. For this purpose it is con-
venient to introduce a change of variables and write:

Figs. (9) to (18) give a representative set of curves illustrating
the form of the propagated current and voltage waves for different
lengths of line, and different values of the line parameters @ and },
or p and oa.

The curves of Figs. (9) and (10) show the current entering the
line in response to a unit e.m.f. applied at time ¢=0. The line is
assumed to be non-leaky (b)=0) and is computed for two different
values of the parameter a. We see that the current instantly jumps to
the value \/ C/L and then begins to die away, the rate at which it dies

begins to die away provided x» and a are such that ax <2. Hf, however,
we are considering a point at which av >2, the current begins to rise
instead of fall after the initial jump, and may attain a maximum
value very large compared with the head before it starts to die away.
This is shown in the curves of Figs. (11), (12) and (13), also computed

for the non-leaky line (b=0). From these curves we see that, as
the length of the line and the parameter a increase, the relative mag-
nitude of the tail, as compared with the head of the wave, increases.
Finally when the line becomes very long, the head of the wave be-

comes negligibly small, and the wave, except in the neighborhood of
its head, becomes very close to that of the corresponding non-inductive

An interesting feature of both current and voltage waves is that
when a sutficient time has elapsed after the arrival of the head of the
wave, the waves become closer and closer to the wave of the cor-
responding non-inductive cable; that is, to the cable having the same
R,C and G. Consequently the inductance plays no part in the
subsidence of the waves to their final values.

ELECTRIC CIRCUIT THEORY
Curves (16), (17) and (18) illustrate the voltage wave for several
conditions. After the arrival of the head, the wave slowly builds
up to its final value. Curve (1S) represents the case where the line
is very nearly distortionless, showing how completely the distorting
tail of the wave is eliminated.
‘6
12
260260 280 300 20 340 360 380
Fig. 15—Propagated current in line; x =200
»
2 L 2 (
b= VG
Multiply ordinates by vy C/L.e
as =
a3
| |
0 | Values of |
0 2 4 6 8 10 2 ‘4 € 18 2
Fig. 16—P voltage fine; WS G
ig. 16—Propagated voltage in line; —-\ G=0.
‘ So far we have confined attention to the current and voltage waves
in response to a unit e.m.f. applied at time ¢=0 to the line terminals.
H Of much greater technical importance is the question of the waves
: In response to a sinusoidal e.m.f. suddenly applied to the line termi-

nals. In order to investigate this important problem it is convenient
to divide the expressions for the current and voltage waves as given
by equations (210-a) and (211-a) into two components. We write
for

where, by definition, J(f) and W(t) are the differences between the
total waves and their heads. The advantage of analyzing the waves
into these components is that the distortion of the waves is due to

J(t) and W(t) respectively, while the first component of (210-b> and
en
(211-b) introduce merely a delay. Thus, if the e.m.f IMpressed
at time 1s the corresponding waves for or r2 x. are
C
he :
es |= é \
to 2]
|
d d
where J’(t) =—J(t) and W(t W(t
dt dt

The integrals of (212) and (213) can be computed and analyzed
in precisely the same way as discussed in connection with the non
inductive cable problem, and are of very much the same character
as the alternating current waves of the cable. In the total waves,
however, as given by (212) and (213), a very essential difference is
introduced by the absence of the first terms, which represent undis
torted waves propagated with velocity v7. Thus, if the impressed

Now the first terms of (214) and (215) are simply the usual steady
state expressions for the current and voltage waves when the fre-
quency is sufficiently high to make the steady-state attenuation
constant equal to a and the phase velocity equal to e. Furthermor
the integral terms become smaller and smaller as the applied fre
quency w/27m is increased. It follows, therefore, that for high fre
quencies the waves assume substantially their tinal steady value
at time ¢=x /v, and that the tails of the waves, or the transient
distortion, becomes negligible. “This is a consequence entirely of the

presence of inductance in the line, and shows its extreme importanc«
in the propagation of alternating waves and the reduction of transien
distortion,

It should be pointed out, however, that if the line is very long and
the attenuation is very high, the integral terms of (214) and (215)
are not negligible unless the applied frequency is correspondingly very
high. For example, on a long submarine cable, the a.c. attenuation
is so large that the first terms of (214) and (215) are very small, and
J(t) is very large compared with e “. Consequently here
there is very serious transient distortion and alternating currents
are therefore not adapted for submarine telegraph signalling.

This discussion may possibly be made a little clearer, without
detailed analysis, if we recall the discussion of alternating current
propagation in the non-inductive cable of the preceding chapter.
Krom that analysis it follows that, when the applied frequency w/27
is suthciently high, the integral term of (214) becomes approximately

J"(t)
and the complete current wave is
sin w(t—x/v)+-—J’(t) (216)
L w
and similarly the voltage wave is
e-**sin w(t—x/v) + W(t). (217)
Now if the total attenuation ax is large the last terms of (216) and
(217), before they ultimately die away, may become very large com-
pared with the first terms, which represent the ultimate steady-state.
Appendix to Chapter V/I, Derivation of Formula (211)

The only troublesome question involved in deriving (211) from
(207) and (209) is that we have to differentiate with respect to x, in
accordance with (207), the discontinuous function F(t). To accom-
plish this we write (209) in the form

where $(f) is defined as a function which is zero for t<x/v and unity
for t2x/v. Clearly this is equivalent to (209) and permits us to deal

with F(t) as a continuous function. Now, in accordance with (207
perform the operation of differentiation upon (209-a): we get

and that the whole contribution to the integral occurs at t=x
With these points clearly in mind, the expression

CHAPTER VIII
PROPAGATION OF CURRENT AND VOLTAGE IN: ARTIFICIAL
LINES AND WAVE FILTERS
The artificial line here considered is a periodic structure, com-
posed of a series of sections connected in tandem, each section con-
sisting of a lumped impedance 3; in series with the line, and a 'umped

impedance z, in shunt across the line. In the artificial line which
we shall consider it will be assumed that the voltage is applied at
the middle of the initial or zeroth section, as shown in Fig. 19. This
termination is chosen because of its practical importance, and be-

cause also of the facet that the mathematical analysis is simplified
thereby. Furthermore any other termination can be regarded and
dealt with as an additional terminal impedance, so there is no essen
tial loss of generality involved.

1. The artificial line is often used as a model of an actual trans
mission line and it is therefore of importance to determine theoretical,
the degree of correspondence between the two.

2. The solution for the corresponding transmission line with con
tinuously distributed constants is derivable from the solution for the
artificial line by keeping the total inductance, resistance, capacity
and leakage constant or finite, and letting the number of sections
approach infinity.

3. The artificial line is very closely related, in its properties and
performance, to the periodically loaded line, and its solution is, to
a first approximation, a working solution for the loaded line.

1. The structure is of great importance in its own right, and when
the impedance elements are properly chosen, constitutes a “wave
filter.”

We shall now derive the operational and symbolic equations which
formulate the propagation phenomena in the artificial line. Let /,
denote the mesh current in the ath section of the line; 7,—, the mesh
current in the (z7—1)"' section, ete. Now write down the expression
for the voltage drop in the #'" section; in accordance with Kirch-
hott's law we get:

ential equation, but the method of solution is essentially the same.
We assume a solution of the form

I, =Ae-"l + Berl (219)
where A, B and VT are independent of , and substitute in (218).
After some simple rearrangements we get

Equation (218) is clearly satistied by the assumed form of solution,
and furthermore leaves the constants 4A and B arbitrary and at our

where 42:.
Now by reference to equation (219) it is easily seen that Pois the
propagation constant of the artificial line, precisely analogous to the

propagation constant y of the smooth line. In terms of 1
pedances 2; and s., the propagation constant of the artificial line ts
determined by (221). This equation may either be regarded as an

operational equation or a symbole equation, depending on whether
the impedances are expressed in terms of the operator Pp or in terms
of tw, where w is 27 times the frequency.

Now return to equation (219) and Jet us assume that the line ts
either infinitely long, or, what amounts to the same thing, that it ts

closed by an impedance which suppresses the retlected wave. We
assume also that a voltage V. is impressed at mid-series position of the
zero” section (w=). Equation (219) becomes

where K is the characteristic impedance of the artificial line (at mid
series position). Hence by (223) and (222)

Before proceeding with the operational equation, and the investi-
gation of transient phenomena in artificial lines, it will be of interest
to deduce from the foregoing the unique and remarkable properties
of wave filters in the steady state. For this purpose we return to
equation (221)

Now suppose that the series impedance 2; is an inductance L and
the shunt impedance zz a capacity C, so that, symbolically,

cos 0=1—3 wLC (227)
0. Consequently if
4 is a real quantity the ratio of the absolute values of the currents in ad-

jacent sections ts unity, and the current is propagated without attenuation.

Inspection of equation (227) shows that @ is real provided the right
hand side lines between +1 and —1: or that w lies between 0 and
2/-\y LC. Consequently this type of artificial line transmits, in the
steady state, sinusoidal currents of all frequencies from = zero to
1 ry LC without attenuation. It is known as the low-pass filter.

If we invert the structure, that is, make the series impedance 2,
a capacity C and the shunt impedance 22 an inductance L, so that

This type of artificial line transmits without attenuation currents
of all frequencies for which the right hand side of (22S8-a) lies between
+1 and —1; that is, all frequencies from infinity to a lower limiting
frequency 1/47 LC, while it attenuates all frequencies below this
range. It is known, on this account, as the high-pass filter.

It is possible by using more complicated impedances to design
filters which transmit a series of bands of frequencies. We cannot,
however, go into the complicated theory of wave filters here, which
has been covered in a series of important papers. One point should
be noted, however: transmission without attenuation implies that
the impedance elements are non-dissipative. Actually, of course,
all the elements introduce some loss, so that in practice the filter
attenuates all frequencies. Careful design, however, keeps the
attenuation very low in the transmission bands.

We shall now derive the indicial admittance formulas for some
representative types of artificial lines and wave filters from the oper-
ational formula

We start with the so-called low-pass filter on account of its sim-
plicity and also its great importance in technical applications. This
type of filter consists of series inductance L and shunt capacity C
The general case which includes series resistance R and shunt leakage
G has been worked out (see Transient Oscillations, Trans. A. 1. ke. k.,
1919). The solution is, however, extremely complicated and will not
be dealt with here. We shall, instead, consider the important and
illuminating case where the series and shunt losses are so related

s) that the preblem is reduced to the solution of the operational
equation for <1 Writing w,=2/+/LC, we have

NZ V/1+(p/w.)? V1+(p p
\ L +. 2
Now refer to formula (72) of the table of integrals; writing y L C=k,
we see by Theorem V that

/ / (
ike, to a unit e.m.f. applied te the initial section (2 =0 From (232
and (234) it follows at once that
+. | dre ] ri
R So Jo
Integrating the second member by parts and noting that .1.(0) =0
this reduces to
231) 1 \
Ay | e Ge’
k,
which is the indicial admittance formula for the quasi-distortionless
low-pass filter, or artificial line.

Before discussing these formulas, it is of interest to derive the
formula for A) by power series expansion. Formula (233) can be
written

) l l
k Pp \ Il+(w p +4 p
This can be expanded in a series in inverse powers of Pp; thus
2) 2n+2 /w 2
Ae ' ( )
32) 271! \p,
(2n+3) (2n+4
nal \ p
Replacing 1p" by fa! in accordance with the Heaviside Rule we get
LIN 2 L'(2n+3 2
2!(2n-+5)! 2
k, ; Phis can be recognized as the power series expansion of (254
The artificial cable is also of interest and practical importance
In this structure the series impedance is a resistance R and the shunt

For large values of » and ¢ this series is difficult to compute or in-
terpret. It can, however, be recognized as the series expansion of the
function

where J,,(2t/RC) is the Bessel function J, of order » and argument
(2¢/RC). This solution, it may be remarked, can be derived directly
by a modification of the integral formula (7).

It is beyond the scope of this paper to consider other types of
artificial lines and wave filters; for a fairly extensive discussion the
reader is referred to ‘Transient Oscillations in Electric Wave-Filters,”’
B.S. T. J., July, 1923. The low-pass wave filter, however, both in
its own right and on account of its close relation to the periodically
loaded line, deserves further discussion.

(‘wt
{° Jon(r) dr (234)
k, 0
while for the quasi-distortionless low-pass wave filter
A | e AT Jo,(r)dr (235)

Computation and analysis of these formulas involve an elementary
knowledge of Bessel functions. The properties necessary for our
purposes are briefly discussed in an appendix to this chapter.

Fig. 21—Low pass wave filter. Indicial admittance of third section (1 =2
Multiply ordinates by vC/L

Values of wt
Fig. 22—Low pass wave filter. Indicial admittance of fifth section (m=4).
g I

that is, the current in response to a steady unit e.m.f. applied at
time ¢=0, are shown in the curves of Figs. 20, 21 and 22, for the
initial or zeroth, the 3rd and the Sth sections, respectively. These

above are sufficient to give a reasonably comprehensive idea of the
general character of these oscillations and their dependence on the
number of sections and the constants of the tilter.

It will be observed that the current is small until a time approxi-
mately equal to 2a w =nyLiC, has elapsed after the voltage is
applied. Consequently the low-pass filter behaves as though cur-
rents were transmitted with a finite velocity. of propagation w, 2
1 y LyCy sections per second. This velocity is, however, only ap-
parent or virtual since in every section the currents are actually
finite for all values of time >0.

After time (=n 1,02 has elapsed the current oscillates about the
value Tok with increasing frequency and diminishing amplitude.
Phe amplitude of these oscillations is approximately

and their instantaneous frequency (measured by intervals between
zeros)

The oscillations are therefore ultimately of cut-otf or critical frequency
w 2r in all sections, but this frequency is approached more and more
slowly as the number of filter sections is increased.

Figs. 238, 24. 25, give the indicial admittance in the 100th, 500th
and 1000th section of the low-pass wave filter. The filter itself seldom

he embodies more than 5 sections Phe case of a large number of se
he tions is of interest, however, because 11 represents a first: approxi

12 +———+—
ip-
le.
a6 + + + —
en
' | |
+
960 980 1000 1020 5 00 0 40 160
Fig. 24—-Low pass wave filter Indicial admittance f 500th se $09
Multiply ordinates by L
Cy
—
to-—
th
mM
a8 t +
near H
|
|
| |
02
| |
|
} | |
960 1980 2000 2020 2040 2060 2080 2120 2140 2160
Fig. 25—-Low pass wave filter. Indicial admittance of 1000th section (2 = 999
Multiply ordinates by vc L
line is ideal and unrealizable, its study is of practical importance
because in this type of line the effect of the discontinuous character
: of the loading of the periodically loaded line is isolated and exhibited

The dotted curves represent the current in the corresponding
smooth line. For the smooth line, the current, as we have seen, is
discontinuous, being identically zero for a time v=” and having an
instantaneous jump to its final value ¥ ‘C/L at vt=n. The current
in the artificial or periodically loaded line differs from that in the cor-
responding smooth line in three important respects: (1) the absence
of the abrupt discontinuous wave front, (2) the presence of super-
posed oscillations, and (3) the absence of a true finite velocity of propaga-
tion. It will be observed, however, that the current in any section
is negligibly small or even sensibly zero until v=, so that the current
is propagated with a virtual velocity 1/+/LC per section. The pres-
ence of a well marked wave front is also evident although this is not
abrupt, as in the smooth line. The effective slope of the wave front
becomes smaller as the current wave travels out on the line, decreasing
noticeably as the number of sections is increased. When the number of
sections becomes large, however, the decrease in the slope is not rapid,
being in the 500*" section about 60 per cent. of that in the 100°" section.

The superposed oscillations are of interest. These are initially
of a frequency depending upon and decreasing with the number of
sections, 2, but in all sections ultimately attaining the frequency

which is the critical or cut-off frequency of the line, above which
steady-state currents are attenuated during transmission and below
which they are unattenuated. When vt is large compared with
the amplitude of these oscillations becomes ./1/zvt so that they
ultimately die away and the current approaches the value 1/C/L
for all sections. The current in the loaded line is thus asymptotic
to the current in the corresponding smooth line and oscillates about
it with diminishing amplitude and increasing frequency.

Since the abscissas of these curves represent values of 2v¢=2t/1/LC,
and the ordinates are to be multiplied by C/L to translate into
actual values, the curves are of universal application for all values
of the constants L and C.

The investigation of the building-up of alternating currents in wave
filters and loaded lines is very important. It depends for the non-
dissipative case on the properties of the definite integrals

ELECTRIC CIRCUIT THEOR) 93
where w=w/w, and w=2z times the applied frequency. The mathe-

matical discussion is, however, quite complicated and will not be
entered into here. The reader, who wishes to follow this further,
is referred to Transient Oscillations, Trans. A. I. E. E.. 1914 and
Transient Oscillations in Electric Wave Filters, B.S. T. J., July, 1923.

The Bessel Functions of the first kind, J..(x) and J,,(x). are detined,
when # is zero or a positive integer, by the absolutely convergent
series

+546 (In +
2.4.6.(2n+ 2) (2n+4)(2n+6)
In the following discussion of the properties of these functions it will
be assumed that the argument x is a pure real quantity.
For large values of the argument (x large compared with 7), the
behavior of the functions is shown by the asymptotic expansions: —

When the argument ts less than the order (0Sx<n) the function
is very small and positive, and is initially zero (except when a=0).
In the neighborhood of x=”, the function begins to build up and
reaches a maximum a little beyond the point x=. Thereafter the
function oscillates with increasing frequency and diminishing ampli-
tude, and ultimately behaves as

This approximate formula is valid only where x>n, its accuracy
increasing with x and with w. For all orders of m it is quite accurate

BUNDANT evidence of many kinds exists to show that each and
every distinct sort of atom is especially adapted to possess
energy, notin any random quantity whatsoever, but in certain peculiar,
detinite, characteristic amounts. An atom having energy in one of
these particular amounts apparently cannot add arbitrary quantities
to its store, nor yield up arbitrary quantities from it; whenever the
atom receives or whenever it gives energy, it receives or gives only
just so much as is exactly sufficient to raise or reduce its) supply
to some one among the others of these distinctive quotas. For
each of the chemical elements there is a great system of these dis-
tinctive energy-values. They are determined chiefly by analyzing
spectra, and for most of the elements—the exceptions being those
of which the spectra are excessively complicated——many of them
have been evaluated very accurately and set down in tables.
The system of distinctive energy-values for any element is a very
important feature of that element; perhaps, indeed, the most im-
portant feature of all.

It is customary to say that when an atom acquires or surrenders
energy, it passes from one into another state; the various states cor-
responding to its various distinctive energy-values are called its
“Stationary States.”’ This is a name which suggests, and is doubtless
meant to suggest, that the energy-value of the atom is but one among
many of its features, all of which change when the energy-value
changes. This is a legitimate idea; theorizing about the atom consists
in speculating about just such features. But the reader will go far
and grievously astray if he lets the name signify to him that many
of them are directly and definitely known. In some few cases there is
good reason to believe that we know the magnetic moment of an
atom in its normal state. Beyond these the energy-values are all
that are known. If the reader chooses everywhere to replace “‘Sta-
tionary State’ by “energy-value’” he will be holding fast to the
physical reality, to the one thing not liable to be compromised by the
future trends of thought.

An atom may pass from one Stationary State to another because
of colliding with an electron or another atom of the same or a different

kind; or by absorbing radiation it may pass from one Stationary
State to another of higher energy; or it may pass spontaneously

the
difference \U between the energy-values of the initial Stationary State
and the final one by the equation

The same equation governs the last case but one. in which it con-
nects the frequency of the absorbed radiation with the enerev-ditfer-
ence between the two Stationary States from and into which the atom
passes. On this equation is founded the method of analyzing spectra
which is the most accurate and most widely applicable method. of
determining the energy-values of Stationarv States. The other
ways in which atoms are caused to pass from one State to another
lead to methods of determining these states, which are almost useless
for accurate measurements, but invaluable as controls.

The energy-values of the various Stationary States of an atom
are interrelated, and sometimes it is possible to express a long sequence
of them by means of a simple or a not very complicated formula.
There are also interrelations between the distinctive enerev-values
for different elements; and this statement is meant to apply also to
atoms from which electrons, one or more, have been detached, which
should be considered as distinct though not as stable elements. There
are unmistakable numerical relations among the Stationarv States
which come into being when atoms are subjected to electric or to
magnetic fields. Finally there is the important principle that the
spontaneous transitions between various pairs of states, which re-
sult in spectrum lines, do not occur equally often; and vet the rela-
tive oftenness or seldomness of their occurrence is itself regulated
by laws. One finds instances in which transitions from a state v4
to a state B, are just twice as common as transitions from A to a state
By close to By. One finds instances in which transitions from a state
A toa state B do not occur at all under usual conditions and an atom
in state A cannot get into state B without touching at some state C
from which A and B are both accessible. It is possible so to arrange
the Stationary States of an atom that by looking at the situations
of any two states in the arrangement, one can tell immediately whether
direct transitions between them do occur or do not: and this arrange-
ment is found to be suited to, even to be demanded by the numerical
interrelations to which I alluded above. Upon these facts the classi-

fications of the Stationary States are founded, and the notations |
which they are named.

The atom-model to which this article is devoted, the atom-medel « |
Rutherford and Bohr, is designed to interpret these facts of t]

terpret. certain experiments —chiefly, though not altogether, exper
ments on the deflections suffered by minute flying charged particle
when they pass through matter — which indicate that an atom consist-
of a positively-charged nucleus with a congeries of electrons around it
Specifically, the results of thesé experiments agree with the notic:
that the Vth element of the Periodic Table consists of a nucleus wit!
positive charge .Ve and .V electrons surrounding it; and this is the
simplest and most satisfying notion with which they do agree. Yer
there is something paradoxical about this atom-medel; for electrons
could neither stand still nor vet revolve permanently in orbits around
a nucleus, if they conformed to the laws of electrostatics. Also
there must needs be something paradoxical about any attempt to
interpret the Stationary States by this model, for there is nothing
inherent in it to make any energy-value preferable to any other
Under these circumstances Bohr's procedure was, resolutely to accept
both paradoxes at once, and to say that the electrons can revolve
permanently in those and just those particular orbits, whereby the
energy of the atom assumes the particular values which are those of
the observed Stationary States. This is easy to sav; but it is not
important, unless one succeeds in showing that those and only those
particular orbits are set apart from all others by some peculiar feature,
are distinguished by conforming to some particular principle, which can
be exalted into a “Law of Nature’ to complement or supersede the
laws of electrostatics. Otherwise the atom-medel would be of no value
Thus in order to make the test of the atom-model, it is necessary
to trace these orbits. One is confronted with this problem of orbit-
tracing: Given: the observed energy-values of the Stationary States;
required: to trace the orbits such that, when the electrons travel in
them, the energy of the atom has these observed values. If this
problem cannot be solved, it is impossible to take the next and es-
sential step of ascertaining whether these particular orbits are dis-
tinguished in any particular way from all the other conceivable ones.
In the case of a single electron revolving about a nucleus, this
problem is sometimes soluble. Tf the mass of the electron is regarded
as invariable, and no outside influences are supposed to act upon the
atom, then the solution is comparatively easy to attain. It was
performed in the Second Part of this article. If an external magnetic

field is superposed, the problem is scarcely more ditheult: if an external
electric field is superp sed, itis ditheult but soluble provided ilwavs
that the mass of the electron be supposed invariable It the mass of
the electron Is supposed to vary with its =peed as the theory of rela
tivity. requires, and as certain experiments suggest, the problem
remains soluble provided that no outside influences act For all
these cases the orbits which vield the observed energy-values have
been traced; and certain features have been shown to be common
to all of these “permitted” orbits, and to no others, so that by these
features the permitted orbits are sel apart from all the rest Inve rsely,
anvone who is told what these features are, and who is sutticientls
adept in dynamics, can trace all the orbits which display them and
calculate the energy-values for these orbits and so predict the energy
values of all of the Stationary States of an atom consisting of a nucleus

and a single electron. Such orbits are known as quantized orbits
The rules whereby they are set apart from all the multitude of orbits
not permitted are the Quantum Condtitons; which some one, it is to
he hoped, will some day succeed in deriving from a general Principle
of Quantization.

The most general way of phrasing these conditions is ditheult to
grasp, and the more intelligible ways are not the most general. The
most general conditions yer formulated are not adequate for all
cases; the completely adequate principle is vet to be discovered
kor the purposes of this summary and of most of what follows, a
very limited expression of the Quantum Conditions will be sufficient.
In the Second Part of this article it was proved that the permitted
orbits of an electron of invariable mass revolving in an inverse-square
field such as is supposed to surround a bare nucleus, are certain
ellipses. Tt was further stated, without proof, that if the electron ot
invariable mass revolves in a central tield which deviates slightly
from an inverse-square field, then the permitted orbits are certain
“rosettes” or precessing ellipses each orbit may be traced by im
agining an ellipse revolving steadily in its own plane around the source
cf the central field (IT will sav the nucleus) at one of its foci. All
the orbits are resettes; the permitted orbits are certain rosettes which
are distinguished from the others by a distinctive feature. One ways
of expressing this feature involves the angular momentum pp, and
the radial momentum p, of the electron. In terms of the mass
of the electron, its distance ¢ from the nucleus, and its angular velocity

4
SOME CONTEMPORA
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icles
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The “principle of quantization” is, that the permitted orbits ar
marked out from all others in that they fulfil these conditions, whic
are the Quantum Conditions:

In these equations each integral is taken around one complete cycl'
of the corresponding variable; # stands for Planck's constant, and
n and k take the values of all positive integers, k never surpassing 1

There is an alternative way of phrasing these quantum-conditions,
which is much easier to visualize; but it emphasizes what are probably
accidental features of the permitted rosettes, rather than fundamental
ones. The rosettes are, as I have said, precessing ellipses; the major
axes 2a and the minor axes 2) of these ellipses are, for the permitted
rosettes

in which # and k& take as before the values of all positive integers,
k not surpassing 7.

Exactly the same principle governs the permitted orbits of | an
electron revolving in a perfect inverse-square central field, but varying
in mass when its speed varies, as the theory of relativity requires
In this case also the orbits are rosettes, and the permitted orbits
are particular rosettes set apart from all the others in that they fulfil
(2) and (3), therefore automatically (4) and (5). The energy-values
of these permitted orbits agree closely with those of the observed
Stationary States of hydrogen and of ionized helium, the atoms of
which are the only atoms believed to consist of a nucleus and one
electron. Inversely, the orbits required to interpret the observed
Stationary States are set apart from all the other conceivable orbits
by the features expressed by (2) and (8), and by (4) and (5). On
these close numerical agreements for hydrogen and ionized helium,
and on other numerical agreements for the same atoms arising when
external fields are applied, the prestige of Bohr’s atom-model is
founded.

The integers 7 and k, the total quantum number and the azimuthal
quantum number, are used as indices to symbolize the various Station-
ary States of hydrogen and ionized helium to which they correspond.
Thus the symbol ‘8,” stands for the Stationary State of either atom,

rosette, or precessing ellipse, for which m=3 and k=2. Orbits for
which k=” are circles; orbits for which <7 are
and the farther k falls below » the more eccentric (narrower) is the
ellipse, although its major axis is independent of k. [I have repeated
all of these statements about precessing ellipses and their quantum-
numbers, because a great part of the speculation about atoms pos-
sessed of more than one electron consists of persevering and obstinate
attempts to interpret their behavior by as nearly as possible the same
ideas.

It is essential to remember also that all the enerey-values of the
Stationary States are reckoned from the “State of the Lonized Atom,”
in which State the energy—i.e., the energy of a system composed of
one atom deprived of an electron, and one electron far away-—is
equated to zero.

N. INTRODUCTION TO THE SPECULATIONS ABOUT ATOMS
WITH MORE THAN ONE ELECTRON

All atoms, except those of hydrogen and ionized helium, possess
more than one electron. There is much evidence of various kinds
for this assertion; and certainly the spectra of these other atoms
cannot be interpreted as those of the first two have been. Thus we
are confronted with the problem of a system composed of a nucleus
and more than one electron. The similarities between the spectra of
hydrogen and ionized helium, and those of other elements, are im-
portant enough to make it desirable to use the same sort of explana-
tion. We imagine the various electrons, when there are two or more,
each to describe certain permitted orbits, set apart from the multi-
tude of other conceivable orbits by peculiar features expressible by a
Principle of Quantization.

Here at the outset we meet with the great hindrance to success
in this problem. It is not possible to determine what features are
common to permitted orbits, for it is not possible even to trace the
permitted orbits. The general problem of tracing the paths of three
or more bodies, attracting or repelling one another according to the
inverse-square law, remains unsolved. Considering that for cen-
turies the related but simpler problem of celestial mechanics has been
under continual and powerful attack, the general problem may fairly
confidently be regarded as insoluble. There is very little hope of ever
dominating it to such an extent, that the spectra of atoms with two
or more electrons can be interpreted exactly by Bohr’s atom-model,

or can be used as strong support to that theory. Tf those two spectra
of hydrogen and ionized helium were unknown, it is unlikely that
the atom-medel would ever have been suggested; it is more than
unlikely that the atom-model could ever have been regarded as
satisfactory. To this day the prestige of the atom-model results
almost entirely from its achievement with those two spectra.

Why then trouble with applying it to the interpretation of other
spectra? Several good reasons can be given. For instance, it may
be that a svstem of several electrons about a nucleus acts in some
respects as a unit that its motion can be considered in some ways
as the motion of a rigid body, that principles of quantization can be
found for the system as a whole, similar to the principles used for
quantizing the smaller and vet perhaps not more consolidated system
which a single electron is. Here and there in the discussion we shall
find indications that this way of thinking is suitable.

Again, one is justified in arguing that if in simple cases a certain
law is proved, and if in complex cases neither that nor any other law
can be proved nor disproved, than we should assume that the law
proved for the simple cases extends over the complex ones. Few
events in this world take place under such conditions that conserva-
tion of energy can be proved to prevail during them; vet, from the
fact that conservation of energy has been verified in whatever events
it has been tested, we do not hesitate to infer that it prevails in all.
Bohr’s model having been so strongly fortified by the data for the
only two atoms for which it can be completely tested, why not assume
it for the others?

And tinally, there is the point that many of the data of experiment
are almost universally expressed in terms of the model, so that the
physical literature of today is almost) incomprehensible without
some knowledge of it. Unfortunate as this is, it shows that the model
is a Valuable aid for visualizing the facts. This justifies any model;
but must not be construed as evidence for it.

It will be expedient to divide the subject substantially under these
following headings.

(a) Lhe IHelium Atom. This, as the case of an atom composed
presumably of a nucleus and two electrons, comes nearest to being
amenable to calculation. Certain) mechanically possible orbits of
the two electrons, possessing the peculiar features of the “permitted”
orbits of a single electron revolving around a nucleus, have been
traced and their energy-values calculated. Not one of them has
given the observed energy-value of a Stationary State of the helium
atom. It is the consensus of opinion that whatever the features

which distinguish the permitted orbits may be. they are not those
which prevail in the hydrogen atom.

(b) Alkali- Metal Atoms. Vor these there is reason to believe that
one electron is normally located far bevond all the others, and may
be supposed to revolve around a “residue” consisting of all the others
and the nucleus. At a ereatl distance, the held due to this residue

of charge +e,—a hydrogen nucleus; for. from a great distance, the
nucleus and the electrons of the residue will seem almost to coincide
in place. Nearer in, the forces due to the electrons of the residue
may be suppesed to compound with that due to the nucleus in suc!
a way that a central field, not varving as the inverse square, results
Thus rosette orbits may be expected (for this reason | quoted the
principle of quantization for such orbits in Section M). An enormous
amount of effort has been spent in constructing central tields, such
that the rosette orbits obeving the quantum-conditions (2) and (3
have nearly the energy-values which the Stationary States of thes
atoms are known to possess. Always, the emission of a spectrum
line is supposed to result from a transition of the outermost or valence
electron from one orbit to another, the electrons of the residue being
scarcely or not at all affected. Such is the general explanation for the
far-reaching and yet imperfect resemblance of the spectra of these
metals to that of hydrogen.

(c) Other Elements. As one passes across the Periodic Table
from left to right along any row, the spectra rapidly lose resemblance
to the hydrogen spectrum. This is taken to mean that the assump
tion used for alkali metals -the assumption that one electron lies far
beyond the others, and executes transitions while the others remain
unaffected — departs progressively further from the truth. Evidence
exists that simultaneous transitions of two electrons occur, and vers
likely yet more drastic rearrangements taking place ex blo:

(d) Building of Atoms by Consecutive ‘Binding’ of Electrons. An
atom composed, when complete, of V3 electrons arranged about t
nucleus bearing the charge 4+-Ze, may have been formed originally in
Z stages by the consecutive advent of Z electrons, the first annexing
itself to the bare nucleus, the second joining itself to the svstem com
posed of the nucleus and the first, and so on until as many have
arrived as the nucleus is able to hold. Each of these stages should
be accompanied by the emission of lines belonging to a particular
spectrum; the ordinary hydrogen spectrum accompanies the forma
tion of a hydrogen atom by the step-by-step binding of an electron
to a nucleus of charge e, the ionized-helium spectrum accompanies

the joining of the first electron to a nucleus of charge +2e, the neutral
helium spectrum the adhesion of the second electron to such a nucleus
Spectra corresponding to the latest four or five stages, in the forma
tion of atoms having many electrons when completed, have been
observed. To a certain extent, but not entirely, an atom with Z
electrons and a nuclear charge Z resembles an atom with Z electrons
and a nuclear charge Z+1. Toa certain extent, therefore, each atom
in the Periodic System may be regarded as resembling the last stage
but one in the formation of the next following atom. This fact is
important in the interpretations of the Periodic Table.

(e) We next take account of the fact that the sequences
of Stationary States, mentioned in the elementary theory and descrip-
tion of spectra, are actually sequences of groups of Stationary States;
and inquire what may be supposed to differentiate the several states
of a group from one another. An elaborate formal theory is based
on the assumption that all of the electrons of what I have called the
“residue” of the atom revolve, if not literally as a rigid block, at least
with a resultant angular momentum which itself is quantized; and
that the outermost electron revolves in its own orbit around. this
residue, the different Stationary States of the group differing from
one another in respect of the inclination of the orbit to the axis of
rotation of the residue. The theory is not quite coherent with what
has gone before; and for that reason the reader should try to separate
its essential qualities from its accidental ones.

(t) dagnetic Properties of Atoms. A magnetic field should treat
a system of electrons revolving around a nucleus in the same way as
it treats one electron, as was said in the Second Part of this article.
One would expect that in this case, if in any, the behavior of complex
atoms would resemble that of the hydrogen atom; yet there is a
striking and inexplicable contrast. This, like the spectrum of the
helium atom, shows that either the quantum conditions governing the
hydrogen atom are not universal, or the expressions hitherto found
for the quantum conditions are too limited. From the responses of
atoms to magnetic fields something is learned about the magnetic
properties of atoms and their residues, some part of which can be
tested by direct experiment; and these experiments include what are
probably, all things considered, the most perplexing and fascinating
ones of recent years.

(g) Interpretation of X-ray Spectra. X-ray spectra are analyzed
as other spectra are, and each absorption and each emission of an
X-ray by an atom is associated with a transition between two Station-
ary States; these ‘X-ray Stationary States’’ however are distin-

guished from the others, by the circumstance that every one of them
involves the absence of an electron from the atom; consequently they
may be described as Stationary States of an atom-residue. There is
reason to believe that each distinct State involves the absence of a
particular one, or of one out of a particular group, of the electrons
bound to the nucleus during the earlier stages of the imagined building
of the atom by successive “binding” of electrons. The speculations
about X-ray spectra consist largely in attempts to correlate the
individual States with absences of particular electrons.

helium atom” as van Vleck calls it, is one of the most tantalizing in
contemporary physics. One feels confident a priori that the same
quantum conditions as suffice so beautifully to constrain the one;
electron atom to vield the hydrogen spectrum should also. suffice,
when applied to the orbits of two electrons, to vield the spectrum of
neutral helium. Yet the various pairs of orbits conforming to these
quantum conditions, which have already been traced, have been
shown (with vast expenditure of intellectual labor by some of the
ablest mathematical physicists of our time) to entail energy-values
for the Stationary States which are hopelessly incorrect.

For instance, one might assume that when the helium atom is in
its Normal State, the two electrons are revolving in a common circular
orbit about the nucleus, being at each instant located at opposite
ends of a diameter; and that this permitted orbit is determined by
the condition that the angular momentum of each electron, or per-
haps that of both together, is h 27. This seems an obvious general-
ization of the Quantum Conditions for hydrogen; but it vields a
false energy-value for the Normal State; and there is nothing more
to be said. Kemble and van Vleck demonstrated that no arrange-
ment in which the two electrons are symmetrically placed relatively
to a line through the nucleus entails the proper energy-value for
the normal state. This still leaves open the possibility that the two
electrons are unsymmetrically placed-——a possibility which to some
people seems repellent enough to be excluded. Born and Heisen-
berg calculated the energy-values corresponding to pairs of orbits
one of which lies far beyond the other at all points, and both of which
are concordant with the Quantum Conditions. These ought to
have agreed with the energy-values of the Stationary States which
are remote from the normal state and near the state of the ionized

atom; but they did not. This result is commonly regarded as the
strongest evidence for the belief that the Quantum Conditions valid
for an atom with one electron are not valid for an atom with two.

The atom-medel favored by Kramers, and hence presumptively by
Bohr, to represent a helium atom in its normal state, involves two
electrons moving in orbits which are not coplanar nor even plane.
Planes tangent to the two surfaces upon which the orbits are traced
intersect each other at 120° along a line passing through the nucleus,
and the electrons pass simultaneously across this line at opposite
crossing-points. “These orbits conform to the Quantum Conditions;
and the resultant of the angular momenta of the two electrons, which
is the angular momentum of the entire atom, is equal to # 27. This
atom-model likewise fails to have the right energy-value for the
normal state.

The alkali metals (lithium, sodium, potassium, rubidium and
caesium) are elements of which the atoms are easily deprived of a
single electron apiece; one electron of each atom is, as the phrase
goes, exceptionally “loosely bound.” Many facts combine to indi-
cate this; for instance, each of these elements enters with violence
into chemical combinations, and the compounds which each forms are
such as to suggest that its atom yields up one electron to the atom or
atoms which join with it. Again, when a salt of one of these metals
is dissolved, the molecules split up and the atoms of the metal are left
wandering around in the solvent minus one electron apiece, while
the atoms of the other element each hold on to one captured electron.
More definite vet is the direct evidence that the ionizing potentials
of the alkali metals are lower than those of any other elements in the
same rows of the Periodic Table, those of rubidium and caesium being
altogether the smallest known. These alkali metals follow, in the
Periodic Table, immediately after the five noble gases helium, neon,
argon, xenon and krypton respectively. These gases are chemically
all but absolutely inert, almost never entering into combinations.
Their ionizing-potentials are higher than those of any other elements
in their respective rows of the periodic table, and those of helium
and neon are the greatest known. The atoms of each of the alkali
metals are much larger than those of the preceding inert gas.

From all these facts the inference is drawn, that the atom of each
inert gas consists of a nucleus and electrons, at least the outermost

ones of which are arranged in a peculiarly stable and symmetrical
fashion (as for instance, in a group of eight at the corners of a cube,
though this is by no means sure); while the atom of the next following
alkali metal consists of just this sort of arrangement or “‘inert-gas
shell.’ now to be known as the “residue” or “kernel,” and of one
additional electron now to be known as the “valence-electron,”” usually
much farther away from the nucleus

If to such an atom-model we apply the doctrine of Stationary
States, we mav infer that for each and every arrangement of the
electrons in the residue, or (to use more general terms) for each condi-
tion of the residue, there is a whole svstem of Stationary States ditter-
ing from one another only in that the valence-electron travels in
different ones among a svstem of quantized orbits. These orbits we
may suppose to conform to the quantum-conditions (2) and (3), at
least until convincing evidence is brought to the contrary. Such in
fact is the interpretation of the svstem of Stationary States, transi
tions between pairs of which are responsible tor the “optical spectrum”
of each alkali metal.

An electron at a very great distance from the kernel of such an
atom will experience an attraction towards it, practically indis-
tinguishable from the attraction which would be exerted by a single
(hydrogen) nucleus of charge +e. One might sav that the (Z—1
electrons surrounding the nucleus of charge + Ze effectively cancel
a portion +(Z—1)e of the nuclear charge; or to use a more common
word, that they “screen” it. As the imagined distant electron moves
inward towards the kernel, the screening will cease to be pertect
An effect should occur analogous to the “stray tield” which penetrates
the meshes of a grid; since the electrons of the kernel do not form a
continuous shell of electricity enclosing the nucleus, the latter should
make itself felt through the interstices, although this effeet may be
diminished by the swift motion of the electrons. All this ts specula-
tion of the wildest kind. The only deduction reasonably sate is this,
that very far outside the kernel the tield will be very nearly the inverse-
square field due to a hydrogen nucleus of charge +e; very near to the
kernel the field will be quite incalculable '; while in between the very
far-out and the very-near-in region, there will be an intermediate
region, in which there may be some chance of finding an adequately
approximate expression for the field. On the existence of such a

Unless it is violently simplified by some agency or restriction of which at present
we know nothing.

rests all the present hope of achieving numerically valid theories in
this division of atomic physics.

One agreement between this theory and certain data may be demon-
strated without making any specific approximation. The farther
away the valence-electron remains from the kernel, the more nearly
identical with the field of a hydrogen nucleus is the field in which it
revolves, the more nearly should it behave like the electron of a hydro-
gen atom. Consider for instance, in a hydrogen atom, the orbit which
vields the Stationary State for which the total quantum-number and
k the azimuthal quantum-number are both equal to 5. This orbit
is a circle of which the radius is 10-7 cm.; far larger than the radius of
any inert-gas atom, presumably a fortiort far larger than the kernel
of any alkali-metal atom. Were the valence-electron of such an
atom to describe this circle, it would pass everywhere in a field very
nearly like that of a hydrogen nucleus, and should very nearly con-
form to the quantum conditions for this field. It follows that an
orbit drawn in the actual field, obeying the quantum-conditions
n=5 and k=5, would be very nearly such a circle with very nearly
the same energy-value. The inference is drawn that for high values
of n and k, the Stationary States of an alkali-metal atom should be
very nearly identical with those of hydrogen. These orbits which lie
far out from the kernel of the alkali metal atom, or from the nucleus
of the hydrogen atom, have small energy-values. It may therefore
be said that if we tabulate the Stationary States of the two atoms in
order of decreasing energy-value, then the farther along the two
tabulations we go, the more nearly should the two systems of Station-
ary States coincide.

This is found to be true, under a limitation. The limitation is an
important aid in interpreting the arrangement of the Stationary States.
It will be recalled from the First Part of this article that the Station-
ary States of the sodium atom are arranged in several sequences (there
illustrated as columns in Fig. 7) known as the s-sequence and the p-
sequence and the d-sequence and the f-sequence and others; and to
these sequences successive values 1,2,3,4.... of a symbol k were ap-
pended. One basis for this classification is that when it is made, the
occurrence or non-occurrence of transitions between any pair of
Stationary States, under normal conditions, can be determined by
applying the “selection-rule” that only such transitions occur as
involve a change of one unit in k. Now there are two reasons for
supposing that the only transitions which can occur are those in
which the Azimuthal Quantum-number of the valence-electron changes
by one unit. Unfortunately it is not possible to introduce these two

reasons with all the necessary background without too long a stoppage
of the main current of this argument.2 I must therefore set it down
as an assertion, that the selection-rule is deducible from the assump
tion that the value of & is the Azimuthal Quantum-number of the
valence-electron; which thus is 1 for all the Stationary States of the
s-sequence, 2 for each State belonging to the p-sequence, 3 for the
d-sequence, and 4 for the f-sequence. The feature common to the
various Stationary States of a sequence is, therefore, the Azimuthal
Quantum-number of the valence-electron—if this atom-model is valid

This being assured, the conclusion is drawn that, since & is higher
for the f-terms than for the d-terms, higher for the d-terms than
for the p-terms and higher for the p-terms than for the s-terms; since,
therefore, the f-orbits are ceteris paribus more nearly circular than
the d-orbits and less inclined to stretch down into the neighbor-
hood of the kernel, the d-orbits more nearly circular than the p-orbits
and the p-orbits more nearly circular than the s-orbits—therefore
the approximation of the sodium terms to the hydrogen terms will
be most nearly perfect for the f (and higher) sequences, less so for
the d, less for the p and worst for the s-terms. This also is veritied.
It reinforces the opinion that the k-values assigned to the various
sequences are actually their azimuthal quantum-numbers.

As the different States of a single sequence share a common Azimu-
thal Quantum-number, they must differ—supposing always that this
atom-model is valid—in their Total Quantum-number. Consecutive
States of a sequence presumably have consecutive values of the
Total Quantum-number (although sometimes one meets with a break
or a jolt in the continuity of a sequence, suggesting a departure from
this rule). The meanings of the Total Quantum-number 2 and of the
Azimuthal Quantum-number & for elliptical orbits are such, that »
can never be less than k. Hence the value of » for the first Station-
ary State of the s-sequence may be unity, or greater; but the values
of for the first terms of the p-sequence, the d-sequence and the f-
sequence may not be less than 2, 3, and 4, respectively.

Strange as it may seem, there is no perfectly satisfactory way of
determining the value of m for all Stationary States. Generally
it happens that the various States of an f-sequence, that of sodium
for example, agree so closely with those States of hydrogen which
form an my sequence, that there is little hesitation in attaching to
each of the f-States the same value of ” as is borne by that State
of the hydrogen atom which coincides with it so nearly. For instance,

? These being (verbum sapienti) the argument associated with the name of Rubin
wicz, and the argument deduced from the Principle of Correspondence.

the first f-State of sodium has very nearly the same energy-value
as the 4, State of hydrogen; the second f-State of sodium nearly
coincides with the 4, State of hydrogen, and so forth along the se-
quence. Hence to the successive States of the f-sequence of the
sodium atom one attaches with confidence the symbols 44, 54, 6;, and
so onward. In some cases this is practicable for the terms of the
d-sequence also; but never for those of the s-sequence. The Stationary
States of the s-sequence depart so far from those of hydrogen, that
one cannot with any security conclude what values of the Total
Quantum Number should be assigned to them. [It used to be assumed
that w#=1 for the first term of the s-sequence and n=2 for the first
term of the p-sequence, and the usual notation for the Stationary
States reflects this supposition; which however is neither necessary
nor probable.

All of the foregoing interpretations are based upon a theory of the
alkali-metal atoms which may be summarized in this way: as the
hydrogen atom is supposed to consist of a nucleus surrounded by an
inverse-square field through which an electron travels always in one
or another of certain orbits determined by quantum-conditions, so
also the alkali-metal atom is supposed to consist of a kernel sur-
rounded by a not-inverse-square field) through which electron
travels always in one or another of certain orbits determined by
identical quantum-conditions. As the Stationary States of the
hydrogen atom correspond each to a certain orbit and are designated
each by certain values of two quantities 7 and k, or for short by a
svmbol a, indicating the features of that orbit, so also the Stationary
States of the alkali-metal atom correspond each to a certain orbit and
are designated each by a symbol ay. For the hydrogen atom we recog-
nize the proper a, for each Stationary State because of the wonderful
numerical agreement between Bohr’s theory and the experimental
values for the energy of each State. For the alkali-metal atom we
can only guess the proper my for each Stationary States from = indi-
cations of much lesser evidential value. We suppose, however, that
k=1,2.34 for the various States of the s, p, d and f sequences, respec-
tively; so that the s-sequence is like the 7, sequence of hydrogen, the
p-sequence like the #2 sequence, and so on. Of the values of m we
are moderately sure for the f and d sequences, quite uncertain for
the terms of the s and p sequences.

One may now wonder whether it is possible to invent a central
field, such that the orbits traced in it according to the quantum-
conditions (2) and (3) would vield a series of energy-values agreeing
with the observed energy-values of the Stationary States of (let me

for the sodium atom involves ten electrons around the nucleus in
addition to the one ‘‘valence” electron for the benetit of which the
field is being devised; and one might expect these ten electrons to bx
rushing around the nucleus in uncoordinated and non-recurring paths
never at any two instants similarly placed and similarly moving,
never at any two instants exerting the same influence upon the valence

electron. Yet the Stationary States of the sodium atom are as
sharply defined as those of the hydrogen atom; and this may
thought to mean that the ten electrons of the kernel are constrained
to a unity and a fixed relationship, like that of the members of a
machine if not like that of the parts of a rigid body, which translates
itself into an influence upon the valence-electron not unlike that of a
central field.

At all events, several physicists working independently in) various
nations have taken the not inconsiderable trouble of devising central
fields to fulfil the condition required; and they appear to have ac
chieved a respectable success. It is not easy to decide what this
success requires the rest of us to believe; perhaps it is formally possibl
to devise a central field to account for amy set of Stationary States:
1am not sure whether this question has been adequately examined
Some have felt confident enough to sav that the results show which of
the Stationary States correspond to orbits of the valence-electron
which “penetrate into the kernel” and which to orbits that remain
in all their circuit quite outside of the kernel. It is to be hoped that
this problem will become clearer in the next few vears. At this point
I will add only, that the orbits traced for the valence-electron are
rosette orbits in which the precession ts very rapid, so that consecutive
loops of a rosette are inclined at a considerable angle to one another
In the model for the hydrogen atom, the consecutive loops of a rosette
orbit lie so close together as to be indistinguishable when drawn to
scale on an ordinary sheet of paper (the separation between them
was much exaggerated in Fig. 3 of the Second Part of this artick
In these atom-models, the orbit looks rather as if it were drawn along
the edges of the blades of an electric fan.

©. INTERPRETATION OF THE OPTICAL SPECTRA
OF OTHER ELEMENTS
As soon as we step from the first column of the Periodic Table into
the second, the obstacles to such a theory as we have hitherto tried

which seems to bear upon the arrangement of the electrons in |},
atoms; but some of it leads to conclusions opposite to those which ||,
remainder suggests.

On the one hand, line-series are discernible in the spectra of cle.
ments in the second and the third columns of the Table, and even
in those of some others; and from these line-series, svstems of Station-
ary States are deduced which resemble those ascertained for the alkali-
metal atoms; and it is natural to extend the same explanation from
that case to these, supposing again that each atom consists of a
nucleus and a certain number of electrons, all but one of which are
tightly bound into a residue, around which the one remaining electron
circulates in one or another of various quantized orbits.

On the other hand, the chemical behavior of these elements does
not confirm this easy classification of the NV electrons of an atom into
(V—1) very-tightly-bound electrons and one which is very loosely
bound. Thus, the atoms of elements of the second and third columns
of the Periodic Table—‘‘alkaline-earth metals” and ‘earth metals,”
as they are called—when floating in water as the fragments of mole-
cules of dissolved salts of these elements, are found to be deprived
of two and of three electrons, respectively; and the composition
of these salts is such as to suggest that the atoms of the other element
or elements involved in them have annexed two or three electrons,
respectively, from the alkaline-earth atom or from the earth-metal
atom. These facts suggest rather that the .V electrons of an alkaline-
earth atom, or of an earth-metal atom, should be classified into (V—2
or (V—8) very-tightly-bound electrons and two or three which are
loosely-bound, respectively. The very tightly bound electrons will be
equal in number to, and presumably arranged like, the electrons of the
atom of the next preceding inert gas. Henceforth I will reserve the
word ‘‘kernel’’ for such a system, and the word “residue” for what
is left behind when one electron is separated in fact or in imagination
from the atom. Thus these two words will not mean the same thing
except in special cases, such as those of the alkali-metal atoms.

Specifically, let us consider the four consecutive elements argon
(inert gas, 18th element of the Periodic Table), potassium (alkali
metal, 19th element), calcium (alkaline-earth metal, 20th element
and scandium (earth-metal, 21st element).

The evidence from chemistry and from electrolysis impels us to
think that the argon atom consists of a nucleus surrounded by (eigh-
teen) electrons tightly bound, in a stable and imperturbable arrange-
ment; that the potassium atom consists of a kernel much like the
argon atom, with one additional electron loosely bound and hence

of kernel and two loosely-bound electrons, t
same sort of kernel and three outer electrons.

The Stationary States of the calcium atom resemble, in their
arrangement, those of the potassium atom sufficiently to make the
same general sort of an explanation desirable,—to make it desirable
to suppose that one electron is loosely-bound and remote from the
nucleus, the other nineteen tightly-bound and near the nucleus; one
loosely-held electron versus nineteen tightly-held ones. But the
evidence from chemistry and electrolysis demands two loosely-held
electrons versus eighteen tightly-held ones.

One might try to evade the dilemma by supposing that the calcium
atom is a sort of three-stage construction, with eighteen electrons
congregated in a kernel around the nucleus, a nineteenth far out bv
comparison with the nucleus, a twentieth far out by comparison with
the nineteenth. For interpreting spectra, the residue of the atom
would be the kernel or “inert-gas shell” and the nineteenth electron,
the valence-electron would be the twentieth. For interpreting chemi-
cal data, the residue of the atom would be the inert-gas shell. This
conception would rescue the interpretation of the calcium spectrum
made after the fashion of the one just expounded for alkali-metal atoms.
It would probably demand a larger atom, or a more shrunken kernel,
than other data will allow.

Or one might suppose that the nineteenth and the twentieth elec-
tron are on the whole about equally remote from the nucleus, and vet
it is possible for one of them to change over between any two of a
vast system of quantized orbits without greatly affecting the other.
There is certain evidence for this conception which I shall presently
narrate.

Or one might suppose that the nineteenth and the twentieth electron
are a system by themselves, and that each Stationary State corresponds
to a particular configuration of this svstem, so that each line of the
spectrum is attributed to a leap not of either electron separately but
of both together. This idea seems to be gaining ground rapidly in
dealing with atoms composed of a kernel and several outer electrons,

three or four or five or six or seven. The preceding notion might ty
brought under it as an especial case. If it is accepted the theory o
atoms other than the alkali-metal atoms will inevitably be mor
complex than the theory mentioned for these in section P.

An interesting feature of some of these spectra discloses that the
residue of the atom may exist in either of two distinct states. It will
be recalled that the energy-values of the Stationary States have been
measured from the state of the ionized atom, to which the energy
value zero is assigned. In this fundamental state, one electron and
the residue of the atom are completely sundered; and the energy-
value of any other Stationary State is the energy required to tear the
electron completely out of the atom when the latter is initially in that
Stationary State. This definition implies that the state attained when
the electron is completely separated from the rest of the atom is de-
terminate and unique. Such must be the case if the atom consists of
an invariable nucleus and one electron, as in hydrogen; but if the
atom contains several electrons, there is no a priori reason for ex-
cluding the possibility that there may be several ‘states of the ionized
atom”; in each of these states one electron will be far away, but the
residue will have as many different arrangements as there are different
states. Extending this idea, one infers that there may be two or
more distinct sets of Stationary States for certain elements, each set
culminating in a different final configuration of the residue,—that
is to say, of the ionized atom.

Several instances of atoms possessing two such distinet families of
Stationary States are known; the most noted is probably that of neon,
but I will describe the case of calcium, lately interpreted by Russell
and Saunders and independently by Wentzel. Two families of terms
“primed” and “unprimed,” had been identified in the spectrum of this
element, and important sequences of each could be followed — sufti-
ciently far to make the extrapolation to the limit not too daring. The
hmits were different, showing that the amount of energy required to
separate an electron from an atom initially in its normal state had
two values differing from one another by 1.72 equivalent volts. Con-
sequently the residue may remain (it is not necessary to assume that
it can long remain) in either of two States differing from one another
(when the extra electron is far away) by this amount.

At this point a very significant numerical agreement enters upon
the scene. The residue of the calcium atom, the ‘onized-calcitum atom,
has itself a spectrum which is known, and from which its system of
Stationary States has been learned and mapped. Like the systems of

pd and other sequences. The Normal State of the ionized-calcium
atom belongs to the s-sequence; following the usual custom it) may
he called the (1, s) State. The State of next lowest enerev-value,
the “next-to-normal”’ State (so to speak belongs to the d sequence,
and may be called the (3, d) State. The energy-ditference between
the (1, s) State and the (3, d) State is 1.69 volts. This agrees within
the error of the experiments with that value 1.72 equivalent volts,
which was found for the energy-ditterence between the two conditions,
in either of which the residue of the calcium atom might be left after
the twentieth electron is abstracted. This agreement shows that the
extraction of the 20th electron from a calcium atom mav leave the
residue either in the (1, s) State or in the (3, d) State.

If now we remember that the ionized-calcium atom is comparable
with the potassium atom (and with alkali-metal atoms generally
having as it does eighteen electrons very tightly bound as a kernel
around the nucleus and one electron loosely held then it is reason-
able to use the same interpretation of its Stationary States as was
expounded in Section P; and to suppose that when the ionized-cal-
cium atom is in the (1, s) State that loosely-held electron is revolving
in a certain 2, orbit, and when the atom is in the (3, d) State the
electron is revolving in a certain m3 orbit. Thus the extraction of
the 20th electron of the calcium atom may be supposed to leave the
19th electron sometimes in the one, sometimes in the other of these
two orbits.

We may now inquire whether the 19th electron will always remain
in its #2, orbit, or in its a, orbit as the case may be, when the 20th
electron reenters the atom, descending from one orbit to another.
Here it is necessary to watch one’s mental steps very closely; for one
is liable to slip into the naive notion of a particular orbit, say for
instance a 33 orbit, as a fixed and permanent railway-track around
which the electron continually runs until something violent derails
it. This could not be true unless (to take this special case) the 2 dth
electron had no influence whatever upon the I@th. Were it so, every
Stationary State of the one family would differ by the same amount.
1.69 equivalent volts, from the corresponding State of the other
family. In fact, the energy-difference between corresponding States
varies from one pair to another. This may well be simply because
the approach of the 20th electron so alters the forces acting upon
the 19th, that its orbit is changed both in geometry and in’ energy-
value, while remaining still identified with the same values of its
quantum-numbers. The experiments neither prove nor disprove
this; it is commonly accepted as true.

It is a very important fact that the atom may pass from a State o
one family to a State of the other,—in terms of the model, that thi
19th electron passes from its m3 orbit to its m orbit, and simultane
ously the 20th electron makes some transition or other of its own
The emitted radiation contains the energy resulting from both change-
simultaneously, fused together without any discrimination.

I next point out that the processes whereby the lines of an optical
spectrum are emitted may be regarded, if this theory of the atom is
valid, as stages in the gradual formation of an atom. Consider the
hydrogen spectrum to begin with; each line is emitted as the atom
passes from one Stationary State to another of lower energy-value,
the state of least energy being the Normal State of the perfected atom
and the state of greatest energy being the condition in which the
atom-residue and its electron are torn apart. The various lines of the
spectrum correspond to various partial steps along the path from the
latter of these states to the former, to various stages of the formation
of a hydrogen atom from two separated parts. The specific conception
of each Stationary State as a definite orbit of the electron about a
nucleus merely reinforces this way of envisaging the process. In the
spectra of ionized helium and of neutral helium, we read the testimony
of the gradual formation of a helium atom out of a nucleus and two
electrons initially quite dissevered. The various lines of the ionized-
helium spectrum correspond to different stages in the advance of an
electron from the state of freedom to the state of most stable asso-
ciation with a nucleus of charge 2e, or in Bohr’s language, to different
stages in the “binding” of an electron by a nucleus of charge 2e. The
various lines of the neutral-helium spectrum correspond to stages in
the “binding” of a second electron by a system composed of a nucleus
of charge + 2e and an electron already bound to it. Thus the two
spectra of helium testify to two consecutive processes in the upbuilding
of a helium atom out of its constituent parts.

The process of building up an atom, by successive adhesions of
electrons to an incomplete electron-system surrounding a nucleus
that is to say, the process of building a system of Z electrons around a
nucleus bearing the charge Ze, out of a system of (Z-b) electrons
surrounding the nucleus, by consecutively adding 6 electrons one
after the other—evidently occurs very profusely in intense high-
current high-voltage discharges in vapours, such as the condensed

park and above all Millikan’s “Vacuum Spark.” To take instances
from the work of Millikan and Bowen, Paschen, and Fowler: in the
.pectra of such discharges lines have been identitied which belong to
atoms for which Z=14 and b} has the several values 1.2.3.4 (four
stages in the building of a silicon atom); and to atoms for which
Z=10+6 and b has the several values 1,2,3,4,5,6. Many of these
spectra of multiply-ionized atoms have not yet been analyzed, but the
work is proceeding rapidly. There is reason to hope that within a
few years we shall be in possession of interpreted spectra not only of
many systems of Z electrons about a nucleus of charge Ze, but also of
many systems of fewer than Z electrons about nuclei of charge + Ze.
This may be highly important, as I will try to show by an illustra-
tion. We will consider two consecutive elements of the periodic table;
sodium (Z=11) and magnesium (Z = 12).

A Mg atom is imagined as 12 electrons around a nucleus of charge
+12 e. It is formed when one electron joins itself to a Mg+ion,
which is composed of Ll electrons about a nucleus of charge +12 e
For this process a spectrum is emitted, the so-called are spectrum of
Mg or “Mgl” spectrum, which is known and analyzed. [t shows that
the normal state of the Mg atom is an s-state (probably of total
quantum-number 3). It is likewise a singlet-and-triplet-spectrum.
The first of these facts is taken to mean that the valence-clectron, or
twelfth electron (the reader will see the reason for this usage, the electron
in question being the last annexed out of the twelve) of the Mg atom
moves in a 3, orbit. The second is taken to mean something or other
about the residue of the atom, as will be shown in section S.

This residue of the atom is itself formed when one electron joins
itself toa Mg+ +ion, which is a group of 10 electrons about a nucleus
of charge +12 e. In this process the so-called spark-spectrum of Mg,
or spectrum, is emitted. Tt is known and analyzed. [t shows
that the normal state of the Mg+ion is an s-state (probably of total
quantum-number 3). [It is a doublet spectrum. The first of these
facts is taken to mean that the valence-electron or eleventh electron of
the Mg+ion, moves in a 3, orbit. The second is taken to mean some-
thing or other about the residue of the Mg++ion.

A very interesting question now arises: is the Mg+ion actually the
same as the residue of the Mg atom?) In other words: when a 12th
electron is added to the group of Ll electrons about a nucleus of
charge+12 e, is the group of eleven left unchanged? Lf so, we have
knowledge about this group from two sources. The character of the
Mgl spectrum (the fact that it is a singlet-and-triplet) spectrum)
teaches something about the group, though what it is is far from

This suggests that it would be a most desirable achievement t
produce the spectra due to groups of (Z—6) electrons congregated
about a nucleus of charge +Ze, for some value of Z (the higher the
better) and all values of 6 from O to (Z~—1). Were this done we
could almost lay claim to having witnessed the creation of an atom
from fundamental particles common to all matter. We could not
quite make this claim, since the nucleus of charge +Ze would still
remain characteristic of that one kind of atom alone; but we should
have made a substantial approach to it. However, there is no im

the spectrum expected for Li++ (ie. for Z=3 and b=2) acts as a
barrier against utterly tearing down the electron-structures of higher
atoms so that they can rebuild themselves before our eves from the
foundations.

The next important question may be introduced in this fashion,
Suppese that nothing were known about the spectrum called Mell,
therefore nothing about the process of adding an eleventh electron
toa group of ten around a nucleus of charge 12 e. Knowledge would
still be available about the process of adding an eleventh electron to
a group of ten about a nucleus of charge 11 e; for this is precisely
the process which creates the neutral sodium atom out of the Na-+ion,
and results in the emission of the Nal spectrum or are spectrum of
sodium. ‘This spectrum is a doublet spectrum, and it shows that the
normal state of the sodium atom is an s-state, probably of total
quantum number 3. This last fact is taken to mean that the eleventh
electron in a group of eleven electrons about a nucleus of charge
+11 e, is revolving in a 3; orbit. Could we have assumed that there-
fore the eleventh electron, in a group of eleven electrons about a
nucleus of charge +12 e, is revolving in a 3; orbit? There is no a
priort certainty of this: but the observations on the MgII spectrum,
as we have seen, confirm it (and also that the residue of the Mg+ion
is like the residue of the Na atom, in causing the next added electron

Were this generally true we could say that each atom in the period
table is like the residue of the next atom following it; and that the
mth electron in the mth atom is revolving in the same sort of orbit
as the outermost electron of the mth atom, for every value of 7 an
for every value of m less than that value of 1.

However, it is not always true. To take another specitic instance,
consider the two elements potassium (Z=19) and calcium (Z = 20
The spectrum KI, which is due to a nineteenth electron joining a
group of 18 about a nucleus of charge +19 e, and the spectrum Call,
which is due to a nineteenth electron joining a group of IS about a
nucleus of charge +20 are dissimilar. The dissimilarity is not
quite so great as to affect the normal states of the two systems, Ko and
Ca+, composed of nuclei of charge 19 e and 20 e each surrounded by
19 electrons; both have as normal state an s-state, apparently of
total quantum-number 4; it is inferred that in each, the d?th electron
revolves in a7, orbit. If we consider, however, the first of the d-states
(to which the total quantum-number 3 is commonly assigned), we
see that in the KI spectrum it has a much larger energy-value than
the Normal State, while in the Call spectrum it has nearly the same
energy-value. A short leap of the imagination leads to the conclusion
that if we could examine the spectrum produced by a 19th electron
joining a group of 18 about a nucleus of charge +21 e, the d-state in
question would have a smaller energy-value than any s-state. In
this case it would be the Normal State itself,’ and we should say that
the 19th electron, in a group of 19 surrounding a nucleus of charge
Ze, revolves in a m, orbit if Z=19 or 20, but in a ny orbit if Z= 21.

This system of 19 electrons around a nucleus of charge 21 e@ is a
doubly-ionized scandium atom, Se++-. Its spectrum has not been
produced, so that the foregoing sentences are still somewhat specu
lative. What gives them value is the inference that scandium marks
a sort of a breach in the regularity of the Periodic System. For
most of the elements in the Periodic System, it can be said that the
atom consists of a residue which is like the atom of the preceding
element, and an additional electron; and that in its turn this atom
resembles the residue of the atom of the element next following. To
this the regular periodicity of the properties of the elements ts ascribed
But when we reach an element of which the atom has a residue dis
tinctly different from the atom of the foregoing element, then the
regular variation of the physical and chemical properties is inte
rupted. Scandium, as a matter of fact, is the first of a group of

In the First Part of this article the impression may have been left that the Normal
State of every atom is an s-state. This is not true; in some known cases the Norma!
State is a p-state, in others an f-state.

elements, the intrusion of which into the Periodic Table brings about
a disruption of the simplicity of its first three rows. There are other
such intrusive groups of elements, notably the celebrated groups of the
rare earths. [It is supposed that wherever such a group commences,
there the residue begins to vary from one atom to the next. The spec-
troscopic evidence is lacking; it is awaited with extreme interest,

The reader will very probably have seen one or more tables of the
distribution of electrons in atoms; tables in which it is stated, for
instance, that the atom of sodium contains two electrons moving in
1, orbits, four in 2; orbits, four in 2s orbits, and one in a 3, orbit;
or more succinctly that it contains “two 1,, four 2;, four 2. and one 3;
electron.” Such tables are built by piecing together bits of evidence,
some of which are such as I have described in this section, while
others are inferences from X-ray spectra, magnetic properties, or
observations of still other kinds. That they are still highly speculative
is confirmed by the fact that they are continually being remodeled.
If we could produce the spectra corresponding to all the stages of
formation of an atom, we should be able to set up a tabulation more
reliable than any vet put together. Even then, however, we should
be contronted with the question whether the addition of a new electron
toa kernel fundamentally alters the distribution of those already there.

Having considered the facts at such length in this section, we are
entitled to consider the theory. In the coupled cases of hydrogen and
ionized helium it was shown by experiment, and rendered plausible
by theory, that the Stationary States of the element with one electron
and a double charge on its nucleus correspond exactly to those of the
clement with one electron and a single charge on its nucleus, and are
endowed with fourfold the energy of these latter. This conclusion
can be extended to cover the case of a valence-electron circulating in
an orbit ata great distance from a kernel composed of (Z—6) electrons
and a nucleus bearing the charge +Ze. The field due to the kernel
will at great distances approximate the field due to a solitary nucleus
bearing the charge be. We have seen already that when b=1 (so
that the total charge on the nucleus balances the total charge of the
electrons, valence-electron included) the Stationary States corres-
ponding to orbits for which » and & are large coincide with Stationary
States of hydrogen. [t follows equally that when )=2, the Stationary
States for which » and & are large have approximately fourfold the
energy of stationary states of hydrogen, and coincide approximately
with Stationary States of ionized helium. This is verified by experi-
ment, and so are the corresponding conclusions for the cases 6=3 and
b= 4,

Heretofore in the Third Part of this article TE have repeated the
procedure adopted in the First Part, simplifving the actual facts by
writing as though the Stationary States of each atom were arranged
in sequences of individual terms, each sequence being distinguished
by a particular value of the Azimuthal Quantum Number Here as
there, it finally becomes necessary to concede the complexity of the
facts, and recognize that the sequences in question are sequences not
of individual terms, but of groups of terms. Thus for instance the
sodium atom possesses a p-sequence, not of individual terms but of
pairs of terms—a pair 2p; and 2p2, then a pair 3p; and 3p., then a
multitude of other pairs. For another instance, the mercury atom
exhibits a p-sequence not of individual terms but of triads of terms
a triad 2p; and 2p2 and 23, then a triad 3p; and 3p. and 3p,, and
then a procession of other triads. These sequences are collected
into systems: an s-sequence and a p-sequence and a d-sequence and
several more constitute a system. There are singlet: svstems and
doublet systems and triplet systems and systems of still higher »ru/si-
plicity; and each kind of system is distinguished by a certain manner
of grouping of the terms which form its various sequences. Note
worthy and peculiar laws govern these groupings; in a doublet svstem,
for instance, the s-sequence consists of individual terms, but all the
others consist of pairs of terms; in a quartet system, the s-sequence
is made up of single terms, the p-sequence of triads of terms, the
remaining sequences of groups of four terms each. From the First
Part of this article reprint a Table showing how the terms are grouped
in systems of all multiplicities from the singlet to the octet. The
numbers opposite the name of each system and under the letters of
the various sequences show how many terms belong to each group
in the various sequences of that system.

leach atom possesses one or more such systems of Stationary States; ‘
and the particular types which an element displays depend in a ia
detinite and fairly clear manner upon the position of the element in 4
the Periodic Table, being in fact one of the most distinctive of the q
periodically-varving qualities. Each atom with an even number of ?
electrons exhibits svstems which are all of odd multiplicity, and each j
atom with an odd number of electrons exhibits systems which are all 4
of even multiplicity; thus magnesium, with twelve electrons, has a |
singlet system and a triplet system, while sodium and once-ionized 4
magnesium, each with eleven electrons, have each a doublet system, 4
and neon with ten has a singlet, a quintet and two triplet systems.‘ :
Remembering what was said about the consecutive binding of elec- :
trons, it will be noticed that these facts show a regular difference be- :
tween the binding of the Vth electron when WV is odd and the binding ‘
of the Vth electron when NV is even. Otherwise expressed, they show 4
that a kernel of .V electrons treats an oncoming member in one or ‘
another of two distinctive ways, according as .V is even or odd. The q
influence of magnetic fields on spectra likewise shows that this com- :
plexity of the Stationary States is a quality not negligible, but primary. 4
The features of the atom-model hitherto described must be supple- ;
mented with some new one if it is to cope with such facts as these. :
We have represented (for example) the sodium atom in its 2p state :
by a “‘valence-electron” cruising with angular momentum 2(h/27) :
in.an orbit around a “kernel” composed of ten electrons and a nucleus. :
But there are two such states instead of one; if the angular momentum ;
of the valence-electron is to be equal to 2(4 27) for each of these, ‘
some other not yet mentioned feature of the atom must discriminate i &
the two. One might, of course, again proceed as we did in discussing :
the “primed terms,” by assuming that the kernel of the atom is in ;
one condition when the atom is in the 2p; state, and in another slightly i
different condition when the atom is in the 2p, state. This would :
probably entail as many different conditions of the kernel as there ;
are pairs of terms in the sodium spectrum —a great number, and yet
| small in comparison with the multitude which would be required by :
other atoms; vet such may be the eventual theory. However, it is :
possible to construct for these facts an atom-model out of two re- i

volving parts, whereby different Stationary States of a group are
represented not by varying the condition of either part separately,

Hydrogen and ionized helium are not included under this rule. Helium shows
a singlet and a doublet system together, a combination which violates the rule as 4
stated, unless the doublet system is really a triplet system in which two states of "
each triad are too close together to be distinguished.

but by varying the relative orientation of the two. Although this
theory has not been harmonized with those which [| have hitherto
recited, it is competent in its own field; and for that reason I present it
We will imagine that the atom is represented by a combination of
two flywheels, two whirling objects, endowed each with angular
momentum. These angular momenta are vectors, pointing along
the directions of the axes of rotation of the respective flywheels, and
having certain magnitudes. 1 will designate them temporarily as Py
and Pr, each symbol standing for a vector generally and also (when
in an equation) for its magnitude. The angular momentum of the
entire atom, which is necessarily constant in magnitude and in direec-
tion so long as the atom its left to itself, is the resultant of Py and
Pria vector, pointing along the direction of the so-called “invariable
axis’ of the atom. I designate it by Pa. The following equation
shows the relation between the magnitudes of these three angular
momenta and the angle 0 between the two ftirst-named, the angle
which describes the relative orientation of the axes of rotation of the

Remembering the successes which in dealing with the spectrum of
hvdrogen have resulted from assuming that the angular momentum
of the entire atom is constrained to take only such values as are
integer multiples Jh/2z of the quantity / 27, we make the same as-
sumption here. We further make the same assumption for each of
the flywheels separately; the magnitudes of the angular momenta P,
and Pr are supposed to take only such values Vh 27 and Rh 27 as
are integer multiples of the same quantity # 27.2 These particular
assumptions, frankly, are foredoomed to failure: but the failure will
be instructive.

Making all these assumptions together, we see that in effect we
have laid constraints upon the angle 0 which measures the relative
orientation of the two flywheels. For if Py is an integer multiple of
h/2n, and Pe is an integer multiple of # 27, then Py which is fully
determined by equation (6) cannot be an integer multiple of & 27
unless 0 is very specially adjusted. To illustrate this by an instance
(which will be clearer if the reader will work it out with arrows on a
sheet of paper): if Py and Pr are each equal to the fundamental
quantity A/2z7, and if P4 must itself be an integer multiple of 4 27:
then cos 0 must take only the values, +1, —3, —1, which vield the

* All that is actually being assumed is, that Py and Py, and Py are all integer

multiples of a common unit; nothing in this section will indicate either #2 or any
other value as the precise amount of that common unit.

values O, 27 3, 7 for 0, which vield the values 24/27, h/27, 0 for P,
Any other integer values for Py (h 27) are unattainable by any
value of Q whatsoever; anv value of 80 not among these three would
vield a value for Py not an integer multiple of 4/22, which is con-
trary to the assumptions. Thus, the assumptions that the atom
is a conjunction of two whirling parts, and that the atom altogether
and each of its two parts separately whirl with angular momenta
which are constrained to be integer multiples of a common factor
these assumptions lead to the conclusion that the relative inclination
of the two revolving parts is constrained to take one or another of a
strictly limited set of values.

This essentially is the model devised by Landé to account for the
complexity of the Stationary States. The several Stationary States
which form a group belonging to a sequence—-in other words, which
share a common value of 2 and a common value of k, like the 2p; and
states of sodium or the 8d), 38d, states of mercury—are supposed
to resemble one another in this, that each of the whirling parts sep-
arately has the same angular momentum in every case; and to differ
from one another in this, that in the several cases the two whirling
parts are differently inclined to one another, so that the angular
momentum of the entire atom differs from one state to the next.
The ditterent Stationary States which share common values of 1 and
k are supposed to correspond to different orientations of the two
parts of the atom and to different values of its angular momentum.

I will now no longer disguise the fact that these whirling parts are,
or at any rate have been, supposed to be precisely the valence-electron
and the residue. To the former we should therefore assign these values
for the angular momentum Py: the value #27 for every state be-
longing to an s-sequence, the value 2h 27 for every p-state, 3h, 27
for every d-state, and so on. Then to the angular momentum Px
of the residue we should assign a suitable constant value; a ‘‘suitable”
value in this case being such a one, as would yield the proper grouping
of terms in the various sequences of the system which the atom under
consideration is known to have. Thus, for an atom-model to represent
sodium with its doublet system we require a value for the angular
momentum of the residue, such as will vield one permitted orientation
when the atom is in an s-state (Pj; =f, 27), and two when it is in any
state for which Py=kh/27 and k is any integer greater than unity.

for, as was shown in the illustrative instance a couple of pages back,
it vields three permitted orientations when Py=h/27, and (as can
casily be sl ) three for each and every other value of Py whict
easily be shown) three for each and every other value of Py which

is an integer multiple of 4 27. Thus it would form an idequate
model for a system of Stationary States in which every group of
terms in every sequence was a triad; but this is not a doublet svstem.
nor even a triplet system, nor any other observed svstem whatever
To make this long story short; it is impossible to simulate anv of the
eight groupings of terms set forth in the eight lines of Table I by

assuming that Py, Pr and P4 are all integer multiples of A 27 (or
of any other common factor
It is in fact necessary to put Py equal, not to # 2r and to 2h 2x

sand p and d states, respectively. This use of “half quantum num-
bers” makes it possible to produce an adequate medel for an atom
possessed of a doublet svstem, by assuming that the angular momen
tum Px of its residue is always / 27, and that its two whirling parts
must always be so inclined to one another that the angular momentum
of the entire atom is an integer multiple of 4 27

kor (to work out one example. and one only) when we make
Pr=h 2r and Py=}(h 27), then the greatest possible resultant
that can be obtained by combining these vectorially is * (A 2x) and
the least possible one is } (h 27); these two extreme values being at
tained when the two component vectors are parallel and when they are
anti-parallel,” respectively. If we permit for the resultant only such
values as are integer multiples of # 27, then there is only one that is
permitted: the value hk, 2x7— for this is the only such value Iving within
the possible range. Next, put Pr=h/ 227 and Py =3 (h 27). All pos

A more drastic use of “half quantum numbers” is required to obtain
an adequate model for atoms showing singlet and triplet and other

7 The diagrams with arrows, offered by Sommerfeld in the fourth edition of hi
classic book, are very helpful in studying these models. Incidentally Sommerfeld’s
alternative way of arriving at the groupings of multiplet terms by compounding
vectors is instructive.

systems of odd multiplicity. Thus to produce a singlet system it is
necessary to put Pr=} (h 27) always; to set Py equal to } (h/ 27) for
all S-states, to} (h 27) for all P-states, and so forth; and to suppose
that the two whirling parts of the atom are constrained to take only
such relative orientations as vield values for P4, the angular momen-
tum of the entire atom, which are odd integer multiples of | (/ /27).
It is easy to see that there is but one such orientation for an s-state,
one for a p-state, and one for any other kind of state. To produce a
triplet, or a quintet, or a septet system, it is necessary to put Pr
or 3 (hk or (h/ 27), respectively; and to retain the
just-stated assumptions about Py and P4.

The question whether these models have any intrinsic truthfulness
has now become acute. If there is any doctrine in contemporary
atomic theory which appears to be multiply tested and approved, it is
surely the doctrine that the angular momentum of the valence-electron
is always an integer multiple of 4/27. Yet in this passage I have
spoken as if this principle had been indifferently and casually dis-
carded, and replaced by a new principle to the effect that the angular
momentum of the valence-electron is alwavs an odd-integer multiple
of } h 2x. It is hard to evade or mitigate this arrant contradiction.

A way out may possibly be found by suggesting that the partition
of an atom into “residue” and ‘“‘valence-electron,”” while appropriate
when calculating energy-values by the method mentioned in Section
P, is not appropriate in this instance; that the two whirling parts of
the atom are respectively a system composed of a part of the residue,
and a system composed of the rest of the residue and the valence-
electron. This seems most admissible for such an atom as magnesium,
consisting as it is supposed of what I have called a “kernel,” and two
additional electrons outside. The two whirling parts may be the kernel
rotating as a unit, and the pair of outer electrons also rotating as a
unit. It may be profitable to push the analysis even further, and to
consider the two outer electrons each as an entity possessed of angular
momentum, their two angular momenta combining with one another
in such a fashion as | have lately described for the two parts of the
atom; this resultant angular momentum of the two may then figure
as the Py emploved in constructing the atom-model. There are
decided possibilities in this way of thinking; but it is doubtful whether
the dithculty about half-quantum-numbers can ever quite be removed.®

* An unfortunate feature of Landé’s model in its original form is that it requires
us to believe that the residue of each atom is different from the completed pre-
ceding atom. For instance since Mg has a singlet and a triplet system, its residue
must have sometimes and sometimes P,=%h

It may be recalled from the First Part of this article that the ditterent
Stationary States of a group, sharing a common value of 7 and a com
mon value of k, are distinguished from one an: ther bv having ditferent

transitions between two states, in each of which 7=0, are likewise
missing. This numeral is correlated with the angular momentum P 4
of the entire atom, in the theory here outlined. For systems of even
multiplicity, Pa is equal to jh 2x; for systems of odd multiplicity,

The various Stationary States of a group differ slight
otherwise, of course, they would never have been discerned. The
energy-value of an atom must be conceived therefore as depending
not merely upon 2 and k, not merely on the rates at which the two
whirling parts are separately spinning, but likewise upon then mutual
orientation and hence upon j.' [In this theory, the dependence of energy
upon orientation must be postulated outright. We shall presently
meet with a case in which the dependence of energy upon orientation
can be foreseen, even in detail.

It appears from all these speculations, that a transition between
two Stationary States is no longer to be cone ceived merely as a simple
leap of an electron from one geometrically-detinite orbit into another.
A leap is indeed supposed to occur, but it is accompanied by a turning-
inward or a turning-outward of the axes of rotation of the two spinning
parts of the atom. The radiation which comes forth isa joint product
of these two processes, in which however no features of either sepat

ately appear; only the net change in the energy «
algebraic sum of the energy-changes due to each process separately,
is radiated as a single fused unit. Nature does not make the separ-

Having used an orientation-theory to interpret the complexity of
the Stationary States, we will now consider an orientation-theory
developed to account for the effect of a magnetic field upon the Station-
ary States. There, it was supposed that the various States belonging
to a single group are distinguished by various orientations of two
spinning portions of an atom, relatively to one another. Here, it will
be supposed that the various States which replace each individual
State, when a magnetic field acts upon the atom, are distinguished

by various orientations of the spinning atom relatively to the fie!
It will presently be seen that the evidence for the orientation-theo:
is much more abundant and more nearly direct, in this case of ma.
netically-excited Stationary States, than in that former case of mi
tiplets. This case in fact was the earliest to which an orientation-
theory was applied; but for it, some quite different form of theory
might have been developed for multiplets. Even here the data ani
the theory are not entirely concordant; but the concordance is sv
extensive, that the discord is sharply localized and identifiable.

Irom the Second Part of this article (Section L) I quote the prin
ciple that an electron (of mass gw) revolving in an orbit with angulai
momentum P is equivalent to a magnet of which the magnetic momen!
M is proportional to P, being

Both P and M are vectors normal to the plane of the orbit and henc
parallel to each other. If several electrons are revolving in divers
plane orbits about the same nucleus, their separate angular momenta
may be summed vectorially into a vector which is the angular mo
mentum of the entire system, and their separate magnetic moments
may likewise be summed vectorially into a vector which is the mag.
netic moment of the entire svstem; and these two summation-vectors
will be parallel to one another, and related by the foregoing equation.
Hence a rigidly-connected revolving framework of electrons—if such
a thing there be—may be treated like a single electron, insofar as the
ratio of magnetic moment to angular momentum is concerned. Wher-
ever in the course of this article we have envisaged electrons, kernels,
or atoms revolving with angular momenta prescribed as integer
multiples of //27 or of $h/27, we might have imagined these as mag-
nets with magnetic moments prescribed as integer multiples of eh /4ruc
or of }eh/4ruc.2 This is not necessary; though the relation between
angular momentum and magnetic moment is derived directly from an
equation valid for perceptible electric currents, it might not be true
for individual electrons. Nevertheless we shall arrive at striking
results, by supposing that it is.

When a magnetic field is applied to a multitude of radiating atoms,
most of the lines of their spectrum are replaced by groups of several
lines each, or “split up”? into several components, as the phrase ts.
This signifies that each of the Stationary States of each atom is ap-
parently replaced by several. One may infer that when an atom is

* The quantity e/4ruc, the presumptive magnetic moment of an electron circu-
lating in an orbit of angular momentum h/2z, is known as the Bohr magneton.

introduced into a magnetic field, each of its Stationary States is modi-
fied into one or another of several new States, differentiated from one
another and from the original State to a small but appreciable extent.
This might arise from some distortion or internal alteration of the
atom by the field; and it will probably be necessary to adopt this view
in some cases. But there is also a simpler effect which the magnetic
field may have upon the apparent energy-values of the Stationary
States, an effect not involving any deformation of the atom by the
field—to wit, an orientation-effect similar to that which was assumed
to account for multiplets. This we proceed to examine

If an atom which is a magnet is floating in a magnetic field, it
experiences a torque which tends to orient it parallel with the tield
By saving that an atom is parallel or oblique to the field, | mean
that the magnetic moment of the atom and therefore also its angular
momentum, are directed parallel or obliquely to the field; and this
usage will be maintained. Owing to this torque it is endowed with
energy due to the field, in addition to its own intrinsic energy; this
additional energy, which depends upon the inclination of the atom to
the field, I shall call its extra magnetic enerey. If the atom turns in
the field, the amount of its extra magnetic energy changes; and if its
magnetic moment suddenly changes, its extra magnetic energy also
changes unless it simultaneously turns by just the right amount to
compensate the change. If at the moment of passing over from one
of its Stationary States to another, its inclination or its magnetic
moment or both are changed; the amount of magnetic energy which it
gains or loses will be added (or subtracted, as the case may be) to the
amount of energy which it gains or loses because of the transition.
The frequency of the radiation sent out or taken in by the atom will
be equal to 1/h times the sum of two energy-changes of distinct
kinds—not, as in the absence of magnetic field, to 1’ times the
energy-difference between the two Stationary States alone. Thus
the effect of a magnetic field upon spectrum lines might be ascribed,
not to any deformation of the atom by the field, but to changes in the
orientations or in the magnetic moments of > the atoms occurring
at the instants when they make their transitions. The question for
us now is, whether the actual details of the observed etfect can be
interpreted in this manner.

Expressing the foregoing statements in formulae, in which MJ
denotes the magnetic moment of an atom, //J the magnetic field, and
a the inclination of the atom to the field, we have for the torque which
the field exerts upon the atom

In this last expression it is tacitly assumed that the extra magnetic
energy is zero when the atom is oriented crosswise (at right angles) to
the field. This is not an arbitrary, but a quite essential convention,
justified from the atom-model.'"") Suppose now that the atom passes
between two stationary states S’ and S”, in which its internal energy,
its magnetic moment and its inclination are denoted by U’, A’, a’
and U”, JM” and a”, respectively. Were there no magnetic field,
the frequency radiated would be

the term Av representing the displacement of the line by the field.
The question is, whether this term can be equated to the observed
displacements.

Consider the most tractable cases, those in which the so-called
“normal Zeeman effect’ is observed. In these cases a line of fre-

quency vy, is replaced by three, of which the frequencies are
Votwll, vo, ve—wll (12)

corresponding to three values for the displacement Av, which are
expressed by

The quantity » occurring in these expressions is a specific numerical
constant. Comparing these with the expressions for Av in (11), we
see that if our model is to be used to interpret the observations, then
for the first of the three observed lines J/’ cos a’ must be greater
than MW” cos a” by the amount / @; for the second, AZ’ cos a’ must
be equal to J/” cos ae”; for the third, AZ’ cos ae must be less than
M” cos ae” by the amount wh.

Another way of putting these statements is, that in order to interpret
the normal Zeeman effect in this manner it must be supposed that
whenever a transition occurs, the projection of the magnetic moment

” The action of the magnetic field upon the revolving electron imparts to it an
extra angular velocity about the direction of the field (the Larmor precession) and
hence an extra kinetic energy which (to first order of approximation) is proportional

to —cos a and is zero when a=7/2. This extra kinetic energy is the extra magnetic
energy AU. It is profitable to_derive the entire theory in this manner.

upon the direction of the field—for this is precisely what / cos @ is
either does not change at all or else changes by +wh. Sometimes
it acts in the first of these wavs, sometimes in the second, sometimes
in the third; but never in any other.

This would result, if the behavior of the atom floating in the mag-
netic field were governed by two rules; first, that it may orient itself
only in certain “‘permitted” directions such that J cos a, the pro-
jection of its magnetic moment upon the tield-direction, assumes
“permitted” values which are integer multiples of wh; second, that
whenever a transition occurs J/ cos @ either retains the value which
it had initially, or else passes to one or the other of the two adjacent
permitted values.

The first of these rules is stated more rigorously than is quite neces-
sary; all that is required is to sav that J cos @ is permitted to take
only such values as belong to an equally-spaced series with intervals
equal to wh. The second rule is necessary.

The theory of the normal Zeeman effect is simply, that the atom
does behave according to these rules. Radiation of thé frequency v
occurs, either when the magnetic moment of the atom does not change
and the atom does not turn, or when the magnetic moment changes
and simultaneously the atom turns just so as to keep the projection
of the magnetic moment on the field-direction constant. We shall
later see that the latter of these two alternatives is the accepted one.
It must be supposed that the atom, so to speak, capsizes when it
emits the frequency v, while floating in a magnetic field; it flops over
at the same moment as it passes from one stationary state to another.
Radiation of the frequency vo+Av or of the frequency v,—Av occurs,
as we shall see, when the magnetic moment of the atom changes; in
some cases the atom capsizes during the process, in others it does not.

I now translate the foregoing rules from the language of magnetic
moments to the language of angular momenta. The first rule is,
that the atom may orient itself only in certain permitted directions
such that P cos a, the projection of the angular momentum upon the
direction of the magnetic field, assumes permitted values which are
consecutively spaced at intervals of (2uc e)wi.

Now it is a fact of experience, that in the cases of the normal Zeeman
effect,

The rule therefore reads, that the projection of the angular momentum
of the atom upon the direction of the magnetic field is constrained to take
certain permitted values, spaced at intervals of h/2r.

We have supposed, in dealing with multiplets, sometimes that the
angular momentum of the entire atom is constrained to take such
values as are integer values of 4/27, and sometimes that it is con-
strained to take such values as are odd-integer multiples of } (h/2z7)."
In either case the permitted values of the angular momentum are
spaced at equal intervals; and as the rule for the component of the
angular momentum along the direction of the field bears the form
which it does, we may well suppose that something in the order of
nature constrains both the angular momentum and its projection to
accept only values which form a sequence spaced always at that
curious interval h, 27.

The total number of permitted orientations will obviously be
limited by the actual magnitude P of the angular momentum. This
being supposed always to be an integer multiple of 3 2/27, let it be
written P=2/(4 h/2x). The permitted orientations are those which
yield a series of values for the projection P cos @ spaced at intervals
h/27; let these be written

of which there are (2/+1) in all. On the other hand, it may be that
the atom is prevented from orienting itself parallel to the field; that
the least permitted angle between the axis of the atom and the direc-
tion of the field is some angle yielding a projection A» intermediate
between P and (P—h,/27). Then there are 2J permitted orientations
altogether.

Summarizing the results of this last paragraph: if the angular
momentum of the atom is an integer multiple 2/(3 h/27) of the fun-

" It was remarked at the beginning of Section S that the evidence to be presented
in that Section would support neither 4/27 nor any other particular numerical value
for the fundamental unit of angular momentum; here, however, we have evidence
for that value.

the atom is permitted to take either (2J+1) or 2/ distinct orienta-
tions in the field; the former number if it is, the latter if it is not
permitted to set itself quite parallel to the field.

It will now be shown that these are by no means idle speculations;
they bear directly upon certain facts accessible to observation. Before
bringing up these facts it is necessary to abandon the policy of speak-
ing exclusively about the “normal” Zeeman effect. This ‘normal’
effect received its adjective because it agrees so excellently with the
original theory devised years before quanta were dreamt of to explain
the effect of magnetic field upon spectra. It is essentially because
of this agreement that it is possible to develop the contemporary theory
of the “normal” effect in a perfectly deductive fashion, using no
new assumptions beyond those general ones of the orientation-theory.
Most spectrum lines, however, are affected by a magnetic field in ways
not compatible with the original theory; which is a consequence of
the fact that the set of new Stationary States, whereby a magnetic
field supplants each original Stationary State, in most cases does not
conform to the laws previously set forth.

The laws to which it generally does conform were read from the
spectra by Landé. The one feature in which the foregoing theory
quite agrees with these laws is its prediction of the total number of
Stationary States. A Stationary State for which the angular mo-
mentum of the atom is determined, by virtue of the theory of multi-
plets which filled the preceding section of this article, as being
2J (jh/2mr), is actually found to be supplanted, when a magnetic
field is impressed upon the atom, by 2J new Stationary States. This
is in agreement with one of the two alternative predictions made a
few paragraphs supra; to wit, with the prediction derived from the
assumption that the atom cannot set itself quite parallel to the
field. This agreement between the orientation-theory of multiplets
and the orientation-theory of Zeeman effect considerably strengthens
both.

The several Stationary States replacing a given original State
are always equally spaced; but the spacing differs in amount
from the value wl/h or ellh/4zuc exhibited when the normal
Zeeman effect occurs, and which we found it possible to deduce
from the simple orientation-theory. The difference is this, that
the actual spacing is a multiple of the value w//h by a factor g
(generally lying between } and 2) which depends upon the original
State:

The only ways hitherto used to accommodate the atom-model
this surprising and inconvenient factor g are tantamount to assumi!

that it enters into the relation between angular momentum J/ and
magnetic moment P? which was derived in Section L and written down
here as equation (7); which relation is accordingly modified without

a very unsatisfying procedure. Lande found it possible to mitigat
this process somewhat and at the same time produce a partial ex-
planation of the formula quoted in the First Part of this article,
whereby g is related to the factors A, R and J which, in the atom-
medel of the two whirling parts, measure the angular momenta of
valence-electron and residue and entire atom respectively in terms of
the common unit 4 27. This explanation involved the postulate
that g=1 for the valence-electron and g=2 for the residue. It would
therefore be necessary to justify, or to postulate without justification,
not a multitude of such relations as (16) with a multitude of unforeseen
values of g, but only a single such relation with a single unforeseen
value of g. This is bad enough, but not so bad as if it were inevitable
to assume that 1/7, P may have a dozen different values in different
cases.

It is now the occasion to recur to the extraordinary experiments
of Gerlach which disclose the magnetic moments of individual atoms
and verify the supposition that certain orientations are permitted
and others inhibited. These experiments having already twice been
mentioned in this series of articles, I shall spend no more space upon
the method than is necessary to say that atoms in a narrow stream
are sent flying across an intense magnetic field with a strong field
gradient, by which they are drawn aside. Were the atoms tiny
magnets oriented randomwise in all directions, the beam would be
broadened into a fan; one edge of the fan would be the path of atoms
oriented parallel to the field, the other edge would be the trajectory
of atoms oriented anti-parallel to the field, while the space between
the edges would be filled by the orbits of atoms pointed obliquely
to the field. Actually Gerlach observed not the whole fan, but two
or several separate diverging pencils of atoms, and between them
vacant regions traversed by none. Certain orientations of atoms to
field were unrepresented in the beam. Here for the first time there is
direct evidence of discrete Stationary States, of quantum permissions
and quantum inhibitions, not deduced from observations upon transi-

tions but drawn forthright from viewing atoms in equilibrium in their
Normal States.

When from the diverging pencils one proceeds to determine the
orientations of the atoms and their magnetic moments, one is con-
fused by a possibility made clear in the foregoing pages, but unsus
pected at the time when the first of these experiments were performed.
I illustrate with the case of silver, the atoms of which flock into two
diverging pencils with a quite vacant space between. At first it was
naturally supposed that one pencil consists of atoms oriented parallel,
the other of atoms oriented anti-parallel to the tield. The detlections
of the two pencils are such, that if this assumption is true then the
numerical value of the magnetic moment of the silver atom agrees
within the error of experiment with the value of eh 4druc —agrees,
therefore, with the notions that the angular momentum of the silver
atom in its normal state is #27 and that the magnetic moment stands
in the right and proper ratio e 2uc to the angular momentum. The
data were supposed to prove these notions. They also agree, how-
ever, with the suppositions that one pencil consists of atoms inclined
at 60° to the field and the other of atoms inclined at 120°; in which
case the magnetic moment of the silver atom would be eh 4ruc,
suggesting that the ratio of magnetic moment to angular momentum
has twice the right and proper value. This inextricable tangling of
the effect of orientation with the effect of magnetic moment makes it
impracticable to deduce quite so much from the data as was at first
thought possible; but plenty still remains. It is found that copper and
gold behave like silver, as was to be expected from their positions
in the Periodic Table. It is found that lead atoms, and (most sur-
prising of all!) cron atoms are not deflected at all; so that either the
magnetic moments of their parts balance one another completely, or
else they all orient themselves crosswise to the field. Nickel, on the
other hand, behaves as though its atoms had each a magnetic moment
surpassing 2eh,4auc, while thallium responds as though that of its
atoms were much less than eh 4ruc. Finally —lest the results seem
too gratifving—it is found that bismuth atoms are deflected in a
manner quite unforeseeable.

There is not time nor space to speak of the other method for de-
termining the magnetic moments of atoms, by measuring the sus-
ceptibilities of great quantities of them in gases or solutions; but the
measurements so made are also very helpful in determining the
magnetic moments of various atoms and ions various groupings,
that is to say, of electrons around nuclei.'"? All such data are of im-

>For the status of such measurements in 1923, the first of this series of articles
may be consulted. (This Journal, September, 1923.)

mense value, and no theory of the atom can be spared from the demai
that it confront them and account for them.

By the term “X-ray” the reader may understand any radiatior
of which the frequency v is so high, that the energy Av of a singk
quantum is several times as great as the energy required to remov
the most-easily-detached electron from an atom; greater, for instance
than 100 equivalent volts, so that the wavelength of the radiation is
less than some 125 Angstrom units. The emission or the absorption
of such radiation by an atom involves too great an energy-change
to be attributed merely to a displacement of the valence-electron or
even to combined displacements of the valence-electron and one or
two others. This definition leaves a sort of “twilight zone” of radia-
tions having frequencies somewhat but not much greater than 1 4
times the ionizing-potential of an atom. Little is known about such
radiations, and in this place they will not be considered.

The absorption of an X-ray quantum by an atom results in the
extrusion of an electron from the atom. The emission of an X-ray
quantum results from the passage of an electron within the residue of
the atom from some original situation to the situation vacated by
the extruded electron, or else into a situation vacated by an electron
which itself has moved elsewhere within the atom. These state-
ments contain the theory of the vast amount of data piled up by
observations upon the emission and absorption of X-rays by matter.

To express the same statements rather differently: X-ray absorp-
tion-spectra and X-ray emission-spectra reveal, when analyzed for
Stationary States in the manner used in analyzing optical spectra,
that each atom with several or many electrons has a considerable
number of Stationary States, distinguished from those we have here-
tofore discovered in that each of them involves the absence of one electron
from the atom. Each of them is therefore strictly an “‘ionized-atom
state,’ and yet there are several of them with extremely different
energy-values. This signifies that the extraction of an electron from
an atom rich in electrons may leave the residue in any one of several
distinct conditions. These distinct conditions are the distinct Sta-
tionary States, transitions between which are responsible for X-ray
spectra. Owing to this striking difference between the Stationary
States hitherto described and these latter, I shall refer to these as the
“X-ray Stationary States’’—not that this name is a_ particularly
good one.

Absorption of an X-ray quantum by an atom, then, results in a
transition of the atom from its normal state to one of the ‘X-ray
Stationary States.” Emission of an X-ray quantum by an atom
results from a transition of the atom from one into another of its
“X-ray Stationary States’’—a transition which begins in a condition
in which the atom lacks one electron, and ends in another condition
in which the atom lacks one electron. To take instances: radiation
of an adequate frequency falling upon an atom in its normal state
may put it over into an X-ray Stationary State known as the L;; state,
an electron being extruded. Radiation of an adequate frequency
(a higher frequency will be required) falling upon a similar atom in
its normal state may put it over into another X-ray Stationary State
of higher energy, known as the A state, an electron being extruded.
The atom in the A state may then spontaneously pass over into the Ly;
state, emitting a radiation belonging to the X-ray emission-spectrum,
its frequency being 1/A times the energy-difference between the K
state and the Ly, state. Later the atom may pass into still another
state, such as the J; state, by emitting radiation of some frequency
v’; the energy of the J/; state is therefore less than that of the Ly,
state by the amount fv’; calculating it thus, and then applying to nor-
mal atoms a stream of electrons or of quanta having energy just
adequate to put them over into this J/; state, we find that this
effect is duly produced.

Thus there is a thoroughgoing analogy between the genesis of
optical spectra by transitions between the optical Stationary States,
and the genesis of X-ray spectra by transitions between the X-ray
Stationary States. The differences between the two kinds of spectra
seem all to derive from the one fundamental difference between the
two kinds of Stationary States; the former do not involve the absence
of an electron from an atom, the latter do. In the optical region,
for instance, we find that an atom in the normal state cannot be put
into a particular excited state by any radiation except one of just
the right frequency v, for which /iv, is equal to the energy-difference
between the normal state and the excited state in question. In
the X-ray region, we find that an atom can be put into the A-state
(for instance) by any radiation of frequency equal to or exceeding
that critical frequency », for which hv, is equal to the energy-difference
between the normal state and the K state. This difference in be-
havior occurs because in the former case a quantum of radiation
having frequency v exceeding », would have no place to put the left-
over energy h(v—yv.), whereas in the latter case this extra energy
can be and is delivered over to the extracted electron as kinetic

energy, with which it flies away. This is known positively; for
extracted electrons can be observed, and their energy measured.

Spontaneous transitions from each of the X-ray Stationary Stat:
oceur to some, but not to all, of the States of lesser energy. Sor
are evidently inhibited; and it is possible to lay down rules of sel
tion, distinguishing those which are permitted. The complicated
system of rules originally proposed has yielded place to a much simple:
one, exactly similar to the one prevailing in the optical spectra. That
is to say: it appears to be possible to assign to each of the X-ray
Stationary States a certain value of a numeral k and a certain value
of a numeral j, such that the only transitions which actually occur
are those in which & changes by one unit and / either changes by on
unit or does not change at all; while transitions between states in
both of which 7=0 are specially excluded. Furthermore, the various
values of k and j thus assigned to the several X-ray Stationary States
are identical with those assigned to the several States constituting a
doublet svstem, such as we have met already in Section S, such as
the sodium atom possesses; so that there is a complete correspondence
between the system of X-ray Stationary States which every atom
rich in electrons possesses, and the doublet system of optical Station-
ary States which only certain atoms possess. A part of this corre-
spondence is expressed in the following Table:

TABLE Il
Values of k: 1 1 2 2 1 2 2 3 3
Values of 7: 1 1 1 2 1 1 2 2 3
Stationary States of
Doublet system: Is 2s 2pi 2pr 3s 3p2
X-ray system: iN L, Li Lin M, My Mir Mi My

No doubt the implications of this close correspondence are deep;
but just what they are is not yet obvious.

The fact that the residue, left behind when an electron is extracted
from an atom, may exist in any one of several distinct States, is quite
naturally interpreted as meaning that the various electrons of the
complete atom are variously situated, or revolving in various distinct
orbits; so that the several X-ray Stationary States differ essentially
in this, that ditterently-located electrons have been removed, leaving
different places untenanted. This notion is easily combined with the
idea that an atom is formed, or at all events behaves as though it
had been formed, by successive self-annexations of electrons to a

secutive adhesions of electrons, each of which settles down into a
peculiar orbit and remains there more or less unperturbed as the later
~omers immigrate one after the other into the svstem. Mav not
then the process of X-ray absorption consist in a powerful intruding
entity, electron or quantum, violently invading the interior regions of
the atom and casting out one or another of the deeper-lying earlier-
added electrons, while the later-added ones nearer or upon the frontier
remain attached?) May not X-ray emission consist in the passage of
one of these latter electrons into the orbit formerly held by its prede-
cessor, now unexpectedly reft away and its place left empty?

Although an affirmative answer to these questions involves a very
literal and concrete conception of electron-orbits, most physicists
make it, and would like to prove it. The chief difficulty lies in the
fact that all information about X-ray Stationary States is primarily
information, not about the prior condition of the electron which is
gone, but about the final condition of the residue which is left behind.

The data show at all events that there are not nearly so mans
conditions of the residue, as there are electrons of the completed atom;
from which it is fairly safe to conclude that the electrons of the atom
are so arranged, that any one of several different electrons may be
removed and the residue be left always in the same condition
therefore, that the electrons are arranged in groups, each electron
being situated essentially like every other of its group. In discussing
the formation of an atom by successive binding of electrons, it was
remarked that several electrons may be bound in orbits each char-
acterized by a common value of » and a common value of &. These
two ideas may coincide; and great efforts are being made to bring
them into entire coincidence. The evidence indicates, for example,
that the first ten electrons bound to a nucleus are divided into four
groups. Absence of an electron from one of these groups entails
that the atom is in the A state; absence of an electron from the second,
third, or fourth group brings it about that the atom is in the L,, or
Ly, or Ly, state, respectively. So much the X-rav data do show
rather definitely; although the actual number of electrons in each
of the four groups cannot yet be deduced. If one could prove a priori
that the first ten electrons annexed by a nucleus settle down into
orbits of four distinct kinds, the achievement would be a great one.
Intimations that something of this sort has been achieved are made
every now and then; but it is diftheult to tell whether the assertions
which are made have been derived cogently from a principle or are
inspired guesswork.

of which was until a couple of years ago regarded as perfectly d
tinct: but at this moment it is beclouded by one of the curious c
tradictions so abundant in the Theory of Atomic Structure. Brief
the typical phenomenon is this: the differences between the energy-
values of the K, Ly, and Ly, states agree notably well with wha:
would be expected if the complete atom contains a few electrons
moving in 1; orbits, a few in 2; orbits and a few in 22 orbits about the
nucleus; and if the K state corresponds to absence of an electron of the
first group, the Ly, state to absence of one out of the second and the
Ly; state to absence of one out of the third. (The reason why calcu-
lations can be made for so indefinitely-phrased a model is this, that
the field due to the highly charged nucleus of a massive atom should
dominate over those of the individual electrons so that it does not
make very much difference how many are supposed to be in each
group.) The natural inference is, that the rest of the atom remains
unchanged or little changed when any one of these electrons is ex
tracted. In this case the Azimuthal Quantum-number of the residue
should differ by one unit for the two states Ly, and Ly. Consulting
Table II, one finds that the quantity called k, which obeys the char-
acteristic selection rule of the Azimuthal Quantum-number, is the
same for Ly, as for Ly. This is an illustration of the collisions be-
tween two sets of inferences which unsettle the supposedly firmest
achievements of this theory.

Of the theory of molecules, a subject large enough for an article
by itself, | can say here nothing more than that it attains some re-
markable successes, achieved by and therefore fortifying some of the
assumptions made in these pages; notably the assumption that
Angular Momentum is a thing required in Nature to assume discrete
values spaced at intervals of h, 27.

The tinal part of this long article has been very unlike the Second
Part, in which an atom-model for the atoms of hydrogen and ionized-
helium was constructed and endowed with certain fundamental qual-
ities, so that it reproduced almost all of the relations of these atoms
to radiation with a truly striking fidelity. This Third Part by con-
trast has been a thing of shreds and patches. Models for many atoms
have been brought forth, but they have not been thoroughly adequate
and they have not been concordant with one another. Some were
designed with the same fundamental qualities as those given to the
model for hydrogen; and scarcely more can be said for any of them,
than that it does not positively clash with the properties of the element
for which it is devised. Others were made competent to deal with a

certain limited set of facts (as the grouping of terms in multiplets
by giving them qualities gravely in discord with those attributed
to the atom-model for hydrogen; and then they proved themselves
surprisingly well able to account for isolated facts of quite a different
sort (as the effect of magnetic fields upon atoms). The presentation
in these pages is naturally very far from complete; had it been com-
plete, it would have filled a book and not an article. But if it had
been complete, the eventual impression would have been the same;
an impression of confusion, vet of a confusion full of hope.

For the “Theory of Atomic Structure” is distinguished especially
by this, that it is not one theory but a multitude of partial theories,
each designed and competent to cover a limited family of the abound-
ing data, each struggling to overlap and to absorb the others. It
may be compared with a cross word-puzzle or a map-puzzle, in which
the beginnings of a solution have been made in half-a-dozen corners
and patches, while wide blank areas adjoin and separate them, and
some of the partial solutions already entered upon the tield may finally
yield to others which can be unified into the perfected pattern. Or
it may be compared with those maps of polar regions, in which here
and there a properly-surveved island or little strip of coastline emerges
from the blankness of the unexplored realms, and some of them are
certainly misplaced relatively to the others and will be shifted on the
map when all the geography is at length made known. Or it may be
compared with the state of a congealing metal, in which a multitude
of little crystals have formed themselves about casual nuclei of crys-
tallization; each is oriented in a different way, and when two of them
grow into contact with one another thev clash and cannot merge,
they stand blocking and thwarting one another. It may be neces-
sary to reliquefy them all and make a new attempt to change the
formless mass into a single crystal.

Meanwhile the work is driven forward with the fervor of discovery
and exploration, in this period which Russell finely called the “Heroic
Age of Spectroscopy,”’ and not of spectroscopy alone. Many, though
not so many as are needed, are busy with determining the Station-
ary States by deciphering the rich and cryptic spectra of some among
the numerous unstudied elements—enormously numerous, taking into
account how many kinds of ionized atoms there are; and others with
the assembling of new photographs of spectra made under the most
varied sorts of excitation, with other aids to discriminating the lines;
and others with the impressing of electric and magnetic tields upon
radiating atoms; and others are engaged in measuring the intensities
of lines. Yet other experimenters are determining the magnetic

moments of various atoms in all the possible ways. Some are seeking
new phenomena which may result from Stationary States and fron
transitions, and occasionally they are rewarded with brilliant example:
such as that vivid demonstration of the atom-magnets which Gerlach
effected, or such as the passage of an atom from one State to anothe:
while it transfers the liberated energy to another particle directly
and produces a chemical change. Others are finding the processes
resulting from the Stationary States manifest on an unearthly scale
within the stars.

The theorists likewise are at work with furious industry. Now and
then a set of data hitherto rebellious is suddenly systematized, usually
in a manner not quite concordant with the other theories holding
other parts of the field. Attempts are made to unify one partial
theory with another, usually unsuccessful. Sometimes an authorita-
tive thinker, despondent over the continuing contradictions, tries to
cut all the knots by declaring that one or another of the conflicting
models is entirely fallacious, and that the numerical agreements on
which it is founded are a delusion and a snare. Another is driven
to concede that the conflicts are destined to endure forever, and
accepts all of the partial theories as equally valid, or else paraphrases
them in ingenious words which veil the contradictions, yet leaving
these essentially unabated. Others, abandoning the general problem,
have returned to the question of the hydrogen atom, and for this they
are trying to rephrase or reshape the Quantum Conditions in a manner
more satisfactory to themselves; sometimes with the aid of new and
unfamiliar forms of mathematics, apparently expecting that when
these become habitual to the human mind, the mystery of the Quan-
tum Conditions will seem simple and clear. That, of course, always
remains a possibility—that the human intellect will accustom itself
so thoroughly to the new systems of ideas that they will cease to
seem incoherent, as the human ear has so accustomed itself to the
harmonic innovations of successive generations of musicians that the
tones which seemed outrageous discords to the audiences of Beethoven
now are to us monotonously sweet. To our minds the various divi-
sions of the Atomic Theory are still discordant. It would not be
fair to leave any other impression of this strange and fascinating
theory; inchoate but full of promise, immature but gathering force,
a fantastic assemblage of failures and successes; irreconcilable with

all other theories, irreconcilable even with itself, and yet perhaps
predestined to refashion all the science of physics in its own image.

Synopsis: The paper is based on radio transmission tests from station
2XB in New York City to two outlying field stations. It is a detailed study
of fading and distortion of radio signals under night time conditions in a par-
ticular region which may or may not be typical.

Night time fading tests using constant single frequencies and bands of
frequencies in whic h the rece iving observe itions were recorded by os illograph
show that the fading is selective. By selective fading it is meant that
ent frequencies do not fade toge ‘ther. From the regularity of the frequency
relation between the frequencies which fade tog ether it is cone luded th it
the selective fading is caused by wave interference. The signals appear to
reach the receiving point by at least two paths of different lengths. The

paths change slowly with reference to each other so that at different time s
the component waves add or neutralize, going through these conditions
progressively, The two major paths by which the interfering waves te ivel

are calculated to have a difference in length of the order of 135 kilometers for
the conditions of the tests. Since this difference is greater than the distance
directly from transmitter to receiver it is assumed that oe path at
must follow a circuitous route, probably reaching upward th rough higher

atmospheric regions. Various theories to explain this are briefly reviewed.

The territory about one of the receiving test sti itions in Connecticut is
found under day time conditions to be the seat of a gigantic fixed wave inter
ference or diffraction pattern caused in part by the shadowing of a group of
high buildings in New York City. The influence of this pattern on night
time fading is discussed. It is considered a contributing but not the control}
ing effect.

Tests using transmission from an ordinary type of broadcasting trans-
mitter show that such transmitters have a dynamic frequency instability or
frequency modulation combined with the amplitude modulation. At night
the wave interference effects which produce selective fading result in  dis-
tortion of the signals when frequency modulation is present. It is shown
that stabilizing the transmitter frequency eliminates this distortion. A
theory explaining the action is given. The distortions predicted by the theory
check with the actual distortions observed.

A discussion of ordinary modulated carrier transmission, carrier sup-
pression, and single side band transmission is given in relation to selective
fading It is shown that the use of a carrier suppression system should
reduce fading.

technique of radio telephone broadcasting is to consolidate and
continue its remarkable progress, is the mechanism of the transmission
of radio signals through space. In many receiving situations
largest apparent defects present in the reproduced signal are those
suffered not in the terminal apnaratus but in transit through space,
and in these cases better methods of utilizing the transmitting medium
In the present
paper we are reporting some investigations in this field of radio trans-
mission which have uncovered a number of interesting facts and have
led to at least one conclusion which is of practical utility.

Night time transmission, which is the usual case in broadcasting,
in many places commonly marred by fading and sometimes by actu
distortion of signals. Often these occur in certain areas not mor
distant from the transmitting station than other areas which enjo:
freedom from such annoyance. Selecting a particular instance of thes
difficulties in an area near New York City which, in so far as can ly
judged at present, is probably a typical instance, we have subjected i
to an intensive experimental study to determine what is the inheren:
nature of the troubles and if possible how they may be alleviated
In doing this it has been necessary to employ novel forms of tests
especially fitted to bring out in a concrete way the phenomena being
investigated.

As the radio art has progressed from spark telegraphy into continu-
ous wave telegraphy and into high quality radio telephone broadcast-
ing, increasing demands have been made on the transmission medium
to deliver at the receiving point a true sample of what was put into it
at the transmitting station. The requirements have grown in rigor
because in telegraphy the end has been to develop increased reliability
of communication at longer ranges and in telephony the medium is
called upon to transmit a highly complex form of intelligence.

Of the requirements placed on the transmission medium by modern
uses, those imposed by telephony are far more exacting than those for
telegraphy. In telegraphy a single frequency, or at most a narrow
band of frequencies sent out intermittently in accordance with a dot
and dash code must reach the receiving station in such shape that it
may be converted into audible sound for aural interpretation or into
current pulses for the operation of relays or recording instruments.
Leaving aside noise, the principal requirement is a sufficient freedom
from fading so that signals can be interpreted or recorded without
interruption. In radio telephony, as at present practiced in broadcast-
ing, there is transmitted a modulated high-frequency wave comprising
a relatively wide band of frequencies, usually at least 10 kilocycles.

Such a modulated high-frequency wave drawn out in the familiar
graphical representation is a comparatively simple-looking thing, but
analyzed into its elements and studied in detail it is revealed as being
an intricate fabric of elemental waves so interwoven with each other
that no one of them can be disturbed without changing in some
degree the complexion of the whole. For perfect results the whole
band must arrive at the receiver with an amplitude continuously
proportional to that leaving the transmitter, or the inflections or expres-
sion of the speech or music will not be correctly reproduced. All the
component frequencies within the band must be unchanged in their
relative amplitudes lest the character of the sounds be altered. Even
the relative phase relations of the various frequencies must be preserved
or, as will be shown later, the interaction of the two side bands in the
receiving detector will result in the partial loss of some of the frequency
components,

It is not long since the time when radio was supposed to be the
perfect medium for voice transmission it being presumed that since the
ether of space (if there be such a thing) was substantially perfect in
its electrical characteristics it must transmit frequency bands carrying
telephone channels without distortion of any kind. This may be true
theoretically of a pure ether but in fact, the ether used for radio com-
munication is filled with a number of things ranging from gaseous
ions down to the solid bed rock of the earth. It is rather to be ex-
pected that these will affect the progress of electromagnetic waves
and we know from experience that they do. Diurnal variations of
attenuation, fading, directional changes, dead spots and the like
are already well known phenomena resulting from the complexity
of our transmission media, although no entirely adequate explana-
tions of their causes have been certainly established. One of the
most recent manifestations of the effects of irregularities in’ trans-
mission through space is in the distortion of the quality of telephone
signals. ‘This was perhaps first noticed in the use of short waves for
broadcasting it being found that frequently the transmission was so
distorted that after detection the signals such as speech and music
were in severe cases almost unrecognizable.

For some time after quality distortion was recognized as a character-
istic of existing short wave transmissions, it was thought that for the
lower broadcasting frequencies at least, it was present only at night
and at relatively very great distances from the transmitter. However,

careful observations demonstrated that there were points relativel
near New York City where quality distortion from several broad
casting stations in the city was marked at night and in at least on
case was detectable even in daytime. When station 2XB the Bel!
Telephone Laboratories’ experimental station at 463 West Street,
New York City, was used to transmit test signals, it was found that
quality distortion could be observed in northern Westchester county
and in southern Connecticut at distances of about 30 to 50 miles from
the transmitter. Fading was also pronounced and it was noted as a
significant fact that distortion was always accompanied by some fading

trials it was noticed that at a particular point near New Canaan,
Connecticut, signals from 2XB were much weaker and more dis-
torted than signals from 2XY, the experimental station of the Amer-
ican Telephone and Telegraph Company at 24 Walker Street, New
York, even though the transmitter at 2XNB was about ten times more
powerful. Daylight field strength measurements at this point showed
that the field strength of 2NB was only one-third that of 2XY. This
led to the rather startling conclusion that there is a ratio of 100 to 1
in the power efficiency of transmission to that particular receiving
point from these two transmitting stations in New York which are
only about one mile apart.

In order to throw some light on this state of affairs a field strength
survey was made by G. D. Gillett which resulted in the field strength
contour map! here reproduced in Fig. 1. The contours on this map
show that there is a series of long nearly parallel hills and valleys of
field strength which, extrapolated, would converge in lower Man-
hattan and which extend out to the northeast as far as it was thought
worth while to follow them. There has occurred to us no better

explanation of this hitherto uncharted form of field strength distribu-_

tion thap that it is a gigantic wave interference pattern. A detailed
discussion of this theory is given in another section of this paper.
The fixed pattern shown by Fig. 1 is definitely present only in the
daytime but that it is fixed is attested by the fact that a second survey
made about a year later checks with the original one quite closely.
At night fading is pronounced in the area covered by the pattern and it
is apparent that some other factors must enter. As a result of an
endeavor to check up the pattern at night it was discovered that
‘This map was prepared by Mr, Gillett using the methods discussed in a paper
Distribution of Radio Waves from Broadcasting Stations Over City Districts,”’

quality distortion was, in general, most evident at places which were,
by day, in the valleys of the field strength diagram and a point in ons
of these valleys near Stamford, Connecticut, was selected for the estab
lishment of a temporary field test station. The interior of this sta

photograph, Fig. 2. At this place apparatus was set up to enable a
study of the nature of the distortion in signals from 2XB. Manv of
the records discussed in’ succeeding paragraphs were taken at this
Stamford field station. Others were taken near Riverhead. Long
Island, which was also found to be well located for such work. Fig. 3

is an outline map showing the relative positions of these field receivin,
stations and the transmitting station.

The reason for settling down at a fixed point in this way was t
attack the problem from a new angle. The field strength survey
and aural observations had yielded much interesting information
but did not appear at that time to shed a great deal of light on the
quality distortion so it was decided to attempt, by an oscillographic

study of received signals sent out under rigorously controlled con-
ditions, to determine just what alterations these signals suffered in
their journey through space.

In finding such distortions the ear is, of course, the primary testing
instrument or indicator of trouble, for, if the trained ear is unable to
detect anything wrong with a received signal in comparison with its
original counterpart it is safe to say that nothing detrimental of import-
ance has happened to it. But the ear is a poor quantitative indicator
and furnishes no permanent or easily analyzed record of its observa-
tions. It is evident that if we are to study quantitatively the char-
acteristics of radio transmission which give rise to quality distortion,

we must devise tests which will disclose changes, of whatever kind.
n the relations between the various component frequencies of the
transmitted band and furnish interpretable permanent records. In

Fig. 3—Outline map showing locations of transmitting station and receiving test
stations

The variable factors in radio transmission which may be directly
controlled are located at the transmitter and receiver. We have as
yet no tangible means of controlling the transmitting medium, but it
can be studied indirectly through the characteristics of the received
signals. Obviously, it is desirable in the interest of simplicity to
stabilize the apparatus variables to the extent that they may be
idealized in considering observed results. Furthermore, at both the
transmitter and receiver, it is desirable to make the antenna arrange-
ments of the simplest form. For our work the normal antenna arrange-
ment at station 2XB was used perforce since any important changes
would have constituted a major operation. It is far from a simple
arrangement, as shown in Fig. 4 which is an outline elevation and plan
of the antenna and building at 463 West Street, New York City. For-
tunately there are no buildings considerably higher than the antenna
within a distance of several wave lengths.

At the receiving test stations both loop and vertical antenna were
used; but in most of the experiments a simple vertical antenna was
employed. It was constructed of brass tubing, 30 feet long, and
guyed in a vertical position. A galvanized iron pipe 12 feet long
was driven in the earth for a ground connection. The vertical re-
ceiving antenna projected through the roof of the test station building

at Riverhead, L. [., as shown in Fig. 5. The receiving antenna wa
not tuned but was connected to the radio receiver through fixed
inductive coupling.

The carrier power in the transmitting antenna normally remains
fairly constant, except for minor variations in voltage of the suppl)
mains, and with a little care on the part of operating personnel, th«

antenna current can be kept within the limits of a | per cent. varia-
tion, which is small compared with the signal fading usually ex-
perienced.

The stabilization of the frequency was of the greatest importance
since in some of the tests it was desired to beat or heterodyne the
signals down to audio frequencies and pass them through narrow
band filters. To provide this stability engineers of the Bell Telephone
Laboratories arranged the 5-kw. transmitter at station 2XB to obtain

its carrier frequency by amplification of the output of a 6LO-ke. piezo-
electric crystal oscillator.

When desired some of the antenna current from the output of the
transmitter was rectified and the resulting current was sent over a
telephone line to the receiving station so that the frequency and wave

Fig. 5—Receiving test station near Riverhead, L. I. showing vertical antenna pro
jecting through roof of building

form of the modulating signal could be seen and photographed at that
point, thus guarding against any possible distortion in the trans-
mitter and enabling a direct “before and after’ comparison to be
made. The telephone circuit was also used for communication
between engineers at the two terminal stations.

At the receiving station double detection receivers and audio fre-
quency amplifiers were employed. These did not have entirely
“flat” transmission characteristics over the audio frequency band,
but in most of the tests this was of no importance. In cases where
it affected the results the making of necessary corrections was a simple
matter. In tests involving beating the received signals down to audio
frequencies through the agency of a local heterodyning frequency,

this was supplied from a shielded vacuum tube oscillator which on
comparison with a standardized piezo-electric oscillator was found
to possess the required stability. The double detection type receivers
were used for no other reasons than their availability and their con-
venience for quantitative work. The beating down oscillator within
the sets and the intermediate frequency step passed through in the
sets by received signals do not figure in the following discussion of
test methods but, of course, in each case the necessary set tuning

VW
| |Frequenc rR
| quency Paper Re
Master
Iscillator| apn
| for Wave Shape
rary

In this work the moving coil type oscillograph was used throughout
for the purpose of making photographic signal records. As  indi-
cated in Fig. 6 two oscillographs with elements connected in series
were employed; one for the purpose of making a continuous record
of the variation in the amplitude of the signal using a slow moving
photographic paper tape and the other to obtain the wave shape of
the signal by means of the usual high speed photographic film drum.
An element of one oscillograph was also used at times to record on
the film drum the wave shape of signals rectifed at the transmitting
antenna and sent over the telephone lines.

In considering these various records perhaps we had best look
first at the simpler ones and then proceed in a more or less orderly

fashion to the more involved ones. The simplest records are fading
records of the unmodulated carrier frequency of 610 ke. At the
receiver the carrier was heterodyned with a local oscillator to pro-
duce a beat tone of about 250 cycles which was fed through amplitiers
to the oscillograph elements.

Fig. 7—Interior view of Riverhead testing station showing recording apparatus

the received carrier signal with time, is given in Fig. 8. [t shows a
typical fading record made at Stamford, Conn., May 16, 1925. The
timing interval on strip 6 is 2.6 seconds.

The feed of the photographic paper tape through the oscillograph
was varied somewhat during the course of the experiments but was
generally in the range of 6 to 12 inches a minute. At this rate the

width corresponding to twice the amplitude of the signal, as both t]
positive and negative half-eveles are recorded. It will be observe
that the outer limits of the band corresponding to the peaks of th
sine wave are darker than the center portion of the record. This i
due to the fact that the rate of change of the movement of the ligh

Fig. 9--Wave form of beat note signal for single-frequency test. Center trace signal
from vertical antenna, upper and lower traces signals from loop antenna receivers

spot on the record is a minimum at the peak of the signal; hence, a
greater quantity of light affects these portions of the record. This
shading effect was very useful in the way it brought out changes in
the distortion of the received signal. This is discussed fully in another
section of the paper. The fuzzy irregular outline on portions of the
records is caused by static and radio noise. The timing marks on the
record allow a measurement of the time interval between points of
minimum signal. Fig. 9 is a sample oscillogram of the wave shape
of a beat note signal recorded by the method described above.
Marked changes in the fading cycle or time interval between points
of minimum signal may occur within a period of a few minutes, and

trom day to day there is often evidenced a modification of the general
haracter and the recurrence of these changes. An example of this
change in a short period of time is well illustrated by the oscillograms
in Fig. 10. Strips 1, 2 and 3 form a continuous record starting at
1:52 a.m.; strips 4, 5 and 6 start at 2:16 a.m.; and strips 7, 8 and 9
start at 2:37 a.m. These are three sections of a continuous record

Fig. 10—Single-frequency fading record, showing variation in rapidity of fading,

selected for the purpose of showing the decrease in the fading period,
in a 45-minute interval. The timing interval on strip 10 which ap-
plies to these records is 5 seconds. In this particular record only
half of the audio signal was recorded, the edge of the strip being the
zero line.

These single frequency fading records do not offer very much to
work on. There is, however, just enough suggestion of regularity
about them to annoy one with the thought that perhaps they may
follow some definite combination of periodicities and with this in
mind we have taken sections of two different records and subjected
them to a harmonic analysis.

So far we have been able to draw no more useful conclusions from
such harmonic analyses than that the heterogeneous scattering of
harmonic values is about what one would expect from the looks of
the curves.

One significant thing about these oscillographic single frequency
fading records is that they show no high speed fading of important

magnitudes. Occasionally one cycle of the beat tone will be some-
what upset by a sudden change in the amplitude but in general n)
changes which consistently distort the wave form were observed.

The slow fading may be considered as a modulation and on this
basis the received signal is seen to be composed of the original con-
stant carrier frequency accompanied by very narrow side bands
occupying at best perhaps a fraction of a cycle.

The next progressive step in the radio transmission studies is
naturally from a single frequency to two or more frequencies trans-
mitted simultaneously. By the use of two crystal oscillators at the
transmitter two separate and distinct radio frequency signals were
transmitted simultaneously. These crystals were ground by the
Bell Telephone Laboratories to oscillate at 610,000 cycles and 609,750
cycles. The amplitudes of these signals at the transmitter were
controllable so that it was possible to make them equal, or one larger
than the other, equivalent to the relative magnitudes usually found
for the carrier and single side-band transmission case. Records
were obtained of the variation of these radio signals, but none is
reproduced here since the information shown by them can be just as
easily abtained from the triple frequency records shown below.

Radio transmission on three frequencies is readily obtained by
modulating the carrier with an audio frequency tone, and observing
the three frequencies separately at the receiver.

where @ is a constant proportional to the percentage modulation.
These three frequencies are not merely a mathematical fiction
but are physically existent as three separate waves bound together
only at their point of origin.
To adequately record them separately by means of the oscillograph
advantage was taken of the fact that a group of frequencies beaten

with a single frequency differing from them by a small amount and
detected may thereby be reduced to audible frequencies without
having their interrelations of phase, amplitude or difference frequency
composition, changed in any respect. For instance if the frequencies
expressed above are beaten with a local constant frequency,
B COS qt T y
the resultant lower or difference frequencies will be
kBAa
cos
+kBA_ sin [(p—gq)t—y¥]
kBAa
eos

Each one of the three components has been changed in amplitude
by the same factor kB representing the efficiency of detection. Each

and each has had its instantaneous phase shifted by an angle —y.
Relative to each other they remain unchanged.

was 608,375 ke. so that the resulting three audio frequencies were
1,875 cycles, 1,625 cycles and 1,375 cycles.

As indicated in Fig. 11 in order to make a record of these signals
they are separated at the receiver by means of band filters. These
filters and others similar in type for other modulating frequencies

were designed and made by the Bell Telephone Laboratories es-
pecially for this work. The band filters used for the purpose of
selecting the carrier and side-band frequencies had a cutoff of 40
Transmission Units 250 cycles from the mid-band frequency.

These cutoffs together with the position in the frequency range
of the pass bands of the filters preclude any troubles from cross modu-
lation of the radio carrier and side bands during the beating down
process. The products of such cross modulation would be fre-
quencies which are multiples of 250 cycles and these cannot pass the
filters. On the other hand the beaten down frequencies will pass
practically intact, since as has been shown by the previously de-
scribed single frequency tests, each of the three frequencies received
although subjected to amplitude modulation by fading, represents
only a very narrow band of frequencies for which the filter pass bands
were of adequate width.

As the modulating tone was carefully calibrated to 250 cycles and
the filters adjusted to transmit the frequencies specified, it was only
necessary to transmit the carrier while adjusting the receiving beating
oscillator. The following procedure for this adjustment was found
to be very successful. A local audio frequency oscillator was set to
the reduced carrier frequency of 1,625 cycles, and its output con-
nected to a telephone receiver. The audio beat note from the radio
signal and local beating oscillator was reproduced by a loud speaker
and its frequency adjusted to zero beat the 1,625-cycle tone from
the telephone receiver.

When this adjustment had been completed the carrier was modu-
lated with the 250-cycle tone, and the side-band signals automatically
pass through their respective filters.

The signals from the outputs of the filters were amplified, and
recorded separately by the three oscillograph elements. The sample
records shown in Fig. 12 are representative.

Strips 1, 2 and 3 are taken from a long record obtained May 7,
1925, 3:22 a.m. The upper trace is a record of the upper side-band
signal, the center trace the carrier, and the lower trace the lower
side-band. Strips 4, 5 and 6 are from a section of a similar type of
record made May 28, 1925, 1:06 a.m., where the carrier was modu-
lated with a 500-cycle tone and different filters were used. In this
record the upper trace is the lower side-band and the lower trace the
upper side-band.

It will be noticed that the timing interruption appears only in the
side-band signals, as the tone was interrupted before modulation
took place, and that the amplitude of the carrier signal is not affected

by the interruption of the modulating tone. This makes it very
easy to identify the side-band signals. These records give an ex
cellent graphic picture of ordinary radio telephone transmission,
bringing out the fact that three truly individual frequencies are
transmitted to repre — one.

In Fig. 12, strips 1, 2, and 3, the relative amplitudes of the three
signals are very stat in snraiatiti to the relative amplitudes of

Fig. 12—Fading record showing individually the fading of carrier and side-ban

frequencies. Made at Riverhead, L. 1. Timing interruptions in side-band signals, 5

the signals as they existed in the ether at the receiving point. Before
this record was made a transmission characteristic of the complete
receiving circuit, including the osciliograph elements, was obtained,
using a local transmitter with modulated carrier for the purpose of
making the measurement. The gain of the audio amplifiers at the
outputs of the filters was adjusted to give substantially uniform
transmission on each of the three frequencies corresponding to the
carrier and side bands of the radio frequency signal.

As shown in Fig. 11, a telegraph key is placed in the circuit of the
center oscillograph element, for the purpose of placing identifying
signals on the records. An example of these identifying signals is

shown in Fig. 12, strip 4, which gives the date and time the record
was started, July 23, 1925, 2:06 a.m. (Eastern daylight saving time).

The record in Fig. 13 is of the carrier and side-band signals with
500-cycle modulation made at Riverhead, L. I., May 25, 1925, 1:25 a.m.
More gain was used in the side-band amplifiers for this record in
order that the effects of fading could be brought out more prominently,
In this record only half of the side-band signals were recorded, the

Fig. 13-—Fading record of carrier and side-band signals, made at Riverhead, L. I.
liming interruptions in side-band signals, 5 seconds apart

zero reference line being at the edge of the strip. The upper trace
is the upper side band, the center the carrier and lower trace the
lower side band. Where the traces of the signals overlap a darker
record is obtained. This record may be confusing at first but if
strip 5 is examined where the amplitudes of the signals are not so
large a better picture of the form of the record will be obtained.

It is obvious from these records that the carrier and side-band
signals do not fade together as a unit. The carrier may pass through
a zero value with still considerable amplitude in the side-band signals
as in strips | and 38. In the first case, strip 1, the three frequencies
successively fade through points of minimum signal in the order lower
side-band, carrier and upper side-band; and in the second case,
strip 3, the three frequencies fade through points of minimum signal
in the reverse order. This is a definite indication of selective fading;
that is, fading is a function of frequency as well as time.

An endeavor to form an explanation of the cause of this selective
action in fading must be largely in the nature of speculation. Fur-
thermore, since our data consist in the results of things which have
happened rather than in any first hand information on the processes
of the happening, the building of an explanation is a synthetic pro-
cess. In general for any given set of facts it is possible to synthesize
a number of explanations. Bearing this philosophy in mind we have
considered various theories in connection with our observations and
have concluded that simple wave interference as a major cause of
the signal variations is at present the most likely ial enatne While
wave interference may be called a major cause it should perhaps also
be called a secondary cause since the assumption of wave interference
presupposes for its origin, primary causation by some physical state or
configuration of the transmission medium. Speculation as to the
nature of this primary cause is one stage further removed from the
data contained in our oscillographic records than is the assumption
of wave interference.

Since it is desirable in the remainder of this discussion to point
out the evidences of wave interference, let us consider brietly the
nature of this phenomenon.

To avoid any possible confusion of terms let it be said that by
“wave interference’? we mean a particular physical phenomenon in
wave transmission and have no reference whatever to static, signals
from other stations, or any other of the forms of radio noise which are
commonly designated by the word “interference” when they hinder
the reception of desired signals.

When two single frequency plane polarized wave trains start out
at the same time from a common source and travel by different
routes to meet again at a distant point the naturg of disturbance at
that point is determined by the relative space phases of the planes of
polarization and time phases of the amplitude of the two arriving
waves.

If we let FE represent the vertical resultant of the electric field,
which would be the only part affecting a simple vertical antenna,
such as we have used in most of our tests, then

E =e, sin 27(Ft+d,) +e. sin 27( (1)
where F is the frequency and d; and dz are the distances along the
respective paths measured in wave lengths and e; and e2 are the
vertical components of the two waves. These two sine terms may
be thought of as two vectors differing in phase.

that is, the difference in length of the two paths must be an exact
whole number of wave lengths. The condition that the two waves
cancel each other giving a field

that is, the difference in length of path must be an exact odd number
of half wave lengths.

Thus if the two components e; and é2 are equal, the resultant vertical
field E will go through values ranging from (e;+e:) down to zero
as the path lengths change relative to each other. If the two waves
do not have exactly the same amplitude, the minimum value of /
will be something more than zero.

Differences in attenuation of the two waves and differences in
their direction of arrival will modify the relative amplitudes of e; and
ég but will not modify the time relations required for minima of the
resultant field unless we assume that at the time of a minimum
neither wave has an appreciable vertical component. Since the
consequences of such an assumption do not accord with our experi-
mental data we have considered that it may be left out of account
in the present discussion.

This is obviously a picture which fits in very well with the simple
single frequency fading records. The major maxima and minima
occur when the conditions of equations (2) and (3) are met and e;
and e, are nearly equal. On the other hand it seems doubtful that
the picture can be so simple. If we suppose two wave paths why
not three or more? Additional paths would add irregularities to
the fading and it would not be necessary to assume as great a degree
of irregularity in the changes in any one path. But with an increasing
number of paths the various arriving waves would tend to average
to a more or less constant mean value and large departures from
this mean would become rare. The fact that the fading signal con-
tinually covers a large range of amplitude, with the maximum many
times the minimum, definitely points toward there being but a very
small number of major paths, probably not more than two.

Considering now the question of selective fading in relation to
wave interference we refer back to equation (2).

If we assume the distances to be measured in any desired units
and call them d,’ and d.’ our equation will still hold provided we divide
each distance by the wave length measured in the same units, thus

may call the frequency spacing interval. That is, with changing
frequency EF will go through maximum values with frequency at a
series of frequencies beginning theoretically with zero and extending
upward in regular spacing to infinity.

The spacing interval is obviously that number of cycles which cor-
responds to the lowest finite frequency in the series, namely, the
frequency for which the distance (d;’—d,’) is just one wave length
since when x =unity equation (4) becomes

By using the same process on equation (3) we find that E has
minimum or zero values at another series of frequencies having the
same spacing interval but lying midway between the frequencies at
which maxima occur.

Thus it is apparent that with fixed path length difference the
amplitude of the field F will be different for different frequencies,
ranging from maxima of (e;+e2) down to minima of zero if the polar-
ization planes and amplitudes of the two vertical components are
equal.

Furthermore, still thinking of equation (1) as representing two
vectors, it is evident that the phase of the resultant field is different
for different frequencies even though these different frequencies
had exactly the same starting phase at the source.

If the paths are changing with time, the field at a given point, as
has already been pointed out, will go through time fluctuations.
Another way to look at this is that there is a space pattern of maxima

and minima and as the paths change the plane section of the pattern
taken by the surface of the earth wanders so that at any one point
the field is continually fading in and out as the maxima and minima
glide by it. Each frequency has its own pattern differing from those
of its neighboring frequencies in such a way that at any given point
the relation between amplitude and frequency is that just discussed

Fig. 14——Plotted curves of signal amplitudes condensing a long fading record, part of
which is shown in Fig. 13. Numbers along time axis correspond to successive 25
second timing interruptions

above. Thus as the paths change and the patterns shift the different
frequencies fade not simultaneously but progressively.

In the above analysis of wave interference it has been assumed
that all frequencies traveled from transmitter to receiver over a
given path in the same elapsed time. This does not mean that they
necessarily follow exactly the same route on this path since they
might follow somewhat different routes of equal length or if their
transmission velocities were different they might follow different
routes of unequal length and still come within the definition of a
“path.” It seems reasonable to assume that over the width of an

STUDIES IN RADIO BROADCAST TRANSMISSION 165
ordinary transmitted band the various frequencies are treated alike
by the medium and the simple assumption that they follow the same
route with the same velocity is justified. If none of these assump-
tions is correct but the departure is not large the effect will be merely
to introduce slight irregularities into the spacing interval and the
general nature of the result will not be changed.

Let us now examine more closely the record, a part of which is
shown in Fig. 13.) A portion of this has been condensed into the
curves of Fig. 14. One unit along the time axes of these curves
represents a 25-second interval.

To obtain these curves the amplitude of the signal has been scaled
off and plotted, ignoring all the minor irregularities. From this
record the relative fading characteristics of these single frequency
signals 500 cycles apart are more easily seen, and it is possible to
contrast the time of occurrence of points of minimum signal for any
pair of them.

For the frequency difference of 500 cycles (610.5-610 and 610—
609.5) these times are obviously quite different but there is no clearly

discernible relation between them. The curves for 1000-cycle dif-
ference (609.5-610.5), however, show a striking relation in that the
maxima and minima of the two are opposed fairly regularly over the
entire 33-minute interval covered by the plot. This means that
when one frequency has a minimum amplitude the other has a mavxi-
mum and vice versa. Certainly this suggests a wave interference
involving only two major paths whose difference in length is such
that the spacing interval is 2,000 cycles. The path difference appears
to be changing somewhat irregularly but at an average rate of the
order of one wave length (or approximately 500 meters) per minute.

Before speculating further on the numerical values which may be
derived from this part of the data we had perhaps best consider some
other records of a somewhat different kind which are better adapted
to provide such values. But first let us reiterate that these are
night-time etfects.

During the day signals substantially uniform in amplitude are
received. An example of the type of transmission obtained in the
daytime is given in Fig. 15, which is a record of the carrier and side-
hand signals received with substantially the same terminal condi-
tions with the exception of the time as that existed when the records
shown in Fig. 12 were made.

The abrupt change in the amplitude of the side-band signals was
due to an intentional change at the transmitter in the input level
of the tone modulating the carrier, and accordingly the amplitude
of the carrier did not change. The timing interval is 5 seconds.

The familiar fading record is limited to two axes, amplitude and
time. So far we have extended this cramped perspective somewhat
by observing as many as three separate fading records spaced at
audio-frequency intervals along the frequency axis. Even these
three narrow lookouts upon the wide range of ether transmission
have indicated amplitude relations along the frequency axis which

Fig. 16—Diagram of system used to obtain records of selective fading or ‘band
fading’’ records

promise to open a new line of attack upon the problem of night-time
fading. But the desirability of knowing what takes place in the
interval unrevealed by these cracks in the fence becomes obvious.
We should like to know the relative amplitude of frequencies over a
wide band, and the change in this relation with time.

Since it is not a simple matter to record simultaneously the ampli-
tude of a large number of waves of frequencies separated by say one
hundred cycles in the radio-frequency range a single frequency in
combination with a frequency stepping device at the transmitter
has been adopted. The circuit arrangement is shown diagrammatic-
ally in Fig. 16. The rotary contactor bringing into the circuit suc-

cessively a total of fifteen small condensers across the main condenser
of the transmitter oscillator shifts the frequency in steps over an
adjustable range. The contactor is rotated at the rate of nine revo
lutions a minute, which is sufficiently slow to show definite ste ps in
the oscillograph record. At the receiving end a local oscillator sup-
plies a radio-frequency wave for beating the incoming frequencies

A long oscillograph record of this stepped frequency gives a sort
of moving picture of the fading for the entire band covered. A
sample of such a record is shown in Fig. 17 with alternate pictures

the two-way traversal of the frequency band successive pictures
are reversed. If a series of such built-up pictures as these could be
taken rapidly on moving picture film, and projected successively
upon a screen we should have before us an animated view of band
fading. And according to the results of experimental investigation
the subject offers a lively theme for such a presentation. The peaks
mid and depressions glide nervously back and forth across the setting.
The successive pictures of Fig. 17 (which, by the way, were selected
for their half-tone reproduction possibilities rather than as first
class examples of the records taken) illustrate a rather leisurely
movement of this sort. These ten built-up photographs cover a
period of slightly more than one minute. In the first seven pictures
a depression appears at the left, while in the last three this depression
seems to have made an exit followed by the simultaneous entrance of
another from the opposite wing of the stage. Evidence of such

organized spacing of the minima is present in all of these night-time
band fading records. As has already been suggested such evidence
has an important significance, but before going into this phase ot
the subject again let us examine a little more in detail the structure
of these band fading records.

The steps in any one picture of Fig. 17 are, as we have said, snap-
shots of the wave amplitude for successively different radio frequen-
cies taken about a quarter of a second apart. The fact that the
fifteen snap-shots used to build up a single picture are not taken
simultaneously causes a skewing of the outlines when movement
of the depressions as shown in Fig. 17 occurs. If, for example, we

Fig. 18—Three dimensional diagram, showing the method of interpreting band fading
records

were to take fifteen separate and successive snap-shots of a mountain
through fifteen long vertical slits side by side it would be possible to
combine the narrow sections so as to form a true picture of the peak.
Now, if by some prodigious act of nature the mountain were shifted
suddenly to one side and back again during the time we were taking
the fifteen successive snap-shots through the vertical slits, the com-
bination of them would form a profile quite different from that ob-
tained when it was stationary. Or if it were simply moved steadily
across the field of vision during the time the snap-shots were being
taken one slope would be made to appear precipitous while the other
would be leveled to a gentle grade in the finally built-up picture.

The character of this skewing, then, and its magnitude depend upon
the rate at which the object being photographed in vertical sections
moves, and the direction of the movement.

In Fig. 18 is shown an imaginary night-time band fading record in
the “assembled” form. Since such a record contains frequency as a
third dimension, in addition to amplitude and time as shown in the
ordinary fading record, our simple fading curve has assumed the
broader aspect of a surface, the selective fading making more or less
parallel ‘valleys’ running across it. The step-frequency svstem of
recording the points amounts to photographing sections of this solid.
The important point to be kept in mind is that these sections are
not perpendicular to the time axis. If they were, the skewing previ-
ously described would not be present. By setting these sections up in
their true relation to the time axis, however, and filling in to produce
a continuous surface such as is shown in Fig. 18 the result is correctly
represented. In order to make a detailed and accurate study of the
band fading records, therefore, it is desirable to construct from the
oscillograph sections the complete surface by the method suggested.

In Fig. 18 the trace of minima crossing the band is shown by J,
M' and M”. Picture sections obtained as our recording apparatus
literally moves back and forth across this frequency band are shown
as (a-b-c-d), (b-c-a’), (a’—b’-c’), etc. It will be evident that the
section P), for example, will, in case a minimum is crossing rapidly,
appear entirely unrelated to section P:. When the minima run
nearly parallel to the time axis (slow changes in transmission condi-
tions) the successive pictures P;, P2, Ps, etc., will reveal their rela-
tion by direct comparison.

Actually to obtain frequency-amplitude sections perpendicular
to the time axis in Fig. 18 would require the simultaneous trans-
mission and reception of a large number of frequencies spaced at
short intervals along the frequency axis. A more practical thought
is to speed up the process and though this seems very simple at first
consideration, it will be shown later to involve a particular kind of
distortion which cannot be separated out as easily as the skewing
encountered by the more deliberated method.

Now that we are familiar with the data, Fig. 19 showing, partially
superimposed in vertical strips, the outlines of successive built-up
pictures of the frequency traverse will be of greater significance.
During the steady periods there appears within the 2,280-cvcle
band covered by these data approximately one complete cycle of
selective fading. The lack of flatness in the audio-frequency-trans-
mission characteristic of terminal apparatus has caused the suppression

of amplitudes toward the right side of these sections. Keeping in
mind also the skewing inherent to this system of presentation during

record. The relative position of these minima gives us an interesting
insight into the nature of the night-time transmission path.

been plotted against time as in Fig. 20.) The widths of the frequency
bands covered in this case are indicated. This picture is essentiall
a bird’s-eye view of band fading records such as are illustrated in
idealized form by Fig. 18, the amplitude axis being perpendicular
to the page. It reveals the presence of minima spaced at more o1

Fig. 20——Plotted curves which condense a long band fading record so as to bring «
the frequency spac ing interval of the selective fading

depressions in regular spacing beyond the scope of our pictures, for
when one minimum slides out of sight another appears to take its
place from the opposite side of the band. The minima traces shown
in broken line were outside the record but were located by extra-
polating the sections.

Other depressions of small amplitude appear to be superimposed
upon the major changes but the present data appear inadequate to
give reliable information concerning them. These minor depressions
seem most evident during periods of rapid change.

The presence of these major minima in regular array bears a marked
similarity to the familiar wave interference case in light and tits in
very nicely with the theory detailed in previous paragraphs. Assume
for a moment the simple case of two transmission paths producing
such an effect and account for the difference in their lengths by pre
suming that one path follows more or less closely along the surface
of the earth while the other seeks higher altitudes and in some fashion
gets back to earth at the receiving station.

The mean frequency difference or spacing interval between suc-
cessive minima for the records given in Fig. 20 is approximately
2,200 cycles. Therefore, the mean wave length difference in length of
path from equation (5) is 277 wave lengths, or 136.5 kilometers.

It is evident that the errant waves following the second path must
have been led a devious route. While this is about all the informa-
tion which can be deduced directly from these data it is interesting
to speculate further with the information along the lines of some of
the theories which have been proposed to account for such wave
deflections. For instance there is the Heaviside layer theory in which
there is supposed to be a more or less well detined reflecting layer in
the upper atmosphere. For this we would visualize our high alti-
tude waves as proceeding in a straight line up to the layer, being
reflected, and striking back to earth at the receiving station.

Since the distance from transmitter to receiver was 110 kilometers
the length of the secondary path was 1104 136.5 or 246.5 kilometers.
By triangulation the height of the assumed reflecting layer may be
determined as very nearly 110 kilometers or equal to the distance
from transmitter to receiver, and the angle of incidence is 26.5 degrees.

As vet no positive information has been acquired concerning the
variation of difference in length of two major night-time transmission
paths with direct distance from the transmitter. If the path differ-
ence is due to reflection from an overhead layer, the expected rela-

When Ad is the difference in length of path, y is the direct distance
and / is the vertical height of the laver.

An investigation of this relation would probably do much to prove
or disprove the reflection theory.

At this point it is well to recall the results of earlier tests in which
it was observed that single frequency waves separated by 1,000 cycles
faded in approximately an inverse relation also indicating a spacing
interval of about 2,000 cycles. The agreement of these earlier records
is particularly noteworthy since about three weeks elapsed before
the more detailed band fading records were made.

Fig. 20 shows a time variation in the frequency position of the
minima which is explained as due to a variation in the difference of
path length. If we indulge in further speculation along the line of
laver phenomena we conclude that the reflecting layer is rising and
falling. It is improbable that the whole layer would rise and fall

together so we conclude that undulations occur along its surface.
These undulations in themselves would cause the length of path of
the wave reflected toward the receiver to undergo a continual change.
They would also introduce minor retlections from surfaces more
distant than that responsible for the major effect which may be
responsible for the more rapid, low amplitude fading which is usually
superimposed upon the slow changes. Obviously, the character of the
fading would in the event that it is caused by undulations along the
reflecting layer, be determined by the amplitude and direction of
movement over the surface.

If, on the other hand, we examine the possibilities of theories such
as those proposed by Nichols and Schelleng, Larmor and others in
which the action of free electrons in the atmosphere is invoked we
might visualize the waves on the second path as following a curved
trajectory. Or we might have the two sets of waves start off to-
gether, become split by double refraction and eventually come. to-
gether again. Perhaps their planes of polarization will have been
rotated. In fact it is possible to build up what appears, we must
confess, to be a highly imaginary explanation in which the wave
interference is accounted for not on the basis of any great difference
in path length but by the assumption that the amount of rotation is
such a function of frequency that a change of about 2,000 evcles adds
or subtracts a complete rotation, and the further assumption that one
set of waves has had its plane of polarization rotated through several
more complete rotations than has the other. The synthetic possi-
bilities are almost endless and we must wait upon further data more
varied in character before the facts can be established. In the present
investigation we have not attempted to determine the mechanism of
the transmission medium except insofar as it could be inferred from
the results of our tests which were aimed at finding out just how radio
signals look after they have been subjected to a trip through this
mechanism.

Returning to the solid band fading record illustrated in Fig. IS,
let us form some conception of the appearance of this figure were it
extended toward the much higher and lower frequencies using as a
basis of this conception the supposition that the existing record is
systematically distorted by wave interference. For a given rate of
change in the physical difference in length of path, such as would be
encountered in the simple reflection case, the rate of movement of
the minima across the band fading pictures would vary directly with
the frequency. Therefore, we can extend the narrow section shown
in Fig. 18 to form a wide band fading record such as is shown in

Fig. 21, wherein we are looking down upon the distorted surface, th«
minima being traced by the light lines. Toward the short wave end
of the band it is evident that a fading record for a single frequency
represented, for example, by a section parallel to the time axis and
perpendicular to the page, a—a’, would show rapid fading, while a
similar record at the long wave end of the range as b—0}’ would give
slow amplitude changes. Such sections representing theoretical

Fig. 21— Theoretical diagram obtained by extrapolating band fading records to show
how the rapidity of fading might be expected to change with the wave-length

single frequency fading records are shown at the bottom of Fig. 21.
The relative fading rates for long and short wave lengths as indicated
by these idealized characteristics, are in accord with general experience

In describing the stepped-frequency method of obtaining band
fading records allusion was made to distortion which might result from
speeding up the process. Suppose that we were to use a very small
rotating condenser in parallel with the main condenser of the trans-
mitter oscillator for changing the frequency, and that this condenser
were capable of changing the frequency sinusoidally about a mean
value. Then we could represent the variation in frequency with time
as is shown by the curve C; in (a) of Fig. 22. Now if the energy

i i transfer from transmitter to receiver takes place over two paths of
d different lengths one wave will constantly lag behind the other.
A" This lag may be measured as a time interval. In Fig. 28 are shown
a two waves, (a) and (b) of constant amplitude but with frequency
a
ssoida

Fig. 22—Curves showing the relative effect of transmission time lag in sinusoidal and
step-by-step methods of frequency variations

n Fig. 23—Diagram showing the effect of frequency modulation
" modulation. The wave (b) representing the indirect wave, it will
r be noticed, lags behind the direct wave represented by (a). The
" amount of this lag is determined by the difference in length of path
p and the transmission velocity. If we were to receive only one wave,

constant amplitude field (providing the high-frequency character-
istic of the receiver is flat over the range of frequency variation).
But when two or more distinct paths exist, the combination at the
receiver becomes complex. This is evident in curve (c) shown in
Fig. 23 which is a direct summation of (a) and (b), and in (d) which
is the envelope of (c). The amplitude is subjected to variations
which did not exist at all in the original wave.

We might set up an equivalent effect right at the receiver by con-
structing two small local oscillators having the same characteristics
as the transmitter oscillator. The two small rotating condensers
would be driven by the same motor but the rotor of one would be
shifted backward in phase relation to the other so as to simulate the
case of transmission lag over the longer path. The relative fre-
quency characteristics of the two may then be represented by curves C,
and C;)’ in (a) of Fig. 22.

The frequency of the signals arriving over devious paths at the
receiver may be put in the form of an equation as,

For a difference in length of path equal to 300 wave lengths at a
frequency of 600,000 cycles per second, for example, the time lag
of one wave behind the other will be equal to 300 600,000 second or
1 2000 second. The lag of one of the condensers behind the other in
the “equivalent” case described above would be then for 30 cycles
per second rotation of the condensers, 30/2000 times 360 degrees
or 5.4 degrees. The lag of 5.4 degrees represents the lag of the con-
denser rotor so that the frequency lag will depend entirely upon the
rate of change of frequency by the rotating condensers at any
given instant.

Now to determine the resultant wave at the receiver we must know
both amplitude and relative phase of the components arriving over
the different paths. The amplitude will be constant, and we shall
assume known, although it may actually follow slow changes with

attenuation or variations in length of path. The relative phase must
be determined from equations (6a) and (6b). Knowing the fre-
quency variation with time we may by integrating the following
equation determine the phase relation at any time (1

Substituting the general relation for F, and F, from equations
(6a) and (6b) we have,

Evidently the relative phase (40) will be the difference between
these two giving,

The equation is not in itself very illuminating, but what it tells us
generally is that if we represent two frequency modulated waves
travelling over paths of different lengths to a distant receiver by
rotating vectors, these vec:ors are constantly shifting their relative
position. The magnitude of the shift at any instant is given by the
varying angle A®. Due to a change in the angle included by the two
vectors their resultant will undergo an amplitude change, the serious-
ness of which we will consider later.

Thus far in the discussion of frequency modulation by means of a
rotating condenser we have assumed sinusoidal changes in frequency.
The ordinary condenser departs considerably from such a perform-
ance. By considering the application of the integral equation for
A@ to such a case it will be recognized that the relative space posi-

tions of the vectors representing the direct and indirect waves will
be subjected to changes at every point where the slope of the fre-
quency-time curve departs from a simple sine relation. The degree
of distortion due to the presence of such irregularities may be con-
siderable.

In Fig. 24 are shown some samples of ‘“‘wobbled”’ carrier frequency
records obtained at Stamford, Connecticut. For these records the

Fig. 24—Sample fast records showing distortion produced by intentional frequency
modulation. a@ day record, 6 and ¢ night records

carrier was wobbled at the rate of about 10 cycles per second. There
is some uncertainty as to the range of frequency variation for these
records although it was probably in the order of a few thousand
cycles. By means of a constant frequency local oscillator the radio-
frequency wave was stepped down in frequency to audio values which
could be amplified and recorded.

The record (a) of Fig. 24 represents stable day-time reception.
The record shows amplitude modulation due to the receiver char-
acteristic alone. If the receiver were, as is desirable, capable of
amplifying all the frequencies present in the received wave in the
same ratio this record would be of constant width. In the sub-
sequent examination of night records we must keep in mind the fact
that the terminal apparatus is responsible for a certain part of the
amplitude modulation. Its influence is readily recognizable.

The night-time records shown in (b) and (c) reveal a distinct dis-
tortion of the envelope aside from that present in the daytime record.
Peaks appear and disappear within time intervals sometimes as short
as a fraction of a second.

The record in Fig. 25 represents a slow picture of the changes shown
in (b) and (c) of Fig. 24. If these wobbled frequency waves are
studied carefully it will be noted that where a single peak stands at
one moment there gradually comes in view another as if it were sliding
from behind the first. The cycle length being about 1/10-second
we may get some idea from this series of the rate at which the changes

Fig. 25—Sample slow record showing distortion produced_by intentional frequency
modulation. Night record

take place. The presence of so many peaks in these records is at-
tributed in part to the fact that the rotating condenser used gave a
frequency change which was far from a simple sinusoidal relation.

Let us now return to the stepped-frequency method of obtaining
the band fading pictures and ascertain why it has certain advantages.
In (b) of Fig. 22 is shown the “equivalent” characteristic for the
stepped condenser. During 1, 2000 of a second (for the conditions so
far assumed) in each step distortion may occur due to transient
conditions, but during the remainder of the quarter second assigned
to each step (for the records so far taken) a steady state is reached.
Thus, theoretically, distortion occurs only during about 1 500 of the
step interval. In (b) of Fig. 22 the lag is greatly exaggerated for pur-
poses of illustration. This means simply that we have maintained
constant frequency for a sufficient length of time to establish, before
taking our picture, a fixed interference condition over the region
including transmitter and receiver at least.

Thus far we have been dealing with the unstable phenomena of
night-time transmission. Our interest has been directed almost

entirely toward variations with time. While the presence of wav
interference has been detected, and the movement of this inter
ference effect across the frequency band has been recorded, littk
effort has been made to form a picture of such interference in its

space relation. A discussion of similar stable, daytime phenomena is i
therefore not out of place, and particularly so in view of an evident
relation of the fickle nocturnal interference phenomena to the steady i
states which follow the appearance of daylight. ‘

In a previously published map of field strength distribution in :
New York City,* it was indicated that the congestion of high buildings .

Fig. 26—Map showing location of radio obstruction on Manhattan Island as de
termined by the intersection of lines between various transmitting points and their
corresponding shadows

; just below Central Park cast a heavy shadow. More recently it has
been determined from observations on a portable transmitter, set
up at various points, that this building center is a consistent per-
former. The position of this obstruction is determined in Fig. 26
wherein only partial contours from maps for the indicated sites are
given to prevent confusion. The intersection of these lines from
transmitter to shadow, falls at approximately 38th Street in the vicinity
of Sixth Avenue.

The dissipation of wave energy at such a point is probably the
composite effect of many adjacent structures. Fig. 27 gives an
elementary idea of how this can occur. The structures filling in

BROADCAST TRANSMISS| \ 1S]
mee each block are, of course, very well connected electrically by means
ter of pipes, cables, ete., with those of adjacent blocks. Between each

ttle oscillating circuit (which is pictured as consisting of two buildings
si with earth connections) there exists a coupling which binds the whole
a is j system together more or less flexibly. Thus the obstacle offered by
ent
rereere —,
ngs r ror
}
:
3 a
i
Fig. 27—Idealized picture of equivalent electrical circuit characteristics of hig!
buildings
a group of buildings might be of a selective nature, and evidently its
frequency characteristic may vary with direction.

Such an aggregate would, in addition to absorbing wave energy,
‘ produce a change in velocity or a refraction of the wave front. Some
eir indication of such an effect will be discussed later. Before leaving the

significance.
‘a From the transmitter a wave front expanding outward and upward
et encounters an obstruction which we shall assume is near the earth
‘i plane. The net result of this encounter is a weakening of the wave
6 over an area nea: this plane, and probably a distortion of the energy-
ba bearing fields. We might then imagine this shadow to be a tunnel-
” like region extending along the earth beyond the obstruction, and as
y having definite vertical as well as horizontal limits.

Fig. 28—Aerial photograph of Manhattan Island showing locations of transmitting
station and obstructing high building area

STUDIES IN RADIO BROADCAST TRANSMISSION 183
Such barriers to wave travel, situated within a short distance from
the source, seem, as we might expect, to have a more extensive and
serious influence upon effective broadcast distribution than similar
obstructions at greater distances.
It will be noticed that the obstruction falls very nearly upon the
direct line from the transmitter to the Stamford testing station.
This will also be evident later after an understanding of Fig. 29.

wherein the position of the “Band Near Sound” represents also the
bearing of the Stamford station. The Riverhead station is not
directly in line with the major obstruction.

In certain sectors of the field strength contour map for station 2XB
there appears to exist a kind of wavy displacement of the contour
lines forming a partial pattern of peaks and depressions side by side.
In general, this pattern must be differentiated from an ordinary
shadow area. A remarkable example of this sort of field distribu-
tion is shown in Fig. | which is one section of a field strength survey
made for station 2XB. These contours are based entirely upon
daytime measurements, and represent a condition which is stable
throughout the daylight period. Considerable difference in’ signal
level is apparent within short distances across the direction of wave
propagation. Two pronounced low. signal channels extend ap-

proximately north-east across this region. These shift with change
in frequency of the transmitted wave. Fig. 30 illustrates the space
relations for such a movement. The full line curve shows a partial cross
section of the contour map of Fig. 1 taken along a line approximately
perpendicular to the direction of transmission 110 wave lengths from
the transmitter. This represents relative field strength values for

Pig 30> -Cross-section of wave interference pattern showing change with frequency
610.0-kilocyele radiation. When the frequency is raised to 635.0
kiloeyecles, there occurs a movement of the peaks and depressions as

is shown by the broken line of Fig. 30.) Apparently the increased
frequency causes these channels to be crowded together.

If we take sections of the field strength contour pattern in Fig. |
and examine carefully the relative amplitude of peaks and depressions
represented by these wavy lines we shall find that the ratio of field
strength of the peaks to that in the depressions increases with dis-
tance from the transmitter. That is, the channels become more
sharply defined as we move away from the transmitter. This ratio
is shown approximately by the curves of Fig. 31. If these peaks
or depressions were simple shadows they would maintain their relative
values at a distance from the source or even tend to ‘heal’ causing
the ratio to fall rather than rise as is actually the case.

Within 14.4 wave-lengths (7.1 km.) of the transmitter the pattern,
so apparent beyond 30 wave-lengths, merges into one deep shadow
a cross-section of which is shown in Fig. 29. The abscissa of this
curve is in degrees measured from the transmitter so that the center
of the two most distinct low field strength channels extending north-

east may be inserted with their true radial relation. The two most
evident in Fig. 1 are shown to be west of the line extending from
transmitter through the center of the obstruction located in Fig. 26.
The presence of Long Island Sound east of the geometrical center
of the shadow has made an extensive survey of this section imprac-

Fig. 31—Plot showing intensity of definition of wave interference pattern

tical. However, a single section taken across the Sound at about
90 wave-lengths from the station shows quite unquestionably the
presence of a low channel about as indicated to the right of the ob-
struction designated in Fig. 29.

We have, therefore, a deep shadow with a more or less orderly
array of maxima and minima within its limits. These maxima
and minima grow more distinct at a distance from the transmitter,
contrary to what we might expect for ordinary shadows. Further-
more, we find that they move as the frequency is changed. These
facts lead to the belief that the phenomena in question are due to
wave interference such as has already been described in connection
with night-time fading, but characterized by very much smaller path
differences. This daytime interference condition is fixed while we
have seen that the nocturnal patterns appear to wander continually.
To explain this more in detail let us return to the shadow and con-
sider the phenomena which might accompany it in a little more
detail.

The study of light has made available much information concern-
ing the subject of wave interference. It is known, for instance, that
the edges of shadows are not sharply discontinuous changes from
light to darkness, but that a series of dark and light bands, called

diffraction fringes, are interposed between the full light and full dark
areas. In our radio case the distance from the source to the obstruc-
tion and the dimensions of the obstruction are both very much smaller,
in comparison with the wave length of the radiation, than for any
ordinary case in light, but apparently the phenomenon is of the
same general nature. By applying the ingenious principle of sec-
ondary sources used by Huyghens we might theoretically determine
the distribution of the field beyond an obstruction placed in the
path of the advancing radio waves. The basis of this principle is

Fig. 32-—Theoretical cross-section of radio shadow and associated wave interference
pattern

the assumption that each elementary part of the advancing wave
may be considered as a tiny transmitter. The effect at any point
behind an obstruction, therefore, becomes the resultant effect, con-
sidering phase as well as amplitude, of the waves from all these minia-
ture sources.

In Fig. 32 the region between vertical lines (a) and (b) represents
the geometrical limits of the cross-section of a well defined shadow
taken some distance behind the obstruction. An analysis of the
resultant field using Huyghens’ construction would show variations
in intensity somewhat as represented by the full line. In other
words the shadow will not be distinct but will have alternate maxima
and minima within its geometrical limits and similar variations
beyond the edges.

It is very likely, of course, that even in case the foregoing specu-
lative analysis of the contour pattern extending north-east of 2XB
is fundamentally correct, a great many other influences than that

of obstruction enter into the final field distribution. Relative atten-
uation of water and land appear to influence the distribution con-
siderably though not as definitely as do steel structures close to the
transmitter. Distinct minima appear both on the Hudson and on
the Sound along radial lines extending from the transmitter.

Probably refraction of the wave front in passing across shore lines
also enters into the shaping of this pattern.

Perhaps as good an elementary picture as any of the phenomena
causing these patterns is that of a ‘“‘dent’’ produced in the wave
front by an encounter with a portion of New York City’s impressive
skyline. Since radio waves travel in a direction perpendicular to
the plane containing the electric and magnetic fields, opposite sides
of this ‘“‘dent’’ would cross over one another with the result that an
interference pattern would appear beyond the obstruction. An
analogous situation exists when a water ripple passes a cluster of
marsh grass which, damping its motion and retarding its progress
causes part of the advancing front to converge and cross beyond the
obstruction.

There is evidently a relation between day patterns such as have
been discussed and night-time conditions. Just what this relation
is offers some further opportunity for conjecture. In the first place
quality distortion in transmission at night was, as previously ex-
plained, observed over parts of the region covered by the pattern
shown in Fig. 1. The worst distortion seemed to be somewhat
associated with the low field strength regions in this daylight survey.
The distortion seemed also to be worse along the low channel ex-
tending in the direction of New Canaan, Conn., and beyond the
100-wave-length circle. It was particularly bad at a distance of
some 140 wave-lengths from the station along this low channel where
the field strength became so low in the daytime as to be unmeasurable
with the set employed for the work. Accompanying the poor quality
were fading and marked directional shifts.

Quality distortion though not so consistently severe at the River-
head station as in the vicinity of Stamford was at times easily detect-
able by audible tests. Due to rapid attenuation of the radio waves
traveling from the site of 2XB across Manhattan and the length of
Long Island the field strength around Riverhead is generally low
with higher levels north and south on the open waters of the Sound
and Ocean respectively. Night-time fading at this point was rep-
resentative of the variety which is usually found at distances of
approximately one hundred miles from a broadcast transmitter.
The situation at Riverhead appears to be somewhat the same as

that which may exist over a large part of the broadcast area at a
distance from the transmitter, while in the Westchester region we
have an extreme and rather special circumstance. Field strength
surveys have shown that there are indications of a daytime inter-
ference pattern over the Riverhead area but this pattern, such as
it is, appears to be irregular and to lack the definition which makes
the Westchester pattern so remarkable.

On the basis of the Westchester data alone we might build up a
theory to the effect that night-time shifts of the stable daylight
pattern were in some way responsible for quality distortion fol-
lowing the departure of daylight. Such a thought applied to the
Riverhead case does not seem so reasonable since here the pattern is
about one-quarter as distinct in terms of the ratio of maxima to
minima values as the Westchester pattern. If, however, we presume
that quality distortion may be expected in areas where daytime
signals arrive considerably attenuated or so interfering as to simulate
such an attenuated condition both situations are satisfied. After a
consideration of the evidence at present available, such a conclusion
seems attractive; that is, a daytime wave interference pattern alone
is only an agency in night-time quality distortion in so far as its
minima in combination with the general shadow effect are responsible
for a low signal directly transmitted. Perhaps, in other words, the
daytime field strength is a measure of direct night-time transmission,
there existing in combination with this direct path at night a second,
variable route of greater effective length. Probably close to the
“indirect”
but shadows or interference may materially modify the ratio.

By receiving simultaneously at several points the signal coming
from a distant transmitter, it ought to be possible to detect the
movement in space of these interference bands we have been dis-
cussing. The question immediately arises as to how far apart these
distributed receivers can be placed without giving us an entirely dis-
continuous and misleading picture. For the first step toward record-
ing space variations, in the vicinity of the Riverhead testing station,
the receivers were spaced 1,16 wave length (30.5 meters), as illus-
trated in Fig. 33. It is necessary in making such determinations to
transmit a single radio frequency, since we have already found that
the interference bands for one component of a modulated wave are
likely to be in a different position than those for another.

In order to receive and record the radio frequency wave it is, as
has already been shown, convenient to use a local oscillator to beat
it down to audible values. Since several oscillators for the separate

employed. This beating oscillator was situated at the testing station
and the receiving antenna at this point was used as a radiator. In
order to prevent overloading, the local receiver, the coupling to the
receiver input coil was balanced to give a minimum of the local signal.

Fig. 34 is a sample of the record obtained. The continuous shadow
band at the top represents the local receiver output. One oscillator

element was used for the other four receivers, their signals being
recorded successively by a commutating device. Incidentally the
interaction between these receivers was checked by observing the
output of any one, while changes were made in the tuning of the
others. The antenna was, however, so nearly aperiodic that no
recognizable distortion or reradiation phenomena could be detected.

Fig. 35 illustrates compactly variations recorded by the oscillo-
graph records (of which Fig. 34 is a sample), for a representative
period of about five minutes. Even within the dimensions of 1/16

Fig. 35—Curves showing single-frequency fading on spaced receivers, condensed from
long record

wave length there appears to exist transient field strength gradients
in the direction of transmission. This is shown by a change in rela-
tive values, in the upper set of curves which represents field strength
at points 1/16-wave length apart in the direction of transmission.
The deviation is particularly noticeable in the relation between
values for the local receiver and the ‘‘West receiver’ which is in the
direction of the transmitting station.

The lower set of curves, representing similar values across the line
of transmission are much more nearly parallel. From the data so far
obtained for the Riverhead testing site, it seems that transient night-
time field strength gradients are more generally evident in the direc-

tion of transmission than perpendicular to this direction. Upon
these limited data one might be tempted to predict the presence of
interference bands across the line of transmission.

The above discussion concerning space relation of field strengths
has been included merely by way of contributing an additional bit
of evidence to the theory that the erratic type of fading ordinarily

Fig. 36-—Single-frequency fading record from vertical antenna and two-loop antenna
crossed at right angles

experienced at night time is due to wave interference. The picture
is very small in terms of wave lengths but considering its content, its
very limits seem to imply wave interference rather than attenuation
alone.

In connection with the wave interference theory thus far suggested
as responsible for a major part of fading Fig. 36 is introduced as
added evidence. The middle record of this group represents am-
plitude changes in the night-time reception of a carrier wave upon
a vertical antenna. The upper and lower records represent the
same for two loops turned at right angles to one another in the hori-
zontal plane. By daytime tests the interaction of this combina-
tion was found to be negligible. Night-time fading recorded simul-
taneously for these three separate receivers occupying as nearly the
same point in space as was possible, show that a high amplitude

signal may be coming in on both loops while the vertical antenna
pick-up approaches zero. Several points of this kind are marked by
arrows below the middle trace in Fig. 36.

There are at least two simple possibilities which might account
for these relations. In case the wave approaches the receiving
point from directly overhead, the vertical antenna would receive a
“zero” signal while the loops would pick up an amount depending
upon the state of polarization. If this be true, the records indicate a
very rapid shift from the vertical direction of reception since the
antenna minima are short lived most of them lasting at best a small
fraction of a second.

On the basis of wave interference it is apparent that two waves
approaching the receiving point in a 90-degree space phase relation
and 180 degrees out-of-time phase could give a maximum signal
on the two loops while that received on the vertical antenna was a
minimum.

A compromise between these two viewpoints is probably a better
guess than either one of them taken alone. That is, the existence
of minima on the vertical antenna at the same moment that a strong
signal is coming in on the loops is perhaps due to the interfering
combination of waves having components in both the vertical and
horizontal planes.

So far the data shown have been limited to the results of observa-
tions taken on special forms of transmission which are simplified
for the purpose of clearly exposing the basic facts. We wish now
to consider some of the more practical aspects of signal distortion.
The first test which we made at our field test station was to record on
slowly moving photographic paper tape and on the high speed film,
the detected audio signal which resulted when the transmitter was
modulated by a pure 26-4-cycle tone.

Fig. 37 is a sample of the general type of audio signal record ob-
tained and Fig. 38 shows copies of the wave shape of the received signal,
at particular times corresponding to the numbers of the oscillograms
on the records in Fig. 37. The abrupt displacement of the timing
trace indicates the point on the long record at which the snap-shot
oscillogram was made. A peculiar characteristic of these records is
the dark shadowy lines weaving back and forth through the band
recording the complete signal. These dark lines correspond to the
kinks in the wave shape shown in Fig. 38. As explained before, the
darkening of the record is caused by the greater quantity of light

trace signal from vertical antenna receiver and lower trace signal from loop antenna
receiver, timing marks 2.6 seconds apart

Fig. 38—Wave form of signals corresponding to numbered positions indicated ¢1

affecting the record at these peak points. At the same time these
observations were made, the wave shape of the signal rectified from
the antenna current at the transmitter was recorded by an oscillo-
graph. These oscillograms showed the signal to be free from dis-
tortion at the transmitter.

The weaving of these shadowy traces together with their width
gives a record of the change in phase and amplitude of the irregu-
larities in the wave shape of the signal. Although the wave shape
of the signal is continually changing, it persists in substantially the
same form for a great many cycles. Thus the record shows that,
in the transmission of this simple tone modulated signal from the
transmitting to the receiving antenna, it has been so modified that
entirely new frequencies appear at the receiver. This receiver was
shown by local tests to be free of any appreciable distortion within
itself. While these new frequencies look like harmonics of the mod-
ulating tone in the snap-shot record it is obvious from the slow record
that they are not true harmonics but that they differ from the har-
monics by a very small amount and are incommensurable with the
modulating tone since they undergo progressive but irregular phase
changes with reference to it.

These records represent in a nutshell the signal distortion problem
as it first presented itself to us. Our work then consisted in raveling
out the complicated relations so that their nature could be ascer-
tained and a theory of the causes established. In this paper, in the
interest of clarity of presentation we have departed considerably
from the actual order of the experimental work but at this point
perhaps the actual order is best to follow for a moment.

With such a weird-looking distortion to analyze, and if possible
eliminate, our first thought was as to whether the terminal apparatus
might not involve unrecognized peculiarities which would be a con-
tributing cause. Local tests and daytime tests of the receiving
system absolved it from doubt and attention was focussed on the
transmitting apparatus.

It was suspected that present day radio telephone transmitters leave
something to be desired in regard to what we may call, for lack of a
better term, their dynamic frequency stability. A very large per-
centage of the transmitters in use throughout the world today pro-
duce amplitude modulation of the carrier by the action of modu-
lating tubes directly upon an oscillating tube circuit. It is to be ex-
pected that the cyclic changes in circuit conditions occurring at the
modulating frequency will have some cyclic effect on the absolute
frequency of the carrier and that this effect will be in the nature of a

wobbling or rapid shifting back and forth in frequency of the ampli
tude modulated carrier. In other words the carrier and side-bands,
without change in their relative frequencies, would be subjected to
“frequency modulation.”

This sort of thing should be clearly distingushed from the slow
wandering of frequency which, for instance, causes beat notes between
carriers of different stations to drift gradually in pitch. What we
have called “dynamic instability” is so rapid (being governed by the
evclic variations of the modulator) that it is difficult to observe by
anv aural method. Since the transmitter being used for our tests

was a member of this almost universal class which employs modulating
elements directly associated with the oscillator elements we deter-
mined to study this aspect of the transmission.

The following test was made to find out the extent of the fre-
quency variation during the period of the modulating cvcle. A
schematic of the testing circuit arrangement is shown in Fig. 39.
The plan was to modulate the carrier with 33 evcles, a tone so low
in frequency that it would not be ethciently transmitted through the
audio frequency amplifier connected to the output of the radio re-
ceiver. Then upon beating the received modulated carrier signal
down to a frequency of about 1,000 cycles, an oscillogram of this
signal would show a 1,000-cycle signal with a 33-cyvcle modulation in
amplitude. Frequency modulation, if present, should then be easily
discernible from the record. This experiment was made for day-
time transmission and oscillograms (A) and (B) shown in Fig. 40
were obtained, one with the frequency of the beating oscillator greater
than the carrier frequency, and the other with the beating oscillator
frequency less than the carrier frequency. Both of these oscillo-
grams show by the change in the frequency of the beat note signal

that frequency modulation occurs in the transmitter circuit. The
.
frequency change is very apparent on the oscillograms when the

lengths of one evele at maximum and minimum amplitudes are com-
pared. The reality of the effect is demonstrated in the two records, |

big. 40 —-Oscillograms showing frequency modulation accompanying amplitude
modulation

the beating frequency is moved in frequency from one side of the
carrier to the other.

The next step was to determine to what extent a stabilization of
the carrier frequency to stop frequency modulation would affect the
distortion of signals. True, master oscillator transmitters capable
of giving the desired stability are not a new thing in the art. Several
such transmitters were built by the Western Electric Company some
vears ago and used successfully ship-to-shore radio telephone
experiments * in which frequency stability was of considerable im-
portance. To modify the ordinary broadcasting transmitter to in-

2 See Fig. 1 and accompanying discussion in: Radio Extension of the Telephone

System to Ships at Sea by Hl. W. Nichols and Lloyd Espenschied Proc, I. R. E., Vol.
IT No.3.

fel ) FEIN EC / JOIN AE
“ce ae reased frequency points with reference to the modulation cycle when
|
Tak

clude this feature involves major mechanical changes and in order to
provide a suitable arrangement for these tests the Bell Telephone
Laboratories engineers merely added to the existing transmitter at
station 2XNB a temporary separate oscillator and high-frequency
amplifier which could be connected to drive the oscillator tubes of
the set as amplifiers. That this was free from frequency modulation
is seen by comparing (C) of Fig. 40 with (A) and (B

Fig. 41-—-Slow record of signal detected from tone modulated transmission with stabil-
ized carrier showing reduction in distortion. Made at Stamford, Conn., Oct. 10, 1924,

with Fig. 37. Fig. 41 like Fig. 37 is the detected result of a signal
which started from the transmitter as a pure tone modulated signal,
but it shows that much of the wave form distortion has disappeared,
there remaining only a residuum which characteristically appears
at the lower amplitudes of the signal. The probable cause of this
residual effect will be discussed later. Tests of speech and music
were concurrent with these findings. Using the normal transmitter,
night-time transmission as received at the test stations was seriously
distorted. When the stabilizing arrangement was emploved this
distortion was apparently eliminated except at the minima of fading.

Having arrived then at this practical result we wished to make
further confirming tests, and tests to determine the whys and where-
fores of the result. We have already detailed the more basic of
these tests in previous sections of this paper and are now ready to
consider the practical distortion records more carefully and to build
up a theory to explain them.

The records shown in Fig. 42 are similar to the records in Fig. 37.
They are shown here to illustrate the difference in the characteristics

of the wave form distortion variation that occurs from day to day.
All these records were made at Stamford, Conn.

Strips 7 and S—Jan. 24, 1925 —-8:00 a.m.
Phere is a marked ditterence in the records obtained on January 23 and
24, which were made at the time an effort was being made to determine

big. 42--Slow record of signal detected from tone modulated transmission taken on
different days showing the changes in the character of the distortion

twisted appearance of the record obtained on January 24 is not very
common in the records obtained. Most of the records have character-
istics similar to those shown in Fig. 387. In the January 24 records
there is a marked change in the characteristic configuration of the
Variation,

In order to obtain a record of the amount of wave form distortion
resulting from frequency modulation present in the detected audio
signal the circuit arrangement shown in Fig. 43 was used. This circuit
was designed to analyze the wave form distortion when a 250-cycle

signal was used to modulate the carrier. Special precautions were taken
to obtain a pure 250-cycle modulating tone. The wave shape of the
signal detected from the carrier at the transmitter was frequently
checked by observations with an oscillograph. The signals detected
from the antenna current at the transmitter, both for the normal

J
Fig. 43-—Diagram of system used to obtain ‘Harmonic’ analysis distortion records

transmitter with frequency modulation and for the stabilized carrie:
transmitter, were practically simple sine waves. The output circuit
of the radio receiver was connected to a group of filters designed to
transmit narrow bands of frequencies straddling the harmonics of
250-cycles.

While below we have referred to the frequencies passing these
filters as “harmonics” it should be borne in mind that they are not
necessarily (rue harmonics since they deviate verv slightly from the
true harmonic relation. The purpose of the test was to procure a record
which would show at a glance the presence or absence of wave form
distortion.

The input circuits of the filters were connected in parallel and the
output circuits separately connected to the audio amplitiers arranged
to operate the oscillograph elements. The input of one amplitier was
arranged so that it could be switched either to the output of the tilter
passing 250-cvcles or the output of the radio receiver. In this way a
record could be obtained of either the whole tone from the receiver or
only the 250-cycle component.

In Fig. 44, Strip 1 is a harmonic analysis record of the audio tone
detected from the carrier and both side bands, transmitted with a
stable carrier frequency. Strip 2 is a section of a record made a few
minutes later when an unstabilized carrier was being used. On this
record the lower trace is the 250-cvcle component, the center trace the

200-cvcele component, and the upper trace the 750-cycle component.
Phe upper and lower traces have their zero lines at the edges of the
strip. This record was made at Riverhead, L. I., April 30, 1925, at
3:35 am. Strip 2 isa section of a record made a few minutes later when
in unstabilized carrier was being used.

The gain in the audio amplifiers connected to the outputs of the
hilters was adjusted to give nearly uniform transmission through the

big. 44--Slow record made with system diagrammed in Fig. 43. Contrasting the
listortion of detected tone transmitted by stabilized and unstabilized carrier frequency

receiving and recording apparatus for the frequencies recorded. Hence
in these records the relative amplitudes of the fundamental and
harmonies of the signal are directly comparable.

Strips 3, 4 and 5 in Fig. 44 are taken from a record made for the
purpose of obtaining a comparison of the wave form distortion sus-
tained by the detected audio signal transmitted by the normal trans-
mitter with frequency modulation present and by a stable frequency
transmitter. In each strip the lower trace is the whole tone from the
output of the radio receiver, the middle trace the second harmonic
500 eycles) and the upper trace the third harmonic (750 cycles).
Strip 8 and half of Strip 4 give the record obtained when the normal!
transmitter was used, and the remainder is the record obtained when
the moditied transmitter was used. There was a few minutes’ difference
in time between the ending of one transmitting condition to the begin-
ning of the next during which the master oscillator control was switched

the making of this record, so that the results obtained from the two
transmitters are directly comparable.

The record of the signal from the normal transmitter shows an
abundance of second and third harmonics, at times equal in amplitude
to that of the whole tone signal. The latter, of course, includes these
harmonics. It will be noted also that dark line shadows run through
the trace of the whole tone, indicating the presence of the wave form
distortion. The signal from the stable frequency transmitter as shown
by the record is practically free from wave form distortion. The

indicate wave form distortion. This record is substantial evidence
that a great deal of the wave form distortion may be eliminated when
the carrier is stabilized. However, the selective fading still remains.

The selective fading we have already explained more or less satis-
factorily and we tind that it does not materially atfect the wave form
of audible trequencies transmitted by a modulated stabilized carrier
unless its changes are more rapid than any we have recorded. The
crippled state of originally perfect tone waves after they have been
transmitted by an unstabilized carrier, we have just observed. Now
let us consider the possible causes of this ditference. The carrier stabil-
ization referred to here, may we repeat, is not stabilization against
slow variations in frequency from second to second or from hour to
hour but rather against rapid variations within the evcle of the modu-
lating frequency.

The reason for such changes over the modulating evcle is that
the variation of the impedance of a vacuum tube across the
oscillating circuit necessarily causes a variation in the nature period
of the oscillation. As a simple case, the circuit in Fig. 45 is given,

when rp is the plate resistance and the remaining constants are given
in the illustration.

Direct modulation by the usual method involves a cyclic change in
the value of plate resistance. Hence, according to the above equa-
tion, there results a cyclic change in frequency which, though rela-
tively small, becomes of the utmost importance when subjected to
the peculiar phenomena of night-time transmission.

By making certain assumptions concerning the nature of frequency
Variation as amplitude modulation takes place, it is possible to work
out distorted waves corresponding to various assumed wave inter
ference conditions at the receiver. Perhaps the most simple and
instructive means for producing these distorted waves is by a graph-
ical method.

The equation for modulation of a high-frequency wave by a single
tone may be written

When 1 represents the unmodulated amplitude of the wave, & is a
factor determined by degree of modulation, v is an angular velocity
of the tone wave and p is the angular velocity of the high-frequency
wave. The amplitude factor in this equation may be considered as a
vector which is undergoing a change in length in accordance with
the term included in the brackets. For the purpose of our analysis
we shall include the angular velocity imparted to this vector by the
last term in the above equation, since we are interested in the en-
velope of the resultant high-frequency wave at the receiver and the
relative phase relations for two waves directly and indirectly trans-
mitted combining to form this resultant. Since both carrier waves
are of the same mean frequency only the relative position need be
considered.

Now in our graphical determinations for the case of two trans-
mission paths different in length, we represent the two effective fields
by vectors varving in length in accordance with the amplitude factor
of equation (15). However, due to the difference in length of path,

In addition to the lag in amplitude there will be a lag in frequency
change over the frequency modulation cvele. This lag which has

Fig. 46—Graphical method of synthesizing distorted wave forms caused by frequency
modulation

already been shown in connection with the analysis of distortion in
certain types of band fading records (see Fig. 22), becomes a change
in the relative phase angle of the vectors under consideration. Thus
our picture finally becomes one of two vectors changing in length,
the changes in one continually lagging the changes in the other, the
two vectors at the same time undergoing what we might term a
relative angular wobble. “

In Fig. 46 these relations are produced graphically. For our pur-
poses we might assume that the vector representing one field is tixed

instant, for example, the directly transmitted field may be repre
sented by a; in this figure. Assuming a difference in length of pat!
we may compute on the basis of the integral equation (13), the rela
tive phase position of the vector representing the indirectly trans
mitted field ®). The relative amplitude of this vector may also |x
determined by substituting Ag in equation (14).

After establishing a sutficient number of vectors to represent. thi
evelic variation we may combine the respective components to obtair

g. 47 Synthetic wave forms showing distortion due to frequency modulation

shown as Ry, Rs, Rs, ete., a broken line being drawn through their
extremities to identify their positions. Now, if we plot these re-
sultants as vertical ordinates in their successive time relation as
shown on the lower right of Fig. 46, we have the envelope of the
resultant wave at the receiver.

When the mean position of the two vectors (a) and (b) in Fig. 27 is
ISO degrees separation, the signal is experiencing a fading minimum.
When they are on the average in phase the amplitude is at a maximum.
We can, therefore, trace a relation between quality distortion and
fading by such an analysis, assuming a constant percentage modula-
tion. Fig. 47 shows a series of high-frequency wave envelopes ob-
tained by this method of graphic analysis. The mean vector rela-
tion is represented by @, and for @, = 180“degrees the fading may be
considered at a minimum. The waves shown in Fig. 47 being en-
velopes of the high frequency will undergo certain changes in the
process of detection. These, however, would only slightly modify
the wave,

STUDIES IN| RADI
For purposes of comparison, a set of oscillograph pictures of rep-
resentative received wave shapes is shown in Fig. 4S These repre-

sent the actual effect. of night-time transmission with frequency
modulation between 463 West Street, New York City and Stamford,

Fig. 48 —Oscillograms showing actual wave forms with distortion re y
quency modulation
to correspond with the order shown in Fig. 47. There exists a striking

similarity. Occasionally, however, the shapes predi ted may depart
considerably from those obtained experimentally. As an example of
such a departure, the record (h) in Fig. 48 has been included. Such
unusual samples may be due to a combination of waves arriving over
more than two paths or it may be that the time variation of the
frequency is far from the simple sinusoid which we have assumed.
As a matter of fact, a critical mathematical treatment of this case
shows that only an approximation of such a sinusoidal condition

possible since as has been shown by Carson,' a frequency modulated
wave of this character consists of an infinite series of fixed frequencies
spaced at regular intervals either side of a “fundamental” carrie:
wave. Obviously only a small part of such a series could get out of
the transmitter or into the receiver due to circuit selectivity. For
the lower modulating frequencies, however, the approximation § in-
volved in the assumption of a simple sinusoidal variation is not far
wrong since the amplitudes of these side frequency components fall
off rapidly as their order in the series increases. While 150 wave
lengths difference in path length has been assumed for the synthesis
of the wave shapes in Fig. 47, this difference may according to the
data obtained amount to much more than this.

It may well be asked why this frequency modulation, since it pro-
duces such marked distortion at night in certain places, does not also
give rise to distortion by day or in locations where transmission is
steady. A full answer to this question would be far from simple. But
in brief it is because the carrier and side-bands shift in absolute fre-
quency together as a unit so that their relative or difference frequencies
which determine the audio signal remain unchanged. Another way to
put it is that the detector operates on the envelope of the high-fre-
quency signals and is blind to the frequencies contained within the
envelope except insofar as they affect the latter. However, since
frequency modulation appreciably widens the frequency band occu-
pied by the radio signals it is to be expected that the tuned circuits
in the receiver would have some reaction on those louder portions of
the signal for which the amplitude modulation and therefore the
frequency modulation is large. The perfection with which broadcast
signals may be received under suitable conditions leads one to believe
that this effect must be small.

It has been shown that serious wave form distortion of the repro-
duced signal may result if frequency modulation occurs with the am-
plitude modulation and the transmission is subjected to night-time
conditions. This distortion from frequency modulation can be elimi-
nated by stabilizing the carrier frequency. There remain some wave
form distortion and the annoying amplitude changes caused by selec-
tive fading which is one of the most serious present day problems in
radio transmission. Let us now consider the nature and cause of this

* See ‘“ Notes on the Theory of Modulation,"’ by John R. Carson, Proc. Institute
of Radio Engineers, February, 1922.

residual wave form distortion and some further consequences of selec-
tive fading under the assumption that there is no frequency modula-
tion involved.

The process of detecting audio signals from radio frequency signals
is, at least in its simpler aspects, well understood, but it may not be
generally appreciated that the action is such that the detected signals
may be greatly modified by changes in the relative amplitudes and
phases of the carrier and side-band components such as may
result from their transmission through the medium. That the ampli-
tudes and phases of the carrier and side-band signals are not neces
sarily received in the same relation that existed as they left the trans-
mitter has been pointed out earlier, in the discussion on selective

fading.
The usual expression for a high-frequency carrier wave of frequency
p/2z modulated by a low-frequency wave of frequency v 27 is
e=A[l+a sin sin pt

where Al is the carrier amplitude, a, the percentage modulation and
@ the starting phase of the modulating tone with reference to the
carrier. Expanded into its components this becomes

cos (pt+it+@ the upper side band
COS (the lower side band
+A, sin pl the carrier)
where ¢,;=¢2=6 and 4A,=A.,=A;=A as the waves leave the trans-

In the receiving set this funct‘on is squared by the action of the
detector and, neglecting direct currents and frequencies above the

original modulating tone and the second term the second harmonic.
From this expression several conclusions can be immediately

factor but the second term does not. Thus, if selective fading erases
the carrier at any time, reducing its amplitude to zero or a small value
the signal, represented by the fundamental tone, practically dis-
appears, even though the side-bands have not faded out, and there remains
only the harmonic. This is the residual distortion shown in Fig. 41
and which can often be heard during a fading out period. It is
caused by the two side-bands beating together in the detector. We
have here exposed a fundamental defect in the usual form of modu-
lated signal transmission. The amplitude of the received signal is
subject to all the whims of the carrier and to paraphrase freely an
old saving we might remark that a signal is no stronger than its
carrier. We may at once conclude that one way to reduce fading
is to suppress the carrier and resupply a constant amplitude carrier
at the receiving station.

Analyzing further the first term of the expression representing the
detected signal, the first part of the bracketed portion results from
beating together in the detector of the carrier and upper side-band
and the second part from the carrier and lower side-band. — It is clear
that one of the side-bands may fade out completely and the other
will still bring in the signal, provided the carrier is not also lost, with
a phase shift to be sure but nevertheless not seriously reduced in
amplitude. In telephony this kind of phase shift is relatively un-
important. Here we have an evident advantage in transmitting both
side-bands since they support each other's frailties. But if the twe
side-bands suffer phase shifts in transmission, as we have earlier
shown may be produced by wave interference, such that @; and @»
differ by w radians or LSO degrees, the two components will cance!
each other provided their amplitudes 4; and A» remain equal. In
other words all three components—carrier and both side-bands——may
arrive at the receiver with full amplitude and vet no signal will be
detected from them except a second harmonic component. This is
obviously a disadvantage of transmitting both side-bands since, at
such an instant, if one of them were eliminated the signal would
reappear.

We conclude that there is, on the basis of such a brief analysis,
not much to choose between single side-band and double side-band
transmission when the carrier is transmitted also.

But if we wish to realize the advantages of carrier suppression a
choice is not difficult. A carrier suppression system in which both
side-bands are transmitted requires that the replacement of the
carrier at the receiving station be done with almost absolute accuracy
as to frequency and phase, a thing which involves very serious prac-

;
\
Ses tical problems. On the other hand if but a single side-band is trans
ue 4 mitted the difficulty is reduced to placing the carrier within a vers
few cycles of its correet: position. The allowable departure will
Ins i depend on a number of things but there is reason to believe that for
1] 3 high quality transmission it must be very small, perhaps no greater
is than two or three cycles.
Ve With the single side-band carrier suppression method, invented by
lu John R. Carson, the radiation is stripped down to the minimum which
Is will fully transmit the telephonic signals and this reduces to a minimum
an the exposure of the signals to the ravages of selective fading. If the
its : spacing interval of the fading is relatively narrow as in the cases we
ng have examined hereinbefore, this form of transmission would not
er i fade seriously in average volume but would be subjected to a continual
changing of its frequency-amplitude characteristic, that is to say
he individual frequency components would fade progressively as the
m minima of the selective fading wandered back and forth across the
1d d frequency range encompassed by the single side-band. If the spacing
ar interval of the fading were very large so that the minima were very
er ; broad of if some other, at present unexplored form of fading which
th covers a wide band at one time were acting, the signal would fade in
1) average volume but the range of its variation would be only the square
n- ; root of that of a carrier transmitted signal, since only the side-band
h i would fade and the locally supplied carrier would remain unchanged.
( : The extent to which these theoretically drawn conclusions may be
" realized in practical application Is vet to be determined but we have a
D» few records bearing upon the matter which at least do not run con-
trary to them.
n All of the transmission tests where the radio signal was beat with a
\ local oscillator and the detected beat note observed, were equivalent
e to single frequency single side-band transmission with carrier sup-
IS pression, the local oscillator functioning as the carrier suppressed at
it the transmitter. In this case, for which a number of records have
(| already been shown, the detected signal is in proportion to the product
of the amplitudes of beating oscillator and received radio signal. The
f. phase of either does not affect the amplitude of the audio signal. Hence,
‘| the only important modification of the original signal is the variation
in the amplitude resulting from selective fading.
7 Unfortunately we have no records in which a direct comparison is
1 made between single side-band transmission with and without carrier
= suppression but the case can be visualized from the record shown in
Fig. 12 or 13. Here each one of the frequencies recorded may be looked
upon as a single side-band frequency which has been detected through

the agency of the resupplied carrier of the beating oscillator used to
bring them down to audio-frequency. If now we were to take two of
these frequencies shown on the record and multiply their amplitudes
together at each point we would obtain the amplitude of the signal
which would result if one of them were a single side-band and the
other its accompanying carrier. It is obvious that the fading varia-
tions would thereby be increased in amplitude and rapidity.

In order to obtain a comprehensive picture of the relative advantages
of radio transmission using a Carrier and one side-band as compared

Fig. 49 —Diagram of system used to obtain records of transmission with carrier and
one side band and carrier and both side-bands

with the common practice of transmitting both side-bands, the follow-
ing tests were made. The schematic diagram of the circuit arrange-
ment is shown in Fig. 49. At the transmitter the carrier and both
side-bands are transmitted and at the receiver they were selected out
by means of filters in the manner previously explained. The signals
from the filters corresponding to the carrier and lower side-band were
applied to the input of a detector circuit and from its output the de-
tected difference signal was selected by a low-pass filter. This signal
was equivalent to that which would be received if only the carrier and
one side-band were transmitted. From the output of the radio receiver
a branch circuit goes to a low-pass filter which transmits only the
signal detected from the carrier and both side bands, suppressing from
this circuit the higher frequency signals corresponding to carrier and
side-bands produced by the beating oscillator and received signals.
By making simultaneously a record of these two signals a direct
comparison is obtained of the effect of selective fading on their am-
plitudes. Fig. 50 shows samples of several such records made at
Riverhead, L. I. The modulating frequency for strips 1, 2 and 3 is
250-cycles, and for strips 4 and 5, 500-cycles. The record on strip :
is shown on account of the peculiar characteristic of the signal fading,
for considerable periods of time remaining at relatively low amplitude.

In these oscillograms the upper trace is the record of the signal from
the carrier and both side-bands, and the lower trace the signal from
the carrier and lower side-band.

Fig. 50-——Slow record comparing the signal detected from carrier and one side-band

with signal detected from carrier and both side-bands. Made at Riverhead, L. 1.

Upper trace carrier + both side-bands, lower trace carrier + one side-band. Strips 1

signal is detected from both side-bands. ‘The amplitude of the signal
from both side-bands in some instances is very small but appreciable
amplitude is still indicated at the same instant for the signal from one
side-band. This is explained as meaning that the side-band phases
were such as to make the component signals 180 degrees out of phase
after detection and that the amplitudes of the components were
practically equal. The reverse situation is also observed where the
amplitude of the signal detected from the lower side-band is zero and
appreciable signal is recorded for the case where both side-bands are
used. This is interpreted to mean that the side-band signal was
eliminated by selective fading. In this event it was, of course, not
contributing to the signal which was detected from both side-band
signals. The recorded signal comes from the other side-band which
evidently was not eliminated at that instant by selective fading.

Visual observations made with the cathode ray oscillograph, which
unfortunately furnishes no permanent record of transient effects, con-
firmed the strip records in regard to the reality of there being side-band
phase variations. From equation (17), it is seen that if these varia-
tions occur the fundamental’of the detected tone signal at the receiver
will not bear a fixed phase relation to that detected from the trans-
mitting antenna current while if there are no such changes the phase
between these two tones would remain constant. The locally detected
tone and the tone detected from the transmitting antenna current and
brought to the receiving station over telephone Wires, were applied
to the two pairs of deflecting plates in the cathode ray oscillograph
Since the detlections caused by these two pairs of plates are at right
angles to each other the resulting Lissajous figure from two sine waves
of the same frequency will be a slanting line, an ellipse or a circle
depending on their phase and amplitude relation. The actual figures
were observed to change progressively through this range of shapes,
the changes following roughly the magnitude and rapidity of the
fading. The efiect of amplitude changes on such figures is quite
distinct from the effect of phase changes and there was no difficulty
in separating out the evidence of large phase changes.

Considering only the above theories and facts there appears to be a
reasonable basis for a conclusion that the best form of radio transmis-
sion for use in broadcasting is single side-band with carrier suppression.
But on practical grounds we do not believe such a conclusion is justified,
The fading and distortions which we have made much of in the pre-
ceding pages are not experienced by the majority of broadcast listeners
when they listen to local stations. To require these listeners to prov ide
themselves with more complicated and expensi\ e receivers, simply to
allow more distant or less favorably situated listeners to obtain better
reception, seems neither reasonable nor desirable. The art offers
several other possible avenues toward improvement much less difficult
of application and it must be remembered that radio broadcasting is
already reaching a degree of standardization and a volume of existing
receiving equipment which rules that changes must come slowly and
without serious prejudice to the existing order.

Subject to the limitations imposed by the scope of our investigations
the following conclusions may be drawn:

Fading can be quite sharply selective as to frequency and the evi-
dence points toward wave interference as the cause.

The evidence for wave interference indicates that some of the energy
of received signals reaches its destinations by a circuitous route and
suggests that this route is by way of upper atmospheric regions.

Quality distortion may result from dynamic instability of the trans-
mitter.

Abstracts of Bell System Technical Papers
Not Appearing in this Journal

New Methods and Apparatus for Testing the Acuity of Tearing.
Harvey Furercuer. This paper presented before the American
Otological Se CIEL, classifies hearing tests in four groups according to

1. Industrial or those made for determining the fitness of a candi-
date for employment. In certain types of work it is particularly
important that a prospective employee meet a definite requirement
for acuity of hearing. Tests made in the army and navy for various
branches of service are conspicuous examples of this kind of test.

2. Educational or those made for determining the degree of hearing
of school children both in the public schools and in the schools for the
deat for the spectal purpose of determining the proper methods to be
used in their education.

It is highly desirable that a single scale be used for representing
the degree of hearing which is independent of the method used and
which has a general application to the four purposes enumerated.
Such a seale is proposed and it is shown how the commonly made
voice test, watch tick, acoumeter, coin click and tuning fork tests
can be expressed in terms of hearing loss units on this seale.

The paper is concluded by summarizing the different methods for
testing the acuity of hearing which are as follows: (1) voice tests,
2) phonograph audiometer, (38) hearing loss for speech calculated
from audiogram, which audiogram may be obtained in three ways,
(a) tuning forks (constant initial amplitude), (b) tuning forks (com-

The Relation Between the Loudness of a Sound and Its Physical
Stimulus. J. C. STEINBERG? Experiments with many types of
sounds have shown that the loudness of a sound is a function of its

sounds whose calculated values L are equal will appear equally loud
to the average normal ear. P; is the r.m.s. pressure of the ith com
ponent of the sound wave. The weight and root factors HW and +,
respectively, are functions of the sensation level, which is svnonyvmous
with the term loudness as formerly used and is detined as

where Pi is the r.m.s. pressure of the ith component when the com-
plex sound is at the threshold of hearing. In case the components in
a narrow band of frequencies An are not resolved their energy must
be integrated to obtain the energy of the equivalent single component.
The root factor 7 is inversely proportional to the ratio of the minimum
perceptible increase in energy to the total energy. For intensities
near the threshold, the weight factors are equal to the reciprocals of
the minimum audible pressures. Curves are given showing the
values for HW" for various frequencies at various sensation levels, also
the values of 7 as a function of S. As the intensity is increased the
weight factors give greater weight to the lower frequencies; hence,
even though the amplitude of the sound wave be increased without
distortion, the ear will perceive both an increase and a distortion.
This effect is due to the non-linearity of the ear.

ear, Where NV is less than 5 or 6 eveles, two kinds of binaural beats
are obtained. Objective binaural beats are heard for most values
of f within the audible frequency range, provided there is the proper
difference in amplitude between the two tones. For telephone re-
ceivers as sound sources, this difference for best beats is about 55 Tl
and for the same receivers supplied with sponge-rubber cushions
about 62 TU. These beats are heard because the louder tone is con-
ducted through the head to the ear of the weaker tone and the two
tones there are about equally loud. Subjective binaural beats are
heard for frequencies below S00 or 1,000 cycles when the tones at the

iwo cars have about the same amplitudes, differing by not more than
25: Fi Data obtained with 22 observers are summarized. The
evidence indicates that these beats are not due to cross conduction
but are of central origin and the result of the sense of binaural local-
ivation of sound by phase. If the beats are slow (ess than 1 per
they are generally recognized as an alternate right and left
localization, though some observers may report one or more intensity
maxima during the beat eyvele. Such maxima are explained as the
result. of one’s interpreting the sound as louder when localization
is more definite. Fast beats (more than | per sec.) are generally
recognized as an intensity fluctuation. They are explained by assum-
ing that the sound appears louder when the phase relations are such
that it is normally best localized in’ the position toward which the
attention is directed. This explanation is supported by observations
made with a constant source rotating around the head of a listener.

Effect of Tension Upon Magnetization and Magnetic Hysteresis in
Permalloy. O. E. and L. McKEENAN.! Wires. ot
five nickel-iron alloys containing 45, 65, 78.5, Sl and S4 per cent.
Ni, 60: em. long and O.1 em. in diameter, were studied by a ballistic
method, for tensions up to 10,000 Ib. per in’ and fields up to satura-
tion (10 to 20 gauss). Permalloy with SI per cent. Ni is nearly in-
different to tension in its magnetic behavior; permalloy with less
nickel is more easily magnetized and has less hysteresis when under
tension, While S4 per cent. permalloy is more dithicultly magnetized
and has greater hysteresis when under tension. The saturation
values are independent of the tension. In 78.5 per cent. permalloy,

under a tension of 3,560 Ib. per in, saturation is reached at only 2
gauss (and is practically complete at 0.2 gauss) and the hysteresis
loss is only SQ ergs per cm.’ per cycle, so small that it may be regarded
as due to slight inhomogeneity rather than to any essential features
of the magnetization process. Relation to crystal oricntation, X-ray
examination proves that this abnormally low loss is not due to any
peculiar orientation of the crystal axes as the crystals are found to be
oriented at random. Miagnetostriction behavior can be deduced
from these results. Above SI per cent. Ni, permalloy contracts like
Ni while below SL per cent. Ni, permalloy expands like Fe.

eter, Was determined experimentally and found to vary from a maxi”
mum of 1.6% 107" to a low value, the changes being like these pre-

A Contribution to the Theory of Ferromagnetism. McIWEEHAN
Relation of permeability and hysteresis to atomic magnetostrictio
In permalloy, it has been found that magnetostriction changes sign
about Sl per cent. Ni, hysteresis losses can be made vanishingly
small near this composition, and these effects are not due to the special
alignment of crystals. It is suggested that in every ferromagneti
material the process of magnetization involves (1) intra-atomic
changes, presumably changes in the orientation of electron orbits,
governed by quantum dynamics and independent of environment;
and (2) inter-atomic changes (stresses and strains). The = inter-
dependence of the inter-atomic changes and the intra-atomic changes
is conveniently described as atomic magnetostriction. On this view,
hysteresis loss and magnetic hardness are due to the energy required
to produce, in succession, the local deformations associated with
changes in the magnetization of single atoms or small groups of atoms.
High initial permeability and low hysteresis loss in) permalloy are
explained as resulting from locally compensatory atomic magneto-
strictions of the nickel and iron atoms in small groups. The tunda-
mental differences in the magnetic behavior of Fe, Ni and Co are
attributed to differences in their atomic magnetostrictions. Other
differences are attributed to differences in the mechanical proper-
ties which alter the energy expended when atomic magnetostriction
takes place.

Induction from Street Lighting Circuits: Effects on Telephone Circuits
R. G. MeCurpy.® Synopsis. This paper discusses) series street
lighting circuits from the point of view of their relations to nearby
telephone circuits. These lighting circuits often have a much greater
inductive influence in proportion to the amount of power transmitted
than have most other types of power distribution or transmission
circuits. This is due to the relatively large distortion in wave shape ot
voltage and current on certain types of these lighting circuits, and to
the unbalanced voltages to ground which occur with series livouts.
Three general types of lighting circuits are discussed. These are a ec,
are circuits, d-¢, are circuits supplied by mercury are rectifiers, and
alternating-current incandescent circuits. Of these, the incandescent
type of circuit, in which the lamps are equipped with individual
series transformers or auto-transformers, is the most important in
this respect. Measures for reducing interference from these circuits
are discussed.

Power Distribution and Telephone Circuits. Inductive and Physical
Relations. H. M. TRUEBLOOD and D. IT. Cone? Consideration of
the relation between power distribution and telephone systems is
naturally involved in the comprehensive review of the problems of
the rapidly expanding power distribution networks in this country.
Pending the completion of studies now being actively carried on in
this comprehensive review, a preliminary and qualitative discussion
is given.

Situations of exposure fall into three groups determined by the
character of the area served. (1) “downtown” districts; (2) residential
urban districts; (3) rural districts. The major problems arise in the
second group. A wide variety of arrangements characterize both
systems, and require consideration.

Among technical features, coethcients of induction for close ex-
posures, shielding action of metallic cable sheaths for both power
and telephone circuits, and “ground potential” effects, are distinctive
problems. Where both classes of circuits are in cable with suitable
precautions as to grounding, interference is rarely to be anticipated.

Noise induction from power-distribution circuits is chiefly from
residuals, which occur on single-phase branches of polyphase circuits,
or where triple harmonics or load-current unbalances are introduced
by grounding neutrals, or where admittances to ground of phase
Wires are unequal. Residual currents are largest in systems having
multiple-grounded neutrals, both load currents and triple harmonics
occurring. Approximate resonance at triple harmonic frequencies
between the inductance of station apparatus and power cable capaci-
tance has characterized several situations. Various single, two and
three-phase arrangements are compared from the induction standpoint.

The closely related matter of unbalances in the telephone plant is
brietly discussed.

BANCROFT GHERARDI, M.E., M.M-.E.. Cornell Universitv. Engi
neering assistant, 1895-09; traffic engineer, 1899, New York Tele-
phone Company; chief engineer, New York and New Jersey Telephone
Company, 1900 06; assistant chief engineer, New York Te lephone
Company, and New York and New Jersey Telephone Company,
1906-07; equipment engineer, American Telephone and Telegraph
Company, 1907-09; engineer of plant, 1909-18; acting chief engineer,
1918-19; chief engineer, 1919-20; vice-president and chief engineer,
1920-——-. Mr. Gherardi’s work in the field of telephony is too well
known to require comment.

RoBERT W. Kina, A.B., Cornell University, 1912; Ph.D., 1915;
assistant and instructor in physics, Cornell, 1913-17; Engineering
Department of the Western Electric Company, 1917-20; Depart-
ment of Development and Research, American Telephone and Tele-
graph Company, 1920-21; Information Department, 192]

WALTER A. SHEWHART, A.B., University of Hlinois, 1913; A.M.,
1914; Ph.D., University of California, 1917; Engineering Depart-
ment, Western Electric Company, 1918S 24; Bell Telephone Labora-
tories, Inc., 1925 —. Mr. Shewhart has been engaged in the study of
the relationship between the microphonic and physicochemical prop-
erties of carbon.

HARVEY FLETCHER, B.S., Brigham Young, 1907; Ph.D... Chicago,
1911; instructor of physics, Brigham Young, 1907-08; Chicago,
1909-10; Professor, Brigham Young, 1911-16; Engineering Depart-
ment, Western Electric Company, 1916 24; Bell Telephone Labora-
tories, Inc., 1925—. During recent vears, Dr. Fletcher has conducted
extensive lavestigations in the fields of speech and audition.

Joun R. Carson, B.S., Princeton, 1907; E.E., 1909; M.S., 1912:
Research Department, Westinghouse Electric and Manufacturing
Company, 1910-12; instructor of physics and electrical engineering,
Princeton, 1912-14; American Telephone and Telegraph Company,
Engineering Department, 1914-15; Patent Department, 1916-17;
Engineering Department, 1918; Department of Development and
Research, 1919 -.) Mr. Carson's work has been along theoretical
lines and he has published many papers on theory of electric circuits

Kart K. Darrow, S.B., University of Chicago, 1911; University
of Paris, 1911-12; University of Berlin, 1912; Ph.D., in physics and
mathematics, University of Chicago, 1917; Engineering Depart
ment, Western Electric Company, 1917-24; Bell Telephone Labora
tories, Inc., 1025. Mr. Darrow has been engaged largely in’ pre
paring studies and analyses of published research in various fields
of physics.

Ratepu Bown, M.E., 1913, M.M.E., 1915, Ph.D., 1917, Cornell
University, Captain Signal Corps, U. S. Army, 1917-19; Depart-
ment of Development and Research, American Telephone and Tele-
graph Company, 1919—. Mr. Bown has been in charge of work
relating to radio transmission development problems.

DrLoss K. Martix, B.S., Polytechnical College of Engineering,
1920; UL S. Navy, 1918S 1919; Department of Development and
Research, American Telephone and Telegraph Company, 1919
Mr. Martin's work has related particularly to radio broadcast trans-
mission.

Ratpn K. Porrer, B.S., Whitman College, 1917; E.E., Columbia
University, 1923; U.S. Army, 1917-19; Department of Development
and Research, American Telephone and Telegraph Company, 1925
Mr. Potter has been engaged in experimental work relating to radio
transmission phenomena.

The Bell System Technical Journal  1926-01: Vol 5 Iss 1