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THE THEORY OF MEASUREMENTS
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THE THEORY OF
MEASUREMENTS
BY
A. DE FOREST PALMER, PH.D.
Associate Professor of Physics in Brown University.
McGRAW-HILL BOOK COMPANY
239 WEST 39TH STREET, NEW YORK
6 BOUVERIE STREET, LONDON, E.G.
1912
COPYRIGHT, 1912,
BY THE
McGRAW-HILL BOOK COMPANY
Stanbopc jjbress
H.GILSON COMPANY
BOSTON, U.S.A.
PREFACE.
THE function of laboratory instruction in physics is twofold.
Elementary courses are intended to develop the power of discrimi-
nating observation and to put the student in personal contact with
the phenomena and general principles discussed in textbooks and
lecture demonstrations. The apparatus provided should be of the
simplest possible nature, the experiments assigned should be for
the most part qualitative or only roughly quantitative, and emphasis
should be placed on the principles illustrated rather than on the
accuracy of the necessary measurements. On the other hand,
laboratory courses designed for more mature students, who are
supposed to have a working knowledge of fundamental principles,
are intended to give instruction in the theory and practice of the
methods of precise measurement that underlie all effective research
and supply the data on which practical engineering enterprises are
based. They should also develop the power of logical argument
and expression, and lead the student to draw rational conclusions
from his observations. The instruments provided should be of
standard design and efficiency in order that the student may gain
practice in making adjustments and observations under as nearly
as may be the same conditions that prevail in original investigation.
Measurements are of little value in either research or engineering
applications unless the precision with which they represent the
measured magnitude is definitely known. Consequently, the stu-
dent should be taught to plan and execute proposed measurements
within definitely prescribed limits and to determine the accuracy
of the results actually attained. Since the treatment of these
matters in available laboratory manuals is fragmentary and often
very inadequate if not misleading, the author some years ago under-
took to impart the necessary instruction, in the form of lectures,
to a class of junior engineering students. Subsequently, textbooks
on the Theory of Errors and the Method of Least Squares were
adopted but most of the applications to actual practice were still
given by lecture. The present treatise is the result of the experi-
257860
VI PREFACE
ence gained with a number of succeeding classes. It has been
prepared primarily to meet the needs of students in engineering
and advanced physics who have a working knowledge of the differ-
ential and integral calculus. It is not intended to supersede but
to supplement the manuals and instruction sheets usually employed
in physical laboratories, Consequently, particular instruments and
methods of measurement have been described only in so far as they
serve to illustrate the principles under discussion.
The usefulness of such a treatise was suggested by the marked
tendency of laboratory students to carry out prescribed work in a
purely automatic manner with slight regard for the significance or
the precision of their measurements. Consequently, an endeavor
has been made to develop the general theory of measurements and
the errors to which they are subject in a form so clear and concise
that it can be comprehended and applied by the average student
with the prescribed previous training. To this end, numerical ex-
amples have been introduced and completely worked out whenever
this course seemed likely to aid the student in obtaining a thorough
grasp of the principles they illustrate. On the other hand, inherent
difficulties have not been evaded and it is not expected, or even
desired, that the student will be able to master the subject without
vigorous mental effort.
The first seven chapters deal with the general principles that
underlie all measurements, with the nature and distribution of the
errors to which they are subject, and with the methods by which
the most probable result is derived from a series of discordant
measurements. The various types of measurement met with in
practice are classified, and general methods of dealing with each
of them are briefly discussed. Constant errors and mistakes are
treated at some length, and then the unavoidable accidental errors
of observation are explicitly defined. The residuals corresponding
to actual measurements are shown to approach the true accidental
errors as limits when the number of observations is indefinitely
increased and their normal distribution in regard to sign and mag-
nitude is explained and illustrated. After a preliminary notion of
its significance has been thus imparted, the law of accidental errors
is stated empirically in a form that gives explicit representation to
all of the factors involved. It is then proved to be in conformity
with the axioms of accidental errors, the principle of the arithmetical
ij and the results of experience. The various characteristic
PREFACE vii
errors that are commonly used as a measure of the accidental errors
of given series of measurements are clearly denned and their signifi-
cance is very carefully explained in order that they may be used
intelligently. Practical methods for computing them are developed
and illustrated by numerical examples.
Chapters eight to twelve inclusive are devoted to a general dis-
cussion of the precision of measurements based on the principles
established in the preceding chapters. The criteria of accidental
errors and suitable methods for dealing with constant and systematic
errors are developed in detail. The precision measure, of the result
computed from given observations, is defined and its significance is
explained with the aid of numerical illustrations. The proper basis
for the criticism of reported measurements and the selection of
suitable numerical values from tables of physical constants or other
published data is outlined ; and the importance of a careful estimate
of the precision of the data adopted in engineering and scientific
practice is emphasized. The applications of the theory of errors to
the determination of suitable methods for the execution of proposed
measurements are discussed at some length and illustrated.
In chapter thirteen, the relation between measurement and re-
search is pointed out and the general methods of physical research
are outlined. Graphical methods of reduction and representation
are explained and some applications of the method of least squares
are developed. The importance of timely and adequate publication,
or other report, of completed investigations is emphasized and some
suggestions relative to the form of such reports are given
Throughout the book, particular attention is paid to methods of
computation and to the proper use of significant figures. For the
convenience of the student, a number of useful tables are brought
together at the end of the volume.
A. DE FOREST PALMER.
BROWN UNIVERSITY,
July, 1912.
CONTENTS.
PAGE
PREFACE v
CHAPTER I.
GENERAL PRINCIPLES 1
Introduction Measurement and Units Fundamental and
Derived Units Dimensions of Units Systems of Units in Gen-
eral Use Transformation of Units.
CHAPTER II.
MEASUREMENTS 11
Direct Measurements Indirect Measurements Classification of
Indirect Measurements Determination of Functional Relations
Adjustment, Setting, and Observation of Instruments Record
of Observations Independent, Dependent, and Conditioned
Measurements Errors and the Precision of Measurements Use
of Significant Figures Adjustment of Measurements Discus-
sion of Instruments and Methods.
CHAPTER III.
CLASSIFICATION OF ERRORS 23
Constant Errors Personal Errors Mistakes Accidental
Errors Residuals Principles of Probability.
CHAPTER IV.
THE LAW OF ACCIDENTAL ERRORS 29
Fundamental Propositions Distribution of Residuals Proba-
bility of Residuals The Unit Error The Probability Curve
Systems of Errors The Probability Function The Precision
Constant Discussion of the Probability Function The Proba-
bility Integral Comparison of Theory and Experience The
Arithmetical Mean.
CHAPTER V.
CHARACTERISTIC ERRORS 44
The Average Error The Mean Error The Probable Error
Relations between the Characteristic Errors Characteristic
Errors of the Arithmetical Mean Practical Computation of
Characteristic Errors Numerical Example Rules for the Use
of Significant Figures.
CHAPTER VI.
MEASUREMENTS OF UNEQUAL PRECISION 61
Weights of Measurements The General Mean Probable Error
of the General Mean Numerical Example.
ix
x CONTENTS
CHAPTER VII.
PAGE
THE METHOD OF LEAST SQUARES 72
Fundamental Principles Observation Equations Normal Equa-
tions Solution with Two Independent Variables Adjustment of
the Angles about a Point Computation Checks Gauss's Method
of Solution Numerical Illustration of Gauss's Method Con-
ditioned Quantities.
CHAPTER VIII.
PROPAGATION OP ERRORS 95
Derived Quantities Errors of the Function Xi X z X 3
. . . X q Errors of the Function ai-Xi 0:2^2 013X3 =h . . .
aqXq Errors of the Function F (Xi, X ? , . . . , Xq) Example
Introducing the Fractional Error Fractional Error of the Func-
tion aX! n > X X 2 n ' X ... X X q n *.
CHAPTER IX.
ERRORS OF ADJUSTED MEASUREMENTS 105
Weights of Adjusted Measurements Probable Error of a Single
Observation Application to Problems Involving Two Unknowns
Application to Problems Involving Three Unknowns.
CHAPTER X.
DISCUSSION OF COMPLETED OBSERVATIONS 117
Removal of Constant Errors Criteria of Accidental Errors
Probability of Large Residuals Chauvenet's Criterion Preci-
sion of Direct Measurements Precision of Derived Measurements
Numerical Example.
CHAPTER XI.
DISCUSSION OF PROPOSED MEASUREMENTS 144
Preliminary Considerations The General Problem The Pri-
mary Condition The Principle of Equal Effects Adjusted Effects
Negligible Effects Treatment of Special Functions Numerical
Example.
CHAPTER XII.
BEST MAGNITUDES FOR COMPONENTS 165
Statement of the Problem General Solutions Special Cases
Practical Examples Sensitiveness of Methods and Instruments.
CHAPTER XIII.
RESEARCH 192
Fundamental Principles General Methods of Physical Research
Graphical Methods of Reduction Application of the Method
of Least Squares Publication.
TABLES 212
INDEX.. 245
LIST OF TABLES.
PAGE
I. DIMENSIONS OF UNITS 212
II. CONVERSION FACTORS 213
III. TRIGONOMETRICAL RELATIONS 215
IV. SERIES 217
V. DERIVATIVES 219
VI. SOLUTION OF EQUATIONS 220
VII. APPROXIMATE FORMULA 221
VIII. NUMERICAL CONSTANTS 222
IX. EXPONENTIAL FUNCTIONS e x AND e~ x 223
X. EXPONENTIAL FUNCTIONS e* 2 AND e~ xZ 224
XI. THE PROBABILITY INTEGRAL P A 225
XII. THE PROBABILITY INTEGRAL P s 226
XIII. CHAUVENET'S CRITERION 226
XIV. FOR COMPUTING PROBABLE ERRORS BY FORMULAE (31) AND (32). 227
XV. FOR COMPUTING PROBABLE ERRORS BY FORMULA (34) 228
XVI. SQUARES OF NUMBERS 229
XVII. LOGARITHMS; 1000 TO 1409 231
XVIII. LOGARITHMS 232
XIX. NATURAL SINES 234
XX. NATURAL COSINES 236
XXI. NATURAL TANGENTS 238
XXII. NATURAL COTANGENTS 240
XXIII. RADIAN MEASURE. . 242
THE
THEOEY OF MEASUREMENTS
CHAPTER I.
GENERAL PRINCIPLES.
i. Introduction. Direct observation of the relative position
and motion of surrounding objects and of their similarities and
differences is the first step in the acquisition of knowledge.
Such observations are possible only through the sensations pro-
duced by our environment, and the value of the knowledge thus
acquired is dependent on the exactness with which we corre-
late these sensations. Such correlation involves a quantitative
estimate of the relative intensity of different sensations and of
their time and space relations. As our estimates become more
and more exact through experience, our ideas regarding the
objective world are , gradually modified until they represent
the actual condition of things with a considerable degree of
precision.
The growth of science is analogous to the growth of ideas.
Its function is to arrange a mass of apparently isolated and un-
related phenomena in systematic order and to determine the in-
terrelations between them. For this purpose, each quantity that
enters into the several phenomena must be quantitatively deter-
mined, while all other quantities are kept constant or allowed
to vary by a measured amount. The exactness of the relations
thus determined increases with' the precision of the measure-
ments and with the success attained in isolating the particular
phenomena investigated.
A general statement, or a mathematical formula, that ex-
presses the observed quantitative relation between the different
magnitudes involved in any phenomenon is called the law of
that phenomenon. As here used, the word law does not mean
1
2 THE THEORY OF MEASUREMENTS [ART. 2
that the phenomenon must follow the prescribed course, but
that, under the given conditions and within the limits of error
and the range of our measurements, it has never been found to
deviate from that course. In other words, the laws of science
are concise statements of our present knowledge regarding
phenomena and their relations. As we increase the range and
accuracy of our measurements and learn to control the condi-
tions of experiment more definitely, the laws that express our
results become more exact and cover a wider range of phenomena.
Ultimately we arrive at broad generalizations from which the
laws of individual phenomena are deducible as special cases.
The two greatest factors in the progress of science are the
trained imagination of the investigator and the genius of
measurement. To the former we owe the rational hypotheses
that have pointed the way of advance and to the latter the
methods of observation and measurement by which the laws of
science have been developed.
2. Measurement and Units. To measure a quantity is to
determine the ratio of its magnitude to that of another quan-
tity, of the same kind, taken as a unit. The number that
expresses this ratio may be either integral or fractional and is
called the numeric of the given quantity in terms of the chosen
unit. In general, if Q represents the magnitude of a quantity,
U the magnitude of the chosen unit, and N the corresponding
numeric we have
Q = NU, (I)
which is the fundamental equation of measurement. The two
factors N and U are both essential for the exact specification of
the magnitude Q. For example: the length of a certain line
is five inches, i.e., the line is five times as long as one inch. It
is not sufficient to say that the length of the line is five; for in
that case we are uncertain whether its length is five inches, five
feet, or five times some other unit.
Obviously, the absolute magnitude of a quantity is independent
of the units with which we choose to measure it. Hence, if we
adopt a different unit U', we shall find a different numeric N'
such that
Q = N'U', (II)
and consequently
NU = N'U',
ART. 2] ' GENERAL PRINCIPLES 3
or $-^- (HI)
Equation (III) expresses the general principle involved in the
transformation of units and shows that the numeric varies in-
versely as the magnitude of the unit; i.e., if U is twice as large
as U', N will be only one-half as large as N'. To take a con-
crete example: a length equal to ten inches is also equal to
25.4 centimeters approximately. In this case N equals ten,
N' equals 25.4, U equals one inch, and U r equals one centi-
N f
meter. The ratio of the numerics -^ is 2.54 and hence the
inverse ratio of the units -, is also 2.54, i.e., one inch is equal to
2.54 centimeters.
Equation (III) may also be written in the form
(IV)
which shows that the numeric of a given quantity relative to the
unit U is equal to its numeric relative to the unit U' multiplied
w
by the ratio of the unit U f to the unit U. The ratio -jj is called
the conversion factor for the unit U f in terms of the unit U.
It is equal to the number of units U in one unit U', and when
multiplied by the numeric of a quantity in terms of U' gives
the numeric of the same quantity in terms of U. The con-
version factor for transformation in the opposite direction, i.e.,
from U to U', is obviously the inverse of the above, or -== In
general, the numerator of the conversion factor is the unit in
which the magnitude is already expressed and the denominator
is the unit to which it is to be transformed. For example:
one inch is approximately equal to 2.54 centimeters, hence the
numeric of a length in centimeters is about 2.54 times its numeric
in inches. Conversely, the numeric in inches is equal to the
numeric in centimeters divided by 2.54 or multiplied by the
reciprocal of this number.
In so far as the theory of mensuration and the attainable
accuracy of the result are concerned, measurements may be made
in terms of any arbitrary unite and, in fact, the adoption oisuch
4 THE THEORY OF MEASUREMENTS [ART. 3
units is frequently convenient when we are concerned only with
relative determinations. In general, however, measurements are
of little value unless they are expressed in terms of generally
accepted units whose magnitude is accurately known. Some
such units have come into use through common consent but most
of them have been fixed by government enactment and their per-
manence is assured by legal standards whose relative magnitudes
have been accurately determined. Such primary standards, pre-
served by various governments, have, in many cases, been very
carefully intercompared and their conversion factors are accu-
rately known. Copies of the more important primary standards
may be found in all well-equipped laboratories where they are
preserved as the secondary standards to which all exact measure-
ments are referred. Carefully made copies are, usually, sufficiently
accurate for ordinary purposes, but, when the greatest precision
is sought, their exact magnitude must be determined by direct
comparison with the primary standards. The National Bureau
of Standards at Washington makes such comparisons and issues
certificates showing the errors of the standards submitted for
test.
3. Fundamental and Derived Units. Since the unit is, neces-
sarily, a quantity of the same kind as the quantity measured, we
must have as many different units as there are different kinds of
quantities to be measured. Each of these units might be fixed
by an independent arbitrary standard, but, since most measur-
able quantities are connected by definite physical relations, it is
more convenient to define our units in accordance with these
relations. Thus, measured in terms of any arbitrary unit, a
uniform velocity is proportional to the distance described in
unit time; but, if we adopt as our unit such a velocity that the
unit of length is traversed in the unit of time, the factor of pro-
portionality is unity and the velocity is equal to the ratio of the
space traveled to the elapsed time.
Three independently defined units are sufficient, in connection
with known physical relations, to fix the value of most of the
other units used in physical measurements. We are thus led to
distinguish two classes of units; the three fundamental units,
defined by independent arbitrary standards, and the derived
units, fixed by definite relations between the fundamental units.
The .magnitude, and to some extent the choice, of the fundamental
ART. 4] GENERAL PRINCIPLES 5
units is arbitrary, but when definite standards for each of these
units have been adopted the magnitude of all of the derived units
is fixed.
For convenience in practice, legal standards have been adopted
to represent some of the derived units. The precision of these
standards is determined by indirect comparison with the standards
representing the three fundamental units. Such comparisons are
based on the known relations between the fundamental and de-
rived units and are called absolute measurements. The practical
advantage gained by the use of derived standards lies in the fact
that absolute measurements are generally very difficult and require
great skill and experience in order to secure a reasonable degree
of accuracy. On the other hand, direct comparison of derived
quantities of the same kind is often a comparatively simple
matter and can be carried out with great precision.
4. Dimensions of Units. The dimensions of a unit is a
mathematical formula that shows how its magnitude is related
to that of the three fundamental units. In writing such formulae,
the variables are usually represented by capital letters inclosed
in square brackets. Thus, [M], [L] and [T]- represent the dimen-
sions of the units of mass, length and time respectively.
Dimensional formulae and ordinary algebraic equations are
essentially different in significance. The former shows the rela-
tive variation of units, while the latter expresses a definite mathe-
matical relation between the numerics of measurable quantities.
Thus if a point in uniform motion describes the distance L in the
time T its velocity V is defined by the relation
V = Y (V)
Since L and T are concrete quantities of different kind, the right-
hand member of this equation is not a ratio in the strict arithmet-
ical sense; i.e., it cannot be represented by a simple abstract num-
ber. Hence, in virtue of the definite physical relation expressed
by equation (V), we are led to extend our idea of ratio to include
the case of concrete quantities. From this point of view, the ratio
of two quantities expresses the rate of change of the first quantity
with respect to the second. It is a concrete quantity of the same
kind as the quantity it serves, to define. As an illustration, con-
sider the meaning of equation (V). Expressed in words, it is " the
6 THE THEORY OF MEASUREMENTS [ART. 4
velocity of a point, in uniform motion, is equal to the time rate at
which it moves through space."
If we represent the units of velocity, length, and time by [7],
[L], and [T\, respectively, and the corresponding numerics by v,
I, and t, we have by equation (I), article two,
F = v(V], L = l(L], T = t[T],
and equation (V) becomes
w-m-i'
or
[V][T] t
Since, by definition, [V] and |~l are quantities of the same kind,
their ratio can be expressed by an abstract number k and equation
(VI) may be written in the form
v = kl, (VII)
which is an exact numerical equation containing no concrete
quantities.
The numerical value of the constant k obviously depends on
the units with which L, T, and V are measured. If we define the
unit of velocity by the relation
ryi-M
[TV
or, as it is more often written,
[F] = [L!T-'] f (VIII)
k becomes equal to unity and the relation (VII) between the
numerics of velocity, length, and time reduces to the simple form
The foregoing argument illustrates the advantage to be gained
by defining derived units in accordance with the physical rela-
tions on which they depend. By this means we eliminate the
often incommensurable constants of proportionality such as k
would be if the unit of velocity were defined in any other way
than by equation (VIII).
ART. 5] GENERAL PRINCIPLES 7
The expression on the right-hand side of equation (VIII) is the
dimensions of the unit of velocity when the units of length, mass,
and time are chosen as fundamental. The dimensions of any
other units may be obtained by the method outlined above when
we know the physical relations on which they depend. The form
of the dimensional formula depends on the units we choose as
fundamental, but the general method of derivation is the same in
all cases. As an exercise to fix these ideas the student should
verify the following dimensional formulae: choosing [M], [L], and
[T] as fundamental units, the dimensions of the units of area,
acceleration, and force are [L 2 ], [LT~ 2 ], and [MLT~ 2 ] respectively.
As an illustration of the effect of a different choice of fundamental
units, it may be shown that the dimensions of the unit of mass is
[FL^T 2 ] when the units of length [L], force [F], and time [T] are
chosen as fundamental. The dimensions of some important
derived units are given in Table I at the end of this volume.
5. Systems of Units in General Use. Consistent systems
of units may differ from one another by a difference in the choice
of fundamental units or by a difference in the magnitude of the
particular fundamental units adopted. The systems in common
use illustrate both types of difference.
Among scientific men, the so-called c.g.s. system is almost
universally adopted, and the results of scientific investigations
are seldom expressed in any other units. The advantage of such
uniformity of choice is obvious. It greatly facilitates the com-
parison of the results of different observers and leads to general
advance in our knowledge of the phenomena studied. The units
of length, mass, and time are chosen as fundamental in this
system and the particular values assigned to them are the centi-
meter for the unit of length, the gram for the unit of mass, and
the mean solar second for the unit of time.
The units used commercially in England and the United States
of America are far from systematic, as most of the derived units
are arbitrarily defined. So far as they follow any order, they
form a length-mass-time system in which the unit of length is the
foot, the unit of mass is the mass of a pound, and the unit of time
is the second. This system was formerly used quite extensively
by English scientists and the results of some classic investigations
are expressed in such units.
English and American engineers find it more convenient to use
8 THE THEORY OF MEASUREMENTS [ART. 6
a system in which the fundamental units are those of length,
force, and time. The particular units chosen are the foot as the
unit of length, the pound's weight at London as the unit of force,
and the mean solar second as the unit of time. We shall see that
this is equivalent to a length-mass-time system in which the units
of length and time are the same as above and the unit of mass is
the mass of 32.191 pounds.
6. Transformation of Units. When the relative magnitude
of corresponding fundamental units in two systems is known, a
result expressed in one system can be reduced to the other with
the aid of the dimensions of the derived units involved. Thus:
let A c represent the magnitude of a square centimeter, A t the
magnitude of a square inch, N c the numeric of a given area when
measured in square centimeters, and Ni the numeric of the same
area when measured in square inches; then, from equation (IV),
article two, we have
But if L c is the magnitude of a centimeter and LI that of an inch,
Ai is equal to Lf, and therefore
Hence, the conversion factor -p for reducing square centimeters
A-i
to square inches is equal to the square of the conversion factor
for reducing from centimeters to inches. Now the dimensions
Li
of the unit of area is [L 2 ], and we see that the conversion factor
for area may be obtained by substituting the corresponding con-
version factor for lengths in this dimensional formula. This is a
simple illustration of the general method of transformation of
units. When the fundamental units in the two systems differ in
magnitude, but not in kind, the conversion factor for correspond-
ing derived units in the two systems is obtained by replacing the
fundamental units by their respective conversion factors in the
dimensions of the derived units considered.
It should be noticed that the fundamental units in the c.g.s.
system are those of length, mass, and time, while on the engineer's
system they are length, force, and time. In the latter system,
ART. 6] GENERAL PRINCIPLES 9
force is supposed to be directly measured and expressed by the
dimensions [F]. Consequently the dimensions of the unit of
mass are [FL~ 1 T 2 ], and the unit of mass is a mass that will acquire
. a velocity of one foot per second in one second when acted upon
by a force of one pound's weight. For the sake of definiteness,
the unit of force is taken as the pound's weight at London, where
the acceleration due to gravity (g) is equal to 32.191 feet per
second per second. Otherwise the unit of force would be variable,
depending on the place at which the pound is weighed.
From Newton's second law of motion we know that the relation
between acceleration, mass, and force is given by the expression
/ = ma.
For a constant force the acceleration produced is inversely pro-
portional to the mass moved. Now the mass of a pound at London
is acted upon by gravity with a force of one pound's weight, and, if
free, it moves with an acceleration of 32.191 feet per second per
second. Hence a mass equal to that of 32.191 pounds acted
upon by a force of one pound's weight would move with an acceler-
ation of one foot per second per second, i.e., it would acquire a
velocity of one foot per second in one second. Hence the unit of
mass in the engineer's system is 32.191 pounds mass. This unit
is sometimes called a slugg, but the name is seldom met with since
engineers deal primarily with forces rather than masses, and are
W
content to write for mass without giving the unit a definite
7
name. This is equivalent to saying that the mass of a body,
expressed in sluggs, is equal to its weight, at London, expressed in
pounds, divided by 32.191.
After careful consideration of the foregoing discussion, it will
be evident that the engineer's length-force-time system is exactly
equivalent to a length-mass-time system in which the unit of
length is the foot, the unit of mass is the slugg or 32.191 pounds'
mass, and the unit of time is the mean solar second. In the latter
system the fundamental units are of the same kind as those of
the c.g.s. system. Hence, if the conversion factor for the unit
of mass is taken as the ratio of the magnitude of the slugg to that
of the gram, quantities expressed in the units of the engineer's
system may be reduced to the equivalent values in the c.g.s.
system by the method described at the beginning of this article.
10 THE THEORY OF MEASUREMENTS [ART. 6
When, as is frequently the case, the engineer's results are expressed
in terms of the local weight of a pound as a unit of force in place
of the pound's weight at London, the result of a transformation
of units, carried out as above, will be in error by a factor equal to
the ratio of the acceleration due to gravity at London and at the
location of the measurements. Unless the local gravitational
acceleration is definitely stated by the observer and unless he
has used his length-force-time units in a consistent manner, it is
impossible to derive the exact equivalent of his results on the
c.g.s. system.
CHAPTER II.
MEASUREMENTS.
IN article two of the last chapter we defined the term " measure-
ment " and showed that any magnitude may be represented by
the product of two factors, the numeric and the unit. The object
of all measurements is the determination of the numeric that ex-
presses the magnitude of the observed quantity in terms of the
chosen unit. For convenience of treatment, they may be classified
according to the nature of the measured quantity and the methods
of observation and reduction.
7. Direct Measurements. The determination of a desired
numeric by direct observation of the measured quantity, with the
aid of a divided scale or other indicating device graduated in
terms of the chosen unit, is called a direct measurement.
Such measurements are possible when the chosen unit, together
with its multiples and submultiples, can be represented by a
material standard, so constructed that it can be directly applied
to the measured quantity for the purpose of comparison, or when
the unit and the measured magnitudes produce proportional
effects on a suitable indicating device.
Lengths may be directly measured with a graduated scale,
masses by comparison with a set of standard masses on an equal
arm balance, time intervals by the use of a clock regulated to
give mean solar time, and forces with the aid of a spring balance.
Hence magnitudes expressible in terms of the fundamental units
of either the c.g.s. or the engineer's system may be directly
measured.
Many quantities expressible in terms of derived units, that can
be represented by material standards, are commonly determined
by direct measurement. As illustrations, we may cite the deter-
mination of the volume of a liquid with a graduated flask and the
measurement of the electrical resistance of a wire by comparison
with a set of standard resistances.
8. Indirect Measurements. The determination of a desired
numeric by computation from the numerics of one or more
11
12 THE THEORY OF MEASUREMENTS [ART. 9
directly measured magnitudes, that bear a known relation to the
desired quantity, is called an indirect measurement.
The relation between the observed and computed magnitudes
may be expressed in the general form
y = Ffa, Xz, x 3 , . . . a, b, c . . . ),
where y, x t , x 2 , etc., represent measured or computed magnitudes,
or the numerics corresponding to them, a, b, c, etc., represent
constants, and F indicates that there is a functional relation
between the other quantities. This expression is read, y equals
some function of xi, x*, etc., and a, b, c, etc. In any particular
case, the form of the function F and the number and nature of the
related quantities must be known before the computation of the
unknown quantities is undertaken.
Most of the indirect measurements made by physicists and
engineers fall into one or another of three general classes, char-
acterized by the nature of the unknown and measured magnitudes
and the form of the function F.
9. Classification of Indirect Measurements.
I.
In the first class, y represents the desired numeric of a magni-
tude that is not directly measured, either because it is impossible
or inconvenient to do so, or because greater precision can be at-
tained by indirect methods. The form of the function F and the
numerical values of all of the constants a, 6, c, etc., appearing in
it, are given by theory. The quantities xi, Xz, etc., represent
the numerics of directly measured magnitudes. In the following
pages indirect measurements belonging to this class will sometimes
be referred to as derived measurements.
As an illustration we may cite the determination of the density
s of a solid sphere from direct measurements of its mass M and
its diameter D with the aid of the relation
M
= F^'
Comparing this expression with the general formula given above,
we note that s corresponds to y, M to xi, D to x a , J to a, TT to 6,
and that F represents the function y^^. The form of the func-
ART. 9] MEASUREMENTS 13
tion is given by the definition of density as the ratio of the mass
to the volume of a body and the numerical constants and w are
given by the known relation between the volume and diameter of
a sphere.
II.
In the second class of indirect measurements, the numerical
constants a, b, c, etc., are the unknown quantities to be computed,
the form of the function F is known, and all of the quantities y,
Xi, x z , etc., are obtained by direct measurements or given by
theory. The functions met with in this class of measurements
usually represent a continuous variation of the quantity y with
respect to the quantities x\, x 2 , etc., as independent variables.
Hence the result of a direct measurement of y will depend on the
particular values of Xi, x 2 , etc., that obtain at the time of the
measurement. Consequently, in computing the constants a, b, c,
etc., we must be careful to use only corresponding values of the
measured quantities, i.e., values that are, or would be, obtained
by coincident observations on the several magnitudes.
Every set of corresponding values of the variables y, Xi, x 2 , etc.,
when used in connection with the given function, gives an algebraic
relation between the unknown quantities a, b, c, etc., involving
only numerical coefficients and absolute terms. When we have
obtained as many independent equations as there are unknown
quantities, the latter may be determined by the usual algebraic
methods. We shall see, however, that more precise results can
be obtained when the number of independent measurements far
exceeds the minimum limit thus set and the computation is made
by special methods to be described hereafter.
The determination of the initial length L and the coefficient of
linear expansion a of a metallic bar from a series of measurements
of the lengths L t corresponding to different temperatures t with the
aid of the functional relation
L t = Lo (1 + at)
is an example of the class of measurements here considered. Such
measurements are sometimes called determinations of empirical
constants.
14
THE THEORY OF MEASUREMENTS [ART. 9
III.
The third class of indirect measurements includes all cases in
which each of a number of directly measured quantities yi, y*, y s ,
etc., is a given function of the unknown quantities Xi, x 2 , X B , etc.,
and certain known numerical constants a, 6, c, etc. In such cases
we have as many equations of the form
y 1 = FI (xi, x 2 , 3 , . . . a, 6, c, . . . ),
2/2 = F 2 (xi, z 2 , $t, . . . a, M, . . . )>
as there are measured quantities yi, y 2 , etc. This number must
be at least as great as the number of unknowns Xi, x 2 , etc., and
may be much greater.
The functions F lt F 2 ,
etc., are frequently dif-
ferent in form and some
of them may not con-
tain all of the un-
knowns. The numeri-
cal constants, appearing
in different functions,
are generally different.
But the form of each
of the functions and
the values of all of the
constants must be
known before a solu-
tion of the problem is
possible.
Problems of this type
are frequently met with
in astronomy and geod-
esy. One of the simplest is known as the adjustment of the
angles about a point. Thus, let it be required to find the most
probable values of the angles Xi, x 2 , and x 3 , Fig. 1, from direct
measurements of yi, y 2 , y 3) . . . y & . In this case the general
equations take the form
FIG.
ART. 11] MEASUREMENTS 15
2/i = xi,
2/2 = xi + x 2 ,
2/4 = X 2 ,
2/5 = 2
2/6 = ,
and all of the numerical constants are either unity or zero. The
solution of such problems will be discussed in the chapter on the
method of least squares.
10. Determination of Functional Relations. When the form
of the functional relation between the observed and unknown
magnitudes is not known, the solution of the problem requires
something more than measurement and computation. In some
cases a study of the theory of the observed phenomena, in con-
nection with that of allied phenomena, will suggest the form of the
required function. Otherwise, a tentative form must be assumed
after a careful study of the observations themselves, generally by
graphical methods. In either case the constants of the assumed
function must be determined by indirect measurements and the
results tested by a comparison of the observed and the computed
values of the related quantities. If these values agree within the
accidental errors of observation, the assumed function may be
adopted as an empirical representation of the phenomena. If
the agreement is not sufficiently close, the form of the function
is modified, in a manner suggested by the observations, and the
process of computation and comparison is repeated until a satis-
factory agreement is obtained. A more detailed treatment of
such processes will be found in Chapter XIII.
11. Adjustment, Setting, and Observation of Instruments.
Most of the magnitudes dealt with in physics and engineering
are determined by indirect measurements. But we have seen
that all such quantities are dependent upon and computed from
directly measured quantities. Consequently, a study of the
methods and precision of direct measurement is of fundamental
importance.
In general, every direct measurement involves three distinct
operations. First: the instrument adopted is so placed that its
16 THE THEORY OF MEASUREMENTS [ART. 12
scale is in the proper position relative to the magnitude to be
measured and all of its parts operate smoothly in the manner and
direction prescribed by theory. Operations of this nature are
called adjustments. Second: the reference line of the instru-
ment is moved, or allowed to move, in the manner demanded by
theory, until it coincides with a mark chosen as a point of reference
on the measured magnitude. We shall refer to this operation as a
setting of the instrument. Third: the position of the index of
the instrument, with respect to its graduated scale, is read. This
is an observation.
As an illustration, consider the measurement of the normal
distance between two parallel lines with a micrometer microscope.
The instrument must be so mounted that it can be rigidly clamped
in any desired position or moved freely in the direction of its
optical axis without disturbing the direction of the micrometer
screw. The following adjustments are necessary: the axis of the
micrometer screw must be made parallel to the plane of the two
lines and perpendicular to a normal plane through one of them;
the eyepiece must be so placed that the cross-hairs are sharply
defined; the microscope must be moved, in the direction of its
optical axis, until the image of the two lines, or one of them if the
normal distance between them is greater than the field of view
of the microscope, is in the same plane with the cross-hairs. The
latter adjustment is correct when there is no parallax between the
image of the lines and the cross-hairs. The setting is made by
turning the micrometer head until the intersection of the cross-
hairs bisects the image of one of the lines. Finally the reading
of the micrometer scale is observed. A similar setting and ob-
servation are made on the other line and the difference between
the two observations gives the normal distance between the two
lines in terms of the scale of the micrometer.
12. Record of Observations. In the preceding article, the
word "observation" is used in a very much restricted sense to
indicate merely the scale reading of a measuring instrument.
This restriction is convenient in dealing with the technique of
measurement, but many other circumstances, affecting the accu-
racy of the result, must be observed and taken into account in a
complete study of the phenomena considered. There is, however^
little danger of confusion in using the word in the two different
senses since the more restricted meaning is in reality only a
ART. 13] MEASUREMENTS 17
special case of the general. The particular significance intended
in any special case is generally clear from the context.
The first essential for accurate measurements is a clear and
orderly record of all of the observations. The record should begin
with a concise description of the magnitude to be measured, and
the instruments and methods adopted for the purpose. Instru-
ments may frequently be described, with sufficient precision, by
stating their name and number or other distinguishing mark.
Methods are generally specified by reference to theoretical treatises
or notes. The adjustment and graduation of the instruments
should be clearly stated. The date on which the work is carried
out and the location of the apparatus should be noted.
Observations, in the restricted sense, should be neatly arranged
in tabular form. The columns of the table should be so headed,
and referred to by subsidiary notes, that the exact significance of
all of the recorded figures will be clearly understood at any future
time. All circumstances likely to affect the accuracy of the
measurements should be carefully observed and recorded in the
table or in suitably placed explanatory notes.
Observations should be recorded exactly as taken from the
instruments with which they are made, without mental computa-
tion or reduction of any kind even the simplest. For example:
when a micrometer head is divided into any number of parts
other than ten or one hundred, it is better to use two columns in
the table and record the reading of the main scale in one and
that of the micrometer head in the other than to reduce the head
reading to a decimal mentally and enter it in the same column
with the main scale reading. This is because mistakes are likely
to be made in such mental calculations, even by the most expe-
rienced observers, and, when the final reduction of the observations
is undertaken at a future time, it is frequently difficult or impos-
sible to decide whether a large deviation of a single observation
from the mean of the others is due to an accidental error of obser-
vation or to a mistake in such a mental calculation.
13. Independent, Dependent, and Conditioned Measure-
ments. Measurements on the same or different magnitudes are
said to be independent when both of the following specifications
are fulfilled: first, the measured magnitudes are not required to
satisfy a rigorous mathematical relation among themselves;
second, the same observation is not used in the computation of
18 THE THEORY OF MEASUREMENTS [ART. 14
any two of the measurements and the different observations are
entirely unbiased by one another.
When the first of these specifications is fulfilled and the second
is not, the measurements are said to be dependent. Thus, when
several measurements of the length of a line are all computed
from the same zero reading of the scale used, they are all dependent
on that observation and any error in the position of the zero mark
affects all of them by exactly the same amount. When the position
of the index relative to the scale of the measuring instrument is
visible while the settings are being made, there is a marked tendency
to set the instrument so that successive observations will be exactly
alike rather than to make an independent judgment of the bisection
of the chosen mark in each case. The observations, corresponding
to settings made in this manner, are biased by a preconceived
notion regarding the correct position of the index and the measure-
ments computed from them are not independent. The impor-
tance of avoiding faulty observations of this type cannot be too
strongly emphasized. They not only vitiate the results of our
measurements, but also render a determination of their precision
impossible.
Measurements that do not satisfy the first of the above speci-
fications are called conditioned measurements. The different
determinations of each of the related quantities may or may not
be independent, according as they do or do not satisfy the second
specification, but the adjusted results of all of the measurements
must satisfy the given mathematical relation. Thus, we may
make a number of independent measurements of each of the
angles of a plane triangle, but the mean results must be so adjusted
that the sum of the accepted values is equal to one hundred and
eighty degrees.
14. Errors and the Precision of Measurements. Owing to
unavoidable imperfections and lack of constant sensitiveness in
our instruments, and to the natural limit to the keenness of our
senses, the results of our observations and measurements differ
somewhat from the true numeric of the observed magnitude.
Such differences are called errors of observation or measurement.
Some of them are due to known causes and can be eliminated,
with sufficient accuracy, by suitable computations. Others are
apparently accidental in nature and arbitrary in magnitude.
Their probable distribution, in regard to magnitude and frequency
ART. 15] MEASUREMENTS 19
of occurrence, can be determined by statistical methods when a
sufficient number of independent measurements is available.
The precision of a measurement is the degree of approximation
with which it represents the true numeric of the observed magni-
tude. Usually our measurements serve only to determine the
probable limits within which the desired numeric lies. Looked
at from this point of view, the precision of a measurement may be
considered to be inversely proportional to the difference between
the limits thus determined. It increases with the accuracy,
adaptability, and sensitiveness of the instruments used, and with
the skill and care of the observer. But, after a very moderate
precision has been attained, the labor and expense necessary for
further increase is very great in proportion to the result obtained.
A measurement is of little practical value unless we know the
precision with which it represents the observed magnitude.
Hence the importance of a thorough study of the nature and dis-
tribution of errors in general and of the particular errors that
characterize an adopted method of measurement. At first sight
it might seem incredible that such errors should follow a definite
mathematical law. But, when the number of observations is
sufficiently great, we shall see that the theory of probability leads
to a definite and easily calculated measure of the precision of a
single observation and of the result computed from a number
of observations.
15. Use of Significant Figures. When recording the nu-
merical results of observations or measurements, and during all
of the necessary computations, the number of significant figures
employed should be sufficient to express the attained precision
and no more. By significant figures we mean the nine digits and
zeros when not used merely to locate the decimal point.
In the case of the direct observation of the indications of instru-
ments, the above specification is usually sufficiently fulfilled by
allowing the last recorded significant figure to represent the
estimated tenth of the smallest division of the graduated scale.
For example: in measuring the length of a line, with a scale
divided in millimeters, the position of the ends of the line would
be recorded to the nearest estimated tenth of a millimeter.
Generally, computed results should be so recorded that the
limiting values, used to express the attained precision, differ by
only a few units in the last one or two significant figures. Thus:
20 THE THEORY OF MEASUREMENTS [ART. 15
if the length of a line is found to lie between 15.65 millimeters and
15.72 millimeters, we should write 15.68 millimeters as the result
of our measurement. The use of a larger number of significant
figures would be not only a waste of space and labor, but also a
false representation of the precision of the result. Most of the
magnitudes we are called upon to measure are incommensurable
with the chosen unit, and hence there is no limit to the number
of significant figures that might be used if we chose to do so; but
experienced observers are always careful to express all observa-
tions and results and carry out all computations with a number
just sufficient to represent the attained precision. The use of
too many or too few significant figures is strong evidence of inex-
perience or carelessness in making observations and computations.
More specific rules for determining the number of significant
figures to be used in special cases will be developed in connection
with the methods for determining the precision of measurements.
The number of significant figure^ in any numerical expression
is entirely independent of the position of the decimal point.
Thus: each of the numbers 5,769,600, 5769, 57.69, and 0.0005769
is expressed by four significant figures and represents the corre-
sponding magnitude within one-tenth of one per cent, notwith-
standing the fact that the different numbers correspond to differ-
ent magnitudes. In general, the location of the decimal point
shows the order of magnitude of the quantity represented and
the number of significant figures indicates the precision with which
the actual numeric of the quantity is known.
In writing very large or very small numbers, it is convenient
to indicate the position of the decimal point by means of a positive
or negative power of ten. Thus: the number 56,400,000 may
be written 564 X 10 5 or, better, 5.64 X 10 7 , and 0.000075 may
be written 75 X W~ or 7.5 X 10~ 5 . When a large number of
numerical observations or results are to be tabulated or used in
computation, a considerable amount of time and space is saved
by adopting this method of representation. The second of the
two forms, illustrated above, is very convenient in making com-
putations by means of logarithms, as in this case the power of
ten always represents the characteristic of the logarithm of the
corresponding number.
In rounding numbers to the required number of significant
figures, the digit in the last place held should be increased by one
ART. 17] MEASUREMENTS 21
unit when the digit in the next lower place is greater than five,
and left unchanged when the neglected part is less than five-
tenths of a unit. When the neglected part is exactly five-tenths
of a unit the last digit held is increased by one if odd, and left
unchanged if even. Thus: 5687.5 would be rounded to 5688 and
5686.5 to 5686.
1 6. Adjustment of Measurements. The results of inde-
pendent measurements of the same magnitude by the same or
different methods seldom agree with one another. This is due to
the fact that the probability for the occurrence of errors of exactly
the same character and magnitude in the different cases is very
small indeed. Hence we are led to the problem of determining
the best or most probable value of the numeric of the observed
magnitude from a series of discordant measurements. The given
data may be all of the same precision or it may be necessary to
assign a different degree of accuracy to the different measure-
ments. In either case the solution of the problem is called the
adjustment of the measurements.
The principle of least squares, developed in the theory of errors
that leads to the measure of precision cited above, is the basis
of all such adjustments. But the particular method of solution
adopted in any given case depends on the nature of the measure-
ments considered. In the case of a series of direct, equally pre-
cise, measurements of a single quantity, the principle of least
squares leads to the arithmetical mean as the most probable, and
therefore the best, value to assign to the measured quantity.
This is also the value that has been universally adopted on a priori
grounds. In fact many authors assume the maximum probability
of the arithmetical mean as the axiomatic basis for the develop-
ment of the law of errors.
The determination of empirical relations between measured
quantities and the constants that enter into them is also based
on the principle of least squares. For this reason, such deter-
minations are treated in connection with the discussion of the
methods for the adjustment of measurements.
17. Discussion of Instruments and Methods. The theory
of errors finds another very important application in the discussion
of the relative availableness and accuracy of different instruments
and methods of measurement. Used in connection with a few
preliminary measurements and a thorough knowledge of the
22 THE THEORY OF MEASUREMENTS [ART. 17
theory of the proposed instruments and methods, it is sufficient
for the determination of the probable precision of an extended
series of careful observations. By such means we are able to
select the instruments and methods best adapted to the particular
purpose in view. We also become acquainted with the parts of
the investigation that require the greatest skill and care in order
to give a result with the desired precision.
The cost of instruments and the time and skill required in
carrying out the measurements increase much more rapidly than
the corresponding precision of the results. Hence these factors
must be taken into account in determining the availableness of a
proposed method. It is by no means always necessary to strive
for the greatest attainable precision. In fact, it would be a
waste of time and money to carry out a given measurement with
greater precision than is required for the use to which it is to be
put. For many practical purposes, a result correct within one-
tenth of one per cent, or even one per cent, is amply sufficient.
In such cases it is essential to adopt apparatus and methods that
will give results definitely within these limits without incurring
the greater cost and labor necessary for more precise deter-
minations.
CHAPTER III.
CLASSIFICATION OF ERRORS.
ALL measurements, of whatever nature, are subject to three
distinct classes of errors, namely, constant errors, mistakes, and
accidental errors.
18. Constant Errors. Errors that can be determined in
sign and magnitude by computations based on a theoretical
consideration of the method of measurement used or on a pre-
liminary study and calibration of the instruments adopted are
called constant errors. They are sometimes due to inadequacy of
an adopted method of measurement, but more frequently to
inaccurate graduation and imperfect adjustment of instruments.
As a simple illustration, consider the measurement of the
length of a straight line with a graduated scale. If the scale is
not held exactly parallel to the line, the result will be too great
or too small according as the line of sight in reading the scale is
normal to the line or to the scale. The magnitude of the error
thus introduced depends on the angle between the line and the
scale and can be exactly computed when we know this angle and
the circumstances of the observations. If the scale is not straight,
if its divisions are irregular, or if they are not of standard length,
the result of the measurement will be in error by an amount
depending on the magnitude and distribution of these inaccuracies
of construction. The sign and magnitude of such errors can
gener o1 ly be determined by a careful study and calibration of the
scai
If M represents the actual numeric of the measured magnitude,
M Q the observed numeric, and Ci, C 2 , C 3 , etc., the constant errors
inherent in the method of measurement and the instruments used,
M = Mo + Ci + C 2 + C 3 + - . (1)
The necessary number of correction terms Ci, G' 2 , C z , etc., is
determined by a careful study of the theory and practical appli-
cation of the apparatus and method used in finding M Q . The
magnitude and sign of each term are determined by subsidiary
23
24 THE THEORY OF MEASUREMENTS [ART. 18
measurements or calculated, on theoretical grounds, from known
data. Thus, in the above illustration, suppose that the scale is
straight and uniformly graduated, that each of its divisions is
1.01 times as long as the unit in which it is supposed to be gradu-
ated, and that the line of the graduations makes an angle a with
the line to be measured. Under these conditions, the number of
correction terms reduces to two: the first, Ci, due to the false
length of the scale divisions, and the second, C 2 , due to the lack
of parallelism between the scale and the line.
Since the actual length of each division is 1.01, the .length of
Mo divisions, i.e., the length that would have been observed on
an accurate scale, is
M l = Mo X 1.01 = Mo + 0.01 Mo = Mo + Ci,
... Ci = + 0.01 Mo.
If the line of sight is normal to the line in making the observa-
tions, the length M 2 that would have been obtained if the scale
had been properly placed is
M 2 = MO cos a = MO + Czj
/. C 2 =-M (l-cosa)=-2M sin 2 ^
and (1) takes the form
M= Mo + 0.01 Mo - 2M sin 2 |>
= M (l+0.01-2sin 2 ^Y
The precision with which it is necessary to determine the cor-
rection terms Ci, C 2 , etc., and frequently the number of these
terms that should be employed depends on the precision with
which the observed numeric M is determined. If M is measured
within one-tenth of one per cent of its magnitude, the several
correction terms should be determined within one one-hundredth
of one per cent of M , in order that the neglected part of the sum
of the corrections may be less than one-tenth of one per cent of
M . If any correction term is found to be less than the. above
limit, it may be neglected entirely since it is obviously useless
to apply a correction that is less than one-tenth of the uncer-
tainty of M .
In our illustration, suppose that the precision is such that we
are sure that M is less than 1.57 millimeters and greater than
ART. 19] CLASSIFICATION OF ERRORS 25
1.55 millimeters, but is not sufficient to give the fourth significant
figure within several units. Obviously, it would be useless to
determine Ci and C% closer than 0.001 millimeter, and if the mag-
nitude of either of these quantities is less than 0.001 millimeter
our knowledge of the true value of M is not increased by making
the corresponding correction. In fact, it is usually impossible
to determine the C's with greater accuracy than the above limit,
since, as in our illustration, M Q is usually a factor in the correction
terms. Hence the writing down of more than the required num-
ber of significant figures is mere waste of labor.
When considering the availableness of proposed methods and
apparatus, it is important to investigate the nature and magni-
tude of the constant errors inherent in their use. It sometimes
happens that the sources of such errors can be sufficiently elimi-
nated by suitable adjustment of the instruments or modification
of the method of observation. When this is not possible the
conditions should be so chosen that the correction terms can be
computed with the required precision. Even when all possible
precautions have been taken, it very seldom happens that the
sum of the constant errors reduces to zero or that the magni-
tude of the necessary corrections can be exactly determined.
Moreover, such errors are never rigorously constant, but present
small fortuitous variations, which, to some extent, are indistinguish-
able from the accidental errors to be described later.
A more detailed discussion of constant errors and the limits
within which they should be determined will be given after we
have developed the methods for estimating the precision of the
observed numeric M.
19. Personal Errors. When setting cross-hairs, or any other
indicating device, to bisect a chosen mark, some observers will
invariably set too far to one side of the center, while others will
as consistently set on the other side. Again, in timing a transit,
some persons will signal too soon and others too late. With
experienced and careful observers, the errors introduced in this
manner are small and nearly constant in magnitude and sign,
but they are seldom entirely negligible when the highest possible
precision is sought.
Errors of this nature will be called personal errors, since their
magnitude and sign depend on personal peculiarities of the
observer. Their elimination may sometimes be effected by a
26 THE THEORY OF MEASUREMENTS [ART. 20
careful study of the nature of such peculiarities and the magnitude
of the effects produced by them under the conditions imposed
by the particular problem considered. Suitable methods for this
purpose are available in connection with most of the investiga-
tions in which an exact knowledge of the personal error is essential.
Such a study is .frequently referred to as a determination of the
"Personal Equation" of the observer.
20. Mistakes. Mistakes are errors due to reading the indi-
cations of an instrument carelessly or to a faulty record of the
observations. The most frequent of these are the following :
the wrong integer is placed before an accurate fractional reading,
e.g., 9.68 for 19.68; the reading is made in the wrong direction of
the scale, e.g., 6.3 for 5.7; the significant figures of a number are
transposed, e.g., 56 is written for 65. Care and strict attention
to the work in hand are the only safeguards against such mistakes.
When a large number of observations have been systematically
taken and recorded, it is sometimes possible to rectify an obvious
mistake, but unless this can be done with certainty the offending
observation should be dropped from the series. This statement
does not apply to an observation showing a large deviation from
the mean but only to obvious mistakes.
21. Accidental Errors. When a series of independent meas-
urements of the same magnitude have been made, by the same
method and apparatus and with equal care, the results generally
differ among themselves by several units in the last one or two
significant figures. If in any case they are found to be identical,
it is probable that the observations were not independent, the
instruments adopted were not sufficiently sensitive, the maximum
precision attainable was not utilized, or the observations were
carelessly made. Exactly concordant measurements are quite as
strong evidence of inaccurate observation as widely divergent
ones.
As the accuracy of method and the sensitiveness of instruments
is increased, the number of concordant figures in the result in-
creases but differences always occur in the last attainable figures.
Since there is, generally, no reason to suppose that any one of the
measurements is more accurate than any other, we are led to
believe that they are all affected by small unavoidable errors.
After all constant errors and mistakes have been corrected, the re-
maining differences between the individual measurements and the true
ART. 22] CLASSIFICATION OF ERRORS 27
numeric of the measured magnitude are called accidental errors.
They are due to the combined action of a large number of inde-
pendent causes each of which is equally likely to produce a posi-
tive or a negative effect. Probably most of them have their
origin in small fortuitous variations in the sensitiveness and
adjustment of our instruments and in the keenness of our senses
of sight, hearing, and touch. It is also possible that the correla-
tion of our sense perceptions and the judgments that we draw
from them are not always rigorously the same under the same
set of stimuli.
Suppose that N measurements of the same quantity have been
made by the same method and with equal care. Let ai, a^, 3,
. . . a N represent the several results of the independent meas-
urements, after all constant errors and mistakes have been elim-
inated, and let X represent the true numeric of the measured
magnitude. Then the accidental errors of the individual measure-
ments are given by the differences,
Ai - ai - X, A 2 = a 2 - X, A 3 = a 3 - X } . . . A^ = a N -X. (2)
The accidental errors AI, A 2 , . . . A# thus denned are sometimes
called the true errors of the observations ai, a 2 , . . . a N .
22. Residuals. Since the individual measurements a\ t a?,
. . . a N differ among themselves, and since there is no reason to
suppose that any one of them is more accurate than any other, it
is never possible to determine the exact magnitude of the numeric
X. Hence the magnitude of the accidental errors A i, A 2 , . . . A#
can never be exactly determined. But, if x is the most probable
value that we can assign to the numeric X on the basis of our
measurements, we can determine the differences
ri = di x, r z = a 2 x, . . . r N = a N x. (3)
These differences are called the residuals of the individual measure-
ments dij 02, . . . a N . They represent the most probable values
that we can assign to the accidental errors AI, A 2 , . . . A# on the
basis of the given measurements.
It should be continually borne in mind that the residuals thus
determined are never identical with the accidental errors. How-
ever precise our measurements may be, the probability that x is
exactly equal to X is always less than unity. As the number
and precision of measurements increase, the difference between
28 THE THEORY OF MEASUREMENTS [ART. 23
the magnitudes x and X decreases, and the residuals continually
approach the accidental errors, but exact equality is never attain-
able with a finite number of observations.
23. Principles of Probability. The theory of errors is an
application of the principles of probability to the discussion of
series of discordant measurements for the purpose of determining
the most probable numeric that can be assigned to the measured
quantity and making an estimate of the precision of the result
thus obtained. A discussion of the fundamental principles of
the theory of probability, sufficient for this purpose, is given in
most textbooks on advanced algebra, and the student should
master them before undertaking the study of the 1 theory of errors.
For the sake of convenience in reference, the three most useful
propositions are stated below without proof.
PROPOSITION 1. If an event can happen in n independent
ways and either happen or fail in N independent ways, the prob-
ability p that it will occur in a single trial at random is given by
the relation
n , A .
p - r w
Also if p' is the probability that it will fail in a single trial at
random,
p = l_p = !_.. ( 5 )
PROPOSITION 2. If the probabilities for the separate occurrence
of n independent events are respectively pi, p%, . . . p n , the prob-
ability PS that some one of these events will occur in a single trial
at random is given by the relation
PS = Pi + Pz + Pz + ' ' ' + P^ (6)
PROPOSITION 3. If the probabilities for the separate occurrence
of n independent events are respectively pi, p 2 , . . . p n , the
probability P that all of the events will occur at the same time is
given by the relation
P = Pi X P2 X X Pn. (7)
CHAPTER IV.
THE LAW OF ACCIDENTAL ERRORS.
24. Fundamental Propositions. The theory of accidental
errors is based on the principle of the arithmetical mean and the
three axioms of accidental errors. When the word " error " is used
without qualification, in the statement of these propositions and
in the following pages, accidental errors are to be understood.
Principle of the Arithmetical Mean. The most probable value
that can be assigned to the numeric of a measured magnitude, on
the basis of a number of equally trustworthy direct measurements,
is the arithmetical mean of the given 'measurements.
This proposition is self-evident in the case of two independent
measurements, made by the same method with equal care, since
one of them is as likely to be exact as the other, and hence it is
more probable that the true numeric lies halfway between them
than in any other location. Its extension to more than two
measurements is the only rational assumption that we can make
and is sanctioned by universal usage.
First Axiom. In any large number of measurements, positive
and negative errors of the same magnitude are equally likely to
occur. The number of negative errors is equal to the number
of positive errors.
Second Axiom. Small errors are much more likely to occur
than large ones.
Third Axiom. All of the errors of the measurements in a
given series lie between equal positive and negative limits. Very
large errors do not occur.
The foundation of these propositions is the same as that of the
axioms of geometry. Namely: they are general statements that
are admitted as self-evident or accepted as a basis of argument by
all competent persons. Their justification lies in the fact that
the results derived from them are found to be in agreement with
experience.
25. Distribution of Residuals. It was pointed out in article
twenty-two that the true accidental errors, represented by A's,
29
30
THE THEORY OF MEASUREMENTS [ART. 26
cannot be determined in practice, but the residuals, represented
by r's, can be computed from the given observations by equation
(3). The A's may be considered as the limiting values toward
which the r's approach as the number of observations is indefinitely
increased. If the residuals corresponding to a very large num-
ber of observations are arranged in groups according to sign and
magnitude, the groups containing very small positive or negative
residuals will be found to be the largest, and, in general, the magni-
tude of the groups will decrease nearly uniformly as the magnitude
of the contained residuals increases either positively or negatively.
Let n represent the number of residuals in any group, and r their
common magnitude, then the distribution of the residuals, in
regard to sign and magnitude, may be represented graphically
by laying off ordinates proportional to the numbers n against
abscissae proportional to the corresponding magnitudes r. The
points, thus located, will be approximately uniformly distributed
about a curve of the general form illustrated in Fig. 2.
The number of residuals in each group will increase with the
total number of measurements from which the r's are computed.
Consequently the ordinates of the curve in Fig. 2 will depend on
the number of observations considered as well as on their accuracy.
Hence, if we wish to compare different series of measurements with
regard to accuracy, we must in some way eliminate the effect of
differences in the number of observations. Moreover, we are not
so much concerned with the total number of residuals of any given
magnitude as with the relative number of residuals of different
magnitudes. For, as we shall see, the acuracy of a series of
observations depends on the ratio of the number of small errors
to the number of large ones.
26. Probability of Residuals. Suppose that a very large
number N of independent measurements have been made and that
AKF.27J THE LAW OF ACCIDENTAL ERRORS 31
the corresponding residuals have been computed by equation (3).
By arranging the results in groups according to sign and magni-
tude, suppose we find HI residuals of magnitude n, n 2 of magni-
tude r 2 , etc., and n\ of magnitude n, n/ of magnitude r 2 , etc.
If we choose one of the measurements at random, the probability
that the corresponding residual is equal to r\ is -^ , since there
are N residuals and n\ of them are equal to r\. In general, if y\, y 2 ,
Hi, 2/2', represent the probabilities for the occurrence
of residuals equal to n, r 2 , . . . n, r 2 , . . . respectively,
When N is increased by increasing the number of measurements,
each of the n's is increased in nearly the same ratio since the
residuals of the new measurements are distributed in essentially
the same manner as the old ones, provided all of the measure-
ments considered are made by the same method and with equal
care. Consequently, the y's corresponding to a definite method
of observation are nearly independent of the number of measure-
ments. As N increases they oscillate, with continually decreas-
ing amplitude, about the limiting values that would be obtained
with an infinite number of observations. Hence the form of a
curve, having y's for ordinates and corresponding r's for abscissae,
depends on the accuracy of the measurements considered and is
sensibly independent of N, provided it is a large number.
27. The Unit Error. The relative accuracy of different
series of measurements might be studied with the aid of the corre-
sponding y : r curves, but since the y's are abstract numbers, and
the r's are concrete, being of the same kind as the measurements,
it is better to adopt a slightly different mode of representation.
For this purpose, each of the r's is divided by an arbitrary con-
stant k, of the same kind as the measurements, and the abstract
numbers y^> -^> etc., are used as abscissae in place of the r's. In
A/ K
the following pages, k will be called the unit error. Its magnitude
may be arbitrarily chosen in particular cases, but, when not
definitely specified to the contrary, it will be taken equal to the
least magnitude that can be directly observed with the instru-
ments and methods used in making the measurements. To
32
THE THEORY OF MEASUREMENTS I ART. 28
illustrate: suppose we are measuring a given length with a scale
divided in millimeters. By estimation, the separate observations
can be made to one-tenth of a millimeter. Hence, in this case
we should take k equal to one-tenth of a millimeter.
If the residuals are arranged in the order of increasing magni-
tude, it is obvious that the successive differences TI r , r? TI
etc., are all equal to k. Hence, if the most probable value of the
measured quantity, x in equation (3), is taken to the same num-
ber of significant figures as the individual measurements, all of
the residuals are integral multiples of k and we have
k
k
28. The Probability Curve. The result of a study of the
distribution of the residuals may be arranged as illustrated in the
following table, where n is the number of residuals of magnitude
r; y is the probability that a single residual, chosen at random, is
of magnitude r; N is the total number of measurements, and k is
the unit error.
r
n
V
r
~k
-r p
n' p
~N~
-P
-n
*
w
-1
"0
no
N
ri
ni
N
+1
rp
n p
w
+P
M
Plotting y against ^ we obtain 2 p discrete points as in Fig. 3.
When N is large, these points, are somewhat symmetrically dis-
tributed about a curve of the general form illustrated by the
dotted line. If a larger number of observations is considered,
ART. 29] THE LAW OF ACCIDENTAL ERRORS
33
some of the points will be shifted upward while others will be
shifted downward, but the distribution will remain approxi-
mately symmetrical with respect to the same curve. In general,
successive equal increments to N cause shifts of continually de-
creasing magnitude; and in the limit, when TV becomes equal to
infinity, and the residuals are equal to the accidental errors, the
points would be on a uniform curve symmetrical to the y Q ordi-
nate. The curve thus determined represents the relation between
the magnitude of an error and the probability of its occurrence
in a given series of measurements. For this reason it is called
the probability curve.
29. Systems of Errors. The coordinates of the probability
curve are y and-r-, since it represents the distribution of the true
accidental errors AI, A 2 , etc., in regard to relative frequency and
magnitude. Since the curve is uniform, it represents not only
the errors of the actual observations, but also the distribution of
all of the accidental errors that would be found if the sensitive-
ness of our instruments were infinitely increased and an infinite
number of observations were made, provided only that all of the
observations were made with the same degree of precision and
entirely independently.
All of the errors represented by a curve of this type belong to a
definite system, characterized by the magnitude of the maximum
ordinate yo and the slope of the curve. Hence, every probability
curve represents a definite system of errors. It also represents
the accidental errors of a series of measurements of definite pre-
cision. Hence, the accidental errors of series of measurements of
different precision belong to different systems, and each series
is characterized by a definite system of errors.
The probability curves A and B in Fig. 4 represent the systems
34
THE THEORY OF MEASUREMENTS [ART. 30
of errors that characterize two series of measurements of different
precision. As the precision of measurement is increased it is
obvious that the number of small errors will increase relatively
to the number of large ones. Consequently the probability of
small errors will be greater and that of large ones will be less in
the more precise series A than in the less precise series B. Hence,
the curve A has a greater maximum ordinate and slopes more
rapidly toward the horizontal axis than the curve B.
30. The Probability Function. The maximum ordinate and
the slope of the probability curve depend on the constants that
appear in the equation of the curve. When we know the form
of the equation and have a method of determining the numerical
value of the constants, we are able to determine the relative pre-
cision of different series of measurements. Since the curve repre-
sents the distribution of the true accidental errors, we are also able
to compare the distribution of these errors with that of the resid-
uals and thus develop workable methods for finding the most
probable numeric of the measured magnitude.
It is obvious, from an inspection of Figs. 3 and 4, that y is a
continuous function of A, decreasing very rapidly as the magni-
tude of A increases either positively or negatively and symmetrical
with respect to the y axis. Hence, the probability curve sug-
gests an equation in the form
(9)
ART. 31] THE LAW OF ACCIDENTAL ERRORS 35
where e is the base of the Napierian system of logarithms, o> is a
constant depending on the precision of the series of measurements
considered, and the other variables have been defined above.
This equation can be derived analytically from the three axioms
of accidental errors, with the aid of several plausible assumptions
regarding the constitution of such errors, or from the principle
of the arithmetical mean. However, the strongest evidence of
its exactness lies in the fact that it gives results in substantial
agreement with experience. Consequently, we will adopt it as an
empirical relation, and proceed to show that it is in conformity
with the three axioms and leads to the arithmetical mean as the
most probable numeric derivable from a series of equally good
independent measurements of the same magnitude.
Equation (9) is the mathematical expression of the law of
accidental errors and is often referred to simply as the law of
errors. Its right-hand member is called the probability function
and, for the sake of convenience, is represented by (A), giving
the relations
2/ = 0(A); ^(A)^' 2 ^. (10)
31. The Precision Constant. The curves in Fig. 4 were
plotted, to the same scale, from data computed by equation (9).
The constant w was taken twice as great for the curve A as for
the curve B, and in both cases values of y were computed for suc-
cessive integral values of the ratio r-- The maximum ordinate of
each of these curves corresponds to the zero value of A and is
equal to the value of co used in computing the y's. The curve
A, corresponding to the larger value of o>, approaches the hori-
zontal axis much more rapidly than the curve B.
Obviously, the constant co determines both the maximum
ordinate and the slope of the probability curve. But we have
seen that these characteristics are proportional to the precision
of the measurements that determine the system of errors repre-
sented. Hence co characterizes the system of errors consid-
ered and is proportional to the precision of the corresponding
measurements. Some writers have called it the precision measure,
but, as it depends only on the accidental errors and takes no
account of the accuracy with which constant errors are avoided
or corrected, it does not give a complete statement of the pre-
36 THE THEORY OF MEASUREMENTS [ART. 32
cision. Consequently the term " precision measure " will be re-
served for a function to be discussed later, and a; will be called the
precision constant in the following pages.
When A is taken equal to zero in equation (9), y is equal to co.
Hence the precision of measurements, so far as it depends upon
accidental errors, is proportional to the probability for the occur-
rence of zero error in the corresponding system of errors. In
this connection, it should be borne in mind that the system of
errors includes all of the errors that would have been found
with an infinite number of observations, and that it cannot be
restricted to the errors of the actual measurements for the pur-
pose of computing o> directly. Indirect methods for computing
a> from given observations will be discussed later.
32. Discussion of the Probability Function. Inspection of
the curves in Fig. 4, in connection with equation (9), is sufficient to
show that the probability function is in agreement with the first
two axioms. Since y is an even function of A, positive and nega-
tive errors of the same magnitude are equally probable, and conse-
quently equally numerous in an extended series of measurements.
Hence the first axiom is fulfilled. Since A enters the function
only in the negative exponent, the probability for the occurrence
of an error decreases very rapidly as its magnitude increases
either positively or negatively. Hence small errors are much more
likely to occur than large ones and the second axiom is fulfilled.
Since the function </> (A) is continuous for values of A ranging
from minus infinity to plus infinity, it is apparently at variance
with the third axiom. For, if all of the errors lie between definite
finite limits L and + L, (A) should be continuous while A
lies between these limits and equal to zero for all values of A
outside of them. But we have no means of fixing the limits
-f- L and L, in any given case; and we note that 0(A) becomes
very small for moderately large values of A. Hence, whatever the
true value of L may be, the error involved in extending the limits
to oo and +00 is infinitesimal. Consequently, </>(A) is in sub-
stantial agreement with the third axiom provided it leads to the
conclusion that all possible errors lie between the limits oo and
+ oo . This will be the case if it gives unity for the probability
that a single error, chosen at random, lies between oo and -f oo .
For, if all of the errors lie between these limits, the probability
considered is a certainty and hence is represented by unity.
ART. 33] THE LAW OF ACCIDENTAL ERRORS
37
33. The Probability Integral. The accidental errors, corre-
sponding to actual measurements, may be arranged in groups ac-
cording to their magnitude in the same manner that the residuals
were arranged in article twenty-eight. When this is done the
errors in succeeding groups differ in magnitude by an amount
equal to the unit error k t since k is the least difference that can
be determined with the instruments used in making the obser-
vations. Hence, if A p is the common magnitude of the errors
in the pth group,
-A = A
(P+2)
-A
(p+i)
or, expressing the same relation in different form,
where a- is an indeterminate quantity that enters each of the
equations because we do not know the actual magnitude of the
A's.
FIG. 5.
Let the probability curve in Fig. 5 represent the system of
errors to which the errors of the actual measurements belong.
Then the ordinates y p , 2/( p +i), 2/( P +2), 2/(p+a) represent the
probabilities of the errors A p , A( p +i>, . . . A( p + e ) respectively.
Since the errors of the actual measurements satisfy the relation
(i), none of them correspond to points of the curve lying between
the ordinates y p , 2/( P + i), . . . 2/( P +). Hence, in virtue of equa-
tion (6), article twenty-three, if we choose one of the measure-
ments at random the probability that the magnitude of its error
lies between A p and A( P + Q ) is
2/CP+8)-
38 THE THEORY OF MEASUREMENTS [ART. 33
Multiplying and dividing the second member by q,
where y pq is written for the mean of the ordinates between y p
and 2/(p+ fl ). From equation (i)
&
Hence,
In the limit, when we consider the errors of an infinite number
of measurements made with infinitely sensitive instruments, every
point of the curve represents the probability of one of the errors
of the system. Consequently, for any finite value of q, Ihe inter-
val between the ordinates y p and y( P +q> is infinitesimal, and all
of the ordinates between these limits may be considered equal.
Hence, in the limit,
p = , y pq = 2/ A =
and (iii) reduces to
=* (A) , (11)
where y% +d * represents the probability that the magnitude of a
single error, chosen at random, is between A and A + dA.
By applying the usual reasoning of the integral calculus, it is
evident that the expression
rf = I /% (A) JA, (12)
/t i/ a
represents the probability that the magnitude of an error, chosen
at random, lies between the limits a and b. The integral in this
expression also represents the area under the probability curve
between the ordinates at T and T. Consequently the probability
in question is represented graphically by the shaded area in Fig. 6.
The probability that an error, chosen at random, is numerically
less than a given error A is equal to the probability that it lies
ART. 33] THE LAW OF ACCIDENTAL ERRORS 39
between the limits A and -J-A. Hence, if we designate this
probability by PA,
A
A
since (A) is an even function of A. Introducing the complete
expression for (A) from equation (10) we obtain
A 2
k jo
For the sake of simplification, put
2 A 2
then
/Y'ett,
Jo
(13)
which is an entirely general expression for the probability PA,
applicable to any system of errors when we know the correspond-
ing values of the constants o> and k. A series of numerical values
of the right-hand member of (13), corresponding to successive
values of the argument t, is given in Table XI, at the end of
this volume. Obviously, this table may be used in computing
the probability PA corresponding to any system of errors, since
the characteristic constants o> and k appear only in the limit of the
integral.
Whatever the values of the constants w and k, the limit vVw T
40 THE THEORY OF MEASUREMENTS [ART. 34
becomes infinite when A is equal to infinity. Hence, in every
system of errors,
* dt = l ) (13a)
where the numerical value is that given in Table XI, for the limit
t equals infinity. Consequently the probability function (A)
leads to the conclusion that all of the errors in any system lie
between the limits <x> and +00, and, therefore, it fulfills the
condition imposed by the third axiom as explained in the last
paragraph of article thirty-two.
34. Comparison of Theory and Experience. Equation (13)
may be used to compare the distribution of the residuals actually
found in any series of measurements with the theoretical distri-
bution of the accidental errors. If N equally trustworthy meas-
urements of the same magnitude have been made, all of the N
corresponding accidental errors belong to the same system, and
the probability that the error of a single measurement is numer-
ically less than A is given by PA in equation (13). Consequently,
if N is sufficiently large, we should expect to find
# A = NP* (iv)
errors less than A. For, if we consider only the errors of the
actual measurements, the probability that one of them is less
than A is equal to the ratio of the number less than A to the total
number. In the same manner, the number less than A 7 should
be
Hence, the number lying between the limits A and A' should be
N* = N* - N*. (v)
These numbers may be computed by equation (13) with the aid
of Table XI, when we know N and the value of the expression
V^co
corresponding to the given measurements. The number,
N r r , of residuals lying between the limits r equals A and r' equals
A' may be found by inspecting the series of residuals computed
from the given measurements by equation (3), article twenty-two.
If N is large and the errors of the given measurements satisfy
the theory we have developed, the numbers N% and N r r ' should
ART. 34] THE LAW OF ACCIDENTAL ERRORS
41
be very nearly equal, since in an extended series of measurements
the residuals are very nearly equal to the accidental errors.
The following illustration, taken from Chauvenet's "Manual
of Spherical and Practical Astronomy," is based on 470 obser-
vations of the right ascension of Sirius and Altair, by Bradley.
The errors of these measurements belong to a system character-
ized by a particular value of the ratio T that has been computed,
by a method to be described later (articles thirty-eight and forty-
two), and gives the relation
VTTCO
k
= 1.8086.
Consequently, to find the theoretical value of PA, corresponding
to any limit A, we take t equal to 1.8086 A in equation (13) and
find the corresponding value of the integral by interpolation from
Table XL
The third column of the following table gives the values of
PA corresponding to the chosen values of A in the first column
and the computed values of t in the second column. The fourth
column gives the corresponding values of N&. computed by equa-
tion (iv), taking N equal to 470. The sixth column, computed
by equation (v), gives the number, Nj[, of errors that should
lie between the limits A and A' given in the fifth. The seventh
column gives the number of residuals actually found between the
same limits.
A
t
^A
^A
Limits
A A'
<
N r
//
0.1
0.1809
0.2019
95
0.0-0.1
95
94
0.2
0.3617
0.3910
184
0.1-0.2
89
88
0.3
0.5426
0.5571
262
0.2-0.3
78
78
0.4
0.7234
0.6937
326
0.3-0.4
64
58
0.5
0.9043
0.7990
376
0.4-0.5
50
51
0.6
1.0852
0.8751
411
0.5-0.6
35
36
0.7
1.2660
0.9266
436
0.6-0.7
" 25
26
0.8
1.4469
0.9593
451
0.7-0.8
15
14
0.9
1.6277
0.9787
460
0.8-0.9
9
10
1.0
1.8086
0.9895
465
0.9-1.0
5
7
00
GO
1.0000
470
l.O-oo
5
8
Comparison of the numbers in the last two columns shows very
good agreement between theory, represented by N%, and expe-
42 THE THEORY OF MEASUREMENTS [ART. 35
rience, represented by N r r f , when we remember that the theory
assumes an infinite number of observations and that the series
considered is finite. Numerous comparisons of this nature have
been made, and substantial agreement has been found in all
cases in which a sufficient number of independent observations
have been considered. In general, the differences between N%
and N^' decrease in relative magnitude as the number of obser-
vations is increased.
35. The Arithmetical Mean. In article twenty-four it was
pointed out, as one of the fundamental principles of the theory
of errors, that the arithmetical mean of a number of equally trust-
wor^hy direct measurements on the same magnitude is the most
probable value that we can assign to the numeric of the measured
magnitude. In order to show that the probability function (A)
leads to the same conclusion, let eft, a 2 , . AT represent the
given measurements, and let x represent the unknown numeric
of the measured magnitude. If the actual value of this numeric
is X, the true accidental errors of the given measurements are
Ai = ai X, A 2 = 02 X, . . . AAT = ax X, (2)
and all of them belong to the same system, characterized by a
particular value .of the precision constant co. The probability
that one of the errors of this system, chosen at random, is equal
to an arbitrary magnitude A p is given by the relation
Since we cannot determine the true value X, the most probable
value that we can assign to x is that which gives a maximum
probability that N errors of the system are equal to the N resid-
uals
TI = ai x, r z = a 2 x, . . . r N = a N x. (3)
This is equivalent to determining x, so that the residuals are as
nearly as possible equal to the accidental errors.
If 2/1, 2/2, ... VN represent the probabilities that a single error
of the system, chosen at random, is equal to r\, r 2 , . . . r N respec-
tively,
2/i = (n), 2/2 = (r 2 ), . . . y N =
Hence, if P is the probability that N of the errors chosen together
ART. 35] THE LAW OF ACCIDENTAL ERRORS 43
are equal to n, r 2 , . . . r N respectively, we have, by equation (7),
article twenty-three,
P = 2/1 X 2/2 X ... X y N
Since the exponent in this expression is negative and -^ is con-
K
stant, the maximum value of P will correspond to the minimum
value of (ri 2 + r 2 2 + . . . -f ?W 2 ). Hence the most probable
value of x is that which renders the sum of the squares of the
residuals a minimum.
In the present case, the r's are functions of a single independent
variable x. Consequently the sum of the squares of the r's will
be a minimum when x satisfies the condition
-f-(ri 2 + r 2 2 + . ... +/VO =0.
(JJU
Substituting the expression for the r's in terms of x from equation
(3) this becomes
(a, - xY + (a 2 - xY + . . . + (a* - z) 2 = 0.
dx( )
Hence, (i - x) + (a 2 - x) + . . . + (a N - x) = 0, (14)
ai -f 2 + + AT
and x = jy-
Consequently, if we take x equal to the arithmetical mean of the
a's in (3), the sum of the squares of the computed r's is less than
for any other value of x. Hence the probability P that N errors
of the system are equal to the N residuals is a maximum, and the
arithmetical mean is the most probable value that we can assign
to the numeric X on the basis of the given measurements.
Equation (14) shows that the sum of the residuals, obtained
by subtracting the arithmetical mean from each of the given
measurements, is equal to zero. This is a characteristic property
of the arithmetical mean and serves as a useful check on the
computation of the residuals.
The argument of the present article should be regarded as a
justification of the probability function 0(A) rather than as a
proof of the principle of the arithmetical mean. As pointed out
above, this principle is sufficiently established on a priori grounds
and by common consent.
CHAPTER V.
CHARACTERISTIC ERRORS.
SEVERAL different derived errors have been used as a measure
of the relative accuracy of different series of measurements. Such
errors are called characteristic errors of the system, and they de-
crease in magnitude as the accuracy of the measurements, on which
they depend, increases. Those most commonly employed are the
average error A , the mean error M, and the probable error E, any
one of which may be used as a measure on the relative accuracy
of a single observation.
36. The Average Error. The average error A of a single
observation is the arithmetical mean of all of the individual errors
of the system taken without regard to sign. That is, all of the
errors are taken as positive in forming the average. Hence, if
N is the total number of errors,
! _
~N~ "W
where the square bracket [ ] is used as a sign of summation, and
the ~~ over the A indicates that, in taking the sum, all of the A's
are to be considered positive.
In accordance with the usual practice of writers on the theory
of errors, the square bracket [ ] will be used as a sign of summa-
tion, in the following pages, in place of the customary sign S.
This notation is adopted because it saves space and renders com-
plicated expressions more explicit.
In equation (15) all of the errors of the system are supposed
to be included in the summation. Hence, both [A] and N are
infinite and the equation cannot be applied to find A directly
from the errors of a limited number of measurements. Conse-
quently we will proceed to show how the average error can be
derived from the probability function, and to find its relation
to the precision constant co. A little later we shall see how A
can be computed directly from the residuals corresponding to a
limited number of measurements.
44
ART. 36] CHARACTERISTIC ERRORS 45
If yd is the probability that the magnitude of a single error,
chosen at random, lies between A and A + dA, and rid is the num-
ber of errors between these limits,
and consequently
n d = Ny d
= N4> (A) ^ (16)
in virtue of equation (11), article thirty-three, where A represents
the mean magnitude of the errors lying between A and A + dA.
Hence, the sum of the errors between these limits is
and the sum of the errors between A = a and A = b is
N
Substituting the complete expression for </>(A) from equation (10)
this becomes
Hence, the sum of the positive errors of the system is
Nu / -*,
-; I Ae kz dA,
k Jo
and the sum of the negative errors is
Nu r
k J -<*
These two integrals are obviously equal in magnitude and opposite
in sign. Consequently the sum of all of the errors of the system
taken without regard to sign is
Ae -^A (17)
7TCO
46 THE THEORY OF MEASUREMENTS [ART. 37
Hence from equation (15),
~ N
and introducing the numerical value of IT,
A =0.3183-- (19)
CO
37. The Mean Error. The mean error M of a single meas-
urement in a given series is the square root of the mean of the
squares of the errors in the system determined by the given
measurements. Expressed mathematically
A^ + A^-f-.* + A^_[A1
N ' N
This equation includes all of the errors that belong to the given
system. Hence, as pointed out in article thirty-six, in regard to
equation (15), it cannot be applied directly to a limited series of
measurements.
By equation (16) the number of errors with magnitudes between
the limits A and A + dA is equal to , . Consequently
/c
the sum of the squares of the errors between these limits is equal
#A 2 4>(A)dA
k
in the last article,
to - .; . Hence, by reasoning similar to that employed
(21)
/
/
2N r A% -, *
since the integrand is an even function of A. Integrating by
parts,
7TCO
The first term of the second member of this equation reduces to
AKT.38] CHARACTERISTIC ERRORS 47
zero when the limits are applied. Putting t 2 for in the
K
second term,
[Al-^P^a- (22)
TT^CO 2 Jo 2 7TC0 2
in virtue of equation (13a). Hence,
N 2*
and
M =
= 0.3989--
CO
(23)
38. The Probable Error. The probable error E of a single
measurement is a magnitude such that a single error, chosen at
random from the given system, is as likely to be numerically
greater than E as less than E. In other words, the probability
that the error of a single measurement is greater than E is equal
to the probability that it is less than E. Hence, in any extended
series of measurements, one-half of the errors are less than E and
one-half of them are greater than E.
The name " probable error," though sanctioned by universal
usage, is unfortunate; and the student cannot be too strongly
cautioned against a common misinterpretation of its meaning.
The probable error is NOT the most probable magnitude of the
error of a single measurement and it DOES NOT determine the
limits within which the true numeric of the measured magnitude
may be expected to lie. Thus, if x represents the measured
numeric of a given magnitude Q and E is the probable error of x,
it is customary to express the result of the measurement in the
form
Q = x E.
This does not signify that the true numeric of Q lies between the
limits x E and x + E, neither does it imply that x is probably
in error by the amount E. It means that the numeric of Q is as
likely to lie between the above limits as outside of them. If a
new measurement is made "by the same method and with equal
care, the probability that it will differ from x by less than E is
equal to the probability that it will differ by more than E.
48
THE THEORY OF MEASUREMENTS [ART. 38
In article thirty-three it was pointed out that the probability
that an error, chosen at random from a given system, lies between
the limits A = a and A = b is represented by the area under the
probability curve between the ordinates corresponding to the
limiting values of A. Hence, the probability that the error of a
single measurement is numerically less than E may be represented
by the area under the probability curve between the ordinates y- E
and y+ E , in Fig. 7, and the probability that it is greater than E by
the sum of the areas outside of these ordinates. Since these two
FIG. 7.
probabilities are equal, by definition, the ordinates correspond-
ing to the probable error bisect the areas under the two branches
of the probability curve.
Since the probability that the error of a single measurement is
less than E is equal to the probability that it is greater than E
and the probability that it is less than infinity is unity, the
probability that it is less than E is one-half. Consequently,
putting A equal to E in equation (13), article thirty-three,
Pw = ~
rw T" 1
e-dt - 2-
\J
From Table XI,
PA = 0.49375 for the limit t = 0.47,
PA = 0.50275 for the limit t = 0.48,
and by interpolation,
P E = 0.50000 for the limit t = 0.47694.
Hence, equation (24) is satisfied when
(24)
= 0.47694,
ART. 39]
and we have
CHARACTERISTIC ERRORS
E
0.47694 k
VTT w
= 0.2691 -
CO
49
(25)
39. Relations between the Characteristic Errors. Elimina-
k
ting- from equations (18), (23), and (25), taken two at a time, we
obtain the relations
(26")
E = 0.4769 VTT -A = 0.8453 -A,
E = 0.4769 V2 M = 0.6745 M,.
which express the relative magnitudes of the average, mean, and
probable errors. These relations are universally adopted in com-
MAE
k k k
FIG. 8.
puting the precision of given series of measurements, and they
should be firmly fixed in mind.
The three equations from which the relations (26) are derived
may be put in the form
A = 0.3183
k co
M _ 0.3989
k co
E = 0.2691
k co
The probability curve in Fig. 8 represents the distribution of
the errors in a system characterized by a particular value of co,
(27)
50 THE THEORY OF MEASUREMENTS [ART. 39
determined by a given series of measurements. The ordinates
AM A E
VA> VM> an d Us correspond to the abscissae -^> -jp and -"& > com "
puted by the above equations. Consequently, y A represents the
probability that the error of a single measurement is equal to
+A, y M the probability that it is equal to +M, and y E the prob-
ability that it is equal to +E. In like manner y- A , y- M , and
y~ E represent the respective probabilities for the occurrence of
errors equal to A, M, and E.
A curve of this type can be constructed to correspond to any
given series of measurements, and in all cases the relative loca-
tion of the ordinates y A , y M) and y E will be the same. It was
pointed out in the last article that the ordinates y E and y- E bisect
the areas under the two branches of the curve. Consequently,
in an extended series of measurements, somewhat more than one-
half of the errors will be less than either the average or the mean
error. Moreover, it is obvious from Fig. 8 that an error equal to
E is somewhat more likely to occur than one equal to either A or M.
Since each of the characteristic errors A, M, and E, bears a
constant relation to the precision constant co, any one of them
might be used as a measure of the precision of a single measure-
ment in a given series, so far as this depends on accidental errors.
The probable error is more commonly employed for this purpose
on account of its median position in the system of errors deter-
mined by the given measurements.
It is interesting to observe that the ordinate y M corresponds to
a point of inflection in the probability curve. By the ordinary
method of the calculus we know that this curve has a point of
inflection corresponding to the abscissa that satisfies the relation
Substituting the complete expression for y
Hence,
ART. 40] CHARACTERISTIC ERRORS 51
is the abscissa of the point of inflection. Comparing this with
equation (23) we see that
and consequently that the ordinates y M and y- M meet the prob-
ability curve at points of inflection.
40. Characteristic Errors of the Arithmetical Mean. Equa-
tion (23) may be put in the form
CO 2 1
where M is the mean error of a single measurement in a series
corresponding to the unit error k and the precision constant w.
Consequently the probability function,
"***&
y = we k y
corresponding to the same series may be put in the form
y = ae 2M *. (i)
If A i, A 2 , . . . AJV are the accidental errors of N direct measure-
ments in the same series, the probability P that they all occur in
a system characterized by the mean error M is equal to the product
of the probabilities for the occurrence of the individual errors in
that system. Hence,
If the individual measurements are represented by a\ t 0,2,
. . . a N , and the true numeric of the measured quantity is X,
Ai = ai - X; A 2 = a z - X\ . . . A# = a N - X,
and, if x is the arithmetical mean of the measurements, the corre-
sponding residuals are
n = ai x', r z = 2 x; . . . r N = a N x.
Consequently, if the error of the arithmetical mean is 5,
X - x = 5,
and
Ai = n - 5; A 2 = r 2 - 5; . . . A# = r N 8.
Squaring and adding,
[A 2 ] = [r 2 ]-25M+ATS 2 ;
(28)
52 THE THEORY OF MEASUREMENTS [ART. 40
since [r] Is equal to zero in virtue of equation (14), article thirty-
five. When this value of [A 2 ] is substituted in (ii), the resulting
value of P is the probability that the arithmetical mean is in
error by an amount 6. For, as we have seen in article thirty-five,
the minimum value of [r 2 ] occurs when x is taken equal to the
arithmetical mean. Consequently, P is a maximum when <5 is
equal to zero and decreases in accordance with the probability
function as 5 increases either positively or negatively.
We do not know the exact value of either X or 5; but, if y a is
the probability that the error of the arithmetical mean is equal
to an arbitrary magnitude 5, the foregoing reasoning leads to the
relation
2M2
But the arithmetical mean is equivalent to a single measurement
in a series of much greater precision than that of the given meas-
urements. Hence, if o> a is the precision constant correspond-
ing to this hypothetical series and M a is the mean error of the
arithmetical mean, we have by analogy with (i)
a*
y a = w a e 2 M 2 . (iv)
Equations (iii) and (iv) are two expressions for the same prob-
ability and should give equal values to y a whatever the assumed
value of 5. This is possible only when
2M ,
and
1 N
~ 2M 2
Hence,
M M
M a = =
VN
Consequently, the mean % error of the arithmetical mean is equal
to the mean error of a single measurement divided by the square
root of the number of measurements.
Since the average, mean, and probable errors of a single meas-
urement are connected by the relations (26), the corresponding
Art. 41] CHARACTERISTIC ERRORS 53
errors of the arithmetical mean, distinguished by th.e subscript
a, are given by the relations
4 = -4=; M a = -^=; E a = -?j=. (29)
VN VN VN
41. Practical Computation of Characteristic* Errors. As
pointed out in article thirty-seven, the square of the mean error
[A 2 1
M is the limiting value of the ratio ^rp when both members
become infinite, i.e., when all of the errors of the given system
are considered. But the errors of the actual measurements fall
into groups, as explained in article thirty-three, and the errors in
succeeding groups differ in magnitude by a constant amount k,
depending on the nature of the instruments used in making the
observations. Consequently, the ordinates, of the probability
curve, corresponding to these errors are uniformly distributed
along the horizontal axis. Hence, if we include in [A 2 ] only the
errors of the actual measurements, the limiting value of the ratio
fA 2 l
L -^- when N is indefinitely increased will be nearly the same as if
all of the errors of the system were included. Since the ratio
approaches its limit very rapidly as N increases, the value of M
can be determined, with sufficient precision for most practical
purposes, from a somewhat limited series of measurements.
If we knew the true accidental errors, the mean error could be
computed at once from the relation
(v)
and, since the residuals are nearly equal to the accidental errors
when N is very large, an approximate value can be obtained by
using the r's in place of the A's. A better approximation can be
obtained if we take account of the difference between the A's
and the r's. From equation (28)
[A 2 ] = [r 2 ] + AT5 2 , (vi)
where 6 is the unknown error of the arithmetical mean. Probably
the best approximation we can make to the true value of 8 is to
set it equal to the mean error of the arithmetical mean. Hence,
from the second of equations (29)
54 THE THEORY OF MEASUREMENTS [ART. 41
Consequently, (vi) becomes
NM 2 = [r 2 ] +
and we have
(30)
Thus the square of the mean error of a single measurement is
equal to the sum of the squares of the residuals divided by the
number of measurements less one.
Combining (30) with the third of equations (26), article thirty-
nine, we obtain the expression
E = 0.6745 V^rj < 31 )
for the probable error of a single measurement. Hence, by equa-
tions (29), the mean error M a and the probable error E a of the
arithmetical mean are given by the relations
and * = - (32)
When the number of measurements is large, the computation
of the probable errors E and E a by the above formulae is some-
what tedious, owing to the necessity of finding the" square of
each of the residuals. In such cases a sufficiently close approx-
imation for practical purposes can be derived from the average
error A with the aid of equations (26). The first of these equa-
tions may be written in the form
[A3 = T [A] 2
N 2 N 2 '
If we assume that the distribution of the residuals is the same as
that of the true accidental errors, a condition that is accurately
fulfilled when N is very large, we can put
N
Consequently,
ART. 41] CHARACTERISTIC ERRORS 55
When the mean error M is expressed in terms of the A's, equation
(30) becomes
[A 2 ]_ M
N ' N-l'
or
[Ag = N [Sp.
[r 2 ] tf- 1 [r]2 '
Consequently
[A? [r?
and, since this ratio is equal to A 2 , we have
== and A = - X (33)
-1) NVN-1
Combining this result with the second of equations (26) and the
third of (29), we obtain
E = 0.8453 . ^ ; E a = 0.8453 - ^ . (34)
VN(N-1)' NVN-1
The above formulae for computing the characteristic errors from
the residuals have been derived on the assumption that the true
accidental errors and the residuals follow the same law of dis-
tribution. This is strictly true only when the number of measure-
ments considered is very large. Yet, for lack of a better method,
it is customary to apply the foregoing formulas to the discussion
of the errors of limited series of measurements and the results
thus obtained are sufficiently accurate for most practical purposes.
When the highest attainable precision is sought, the number of
observations must be increased to such an extent that the theo-
retical conditions are fulfilled.
The choice between the formulae involving the average error
A and those depending on the mean error M is determined largely
by the number of measurements available and the amount of
time that it is worth while to devote to the computations. When
the number of measurements is very large, both sets of formulae
lead to the same values for the probable errors E and E a , and
much time is saved by employing those depending on A. For
limited series of observations a better approximation to the true
values of these errors is obtained by employing the formulae in-
volving the mean error. In either case the computation may be
56
THE THEORY OF MEASUREMENTS [ART. 42
facilitated by the use of Tables XIV and XV at the end of this
volume. These tables give the values of the functions
0.6745 0.8453 0.8453
0.6745
VN(N-1)'
and
NVN-l'
corresponding to all integral values of N between two and one
hundred.
42. Numerical Example. The following example, represent-
ing a series of observations taken for the purpose of calibrating
the screw of a micrometer microscope, will serve to illustrate the
practical application of the foregoing methods. Twenty inde-
pendent measurements of the normal -distance between two
parallel lines, expressed in terms of the divisions of the micrometer
head, are given in the first and fourth columns of the following
table under a.
a
r
r i
a
r
r 2
194.7
+0.53
0.2809
194.3
+0.13
0.0169
194.1
-0.07
0.0049
194.3
+0.13
0.0169
194.3
+0.13
0.0169
194.0
-0.17
0.0289
194.0
-0.17
0.0289
194.4
+0.23
0.0529
193.7
-0.47
0.2209
194.5
+0.33
0.1089
194.1 -0.07
0.0049
193.8
-0.37
0.1369
193.9 -0.27
0.0729
193.9
-0.27
0.0729
194.3 +0.13
0.0169
193.9
-0.27
0.0729
194.3 +0.13
0.0169
194.8
+0.63
0.3969
194.4 +0.23
0.0529
193.7
-0.47
0.2209
194.17
5.20
1.8420
.r
[r 2 ]
Since the observations are independent and equally trust-
worthy, the most probable value that we can assign to the numeric
of the measured magnitude is the arithmetical mean x; and we
find that x is equal to 194.17 micrometer divisions. Subtracting
194.17 from each of the given observations we obtain the residuals
in the columns under r. The algebraic sum of these residuals is
equal to zero as it should be, owing to the properties of the arith-
metical mean. The sum without regard to sign, [r], is equal to
5.20. Squaring each of the residuals gives the numbers in the
columns under r 2 and adding these figures gives 1.8920 for the
sum of the squares of the residual [r 2 ].
Taking N equal to twenty, in formulae (33) and (34), we find
the average and probable errors
ART. 42] CHARACTERISTIC ERRORS 57
= =b 0.267; A a = Ar ^ = 0.0596,
NVN-l
E = 0.8453 7== = 0.226; # = 0.8453 ^-^ = = 0.0504,
where the numerical results are written with the indefinite sign
since the corresponding errors are as likely to be positive as nega-
tive.
When formulae (30), (31), and (32) are employed we obtain the
mean errors,
and the probable errors
E = 0.6745
The values of the probable errors E and jEk, computed by the
two methods, agree as closely as could be expected with so small
a number of observations. Probably the values d= 0.210 and
0.047, computed from the mean errors M and M a , are the more
accurate, but those derived from the average errors A and A a are
sufficiently exact for most practical purposes. An inspection of
the column of residuals is sufficient to show that eleven of them
are numerically greater, and nine are numerically less than either
of the computed values of E. Consequently, both of these values
fulfill the fundamental definition of the probable error of a single
measurement as nearly as we ought to expect when only twenty
observations are considered.
If we use D to represent the measured distance between the
parallel lines, in terms of micrometer divisions, we may write
the final result of the measurements in the form
D = 194.170 =t 0.047 mic. div.
This does not mean that the true value of D lies between the
specified limits, but that it is equally likely to lie between these
limits or outside of them. Thus, if another and independent
series of twenty measurements of the same distance were made
58 THE THEORY OF MEASUREMENTS [ART. 43
with the same instrument, and with equal care, the chance that
the final result would lie between 194.123 and 194.217 is equal to
the chance that it would lie outside of these limits.
Equation (25), article thirty-eight, may be written in the form
-co 0.4769
Taking E equal to 0.210, we find that
v = 2.271
k
for the particular system of errors determined by the above meas-
urements. Consequently, the probability for the occurrence of an
error less than A in this system is, by equation (13), article thirty-
three,
2.271.A
and, since there are twenty measurements, we should expect to
find 20 PA errors numerically less than any assigned value of A.
The values of PA, corresponding to various assigned values of
A, can be easily computed with the aid of Table XI and applied,
as explained in article thirty-four, to compare the theoretical
distribution of the accidental errors with that of the residuals
given under r in the above table. Such a comparison would have
very little significance in the present case, however it resulted,
since the number of observations considered is far too small to
fulfill the theoretical requirements. But it would show that,
even in such extreme cases, the deviations from the law of errors
are not greater than might be expected. The actual comparison
is left as an exercise for the student.
43. Rules for the Use of Significant Figures. The funda-
mental principles underlying the use of significant figures were
explained in article fifteen. General rules for their practical ap-
plication may be stated in terms of the probable error as follows:
All measured quantities should be so expressed that the last
recorded significant figure occupies the place corresponding to the
second significant figure in the probable error of the quantity
considered.
The number of significant figures carried through the compu-
ART. 43] CHARACTERISTIC ERRORS 59
tations should be sufficient to give the final result within one unit
in the last place retained and no more.
For practical purposes probable errors should be computed to
two significant figures.
The example given in the preceding article will serve to illus-
trate the application of these rules. The second significant figure
in the probable error of the arithmetical mean occupies the third
decimal place. Consequently, the final result is carried to three
decimal places, notwithstanding the fact that the last place is
occupied by a zero. It would obviously be useless to carry out
the result farther than this, since the probable error shows that
the digit in the second decimal place is equally likely to be in
error by more or less than .five units. If less significant figures
were used, the fifth figure in computed results might be vitiated
by more than one unit.
In order to apply the rules to the individual measurements, it
is necessary to make a preliminary series of observations, under
as nearly as possible the same conditions that will prevail during
the final measurements, and compute the probable error of a
single observation from the data thus obtained. Then, if possible,
all final measurements should be recorded to the second significant
figure in this probable error and no farther. It sometimes happens,
as in the above example, that the graduation of the measuring
instruments used is not sufficiently fine to permit the attainment
of the number of significant figures required by the rule. In such
cases the observations are recorded to the last attainable figure,
.or, if possible, the instruments are so modified that they give
the required number of figures. Thus, in the example cited, the
second significant figure in the probable error of a single measure-
ment is in the second decimal place, but the micrometer can
be read only to one-tenth of a division. Hence the individual
measurements are recorded to the first instead of the second
decimal place. In this case the accuracy attained in making the
settings of the instrument was greater than that attained in
making the readings, and an observer, with sufficient experience,
would be justified in estimating the fractional parts to the nearest
hundredth of a division. A better plan would be to provide the
micrometer head with a vernier reading to tenths or hundredths of
a division. In the opposite case, when the accuracy of setting is
less than the attainable accuracy of reading, it is useless to record
60 THE THEORY OF MEASUREMENTS [ART. 43
the readings beyond the second significant figure in the probable
error of a single observation.
For the purpose of computing the residuals, the arithmetical
mean should be rounded to such an extent that the majority of
the residuals will come out with two significant figures. This
greatly reduces the labor of the computations and gives the calcu-
lated characteristic errors within one unit in the second significant
figure.
CHAPTER VI.
MEASUREMENTS OF UNEQUAL PRECISION.
44. Weights of Measurements. In the preceding chapter
we have been dealing with measurements of equal precision, and
the results obtained have been derived on the supposition that
there was no reason to assume that any one of the observations
was better than any other. Under these conditions we have
seen that the most probable value that we can assign to the
numeric of the measured magnitude is the arithmetical mean of
the individual observations. Also, if M and E are the mean and
probable errors of a single observation, M a and E a the mean and
probable errors of the arithmetical mean, and A/" the number of
observations, we have the relations
# = 0.6745 M; ' E a = 0.6745 M n ,
M E
v
(35)
The true numeric X of the measured magnitude cannot be
exactly determined from the given observations, but the final
result of the measurements may be expressed in the form
X = x E a ,
which signifies that X is as likely to lie between the specified
limits as outside of them.
Now suppose that the results of m independent series of meas-
urements of the same magnitude, made by the same or different
methods, are given in the form
X = xi E lt
X = x% it EZ,
X = x m d= E m .
61
62 THE THEORY OF MEASUREMENTS [ART. 44
What is the most probable value that can be assigned to X on
the basis of these results? Obviously, the arithmetical mean of the
x's will not do in this case, unless the E's are all equal, since the
x's violate the condition on which the principle of the arithmetical
mean is founded. If we knew the individual observations from
which each of the x's were derived, and if the probable error of
a single observation was the same in each of the series, the most
probable value of X would be given by the arithmetical mean of
all of the individual observations. Generally we do not have the
original observations, and, when we do, it frequently happens that
the probable error of a single observation is different in the differ-
ent series. Consequently the direct method is seldom applicable.
The E's may differ on account of differences in the number of
observations in the several series, or from the fact that the prob-
able error of a single observation is not the same in all of them, or
from both of these causes. Whatever the cause of the difference,
it is generally necessary to reduce the given results to a series of
equivalent observations having the same probable error before
taking the mean. For it is obvious that a result showing a small
probable error should count for more, or have greater weight,
in determining the value of X than one- that corresponds to a
large probable error, since the former result has cost more in time
and labor than the latter.
The reduction to equivalent observations having the same
probable error is accomplished as follows: m numerical quanti-
ties wi, w 2 , . . . w m , called the weights of the quantities Xi, x 2 ,
. . . x m , are determined by the relations
E* E a 2 E*
W ^E?> W *=Ef'> ' ' Wm =E^' (36)
where E a is an arbitrary quantity, generally so chosen that all
of the w's are integers, or may be placed equal to the nearest
integer without involving an error of more than one or two units
in the second significant figure of any of the E's. In the following
pages E 8 will be called the probable error of a standard observa-
tion. Obviously, the weight of a standard observation is unity
on the arbitrary scale adopted in determining, the w's; for, by
equations (36),
ART. 45] MEASUREMENTS OF UNEQUAL PRECISION 63
Such an observation is not assumed to have occurred in any of
the series on which the x's depend, but is arbitrarily chosen as a
basis for the computation of the weights of the given results.
By comparing equations (35) and (36), we see that E\ is equal
to the probable error of the arithmetical mean of w\ standard
observations. But it is also the probable error of the given
result XL Consequently x\ is equivalent to the arithmetical
mean of wi standard observations. Similar reasoning can be
applied to the other E's and in general we have
Xi = mean of w\ standard observations,
x 2 = mean of w 2 standard observations,
x m = mean of w m standard observations.
(i)
The weights Wi, w 2} . . . w m are numbers that express the rela-
tive importance of the given measurements for the determination
of the most probable value of the numeric of the measured mag-
nitude. Each weight represents the number of hypothetical
standard observations that must be combined to give an arith-
metical mean with a probable error equal to that of the given
measurement.
45. The General Mean. From equations (i) it is obvious
that
= the sum of Wi standard observations,
= the sum of w z standard observations,
w m x m = the sum of w m standard observations,
and, consequently,
-f + w m x m
is equal to the sum of w\ + ^2 + . . + W TO standard observa-
tions. Since the probable error E 8 is common to all of the
standard observations, they are equally trustworthy and their
arithmetical mean is the most probable value that we can assign
to the numeric X on the basis of the given data. Representing
this value of X Q we have
_ WiXi + W 2 X 2 + * + W m X m X Q( _V
Wl+W2 + . . . + Wm
The products W&1, etc., are called weighted observations or meas-
64 THE THEORY OF MEASUREMENTS [ART. 45
urements, and x is called the general or weighted mean. The
weight W Q of X Q is obviously given by the relation
wo = wi + w 2 + - + w m , (38)
since X Q is the mean of w standard observations.
Equation (37) for the general mean can be established inde-
pendently from the law of accidental errors in the following manner:
Let coi, o> 2 , . . . w m represent the precision constants correspond-
ing to the probable errors EI, E z , E m , and let w s be an
arbitrary quantity connected with the arbitrary quantity E 8 by
the relation
# 8 = 0.2691 -
fc>
Then, by equations (25) and (36),
i 2 C0 2 2 CO TO 2
Wl = ~^> W2 = l^> *- IF- (39)
If XQ is the most probable value of the numeric X, the residuals
corresponding to the given aj's are
ri = xi XQ', r 2 = x z XQ', . . . r m = x m x .
The probability that the true accidental error of x\ is equal to r\
s
in virtue of equations (39). Similarly, if 2/1, 2/2, Vm are the
probabilities that r\, r 2 , . . . r m are the true accidental errors of
x m}
OJ.2
T-TT
2/2 = co 2 e
Hence, if P is the probability that all of the r's are simultaneously
equal to true accidental errors, we have
w z
-Tr--
P = (wio> 2 . . . ov)e
and the most probable value of X is that which renders P a
maximum. Obviously, the maximum value of P occurs when
ART. 45] MEASUREMENTS OF UNEQUAL PRECISION 65
(wirf + w 2 r 2 2 + . . . + w m r m 2 ) is a minimum. Consequently the
most probable value X Q is given by the relation
^T (wiri 2 + w 2 r 2 2 + + w m r m 2 ) = 0.
Substituting the values of the r's and differentiating this becomes
Wi (Xi XQ) + W 2 (X 2 XQ) + W m (x m XQ) = 0.
Hence,
WiXi + W 2 X 2 + + W m X m
XQ ; : : j
as given above.
If we multiply or divide the numerator and denominator of
equation (37) by any integral or fractional constant, the value
of #o is unaltered. Hence, from (36), it is obvious that we are at
liberty to choose any convenient value for E a) whether or not it
gives integral values to the w's. Equations (36) also show that
the weights of measurements are inversely proportional to the
squares of their probable errors and consequently we may take
#! 2 E? EJ
w 2 = wi-^-', w 3 = w 1 ^-; . . . w m = wi-^-- (40)
Etf 1 &m
Hence, if we choose, we can assign any arbitrary weight to one of
the given measurements and compute the weights of the others
by equation (40).
The foregoing methods for computing the weights w\, w 2 , etc.,
are applicable only when the given measurements x\, x 2 , etc., are
entirely free from constant errors and mistakes. When this
condition is not fulfilled the method breaks down because the
errors of the x's do not follow the law of accidental errors. In
such cases it is sometimes possible to assign weights to the given
measurements by combining the given probable errors with an
estimate of the probable value of the constant errors, based on a
thorough study of the methods by which the x's were obtained.
Such a procedure is always more or less arbitrary, and requires
great care and experience, but when properly applied it leads to a
closer approximation to the true numeric of the measured magni-
tude than would be obtained by taking the simple arithmetical
mean of the x's. Since it involves a knowledge of the laws of
propagation of errors and of the methods for estimating the pre-
66 THE THEORY OF MEASUREMENTS [ART. 46
cision attained in removing constant errors and mistakes, it can-
not be fully developed until we take up the study of the under-
lying principles.
46. Probable Error of the General Mean. When the given
x's are free from constant errors and the E's are known, the weights
of the individual measurements are given by (36), and the weight
W of the general mean is given by (38). Consequently, if E is
the probable error of the general mean, we have by analogy with
equations (36)
1*0=14 and #0=-- (41)
If we choose, E may be expressed in terms of any one of the E's
in place of E 8 . Thus, let E n and w n be the probable error and
the weight of any one of the x's, then by (36)
E >
W
and eliminating E a between this equation and (41) we have
(42)
When the weights are assigned by the method outlined in the
last paragraph of the preceding article, or when, for any reason,
the w's are given but not the E's, (41) and (42) cannot be applied
until E a or E n has been derived from the given x's and w's. If
the number of given measurements is large, the value of E 8 corre-
sponding to the given weights can be computed with sufficient
precision by the application of the law of errors as outlined below.
If the number of given measurements is small, or if constant
errors and mistakes have not been considered in assigning the
weights, the following method gives only a rough approximation
to the true value of E s , and consequently of E Q) since the condi-
tions underlying the law of errors are not strictly fulfilled. It will
be readily seen that while E 8 may be arbitrarily assigned for the
purpose of computing the weights, when the E's are given, its
value is fixed when the weights are given.
Let xi, z 2 , . . . x m represent the given measurements and
Wi, ^ 2 , ... w m , the corresponding weights. Then, if o? 8 repre-
ART. 46] MEASUREMENTS OF UNEQUAL PRECISION 67
sents the precision constant of a standard observation, and wi
that of an observation of weight w\, we have by (39)
Consequently, if 2/ A is the probability that the error of x i is equal
to A,
and, by equation (11), article thirty-three, the probability that
the error of x\ lies between the limits A and A + dA is
Now, WiA 2 is the weigh ted square of the error A, and in the follow-
ing pages the product VwA will be called a weighted error. Hence,
if we put d = VwjA, and dd = Vw { dA, we have for the probability
that the weighted error of Xi lies between the limits 5 and d -\- dd
Since the same result would have been obtained if we had started
with any other one of the x's and w's, it is obvious that this equa-
tion expresses the probability that any one of the x's, chosen at
random, is affected by a weighted error lying between the limits
5 and d + dd. But, if rid is the number of #'s affected by weighted
errors lying between these limits, and m is the total number of
as's, we have also
or
Hence, the sum of the squares of the weighted errors lying between
5 and 5 -f- dd is given by the relation
S2 u s - TO- ,* ,
= m8 2 -re dS,
=
" m
68 THE THEORY OF MEASUREMENTS [A RT . 46
and, by the method adopted in articles thirty-six and thirty-seven,
we have
[g] = 2 a), r
m A: Jo
where [5 2 ] is supposed to include all possible weighted errors
between the limits plus and minus infinity. Introducing the
values of the S's in terms of the w's and A's this becomes
m m
which is an exact equation only when the number of measure-
ments considered is practically infinite.
If M 8 is the mean error of a standard observation, we have from
equation (23)
Hence, from equation (26)
. = 0.6745
Now, we do not know the true value of the A's and the number of
given measurements is seldom sufficiently large to fulfill the con-
ditions underlying this equation. But we can compute the gen-
eral mean X Q and the residuals
Ti = Xi XQ] r 2 = X 2 XQ] . . . T m = X m X ,
and, by a method exactly analogous to that of article forty-one,
it can be shown that the best approximation that we can make is
given by the relation
[wr 2 ]
m m 1
Hence, as a practicable formula for computing E 8 , we have
E a = 0.6745 V-T' (43)
~ m 1
and consequently E is given by the relation
Eo = 0.6745V... r ,,'
in virtue of equation (41).
ART. 47] MEASUREMENTS OF UNEQUAL PRECISION 69
When the probable errors of the given measurements are
known, and the weights are computed by equation (36), the value
of E 8 computed by equation (43) will agree with the value arbi-
trarily assigned, for the purpose of determining the w's, provided
the x's are sufficiently numerous and free from constant errors
and mistakes. The number of measurements considered is
seldom sufficient to give exact agreement, but a large difference
between the assigned and computed values of E 8 is strong evidence
that constant errors have not been removed with sufficient pre-
cision. On the other hand, satisfactory agreement may occur
when all of the x's are affected by the same constant error. Con-
sequently such agreement is not a criterion for the absence of
constant errors, but only for their equality in the different meas-
urements.
47. Numerical Example. As an illustration of the applica-
tion of the foregoing principles, consider the micrometer measure-
ments given under x in the following table. They represent the
results of six series of measurements similar to that discussed in
article forty-two, the last one being taken directly from that
article. The probable errors, computed as in article forty-two,
are given under E. They differ partly on account of differences
in the number of observations in the several series, and partly
from the fact that the individual observations were not of the
same precision in all of the series. The squares of the probable
errors multiplied by 10 4 are given under E 2 X 10 4 to the nearest
digit in the last place retained. It would be useless to carry them
out further as the weights are to be computed to only two signifi-
cant figures.
X
E
E* X 10*
w
^5?
w
194.03
0.066
44
11
0.066
193.79
0.12
144
3
0.127
194.15
0.091
83
6
0.090
193.85
0.11
121
4
0.110
194.22
0.099
98
5
0.098
194.17
0.047
22
22
0.047
Taking E a equal to 0.22 gives E 8 * X 10 4 equal to 484, and by
applying equation (36), we obtain the weights given under w to
the nearest integer. Inverting the process and computing the
70
THE THEORY OF MEASUREMENTS [ART. 47
E's from the assigned w's and E 8 gives the numbers in the last
column of the table. Since these numbers agree with the given
E's within less than two units in the second significant figure, we
may assume that the approximation adopted in computing the
w's is justified. If the agreement was less exact and any of the
differences exceeded two units in the second significant figure, it
would be necessary to compute the w's further, or, better, to adopt
a different value for E 8 , such that the agreement would be suffi-
cient with integral values of the w's.
For the purpose of computation, equation (37) may be written
in the form
X Q = C +
- C) + w, (x 2 - C) +
W m (X m C)
where C is any convenient number. In the present case 193 is
chosen, and the products w (x 193) are given in the first column
of the following table.
w (x - 193)
T
r2 X 10<
wr* X 10<
11.33
-0.065
42
462
2.37
-0.305
930
2790
6.90
+0.055
30
180
3.40
-0.245
600
2400
6.10
+0.125
156
780
25.74
+0.075
56
1232
55.84
7844
Substitution in the above equation for the general mean gives
and this is the most probable value that we can assign to the
numeric of the measured magnitude on the basis of the given
measurements.
By equation (38) the weight, w , of the general mean is 51.
Hence equation (41) gives
0.22
/=
V51
0.031
for the probable error of x . Selecting the first measurement
ART. 47] MEASUREMENTS OF UNEQUAL PRECISION 71
since its weight corresponds exactly to its probable error, equa-
tion (42) gives
Eo = 0.066 i/ = 0.031.
51
If the second, third, or fifth measurement had been chosen, the
results derived by the two formulae would not have been exactly
alike; but the differences would amount to only a few units in the
second significant figure, and consequently would be of no prac-
tical importance. However, it is better to proceed as above and
select a measurement whose weight corresponds exactly with its
probable error as shown by the fifth column of the first table
above.
The residuals, computed by subtracting x from each of the
given measurements, are given under r in the second table; and
their squares multiplied by 10 4 are given, to the nearest digit in
the last place retained, under r 2 X 10 4 . The last column of the
table gives the weighted squares of the residuals multiplied by
10 4 . The sum, [wr 2 ], is equal to 0.784. Hence by equation (43)
E 8 = 0.6745 1/ ' 784 = =t 0.27,
o
and by equation (44)
JB, = 0.6745 J^- = 0.037.
51 X o
These results agree with the assumed value of E 8 and the pre-
viously computed value of E as well as could be expected when
so small a number of measurements are considered. Conse-
quently we are justified in assuming that the given measurements
are either free from constant errors or all affected by the same
constant error.
In practice the second method of computing E Q is seldom used
when the probable errors of the given measurements are known,
since its value as an indication of the absence of constant errors
is not sufficient to warrant the labor involved. When the prob-
able errors of the given measurements are not known it is the
only available method for computing EQ and it is carried out here
for the sake of illustration.
CHAPTER VII.
THE METHOD OF LEAST SQUARES.
48. Fundamental Principles. Let Xi, X 2 , . . . X g , and FI,
Y 2 , . . . Y n represent the true numerics of a number of quan-
tities expressed in terms of a chosen system of units. Suppose
that the quantities represented by the Y's have been directly
measured and that we wish to determine the remaining quantities
indirectly with the aid of the given relations
YZ = FZ (Xl, Xz, . . . X q ),
Y n = F n (Xi,Xz, . . X q ).
(45)
The functions FI, F 2 , . . . F n may be alike or different in form
and any one of them may or may not contain all of the X's, but
the exact form of each of them is supposed to be known.
If the F's were known and the number of equations were equal
to the number of unknowns, the X's could be derived at once
by ordinary algebraic methods. The first condition is never ful-
filled since direct measurements never give the true value of the
numeric of the measured quantity. Let s i; s 2 , . . . s n represent
the most probable values that can be assigned to the F's on the
basis of the given measurements. If these values are substituted
for the F's in (45), the equations will not be exactly fulfilled and
consequently the true value of the X's cannot be determined. The
differences
Fi(Xi,X Z) . . . X q )-si = k
Fz(Xi,Xz, . . . Xq)-s 2 = k
*, . . . X q )-s n = A n
(46)
represent the true accidental errors of the s's.
Let Xi, Xz, . . . x q represent the most probable values that we
can assign to the X's on the basis of the given data. Then, since
72
ART. 48] THE METHOD OF LEAST SQUARES 73
the s's bear a similar relation to the Y's } equations (45) may be
written in the form
Fi (Xi, X 2) . . . X q ) = S b
F 2 (xi, x 2) . . . x q ) = s 2}
F n (xi, x 2} . . . x q ) = s n ,
(47)
where the functions F i} F 2 , etc., have exactly the same form as
before. When the number of s's is equal to the number of x's,
these equations give an immediate solution of our problem by
ordinary algebraic methods; but in such cases we have no data
for determining the precision with which the computed results
represent the true numerics Xi, X 2) etc.
Generally the number of s's is far in excess of the number of
unknowns and no system of values can be assigned to the x's
that will exactly satisfy all of the equations (47). If any assumed
values of the x's are substituted in (47), the differences
^1 (Xi, X 2) . . . X q ) Si = 7*1,
F 2 (xi, x 2) . . . x q ) - s 2 = r 2 ,
F n (Xi, X 2 , . . . X q ) - S- n = T n
represent the residuals corresponding to the given s's. ^Obviously, f
the most probable values that we can assign to the x's will be
those that give a maximum probability that these residuals are
equal to the true accidental errors AI, A 2 , etc.
If the s's are all of the same weight, the A's all correspond to
the same precision constant co. Consequently, as in article thirty-
five, the probability that the A's are equal to the r's is
and this is a maximum when
ri 2 + r 2 2 + . . . + r n 2 = [r 2 ] = a minimum. (49)
Hence, as in direct measurements, the most probable values that
we can assign to the desired numerics are those that render the
sum of the squares of the residuals a minimum. For this reason
the process of solution is called the method of least squares.
74 THE THEORY OF MEASUREMENTS [ART. 49
Since the r's are functions of the q unknown quantities x i} x 2)
etc., the conditions for a minimum in (49) are
provided the x's are entirely independent in the mathematical
sense, i.e., they are not required to fulfill any rigorous mathe-
matical relation such as that which connects the three angles of
a triangle. The equations (47) are not such conditions since the
functions F i} F 2 , etc., represent measured magnitudes and may
take any value depending on the particular values of the x's that
obtain at the time of the measurements. When the r's are re-
placed by the equivalent expressions in terms of the x's and s's as
given in (48), the conditions (50) give q, and only g, equations
from which the x's may be uniquely determined.
If the weights of the s's are different, the A's correspond to
different precision constants coi, 0)2, . . . , co n given by the rela-
tions
where w a is the precision constant corresponding to a standard
measurement, i.e., a measurement of weight unity; and wi, w 2 ,
. . . , w n are the weights of the s's. Under these conditions, as
in article forty-five, the most probable values of the re's are those
that render the sum of the weighted squares of the residuals a
minimum. Thus, in the case of measurements of unequal weight,
the condition (49) becomes
wiri 2 f w 2 2 + + MV 2 = [wr 2 ] = a minimum, (51)
and conditions (50) become
A M = ; ^M = 0; ... A M = . (52)
49. Observation Equations. The equations (50) or (52) can
always be solved when all of the functions FI, F 2) . . . F n are
linear in form. Many problems arise in practice which do not
satisfy this condition and frequently it is impossible or incon-
venient to solve the equations in their original form. In such
cases, approximate values are assigned to the unknown quantities
and then the most probable corrections for the assumed values
are computed by the method of least squares. Whatever the form
ART. 50] THE METHOD OF LEAST SQUARES
75
of the original functions, the relations between the corrections can
always be put in the linear form by a method to be described in a
later chapter.
When the given functions are linear in form, or have been
reduced to the linear form by the device mentioned above, equa-
tions (47) may be written in the form
+ to +
+ to +
+ piX q = si,
= s 2 ,
p n x q =
(53)
where the a's, 6's, etc., represent numerical constants given either
by theory or as the result of direct measurements. These equa-
tions are sometimes called equations of condition; but in order
to distinguish them from the rigorous mathematical conditions,
to be treated later, it is better to follow the German practice and
call them observation equations, "Beobachtungsgleichungen."
By comparing equations (47), (48), and (53), it is obvious that
the expressions
+ to + CiX 3 +
-f to + c 2 x 3 +
b n x
c n x 3
s 2 = r 2 ,
p n x q - s n = r n
(54)
give the resi'duals in terms of the unknown quantities x\, x z , etc.,
and the measured quantities si, s 2 , etc.
50. Normal Equations. In the case of measurements of
equal weight, we have seen that the most probable values of the
unknowns x\, x 2 , etc., are given by the solution of equations (50)
provided the x's are independent. Assuming the latter condition
and performing the differentiations we obtain the equations
dr, dr.
dr 3
dx t
(0
76
THE THEORY OF MEASUREMENTS [ART. 50
Differentiating equations (54) with respect to the x's gives
dri _ dr 2 _
~dx\ ~ ai ' dxi ~~
dx c
= a n ,
= b n ,
dr 2
and hence equations (i) become
r 2 a 2 +
i + r 2 6 2 + .
. drn
' dx q
+ r n a n = 0,
+ r n b n = 0,
(ii)
(iii)
- . . . + r n p n = 0.
Introducing the expressions for the r's in terms of the x's from
equations (54) and putting
[aa] = didi -{- a 2 a 2 -|- a 3 a 3 ~h ~h d n d n}
w>
[as] = diSi + a 2 s 2 + a 3 s 3 +
[bd] = bidi + 6 2 a 2 + b s d s +
[66] = &!&! + 6 2 6 2 + 6 3 6 3 +
[be] = 6iCi + 6 2 c 2 + 6 3 c 3 +
a n s n ,
6 n a n = [ab]j
b n b n ,
6 n c n
(55)
equations (iii) reduce to
[aa] x-i + [ab] x z + [ac] x 3
[ac]
[be] x 2 + [cc] x 3
[bp]x q =[bs],
[CP] X* = N,
(56)
giving us q, so-called, normal equations from which to determine
the q unknown x's.
Since the normal equations are linear in form and contain only
numerical coefficients and absolute terms, they can always be
solved, by any convenient algebraic method, provided they are
entirely independent, i.e., provided no one of them can be ob-
tained by multiplying any other one by a constant numerical
ART. 50] THE METHOD OF LEAST SQUARES 77
factor. This condition, when strictly applied, is seldom violated
in practice; but it occasionally happens that one of the equations
is so nearly a multiple or submultiple of another that an exact
solution becomes difficult if not impossible. In such cases the
number of observation equations may be increased by making
additional measurements on quantities that can be represented
by known functions of the desired unknowns. The conditions
under which these measurements are made can generally be so
chosen that the new set of normal equations, derived from all of
the observation equations now available, will be so distinctly
independent that the solution can be carried out without difficulty
to the required degree of precision.
By comparing equations (53) and (56), it is obvious that the
normal equations may be derived in the following simple manner.
Multiply each of the observation equations (53) by the coefficient
of xi in that equation and add the products. The result is the
first normal equation. In general, q being any integer, multiply
each of the observation equations by the coefficient of x q in that
equation and add the products. The result is the gth normal
equation. The form of equations (56) may be easily fixed in
mind by noting the peculiar symmetry of the coefficients. Those
in the principal diagonal from left to right are [aa], [66], [cc], etc.,
and coefficients situated symmetrically above and below this
diagonal are equal.
When the given measurements are not of equal weight, the
observation equations (53), and the residual equations (54) remain
unaltered, but the normal equations must be derived from (52)
in place of (50). Since the weights Wi, w 2 , etc., are independent
of the x's, if we treat equations (52) in the same manner that we
have treated (50), we shall obtain the equations
* + w n r n a n = 0,
'. .4 Wn&n = 0,
(iv)
+ Wtfzpz + ' ' ' + W n r n p n = 0,
in place of equations (iii). Hence, if we put
[iWia] = Wididi -f~ WzClzCLz ~\~ ' ' ' ~\~ W n d n CLnj
(57)
[was] = WidiSi + w&zSz + + w n a n s nj
' -\-WnpnPn,
78
THE THEORY OF MEASUREMENTS [ART. 51
the normal equations become
[waa] xi + [wab] x 2 + [wac] z 3
[wab] Xi + [wbb] x z + [wbc] x z
[wac] X! + [wbc] x 2 + [wcc] x z
+ [wap] x q = [was],
+ [wbp] x q = [wbs],
+ [wcp] x q = [wcs],
(58)
[wap]xi + [wbp]x 2 + [wcp]x$ + + [wpp]x q = [wps].
These equations are identical in form with equations (56), and
they may be solved under the same conditions and by the same
methods as those equations. Consequently, in treating methods
of solution, we shall consider the measurements to be of equal
weight and utilize equations (56). All of these methods may be
readily adapted to measurements of unequal weight by substitut-
ing the coefficients as given in (57) for those given in (55).
51. Solution with Two Independent Variables. When only
two independent quantities are to be determined the observation
equations (53) become
"
= s,
and the normal equations (56) reduce to
[aa] Xi + [ab] x 2 = [as],
[ab] X! + [bb] x 2 = [bs].
Solving these equations we obtain
[bb] [as] - [ab] [bs]
[aa] [bb] - [ab] 2
_ [aa] [bs] [ab] [as]
[aa] [bb] - [ab] 2
As an illustration, consider the determination of the length Z/
at C., and the coefficient of linear expansion a of a metallic
bar from the following measurements of its length L t at temper-
ature t C.
(56a)
(59)
t
L t
C.
20
mm.
1000.36
30
1000.53
40
1000.74
50
1000.91
60
1001.06
Ara.51] THE METHOD OF LEAST SQUARES
79
or
Within the temperature range considered, L t and t are connected
with LO and a by the relation
L t = Lo (1 + at),
L t = Lo + L at, (v)
and a set of observation equations might be written out at once
by substituting the observed values of L t and t in this equation.
But the formation of the normal equations and the final solution
is much simplified when the coefficients and absolute terms in the
observation equations are small numbers of nearly the same order
of magnitude. To accomplish this simplification, the above func-
tional relation may be written in the equivalent form
and if we put
it becomes
L t - 1000 = Lo - 1000 + WL<xx
L t - 1000 = s; JQ = 6,
LO 1000 = Xi] 10 LOCK = Xz,
Xi -J- 6^2 = s.
(vi)
Using this function, all of the a's in equation (53a) become equal
to unity and the 6's and s's may be computed from the given
observations by equations (vi).
the observation equations are
xi + 2 z 2 =
xi + 3 x 2 =
Hence, in the present case,
.36,
.53,
x l +x 2 = .74,
zi + 5z 2 = .91,
Xl + 6x 2 = 1.06.
For the purpose of forming the normal equations, the squares
and products of the coefficients and absolute terms are tabulated
as follows :
Obs.
aa
ab
as
bb
bs
1
2
0.36
4
0.72
2
3
0.53
9
1.59
3
4
0.74
16
2.96
4
5
0.91
25
4.55
5
6
1.06
36
6.36
5
20
3.60
90
16.18
[aa]
W
[as]
[bb]
[bs]
Substituting these values of the coefficients in (56a) gives the
normal equations
80
THE THEORY OF MEASUREMENTS [ART. 51
= 3.60,
= 16.18,
and by (59) we have
_ 90 X 3.60 - 20 X 16.18
5 X 90 - 400
5 X 16.18 - 20 X 3.60
= 0.008,
= 0.178.
5 X 90 - 400
From these results, with the aid of relations (vi), we find
Lo = xi + 1000 = 1000.008,
L a = ^ = 0.0178,
0.0178
= 0.0000178,
and finally
L t = 1000.008 (1 + 0.0000178 1) millimeters. (vii)
The differences between the values of L t computed by equation
(vii), and the observed values give the residuals. But they can
be more simply determined by using the above values of x\
and Xz in the observation equations and taking the difference
between the computed and observed values of s. Thus, if s'
represents the computed value and r the corresponding residual
s' = 0.008 + 0.178 6,
and r = s f s.
With the values of s and 6 used hi the observation equations we
obtain the residuals as tabulated below:
s'
8
r
7-2 X 10*
0.364
0.542
0.720
0.898
1.076
0.36
0.53
0.74
0.91
1.06
+0.004
+0.012
-0.020
-0.012
+0.016
0.16
1.44
4.00
1.44
2.56
[r 2 ] = 9.60XlO~ 4
Since the above values of x\ and x 2 were computed by the method
of least squares, the resulting value of [r 2 ], i.e., .000960, should be
less than that obtainable with any other values of x\ and x%.
That this is actually the case may be verified by carrying out the
computation with any other values of x\ and x z .
ART. 52] THE METHOD OF LEAST SQUARES
81
52. Adjustment of the Angles About a Point. As an illus-
tration of the application of the method of least squares to the
solution of a problem involving more than two unknown quanti-
ties, suppose that we wish to determine the most probable value
of the angles AI, AZ, and A 3 , Fig. 9, from a series of independent
measurements of equal weight on the angles Mi, M 2 , . . . M 6 .
If the given measurements were all exact, the equations
AI = Mi; AZ = M 2 ; A 3 = M 3 ;
AI-\- AZ = M 4 ; AI + AZ -{-As = MS; and Az -\- As = Me,
would all be fulfilled identically. In practice this is never the
case and it becomes
necessary to adjust the
values of the A's so that
the sum of the squares
of the discrepancies will
be a minimum. The
adjustment may be ef-
fected by adopting the
above equations as ob-
servation equations and
proceeding at once to
the solution for the A's
by the method of least
squares. But the ob-
served values of the M's
usually involve so many
significant figures that
the computation would
be tedious. It is better
to adopt approximate
values for the A's and then compute the necessary corrections by
the method of least squares.
For this purpose, suppose we adopt MI, M 2 , and M 3 as approxi-
mate values of A\, A 2 , and A s respectively and let xi, Xz, and x 3
represent the corrections that must be applied to the M's in order
to give the most probable values of the A's. Then, putting
AI = MI + xi, AZ = MZ + Xz, and A 3 = M 3 + # 3 , (viii)
the above equations become
FIG. 9.
82
THE THEORY OF MEASUREMENTS [ART. 52
+ x 2
= 0,
= 0,
= 0,
= M 4 - (Af ! + M 2 ),
To render the problem definite, suppose that the following
values of the M's have been determined with an instrument read-
ing to minutes of arc by verniers:
Mi = 10 49'.5, M 4 = 45 24'.0,
M 2 = 34 36'.0, M 6 = 60 53'.5,
M 3 = 15 25'.5, M 6 = 50 O'.O.
Substituting these values in the above equations we obtain
xi = 0,
x 2 = 0,
2'.5,
Adopting these as our observation equations and comparing with
(53) we obtain the coefficients and absolute terms tabulated below:
Oba.
a
b
c
s
1
1
2
1
3
1
4
1
1
-1.5
5
1
1
1
2.5
6
1
1
-1.5
The squares and products of the coefficients and absolute terms
may be tabulated, for the purpose of forming the normal equations,
as follows :
M
ab
ac
as
66
be
bs
cc
cs
1
1
1
1
1
1
-1.5
2.5
1
1
1
1
1
2
[be]
-1.5
2.5
-1.5
1
1
1
2.5
-1.5
3
[aa]
[ab]
1
[ac]
1
fas]
4
[66]
-0.5
[6s]
3
[cc]
1
[cs]
ART. 53] THE METHOD OF LEAST SQUARES 83
Substituting these values in (56) the three normal equations
become
-0.5,
1 xi -f 2 x 2 + 3 z 3 = 1,
and solution by any method gives
xi = 0.625; x 2 = - 0.75; x 3 = 0.625.
With these results together with the given values of MI, Mz,
and M 3 we obtain from equations (viii)
A! = 10 50M25,
A 2 = 34 35'.25,
A 3 = 15 26M25.
In a problem so simple as the present the normal equations are
generally written out at once from the observation equations by
the rule stated in article fifty, without taking the space and time
to tabulate the coefficients, etc. But, until the student is thor-
oughly familiar with the process, it is well to form the tables as
a check on the computations and to make sure that none of the
coefficients or absolute terms have been omitted. For this reason
the tabulation has been given in full above and the student is
advised to carry out the formation of the normal equations by
the shorter method as an exercise.
53. Computation Checks. When the number of unknowns
is greater than two and a large number of observation equations
are given with coefficients and absolute terms involving more than
two significant figures, the formation of the normal equations is
the most tedious and laborious part of the computations. It is,
therefore, advantageous to devise a means of checking the com-
puted coefficients and absolute terms in the normal equations
before we proceed to the final solution.
For this purpose compute the n quantities t\ t ^2, ... t n by the
equations
ai + &i -f ci + - + pi = ti,~
02 + &2 4- c 2 -f + pz = h,
On + & + C n -f - + p n =
(60)
84 THE THEORY OF MEASUREMENTS [ART. 54
where the a's, b's, etc., are the coefficients in the given observa-
tion equations. Multiply the first of equations (60) by Si, the
second by s 2 , etc., and add the products. The result is
[as] + [bs] + [cs] + + \ps] = [ts]. (61)
In the same way, multiplying by the a's in order and adding, then
by the b's in order and adding, etc., we obtain the following rela-
tions
[aa] + [db] + [ac] + .-. + [ap] = [at],
[ab] + [bb] + [be] + + [bp] = [bt],
[ac] + [be] + [cc] + + [cp] = [ct], (62)
[ap] + \bp] + [ep] + . . . + \pp] = \pt].
If the absolute terms in the normal equations have been accu-
rately computed, equation (61) reduces to an identity. If the
coefficients have been accurately computed equations (62) all
become identities. Hence (61) is a check on the computation of
the absolute terms and equations (62) bear the same relation to
the coefficients. The extra labor involved in computing the quan-
tities [ts] t [at], . . . , [pt] is more than repaid by the added confi-
dence in the accuracy of the normal equations.
When all attainable significant figures are retained throughout
the computations, the checks (61) and (62) should be identities.
In practice the accuracy of the measurements is seldom sufficient
to warrant so extensive a use of figures, and, consequently, the
squares and products, aa, ab, . . . as, at, etc., are rounded to such
an extent that the computed values of the x's will come out with
about the same number of significant figures as the given data.
Judgment and experience are necessary in determining the number
of significant figures that should be retained in any particular
problem and it would be difficult to state a general rule that
would not meet with many exceptions. When the computed
coefficients and absolute terms are rounded, as above, the checks
may not come out absolute identities, but they should not be
accepted as satisfactory when the discrepancy is more than two
units in the last place retained.
54. Gauss's Method of Solution. When the normal equa-
tions (56) are entirely independent, they may be solved by any
of the well-known methods for the solution of simultaneous
linear equations and lead to unique values of the unknown quan-
ART. 54] THE METHOD OF LEAST SQUARES 85
titles xi, x 2) etc. Gauss's method of substitution is frequently
adopted for this purpose since it permits the computation to be
carried out in symmetrical form and provides numerous checks
on the accuracy of the numerical work. The general principles
of the method will be illustrated and explained by completely
working out a case in which there are only three unknowns.
Since the process of solution is entirely symmetrical, it can be
easily extended for the determination of a larger number of
unknowns, but too much space would be required to carry through
the more general case here.
When only three unknowns are involved, the normal equations
(56) and the check equations (60) and (61) may be completely
written out in the following form, the computed quantities and
equations being placed at the left, and the checks at the right.
[aa] xi + [ab] x 2 + [ac] x 3 = [as]. [aa] + [ab] + [ac] = [at].
[ab] xi + [bb] x 2 + [be] x 3 = [bs]. [ab] + [bb] + [be] = [bt].
[ac] xi + [be] x 2 + [cc] x 3 = [cs]. [ac] + [be] + [cc] = [ct].
[as] + [bs]+[cs] =[st].\
Solve the first equation on the left for xi y giving
[as] [ab] [ac]
Xi = 7 7 f 1 X 2 f 1 X$.
[aa] [aa] [aa\
Compute the following auxiliary quantities:
(63)
[56] _ P4 [ &] = [bb 1], [bt] - pi M = [^ ' 1L
L aa]
[a61
[6c]-
M L " M
~ M = [6s ' 1] ' M ~ N1 = [st
As a check on these computations we notice that
[bb 1] + [be - 1] = [bb] + [be] - |^| ([ab] + [c]),
[aaj
= [bt] - lab] - ([at] - [aa]),
86 THE THEORY OF MEASUREMENTS [ART. 54
In a similar way we may show that we should have
[6c-l] + [cc-l] = [cM] and [6s- 1] + [cs- 1] = [st- 1].
Substituting (64) in the last two of (63) and placing the above
checks to the right, we have the equations
[bb -I]x 2 + [be 1] x s = [bs 1], [bb 1] + [be- 1] = [fa 1],
[be -I]x 2 + [cc 1] z 3 = [cs 1], [be 1] + [cc- 1] = [ct 1], (65)
[6s-l] + [cs.l] = [s*. 1],.
which show the same type of symmetry as (63), but contain only
two unknown quantities. Solve the first of (65) for x 2 giving
__
* 2 ~[6&.l] [bb-lf 3 '
and compute the following auxiliaries:
[<*!] - [l^jlfc-1] = [-2], [cM] - l ~^ } [bt. 1} = let- 2],
(cs 1} - |^jj [bs 1} - [cs 2], [st 1] - |^|j (bt 1] = [* 2}.
By a method similar to that used above we can show that we
should have
[cc 2] = [ct 2] and [cs 2] = [st 2].
Hence, substituting (66) in the last of (65), we have
[cc 2] x 3 = [cs 2], [cc 2] = [ct 2],
[cs.2] = N-2],
and consequently
[cs 2] _.
*"fc^t' (67)
Having determined the value of x 3 from (67), x% may be cal-
culated from (66), and then Xi from (64).
A very rigorous check on the entire computation is obtained as
follows: using the computed values of Xi, x z , and z 3 in equations
(54), derive the residuals
(68)
- s 2 ,
T n = d n Xi ~|- O n X 2 ~\- C n Xs S n ,
and then form the sums
[rr] = n 2 + r 2 2 + r 3 2 + - - - + r n 2 ,
[SS] = Si 2 + S 2 2 + S 3 2 + + n 2 .
ART. 55] THE METHOD OF LEAST SQUARES 87
If the computations are all correct, the computed quantities will
satisfy the relation
W = M-[a S ]-M[6 S .l]- [cs . 2] . (69)
To prove this, multiply the first of (68) by ri, the second by r%,
etc., and add the products. The result is
[rr] = [ar] Xi + [br] x 2 -f [cr] 3 - [sr].
But from equations (iii), article fifty,
[ ar ] = [br] = [cr] = 0,
consequently
[rr]=- []. (70)
Multiply each of equations (68) by its s; add, taking account of
(70), and we obtain
[rr] = [ss] - [as] Xi - [6s] x z - [cs] x z .
Eliminating x\, X 2 , and z 3 , in succession with the aid of (64), (66),
and (67) we find
[rr] = [ss] - [as] - [6s 1] x 2 - [cs 1] x 9 ,
and finally
r i r i l as ] r i [& s * 1] n n t cs ' 2] r Ol
[rr] = M ~ y M - I667i] [6s ' 1] - RT2] [cs ' 2] '
which is identical with (69).
55. Numerical Illustration of Gauss's Method. The fore-
going methods are most frequently used for the adjustment of
astronomical and geodetic observations, and their application to
particular problems is fully discussed in practical treatises on
such observations. The physical problems, to which they are
applicable, usually involve the determination of an empirical
relation between mutually varying quantities. Such problems
will be discussed at some length in Chapter XIII, and the corre-
sponding observation equations will be developed.
It would require too much space to carry out the complete dis-
cussion of such a problem, in this place, with all of the observa-
tions made in any actual investigation. But, for the purpose of
illustration, the most probable values of xi, Xz, and x 3 will be
88
THE THEORY OF MEASUREMENTS [ART. 55
derived, from the following typical observation equations, by
Gauss's method of solution:
+ 2x 2 + 0.4z 3 =
+ 4x 2 + 1.6x3 =
+ 6 x 2 + 3.6 z 8 =
+ 8x 2 + 6.4x 3 =
+10x 2 +10.0^3 =
0.24,
- 1.18,
- 1.53,
- 0.69,
1.20,
4.27.
Since the coefficient of xi is unity in each of these equations,
the products aa, ab, aCj as, and at are equal to a, 6, c, s, and t,
respectively. Consequently the first five columns of the follow-
ing table show the coefficients, absolute terms, and check terms
(t = a + b + c) of the observation equations as well as the
squares and products indicated at the head of the columns. The
sums [aa], [ab], etc., are given at the foot of the columns and the
checks, by equations (61) and (62), are given below the tables.
In the present case, the coefficients are expressed by so few signifi-
cant figures that it is not necessary to round the computed products
and consequently the checks come out identities.
aa
ab
ac
as
at
bb
be
2
4
6
8
10
0.0
0.4
1.6
3.6
6.4
10.0
0.24
-1.18
-1.53
-0.69
1.20
4.27
1.0
3.4
6.6
10.6
15.4
21.0
4
16
36
64
100
0.0
0.8
6.4
21.6
51.2
100.0
6
M
30
[ab]
22.0
M
2.31
[as]
58.0
M
220
m
180.0
[be]
Check: [
aa] + [ab] + [c
ic] = 58.0.
bs
cc
cs
bl
ct
st
0.00
-2.36
-6.12
-4.14
9.60
42.70
0.00
0.16
2.56
12.96
40.96
100.00
0.00
-0.472
-2.448
-2.484
7.680
42.700
0.0
6.8
26.4
63.6
123.2
210.0
0.00
1.36
10.56
38.16
98.56
210.00
0.24
- 4.012
-10.098
- 7.314
18.480
89.670
39.68
[bs]
156.64
[cc]
44.976
[cs]
430.0
M
358.64
[ct]
86.966
[st]
Checks: [ab] + [66] + [be] = 430.0
[ac] + [be] + [cc] = 358.64
M + [6s] + [cs]= 86.966
ART. 55] THE METHOD OF LEAST SQUARES
89
The normal equations and their checks might now be written
out in the form of equations (63), but, since the coefficients and
other data necessary for their solution are all tabulated above, it
is scarcely worth while to repeat the same data in the form of
equations. The computation of the auxiliaries [bb 1], [be 1],
etc., and the final solution for x i} x 2) and # 3 by logarithms is best
carried out in tabular form as illustrated on pages 90 and 91.
The meaning of the various quantities appearing in these tables, and
the methods by which they are computed, will be readily under-
stood by comparing the numerical process with the literal equa-
tions of the preceding article. When the letter n appears after a
logarithm it indicates that the corresponding number is to be taken
negative in all computations.
The computation of the residuals by equations (68) and the
final check by (69) is carried out in the following table, where
Scale, is written for the value of the expression axi + bx 2 + cxs,
when the computed values of x\, x 2 , and x 3 are used and s bs. is
the corresponding value of s in the observation equations. Thus
+-
Si = Si calc. ~ Si obs.-
I*
S ObB.
r
*Xio.
ss
0.245
-1.195
-1.512
-0.709
1.215
4.264
0.24
-1.18
-1.53
-0.69
1.20
4.27
+0.005
-0.015
+0.018
-0.019
+0.015
-0.006
25
225
324
361
225
36
0.0576
1.3924
2.3409
0.4761
1.4400
18.2329
.001196
[rr]
23.9399
[as] r , [6s 1] , .,, [cs 2]
:s-2]
52 = 23.9387
0.0012
[aa\ [oo i\ [cc A\
0.8893 + 11.3042 + 11.74
Final check by (69): [rr]
Since the checks are all satisfactory, we are justified in assum-
ing that the computations are correct. Hence the most probable
values of the unknowns, derivable from the given observation
equations, are
xi = 0.245; x 2 = - 1.0003; z 3 = 1.4022,
90
THE THEORY OF MEASUREMENTS [ART. 55
jfl
3
r [
is
co ^^
O
, ^
_|- Oi
1 ^
g
rH
- , IQ
r^^
4
1
i
CO
00 ^
1-H
o c^ o o
HII II II
t^ CO
O CD CO
^^ t^ CD 1>-
GO CO <M O
HII II II
H"
"e "e
II II II
^r " ^^
tj ~Q O r ~ l
" "e o ^
I
'|^
be *
bfi ^
bC ^
c\j d II-H
OO
OO <N >O CO
CD CD O i i
Oi i i 00
CO rH rH (M
II II II II
05 ""ST 5P ^H 1
o o -2 .
CD GO O CD
b bfi
-2 -2
-STe-i 3
1111 i -
I
CO 1C
<M i i O
0^00
dodo
S <N ^^
CD CD CD t^*
CD Oi
O O >O
r-< i 1 OO !>
II II II
II H II II
II II II II
o W 53
5" o^
0^
bfi M
mi^
ffi 1
M
bD *
00
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H H H
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11111
1 1 1?
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^l^
1
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11
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ART. 55] THE METHOD OF LEAST SQUARES 91
03
M
<N CY^
<M CO
(0
O Oi
Ji
3 ^
6
10
"
CO
CO
co
CO i ( CO
CO CO CO
CO 1 CO
OS T-( OS
CO ^ ^ CO
o c^i
o d o 10
^ T-H CO OO
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II II
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II II II II
8
1 1 T 1
| i fffl
ifL rSl
"o S -^ "S
"S S - "5
^ ^ 8 8
' ' '~~ ' c
d d H r-i d
bfi
^ ^r 03
' 1- _ ( ' ^ ( Q^
1
j3
B .
^ s
CO rO
II II II II ||
I
M
r-O -O *T? O
b- b-
l>.
^ (N
1 " ' co Uo C3 ^
CO
co OS CO
tO "^t 1 T 1 CO
ill
jjO o3
bfi
^O
?7 7
CO T-I OO 00
CO <M
II II II II
d d i-i
II II II
T- ' 1 ' T-H T-l
TH TH ^ (M
<M H H
=0 =0 oo -O
*O "O ^j ^^
n i 2
bfi bfi
' T ' c3
bfi
-2
TH II
O
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bfi
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g 1 g
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OS OO O OS
:
d-
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d
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II
T i T i ^n" 1 i i
^ ^ ^o ^
cT
e
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^J o 1 o
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8
| 2 8 jo^g
bfi bfi
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^T^T
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CO CO C^ !> ^H C*l CO
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^^ ^^ *-O ^^ O ^D I^H
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w
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TH'
W) o3 bfi o3
II II
Tf
.
92 THE THEORY OF MEASUREMENTS [ART. 56
and the corresponding empirical relation becomes
s = 0.245 a - 1.0003 6 + 1.4022 c.
A small number of observation equations with simple coefficients
have been chosen, in the above illustration, partly to save space
and partly in order that the computations may be more readily
followed. In practice it would seldom be worth while to apply
the method of least squares to so small a number of observations
or to adopt Gauss's method of solution with logarithms when the
normal equations are so simple. When the number of observa-
tions is large and the coefficients involve more than three or four
significant figures, the method given above will be found very
convenient on account of the numerous checks and the symmetry
of the computations. In order to furnish a model for more
complicated problems, the process has been carried out completely
even in the parts where the results might have been foreseen
without the use of logarithms.
56. Conditioned Quantities. When the unknown quantities,
Xi, Xz, etc., are not independent in the mathematical sense, the
foregoing method breaks down since the equations (50) no longer
express the condition for a minimum of [rr]. In such cases the
number of unknowns may be reduced by eliminating as many of
them as there are rigorous mathematical relations to be fulfilled.
The remaining unknowns are independent and may be deter-
mined as above. The eliminated quantities are then determined
with the aid of the given mathematical conditions.
For the purpose of illustration, consider the case of a single
rigorous relation between the unknowns, and let the correspond-
ing mathematical condition be represented by the equation
0(x lt x,, . . . , x q ) =0. (71)
As in the case of unconditioned quantities, the observation equa-
tions (53) are
+ C&s + + piX q = Si,
c n x 3 p n x q =
The solution of (71) for x\, in terms of Xz, x s , . . ., x q , may be
written in the form
xi=f(xz,x a , ..*,*) (72)
ART. 56] THE METHOD OF LEAST SQUARES 93
Introducing this value of xi, equations (53) become
+ ClX* + * + PlX q = Si,
+ C 2 X 3 + + P2Z a = S 2 ,
4- c n z 3 + + p n x q = s.
Since the form of 6 is known, that of / is also known. Hence, by
collecting the terms in x%, x S} etc., and reducing to linear form,
if necessary, we have
bixz + ci'x s + + p\x q = s/,
The x's in these equations are independent, and, consequently,
they may be determined by the methods of the preceding articles.
Using the values thus obtained in (71) or (72) gives the remaining
unknown x\. The #'s, thus determined, obviously satisfy the
mathematical condition (71) exactly, -and give the least magnitude
to the quantity [rr] that is consistent with that condition. They
are, consequently, the most probable values that can be assigned
on the basis of the given data.
As a very simple example, consider the adjustment of the
angles of a plane triangle. Suppose that the observed values of
the angles are
si = 60 1'; s 2 = 59 58'; s 3 = 59 59'.
The adjusted values must satisfy the condition
xi + x 2 + x* = 180,
or
xi = 180 - x 2 - x 3 .
Eliminating Xi from the observation equations,
xi = Si' t Xz = s 2 ; and x s s 3 ;
and substituting numerical values we have
x z +x 3 = 119 59',
x 2 = 59 58',
x 3 = 59 59'.
The corresponding normal equations are
2z 2 + z 3 = 179 57',
= 179 58',
94 THE THEORY OF MEASUREMENTS [ART. 56
from which we find
x 2 = 59 58'.7 and x s = 59 59'.7.
Then, from the equation of condition,
xi = 60 1'.6.
When there are two relations between the unknowns, expressed
by the equations
01 (xi, x t , . . . , x q ) = 0,
02 (xi, x 2 , . . . , x q ) = 0,
they may be solved simultaneously for xi and x 2 , in terms of the
other x's, in the form
xi = fi(x 3 , xt, . . . , x q ),
x z = /2(z3, $,..., x q ).
Using these in the observation equations (53) we obtain a new set
of equations, independent of x\ and x* t that may be solved as
above. It will be readily seen that this process can be extended
to include any number of equations of condition.
When the number of conditions is greater than two, the compu-
tation by the above method becomes too complicated for practical
application and special methods have been devised for dealing
with such cases. The development of these methods is beyond
the scope of the present work, but they may be found in treatises
on geodesy and practical astronomy in connection with the prob-
lems to which they apply.
CHAPTER VIII.
PROPAGATION OF ERRORS.
57. Derived Quantities. In one class of indirect measure-
ments, the desired numeric -X" is obtained by computation from
the numerics Xi, X z , etc., of a number of directly measured mag-
nitudes, with the aid of the known functional relation
X = F(X 1 ,X i , . . . ,X q ).
We have seen that the most probable value that we can assign to
the numeric of a directly measured quantity is either the arith-
metical mean of a series of observations of equal weight or the
general mean of a number of measurements of different weight.
Consequently, if x\, Xz, . . , x q represent the proper means of
the observations on Xi, X 2 , . . . , X q the most probable value
x that we can assign to X is given by the relation
x = F (xi, x z , . . . , x q )
where F has the same form as in the preceding equation.
Obviously, the characteristic errors of x cannot be easily deter-
mined by a direct application of the methods discussed in Chapters
V and VI, as this would require a separate computation of x from
each of the individual observations on which Xi, Xz, etc., depend.
Furthermore, it frequently happens that we do not know the
original observations and are thus obliged to base our computa-
tions on the given mean values, x\, Xz, etc., together with their
characteristic errors.
Hence it becomes desirable to develop a process for computing
the characteristic errors of x from the corresponding errors of
Xij xz, etc. For this purpose we will first discuss several simple
forms of the function F and from the results thus obtained we
will derive a general process applicable to any form of function.
58. Errors of the Function Xi Xz X 3 =t . . . X q .
Suppose that the given function is in the form
X = Xi + X 2 , or X = Xi - X 2 .
These two cases can be treated together by writing the function in
the form
X = X\ db Xz,
95
96 THE THEORY OF MEASUREMENTS [ART. 58
and remembering that the sign indicates two separate problems
rather than, as usual, an indefinite relation in a single problem.
If the individual observations on Xi are represented by ai, a 2 ,
. . . , a n , and those on X 2 by 61, 6 2 , . . . , b n , we have
n n
and the most probable value of X is given by the relation
x = Xi xz.
From the given observations we can calculate n independent
values of X as follows :
Ai = ai &i, A 2 = az d= 6 2 , . . . , A n = a w db 6 n ,
and it is obvious that the mean of these is equal to x. The true
accidental errors of the a's are
Aai = oi Xi, Aa 2 = a z Xi, . . . , Aa n = a n Zi;
those of the 6's are
Ah = 61 - Z 2 , A6 2 = 6 2 - Z 2 , . . . , A6 n = b n - X 2 ;
and those of the A's are
^A l =A 1 -X ) &A 2 =A 2 -X, . . . , &A n =A n -X.
We cannot determine these errors in practice, since we do not
know the true value of the X's, but we can assume them in literal
form as above for the purpose of finding the relation between the
characteristic errors of the x's.
Combining the equations of the preceding paragraph with the
given functional relation, we have
AA X = (ai 60 - (Zi Z 2 )
= (a! - ZO (61 - Xz)
= Aai A&i,
and similar expressions for the other A A's. Consequently
(AAO 2 = (AaO 2 d= 2 AaiA&i + (A6i) 2 ,
(AA 2 ) 2 = (Aa 2 ) 2 d= 2 Aa 2 A6 2
(AA n ) 2 = (Aa n ) 2 2 ka n tU) n
Adding these equations, we find
[(AA) 2 ] = [(Aa) 2 ] 2 [AaA6] + [(A6) 2 ].
ART. 58] PROPAGATION OF ERRORS 97
Since A a and A b are true accidental errors, they are distributed
in conformity with the three axioms stated in article twenty-four.
Consequently equal positive and negative values of Aa and A6
are equally probable and the term [AaA6] would vanish if an
infinite number of observations were considered. In any case it
is negligible in comparison with the other terms in the above
equation. Hence, on dividing through by n, we have
[(AA)1 = [(Aa)l [(A6)*]_
n n n
and by equation (20), article thirty-seven, this becomes
M A 2 = M a 2 + M b 2 , (73)
where M A is the mean error of a single A, M a that of a single a,
and M b that of a single b. Since x, xi, and z 2 are the arithmetical
means of the A's, a's, and 6's, respectively, their respective mean
errors, M , MI, and M 2 , are given by the relations
M 2 M 2 Tlf i 2
M* = ^, itf-=, and M, = ^-
n n n
in virtue of equations (29), article forty. Consequently, by (73)
M 2 = Mi 2 + M 2 2 ,
or M = VMi 2 + M 2 2 . (74)
Since the mean and probable errors, corresponding to the same
series of observations, are connected by the constant relation (26),
article thirty-nine, we have also
+ Ef, (75)
where E, EI, and E z are the probable errors of x, x\, and #2,
respectively.
It should be noticed that the ambiguous sign does not appear
in the expressions for the characteristic errors. The square of
the error of the computed quantity is equal to the sum of the
squares of the corresponding errors of the directly measured quan-
tities; whether the sign in the functional relation is positive or
negative. Thus the error of the sum of two quantities is equal
to the corresponding error of the difference of the same two quan-
tities.
Now suppose that the given functional relation is in the form
X = Xi d= X 2 X t .
98 THE THEORY OF MEASUREMENTS [ART. 59
The most probable value of X is given by the relation
x = xi x z x 3y
where the notation has the same meaning as in the preceding
case. Represent x\ x z by x p , then
a; = x p =t z 3 ,
and, by an obvious extension of the notation used above, we have
M P 2 = Mi 2 + M 2 2 ,
M z = M P 2 + M 3 2
= Mi 2 + M 2 2 + M 3 2 .
Passing to the more general relation
X = Xi X 2 X 3 - - - X,,
we have a; = 1 db # 2 x 3 z fl ,
and, by repeated application of the above process,
M 2 = M M 2 MJ + - - + M 3 2 , )
+ -E-
Thus the square of the error of the algebraic sum of a series of
terms is equal to the sum of the squares of the corresponding
errors of the separate terms whatever the signs of the given terms
may ba
59. Errors of the Function a\Xi =t 0:2^2 db a s X 3 =b - a q X q .
Let the given functional relation be in the form
X =
where a\ is any positive or negative, integral or fractional, con-
stant. The most probable value that we can assign to X on the
basis of n equally good independent measurements of X is
x = aiXi,
where Xi is the arithmetical mean of the n direct observations
ai, a 2 , a s , . . . , a n .
The n independent values of X obtainable from the given obser-
vations are
AI ami, Az aids, . . . , A n = a\a n .
The accidental errors of the a's and A's are
Aai = a\ Xij Aa 2 = a 2 X\ t . . . , Aa n = a n X\,
and
A4i = Ai - X, A^ 2 = A t -X, . . . , AA n = A n -X.
ART. 60] PROPAGATION OF ERRORS 99
Combining these equations we find
and similar expressions for the other AA's. Consequently
(AAO 2 = ai 2 (Aax) 2 ,
and [(AA) 2 ] = ai [(Aa) 2 ].
If M and Af i are the mean errors of x and xi t respectively,
and Jf,..I3.
Hence M 2 = onWi 2 , (77)
and, since the probable error bears a constant relation to the
mean error,
E 2 = a^! 2 . (78)
When the given functional relation is in the more general form
X = aiXi =b 0:2^2 =b 0.3X3 =b otqXqj
we have
x =
where the a;'s are the most probable values that can be assigned
to the X's on the basis of the given measurements. Applying
(77) and (78) to each term of this equation separately and then
applying (76) we have
t
E 2 =
where the ATs and E's represent respectively the mean and prob-
able errors of the x's with corresponding subscripts.
60. Errors of the Function F (X l} X 2 , . . . , X q ).
We are now in a position to consider the general functional
relation
X = F (Xi, Xz, . . . , X q ),
where F represents any function of the independently measured
quantities Xi, X 2 , etc. Introducing the most probable values of
the observed numerics, the most probable value of the computed
numeric is given by the relation
x = F fa, x 2) . . . , Xq). (80)
This expression may be written in the form
& l ), (Z 2 -f-5 2 .. . . . , (* + ,)!, 0)
100 THE THEORY OF MEASUREMENTS [ART. 60
where the I's represent arbitrary constants and the.S's are small
corrections given by relations in the form
Obviously, the errors of the 5's are equal to the errors of the corre-
sponding x's. For, if Mi, Ms, and MI are the errors of Xi, 5i, and
Zi, respectively, we have by equation (74)
M s * = Mi 2 + Mf.
But MI is equal to zero, because I is an arbitrary quantity and any
value assigned to it may be considered exact. Consequently
Mi 2 . (ii)
Since the I's are arbitrary, they may be so chosen that the
squares and higher powers of the-5's will be negligible in compari-
son with the 8's themselves. Hence, if the x's are independent,
(i) may be expanded by Taylor's Theorem in the form
dF d , \ **
where = F (z, z, . . . , x) = >
and the other differential coefficients have a similar significance.
When the observed values of the x's are substituted in these
coefficients, they become known numerical constants.
The mean error of F (li, Z 2 , . . . , l q ) is equal to zero, since it
is a function of arbitrary constants; and the mean errors of the
5's are equal to the mean errors of the corresponding x's by (ii).
Consequently, if M, Mi, M 2 , . . . , M q represent the mean errors
of x, Xi, x z , . . . , x q , respectively, we have by equation (79)
/dF - . V , fdF , , V ,
= F~ MI ) + brr^ 2 ) +
\dxi I \dx 2 I N ~~ , , .
(OL)
where the E's represent the probable errors of the x's with corre-
sponding subscripts.
Equations (81) are general expressions for the mean and prob-
able errors of derived quantities in terms of the corresponding
errors of the independent components. Generally x\ t x 2 , etc.,
ART. 61] PROPAGATION OF ERRORS 101
represent either the arithmetical or the general means of series of
direct observations on the corresponding components, and EI, E z ,
etc., can be computed by equations (32) or (41). In some cases,
the original observations are not available but the mean values
together with their probable errors are given.
For the purpose of computing the numerical value of the differ-
r\Tj1 r\Tj1
ential coefficients -r ; > etc., the given or observed values of
oXi 0X2
the components x i} x 2) etc., may generally be rounded to three
significant figures. This greatly reduces the labor of computa-
tion and does not reduce the precision of the result, since the E's
and M's are seldom given or desired to more than two significant
figures.
61. Example Introducing the Fractional Error. The prac-
tical application of the foregoing process is illustrated in the follow-
ing simple example: the volume V of a right circular cylinder is
computed from measurements of the diameter D and the length L,
and we wish to determine the probable error of the result. In
this case, V corresponds to x, D to xi, L to x 2) and the functional
relation (80) becomes
Also, if EV, E D , and EL are the probable errors of V, D, and L,
respectively, the second of equations (81) becomes
where
sv
and
dV d /I \ 1 n2
-r^F- = ^F \ -7 TTL) L ] = -TrD*.
dL dL\4 / 4
Hence
The computation can be simplified by introducing the frac-
TTT
tional error -^~- Thus, dividing the above equation by
we have
^ =4 ^! + ^
T7"O 7~^9 I T O
102 THE THEORY OF MEASUREMENTS [ART. 62
or, writing PV, PD, and PL for the fractional errors,
Py 2 = 4 Pz> 2 + P L \
P V
and finally
E v = FP F = V
A similar simplification can be effected, in dealing with many
other practical problems, by the introduction of the fractional
errors. Consequently it is generally worth while to try this ex-
pedient before attempting the direct reduction of the general
equation (81).-
In order to render the problem specific, suppose that
D = 15.67 0.13 mm.,
L = 56.25 d= 0.65 mm.,
then V = 10848
PD = = =
P L = ^ = ^ = .0116; Pz, 2 = 135 X 10- 6 ,
= 0.020,
E v = VTV = 220 mm
Hence
7= 10.85 0.22 cln. 3
62. Fractional Error of the Function aX^ 1 X Z 2 U2 X
X a n5 .-
Suppose the given relation is in the form
X = F(X l ) =aXi
where a and n are constants and the =fc sign of the exponent n is
used for the purpose of including the two functions aXi +n and
aX-r^ in the same discussion. In this case equation (80) becomes
x = axi n ,
and the second of (81) reduces to
But
_=_
Consequently
ART. 62] PROPAGATION OF ERRORS 103
If P and PI are the fractional errors of x and xi, respectively, we
have
E*
-
Hence
i P = nP,. (82)
If we replace n by in the above argument, (80) becomes
_
x = aXi m ,
and we find
m
Hence the fractional error of any integral or fractional power of
a measured numeric is equal to the fractional error of the given
numeric multiplied by the exponent of the power.
If the given function is in the form of a continuous product
X = aX l X X, X X X qt
(80) becomes x = axi X x 2 X X x q .
dF
Hence = ax z X x 3 X X x g ,
ox\
I dF 1
and - =
Hence, by (81),
JP _ Ei 2 EJ Eg 2
r z ~ 7~2 ~f~ ~~2 ~r T >
Js JL>1 JU2 Lq
and, if P, PI, P 2 , . . . , P q represent the fractional errors of the
#'s with corresponding subscripts,
Combining the above cases we obtain the more general rela-
tion
X = aXi 1 X Xz 2 X * * X X q ,
and the corresponding expression for (80) is
Applying (82) to each factor separately and then applying (83) to
the product, we find
f - - - +nfPf. (84)
104 THE THEORY OF MEASUREMENTS [ART. 62
For the sake of illustration and to fix the ideas this result may
be compared with the example of the preceding article. If we
put x = V, Xi = D, HI = 2, x 2 = L, n 2 = 1, a = -7 , P = Py,
PI = PD, and PZ = PL the above expression for x becomes
V = %TrD 2 L,
and (84) becomes
Occasionally it is convenient to express the probable error in
the form of a percentage of the measured magnitude. If E and
p are respectively the probable and percentage errors of x,
p= 100 - = 100 P. (85)
x
Consequently (84) may be written in the form
P 2 = niW + n 2 2 p 2 2 + + nfp*, (84a)
where pi, p 2 , . . . , p q are the percentage errors of Xi, x 2 , . . . , x q ,
respectively
CHAPTER IX.
ERRORS OF ADJUSTED MEASUREMENTS.
WHEN the most probable values of a number of numerics
Xi, X 2 ,etc., are determined by the method of least squares, the
results Xi, x 2 ,etc., are called adjusted measurements of the quan-
tities represented by the X's. In Chapter VII we have seen how
the x's come out by the solution of the normal equations (56) or
(58), and how these equations are derived from the given obser-
vations through the equations (53). In the present chapter we
will determine the characteristic errors of the computed x's in
terms of the corresponding errors of the direct measurements on
which they depend.
63. Weights of Adjusted Measurements. When there are q
unknowns and the given observations are all of the same weight,
the normal equations, derived in article fifty, are
[aa] Xi + [ab] x 2 + [ac] x 3 + - + [ap] x q = [as],
[db] x, + [66] x 2 + [6c] *,+ + [bp] x q = [bs], (56)
[ap] xi + [bp] x z + [cp] x 3 + + [pp] x q = [ps].
Since these equations are independent, the resulting values of the
x's will be the same whatever method of solution is adopted. In
Chapter VII Gauss's method of substitution was used on account
of the numerous checks it provides. For our present purpose
the method of indeterminate multipliers is more convenient as it
gives us a direct expression for the x's in terms of the measured
s's. Obviously this change of method cannot affect the errors of
the computed quantities.
Multiply each of equations (56) in order by one of the arbitrary
quantities AI, A 2 , . . . , A q and add the products. The result-
ing equation is
(86)
+ ([db] A 1 + [bb] A, + + [bp] A q ) x 2
+ >
= [as] A l + [6s] A 2 + + [ps] A q .
105
[ob] A, + [66] A* + + [6p] A q = 0,
106 THE THEORY OF MEASUREMENTS [ART. 63
Since the A's are arbitrary and q in number, they can be made to
satisfy any q relations we choose without affecting the validity
of equation (86). Hence, if we determine the A's in terms of the
coefficients in (56) by the relations
(g7)
equation (86) gives an expression for x\ in the form
xi = [as] Ai + [&*] 4i +!-'+ \ps]A t . (88)
If we repeat this process q times, using a different set of multipliers
each time, we obtain q different equations in the form of (86).
In each of these equations we may place the coefficient of one of
the x's equal to unity and the other coefficients equal to zero, giv-
ing q sets of equations in the form of (87) for determining the q sets
of multipliers. Representing the successive sets of multipliers by
A's, B's, C"s, etc., we obtain (88), and the following expressions
for the other x's :
x 2 = [as] Bi + [bs] ft +...;+ \p 8 ] B q ,
x 3 = [as] Ci + [6s] C 2 + + \ps] C q ,
x q = [as] P! + [6s] P 2 + + \ps] P q .
From equations (87), it is obvious that the A's do not involve
the observations Si, s 2 , etc. Consequently (88) may be expanded
in terms of the observations as follows:
Xi = ctiSi + a z s 2 -f + ctgS q , (89)
where the a's depend only on the coefficients in the observation
equations (53) and are independent of the s'a. Since we are con-
sidering the case of observations of equal weight, each of the s's
in (89) is subject to the same mean error M 8 . Her e, if MI is
the mean error of Xi, we have by equations (79), article fifty-nine,
Mx 2 = ai 2 M s 2 + 2 2 M a 2 + - + a n 2 M, 2
= M M, 2 .
But, if Wi is the weight of x\ in comparison with that of a single s,
we have by (36), article forty-four,
Wl w i (90)
Mi 2 [act]
ART. 63] ERRORS OF ADJUSTED MEASUREMENTS 107
since the ratio of the mean errors of two quantities is equal to the
ratio of their probable errors.
Comparing equations (88) and (89), with the aid of equations
(55), article fifty, we see that
biA 2 + +piA q ,
(i)
a n =
p n A q .
Multiply each of these equations by its a and add the products,
then multiply each by its b and add, and so on until all of the
coefficients have been used as multipliers. We thus obtain the
q sums [aa], [ba], . . . , [pa], and by taking account of equations
(87) we have
[aa] = 1, >
[ba] = [ca] = . = [pa] = 0. )
Hence, if we multiply each of equations (i) by its a and add the
products, we have
[aa] = A i.
Consequently equation (90) becomes
A l
(91)
The weights of the other x's may be obtained, by an exactly
similar process, from equations (88a). The results of such an
analysis are as follows:
M
M a 2 P t
(91a)
Obviously the coefficients of the sums [as], [bs], etc., in equa-
tions (88) and (88a) do not depend upon the particular method by
which the normal equations are solved, since the resulting values
of the x's must be the same whatever method is used. Conse-
quently, if the absolute terms [as], [bs], . . . , [ps] are kept in literal
form during the solution of the normal equations by any method
whatever, the results may be written in the form of equations
108 THE THEORY OF MEASUREMENTS [ART. 64
(88) and (88a); and the quantities AI, B 2) etc., will be numerical
if the coefficients [aa], [ab], . . . , [bb], . . . , [pp] are expressed
numerically.
Hence, in virtue of (91) and (91 a), we have the following rule
for computing the weights of the z's.
Retain the absolute terms of the normal equations in literal
form, solve by any convenient method, and write out the solution
in the form
a?i = [as] A! + [bs] A 2 + [cs] A 3 + - + \ps] A qt
x 2 = [as] B l + [bs] B 2 + [cs] B 3 + - - - + \ps] B q ,
x q = [as] P 1 + [bs] P 2 + [cs] P, + - - + [ps] P q .
Then the weight of x\ is the reciprocal of the coefficient of [as] in
the equation for x\, the weight of x 2 is the reciprocal of the co-
efficient of [bs] in the equation for x%, and in general the weight of
x q is the reciprocal of the coefficient of [ps] in the equation for x q .
As an aid to the memory, it may be noticed that the coefficients
AI, B 2 , Cs, . . . , P q , that determine the weights, all lie in the
main diagonal of the second members of the above equations.
When the number of unknowns is greater than two, the labor of
computing all of the A's, B's, etc., would be excessive, and conse-
quently it is better to determine the x's by the methods of Chap-
ter VII. The essential coefficients AI, B 2 , C 3 , . . . , P q can be
determined independently of the others by the method of deter-
minants as will be explained later.
If the given observations are not of equal weight, the weights
of the x's may be determined by a process similar to the above,
starting with normal equations in the form of (58), article fifty.
The result of such an analysis can be expressed by the rule stated
above if we replace the sums [as], [bs], . . . , [ps] by the weighted
sums [was], [wbs], . . . , [wps], the notation being the same as in
article fifty.
64. Probable Error of a Single Observation. By definition,
article thirty-seven, the mean error M 8 of a single observation is
given by the expression
_ Af + A^+.-.+A.' _ [AA] , (iii)
n n
where the A's represent the true accidental errors of the s's.
When the number of observations is very great, the residuals given
ART. 64] ERRORS OF ADJUSTED MEASUREMENTS 109
by equations (54) may be used in place of the A's without causing
appreciable error in the computed value of M 8 . But, in most
practical cases, n is so small that this simplification is not admis-
sible and it becomes necessary to take account of the difference
between the residuals and the accidental errors.
Let Ui, u 2 , . . . , u q represent the true errors of the x's ob-
tained by solution of the normal equations (56). Then the true
accidental error of the first observation is given by the relation
Ol (Xi + Ui) + 61 (X 2 + U 2 ) + + Pl (X q + U q ) - Si = Ai.
But, by the first of equations (54),
aiXi + 6ix 2 -f cix s + + pix q si = ri,
where r\ is the residual corresponding to the first observation.
Combining these equations and applying them in succession to
the several observations, we obtain the following expressions for
the A's in terms of the r's:
ri + aiui + biu 2 + CiU 3 + - + piUq = Ai,
A 2 ,
,.*
+ b n u 2 + c n u 3 + + p n u q = A n .
Multiply each of these equations by its r and add; the result is
[rr] + [ar] HI + [br] u 2 + [cr] u 3 + + [pr] u q = [Ar].
But by equations (iii), article fifty,
[ar] = [br] = [cr] = = for] = 0, (v)
and, consequently,
[rr] = [Ar]. (vi)
Multiply each of equations (iv) by its A and add. Then, taking
account of (vi), we have
[rr] + [aA] Ul + [6A] u 2 + + [pA] u q = [AA]. (vii)
In order to obtain an expression for the u's in terms of the A's,
multiply each of equations (iv) by its a and add, then multiply
by the b's in order and add, and so on with the other coefficients.
The first term in each of these sums vanishes in virtue of (v), and
we have
[aa] ui + [ab] w 2 + + [ap] u q = [aA],
[db] Ul + [bb] u, + + \bp] u q = [6A],
lap] ui + [bp] u 2 + - - - + [pp] u q =
(viii)
110 THE THEORY OF MEASUREMENTS [Am. 64
These equations are in the same form as the normal equations (56)
with the z's replaced by u's and the s's by A's. Hence any solu-
tion of (56) for the x's may be transposed into a solution of (viii)
for the u's by replacing the s's by A's without changing the coeffi-
cients of the s's. Consequently, by (89), we have
and similar expressions for the other u's.
The coefficients of the u's in (vii) expand in the form
[aA] = aiAi + a 2 A 2 + + a n A n .
Hence
[aA] ui = aiaiAi 2 + a 2 2 A 2 2 +.+ a n a n A n 2 ,
Since positive and negative A's are equally likely to occur, the
sum of the terms involving products of A's with different subscripts
will be negligible in comparison with the other terms. The sum
of the remaining terms cannot be exactly evaluated, but a suffi-
ciently close approximation is obtained by placing each of the A 2 's
equal to the mean square of all of them, - - -* Consequently, as
the best approximation that we can make, we may put
n
But, by equations (ii), [aa] is equal to unity. Hence
[aA] - M.
iv
Since there is nothing in the foregoing argument that depends on
the particular u chosen, the same result would have been obtained
with any other u. .Consequently, in equation (vii), each term that
involves one of the u's must be equal to - - !i and, since there
tv
are q such terms, the equation becomes
Hence, by equation (iii),
and
ART. 64] ERRORS OF ADJUSTED MEASUREMENTS 111
where the r's represent the residuals, computed by equations (54) ;
n is the number of observations ; and q is the number of unknowns
involved in the observation equations (53). In the case of direct
measurements, the number of unknowns is one, and (92) reduces
to the form already found in article forty-one, equation (30), for
the mean error of a single observation.
When the observations are not of equal weight, the mean error
M 8 of a standard observation, i.e. an observation of weight
unity, is given by the expression
2 =
n
where the w's are the weights of the individual observations.
Starting with this relation in place of (iii) and making correspond-
ing changes in other equations, an analysis essentially like the
preceding leads to the result
Ma = ^'^-, (93)
T n q
which reduces to the same form as (92) when the weights are all
unity.
Introducing the constant relation between the mean and probable
errors, we have the expressions
E 8 = 0.6741/-M- , (94)
V n q
for the probable error of a single observation in the case of equal
weights, and
E 8 = 0.674\/-^i, (95)
V n q
for the probable error of a standard observation in the case of
different weights.
Finally, if M k , E k , and w k represent the mean error, the probable
error, and the weight of x k , any one of the unknown quantities,
we may derive the following relations from the above equations
by applying equations (36), article forty-four:
M s
- = 7= V '
A/in. T n o
(96)
112
THE THEORY OF MEASUREMENTS [ART. 65
when the weights of the given observations are equal, and
M k = -^= = L Y/-^->
v Wk vWk n ~ Q
E, 0.674
Ek = / - =
(97)
~ 2
when the weights of the given observations are not equal.
65 . Application to Problems Involving Two Unknowns . When
the observation equations involve only two unknown quantities,
the solution of the normal equations is given by (59), article
fifty-one, in the form
_ [66] [as] - [ab] [bs]
[aa] [bb] - [ab] 2 '
_ [aa] [bs] [ab] [as]
[aa] [bb] - [ab] 2
By the rule of article sixty-three, the weight of Xi is equal to the
reciprocal of the coefficient of [as] in the equation for Xi, and the
weight of #2 is equal to the reciprocal of the coefficient of [bs] in
the equation for x 2 . Hence, by inspection of the above equations,
we have
[aa] [bb] - [ab] 2
_
W 2 =
[bb]
[aa] [bb] - [ab] 2
[aa]
(98)
Since there are only two unknown quantities, and the observa-
tions are of equal weight, equation (92) gives the mean error of a
single observation when q is taken equal to two. Hence
(99)
where n is the number of observation equations and [rr] is the
sum of the squares of the residuals that are obtained when the
computed values of Xi and Xz are substituted in equations (53a),
article fifty-one.
Combining equations (98) and (99) with (96), we obtain the
following expressions for the probable errors of Xi and x 2 :
0.674
E 2 = 0.674
v/
v/
[66]
[aa][bb] - [ab] 2 n-2
[aa]
[rr>
[aa] [bb] - [ab] 2 n-2
(100)
ART. 65] ERRORS OF ADJUSTED MEASUREMENTS 113
For the purpose of illustration, we will compute the probable
errors of the values of x\ and x 2 obtained in the numerical prob-
lem worked out in article fifty-one. Referring to the numerical
tables in that article, we find
[aa] = 5; [ab] = 20; [bb] = 90; n = 5;
[rr] = 9.60 X 1Q- 4 .
Hence, by equations (100),
*'
V /
5X90-400
By equations (vi), article fifty-one, the length L of the bar at
C., and the coefficient of linear expansion a are given by the
relations
L = iooo + si; a = -L.*.
10 -L70
Since L is equal to #1 plus a constant, its probable error is equal
to that of Xi by the argument underlying equation (ii), article
sixty. Hence
EL. = E! = =fc 0.016.
To find the probable error of a, we have by equations (81), article
sixty,
But, since L is very large in comparison with x 2 , the second term
on the right-hand side is negligible in comparison with the first.
Consequently, without affecting the second significant figure of
the result, we may put
= Ei X 10- 4 = =fc 0.038 X 10- 5 .
Hence the final results of the computations in article fifty-one may
be more comprehensively expressed in the form
L Q = 1000.008 db 0.016 millimeters,
a = (1.780 db 0.038) X 10~ 5 ,
114
THE THEORY OF MEASUREMENTS [AET. 66
when we wish to indicate the precision of the observations on
which they depend.
66. Application to Problems Involving Three Unknowns. The
normal equations, for the determination of three unknowns, take
the form
[aa] Xi + [ah] x 2 + [ac] x 3 = [as],
[ac] xi + [be] x 2 + [cc] x 3 = [cs].
Solving by the method of determinants and putting
we have
[as]
x 2 = [as
[as]
Hence, by the rule of article sixty-three,
D
Wl [bb][cc] -[be] 2 '
= D
2 ~~ [aa] [cc] [ac] 2 '
D
[aa][bb]-[ab]*'
[aa]
[ab]
[ac]
[ab] [ac
[66] [be
[be] [cc
]
= A
[bb] [be]
[be] [cc]
1 J
[be] [cc]
[06] [ac]
4
-[cs]
[06] [ac]
[bb] [be]
t
D
D
D
[ac] [cc]
[06] [6c]
- + [6s]
[aa] [ac]
[ac] [cc]
-
-[cs]
[ab] [be]
[aa] [ac]
,
D
D
D
[ah] [66]
[ac] [6c]
+ N-
[ac] [be]
[aa] [ab]
+ [cs]
[aa] [ab]
[ab] [bb]
D
D
D
w s =
(ix)
(x)
The determinant D can be eliminated from equations (x), if
we can obtain an independent expression for any one of the w's.
The solution of the normal equations by Gauss's Method in
article fifty-four led to the result
-
X3 ~
[cc'2]
ART. 66] ERRORS OF ADJUSTED MEASUREMENTS 115
The auxiliary [cc 2] is independent of the absolute terms [as],
[6s], and [cs]. The auxiliary [cs 2] may be expanded as follows:
[oc] r , [6cl] ( , [ab]
[6cl] ( ,
~ PTTJ \ M -
Hence the coefficient of [cs] in the above expression for x$ is
r - ~y, and, consequently, the weight of x$ is equal to [cc2].
[CC ZJ
Substituting this value for w s in the third of equations (x) and
eliminating D from the other two we have
[aa] [bb 1]
[66
(101)
w 3 = [cc 2],
where the auxiliary quantities [66 1], [cc 1], and [cc 2] have the
same significance as in article fifty-four.
The weights of the x's having been determined by equations
(101), their probable errors may be computed by equations (96).
In the present case q is taken equal to three, since there are three
unknowns, and the r's are given by equations (68).
In the numerical illustration of Gauss's Method, worked out in
article fifty-five, we found the following values of the quantities
appearing in equations (96) and (101):
[aa] = 6; [66] = 220; [6c] = 180; [cc] = 157;
[66 1] = 70; [cc 1] = 76.0; [cc 2] = 5.97;
[rr] = 0.00120; n = 6; q = 3.
These values have been rounded to three significant figures, when
necessary, since the probable errors of the #'s are desired to only
two significant figures. Substituting in equations (101) we have
Wl = 6X7 _ 2 5.97 -1.17,
220 X 157 - 180
70
^2 = y^5.97 = 5.50,
w 3 = 5.97,
116 THE THEORY OF MEASUREMENTS [ART. 66
From equation (94)
\E.
and, by equations (96),
a = 0.674 1/ ' 0012 = 0.0135,
0.0135
. O.UUoo.
Consequently the precision of the measurements, so far as it
depends on accidental errors, may be expressed by writing the
computed values of the x's in the form
xi = 0.245 0.012,
X2 =- 1.0003 0.0057,
z 3 = 1.4022 0.0055.
Since the last significant figure in each of the x's occupies the same
place as the second significant figure in the corresponding prob-
able error, it is evident that the proper number of figures were
retained throughout the computations in article fifty-five.
CHAPTER X.
DISCUSSION OF COMPLETED OBSERVATIONS.
67. Removal of Constant Errors. The discussion of acci-
dental errors and the determination of their effect on the result
computed from a given series of observations, as carried out in the
preceding chapters, are based on the assumption that the meas-
urements are entirely free from constant errors and mistakes.
Hence the first matter of importance, in undertaking the reduction
of observations, is the determination and removal of all constant
errors and mistakes. Also, in criticizing published or reported
results, judgment is based very largely on the skill and care with
which such errors have been treated. In the former case, if suit-
able methods and apparatus have been chosen and the adjust-
ments of instruments have been properly made, sufficient data is
usually at hand for determining the necessary corrections within
the accidental errors. In the latter case we must rely on the dis-
cussion of methods, apparatus, and adjustments given by the
author and very little weight should be given to the reported
measurements if this discussion is not clear and 'adequate.
No evidence can be obtained from the observations themselves
regarding the presence or absence of strictly constant errors.
The majority of them are due to inexact graduation of scales,
imperfect adjustment of instruments, personal peculiarities of the
observer, and faulty methods of manipulation. They affect all
of the observations by the same relative amount. Their detec-
tion and correction or elimination depend entirely on the judg-
ment, experience, and care of the observer and the computer.
When the same magnitude has been measured by a number of
different observers, using different methods and apparatus, the
probability that the constant errors have been the same in all of
the measurements is very small. Consequently if the corrected
results agree, within the accidental errors of observation, it is
highly probable that they are free from constant errors. This is
the only criterion we have for the absence of such errors and it
117
118 THE THEORY OF MEASUREMENTS [ART. 67
breaks down in some cases when the measured magnitude is not
strictly constant.
Sometimes constant errors are not strictly constant but vary
progressively from observation to observation owing to gradual
changes in surrounding conditions or in the adjustment of instru-
ments. The slow expansion of metallic scales due to the heat
radiated from the body of the observer is an illustration of a
progressive change. Such variations are usually called systematic
errors. They may be corrected or eliminated by the same methods
that apply to strictly constant errors when adequate means are
provided for detecting them and determining the magnitude of
the effects produced. When their range in magnitude is compara-
ble with that of the accidental errors, their presence can usually be
determined by a critical study of the given observations and their
residuals. But, if they have not been foreseen and provided for
in making the observations, their correction is generally difficult
if not impossible. In many cases our only recourse is a new series
of observations taken under more favorable conditions and accom-
panied by adequate means of evaluating the systematic errors.
A general discussion of the nature of constant errors and of the
methods by which they are eliminated from single direct observa-
tions was given in Chapter III. These processes will now be con-
sidered a little more in detail and extended to the arithmetical
mean of a number of direct observations. Let a\ t d 2 , a s , . . . , a n
represent a series of direct observations after each one of them
has been corrected for all constant errors. Then the most prob-
able value that can be assigned to the numeric of the measured
magnitude is the arithmetical mean
x = q i + fl2 + +a n /jx
IV
Now suppose, that the actual uncorrected observations are 01, o 2 ,
o 3 , , o n , then
ai = 01 + cj + cj' + cj" + + ci<*> = 01 + [cj,
a 2 = o 2 + cj + c 2 " + cj" + + c 2 ("> = o 2 +
C*n = O n + C n ' + C" + C n '" + + cj* = O n + [c
where the c's represent the constant errors to be eliminated and
may be either positive or negative. There are as many c's in
each equation as there are sources of constant error to be consid-
ART. 67] DISCUSSION OF COMPLETED OBSERVATIONS 119
ered. Usually, when all of the observations are made by the
same method and with equal care, the number of c's is the same
in all of the equations. Substituting (ii) in (i)
J . = 0l + 02+ +. [Cj + [cj+ - - +[ftj
n n '
When there are no systematic errors
Cl = Cz = C 3 ' =
Cl " = C 2 " = C," = = Cn " = C ",
= C 3 ' = * = Cn
Consequently
[ci] = [c z ] = [c 3 ] = = [c n ] = [c], (iv)
and we have
x = + [c]
n
= Om + c' + c" + c"' + -f c<>, (102)
where o m is written for the mean of the actual observations.
Hence, when all of the observations are affected by the same con-
stant errors, the corrections may be applied to the arithmetical
mean of the actual observations and the resulting value of x will
be the same as if the observations were separately corrected before
taking the mean.
The residuals corresponding to the corrected observations ai,
a 2 , a 3 , . . . , a n are given by equations (3), article twenty-two.
Replacing x and the a's by their values in terms of o m and the
o's as given in (102) and (ii), and taking account of (iv), equations
(3) become
ri = di X = Oi+ [Ci] - Om- [C\ = 01 - O m ,
r 2 = a 2 x = o 2 + [c 2 ] o m [c] = o 2 o m , (103)
r n = a n - X = O n + [C n ] -Om- [c] = O n - O m .
Consequently, when there are no systematic errors, the residuals
computed from the o's and o m will be identical with those com-
puted from the a's and x. Hence, if the uncorrected observations
are used in computing the probable error of x, by the formula
/ W
E = 0.674\/ / J 1X >
V n (n 1)
120 THE THEORY OF MEASUREMENTS [ART. 67
the result will be the same as if the corrected observations had
been used; and, as pointed out above, the observations and their
corresponding residuals give no evidence of the presence of strictly
constant errors.
When the constant errors affecting the different observations
are different or when any of them are systematic in character,
equation (iv) no longer holds, and, consequently, the simplifica-
tion expressed by (102) is no longer possible. In the former case
the observations should be individually corrected before the mean
is taken. The same result might be obtained from equation (iii),
but the computation would not be simplified by its use. In the
latter case the several observations are affected by errors due to
the same causes but varying progressively in magnitude in response
to more or less continuous variations in the conditions under
which they are made.
In equations (ii) the c's having the same index may be con-
sidered to be due to the same cause, but to vary in magnitude
from equation to equation as indicated by the subscripts. The
arithmetical means of the errors due to the same causes are
, _ Ci' + C 2 ' + + C n '
Cm '~ ~
_
Cm -
n
and the mean of the observations is
01 + 02 + ' ' '
O m =
n
Substituting (ii) in (i) and taking account of the above relations
we have
X = O m + C m ' + C m " + ' ' ' + C w <> . (104)
Hence, in the case of systematic errors, the most probable value
of the numeric of the measured magnitude may be obtained from
the mean of the uncorrected observations by applying mean cor-
rections for the systematic errors. When all of the errors are
strictly constant equation (104) becomes identical with (102)
because all of the errors having the same index are equal. Obvi-
ART. 68] DISCUSSION OF COMPLETED OBSERVATIONS 121
ously it also holds when part of the c's are strictly constant and the
remainder are systematic.
If we use the value of x given by (104) in place of that given
by (102) in the residual equations (103), the c's will not cancel.
Hence, if any of the constant errors are systematic in nature,. the
residuals computed from the o's and o m will be different from
those computed from the a's and x; and, consequently, they will
not be distributed in accordance with the law of accidental errors.
In practice it is generally advisable to correct each of the ob-
servations separately before taking the mean rather than to use
equation (104), since the true residuals are required in computing
the probable error of x, and they cannot be derived from the un-
corrected observations. Whenever possible the conditions should
be so chosen that systematic errors are avoided and then the
necessary computation can be made by equations (102) and (103).
68. Criteria of Accidental Errors. We have seen that the
residuals computed from observations affected by systematic errors
do not follow the law of accidental errors. Hence, if it can be
shown that the residuals computed from any given series of obser-
vations are distributed in conformity with the law of errors, it is
probable that the given observations are free from systematic
errors or that such errors are negligible in comparison with the
accidental errors. Observations that satisfy this condition may
or may not be free from strictly constant errors, but necessary
corrections can be made by equation (102) and the probable error
of the mean may be computed from the residuals given by
equation (103).
Systematic errors should be very carefully guarded against in
making the observations, and the conditions that produce them
should be constantly watched and recorded during the progress
of the work. After the observations have been completed they
should be individually corrected for all known systematic errors
before taking the mean. The strictly constant errors may then
be removed from the mean, but before this is done it is well to
compute the residuals and see if they satisfy the law of accidental
errors. If they do not, search must be made for further causes
of systematic error in the conditions surrounding the measure-
ments and a new series of observations should be made, under
more favorable conditions, whenever sufficient data for this pur-
pose is not available.
122 THE THEORY OF MEASUREMENTS [ART. 68
Residuals, when sufficiently numerous, follow the same law of
distribution as the true accidental errors. Consequently system-
atic errors and mistakes might be detected by a direct comparison
of the actual distribution with the theoretical, as carried out in
article thirty-four, provided the number of observations is very
large. However, in most practical measurements, the residuals
are not sufficiently numerous to fulfill the conditions underlying
the law of errors, and a considerable difference between their
actual and theoretical distribution is quite as likely to be due to^
this fact as to the presence of systematic errors. Whatever the
number of observations, a close agreement between theory and
practice is strong evidence of the absence of such errors but it is
seldom worth while to carry out the comparison with less than
one hundred residuals.
When the residuals are numerous and distributed in the same
manner as the accidental errors, the average error of a single
observation, computed by the formula
Vn(n- 1)'
and the mean error, computed by the formula
satisfy the relation
M = 1.253 A.
Also the formulae
E = 0.8453 A and E = 0.6745 M
give the same value for the probable error of a single observation.
When the number of observations is limited, exact fulfillment of
these relations ought not to be expected, but a large deviation
from them is strong evidence of the presence of systematic errors
or mistakes. Unless the number of observations is very small,
ten or less, the relations should be fulfilled within a few units in
the second significant figure, as is the case in the numerical example
worked out in article forty-two.
Obviously the arithmetical mean is independent of the order
in which the observations are arranged in taking it, but the order
of the residuals in regard to sign and magnitude depends on the
order of the observations. When there are systematic errors and
the observations are arranged in the order of progression of their
ART. 68] DISCUSSION OF COMPLETED OBSERVATIONS 123
cause, the residuals will gradually increase or decrease in absolute
magnitude in the same order; and, if the systematic errors are
large in comparison with the accidental errors, there will be but
one change of sign in the series. Thus, if the temperature is
gradually rising while a length is being measured with a metallic
scale and the observations are arranged in the order in which they
are taken, the first half of them will be larger than the mean and
the last half smaller, except for the variations caused by accidental
errors. For the purpose of illustration, suppose that the observa-
tions are
1001.0; 1000.9; 1000.8; 1000.7; 1000.6; 1000.5; 1000.4.
The mean is 1000.7 and the residuals
+ .3; +.2; +.1; .0; -.1; -.2; -.3
decrease in absolute magnitude from left to right, i.e., in the order
in which the observations were made. There are five cases in
which the signs of succeeding residuals are alike and one in which
they are different; the former cases will be called sign-follows and
the^latter a sign-change. This order of the residuals in regard to
magnitude and sign is typical of observations affected by sys-
tematic errors when they are arranged in conformity with the
changes in surrounding conditions. Since such changes are usually
continuous functions of the time, the required arrangement is
generally the order in which the observations are taken.
Such extreme cases as that illustrated above are seldom met
with in practice owing to the impossibility of avoiding accidental
errors of observation and the complications they produce in the
sequence of residuals. Generally the systematic errors that are
not readily discovered and corrected before making further re-
ductions are comparable in magnitude with the accidental errors.
Consequently they cannot control the sequence in the signs of
the residuals but they do modify the sequence characteristic of
true accidental errors.
In any extended series of observations there should be as many
negative residuals as positive ones, since positive and negative
errors are equally likely to occur. After any number of observations
have been made, the probability that the residual of the next obser-
vation will be positive is equal to the probability that it will be nega-
tive, since the possible number of either positive or negative errors
is infinite. Consequently the chance that succeeding residuals
124 THE THEORY OF MEASUREMENTS [ART. 69
will have the same sign is equal to the chance that they will have
different signs. Hence, if the residuals are arranged in the order
in which the corresponding observations were made, the number
of sign-follows should be equal to the number of sign-changes.
The residuals, computed from limited series of observations,
seldom exhibit the theoretical sequence of signs exactly because
they are not sufficiently numerous to fulfill the underlying condi-
tions. Nevertheless, a marked departure from that sequence
suggests the presence of systematic errors or mistakes and should
lead to a careful scrutiny of the observations and the conditions
under which they were made. If the disturbing causes cannot be
detected and their effects eliminated, it is generally advisable to
repeat the observations under more favorable conditions. The
numerical example, worked out in article forty-two, may be cited
as an illustration from practice. The observations were made in
the order in which they are tabulated, beginning at the top of the
first column and ending at the bottom of the fourth column. In
the second and fifth columns we find ten positive and ten negative
residuals. The number of sign-follows is ten and the number of
sign-changes is nine. This is rather better agreement with the
theoretical sequence of signs than is usually obtained with so few
residuals. It indicates that the observations were made under
favorable conditions and are sensibly free from systematic errors
but it gives no evidence whatever that strictly constant errors
are absent.
Although the foregoing criteria of accidental errors are only
approximately fulfilled when the number of observations is lim-
ited, their application frequently leads to the detection and elimi-
nation of unforeseen systematic errors. The first method is rather
tedious and of little value when less than one hundred obser-
vations are considered, but the last two methods may be easily
carried out and are generally exact enough for the detection of
systematic errors comparable in magnitude with the probable error
of a single observation.
69. Probability of Large Residuals. In discussing the dis-
tribution of residuals in regard to magnitude, the words large and
small are used in a comparative sense. A large residual is one that
is large in comparison with the majority of residuals in the series
considered. Thus, a residual that would be classed as large in a
series of very precise observations would be considered small in
ART. 69] DISCUSSION OF COMPLETED OBSERVATIONS 125
dealing with less exact observations. Consequently, in expressing
the relative magnitudes of residuals, it is customary to adopt a
unit that depends on the precision of the measurements considered.
The probable error of a single observation is the best magnitude
to adopt for this purpose, since it is greater than one-half of the
errors and less than the other half. If we represent the relative
magnitude of a given error by S, the actual magnitude by A, and
the probable error of a single observation by E,
S = |- (105)
The relative magnitudes of the residuals may be represented in
the same way by replacing the error A by the residual r. It is
obvious that values of S less than unity correspond to small re-
siduals and values greater than unity to large residuals in any
series of observations.
In equation (13), article thirty-three, the probability that an
error chosen at random is less than a given error A is expressed
by the integral
*/~ A
o / v j
PA = -^= e-*dt. (13)
V-n-Jo
Equation (25), article thirty-eight, may be put in the form
V ** k
& = 7= ->
VTT a?
where $ is written for the numerical constant 0.47694. Hence,
introducing (105),
and (13) becomes
P 8 = 'eft. (106)
Obviously this integral expresses the probability that an error
chosen at random is less than S times the probable error of a
single observation. It is independent of the particular series to
which the observations belong and its values, corresponding to
a series of values of the argument S, are given in Table XII.
Since all of the errors in any system are less than infinity, Poo
is equal to unity. Hence the probability that a single error,
126
THE THEORY OF MEASUREMENTS [ART. 69
chosen at random, is greater than S times E is given by the rela-
tion
Qs = 1 - Pa- (V)
Now the residuals, when sufficiently numerous and free from
systematic errors and mistakes, should follow the same distri-
bution as the accidental errors. Hence, if n s is the number of
residuals numerically greater than SE and N is the total number
in any series of observations, we should have
Qs = T?" (vi)
Since the numerical value of P 8 , and consequently that of Q 8
depends only on the limit S and is independent of the precision
of the particular series of measurements considered, the ratio jj. >
corresponding to any given limit S, should be the same in all
cases. Consequently, if N observations have been made on any
magnitude and by any method whatever, n 8 of them should corre-
spond to residuals numerically greater than SE. Conversely, if
we assign any arbitrary number to n a , equation (vi) defines the
number of observations that we should expect to make without
exceeding the assigned number of residuals greater than SE.
Hence, if N a is the number of observations among which there
should be only one residual greater than S times the probable
error of a single observation, we have, by placing n s equal to
one in (vi), and substituting the value of Q 8 from (v),
*--r^>r (107)
The fourth column of the following table gives the values of N a ,
to the nearest integer, corresponding to the integral values of the
limit S given in the first column. The values of P 8 in the second
column are taken from Table XII, and those of Q 8 in the third
column are computed by equation (v).
S
P.
e.
N s
1
0.50000
0.50000
2
2
0.82266
0.17734
6
3
0.95698
0.04302
23
4
0.99302
0.00698
143
5
0.99926
0.00074
1351
ART. 70] DISCUSSION OF COMPLETED OBSERVATIONS 127
To illustrate the significance of this table, suppose that 143
direct observations have been made on any magnitude by any
method whatever. The probable error E of a single observation
in this series may be computed from the residuals by equation (31)
or (34). Then, if the residuals follow the law of errors, not more
than one of them should be greater than four times as large as E.
If the number of observations had been 1351, we should expect
to find one residual greater than five times E, and on the other
hand if the number had been only twenty-three, not more than
one residual should be greater than three times E.
Although the probability for the occurrence of large residuals
is small, and very few of them should occur in limited series
of observations, their distribution among the observations, in
respect to the order in which they occur, is entirely fortuitous.
A large residual is as likely to occur in the first, or any other,
observation of an extended series as in the last observation. Con-
sequently the limited series of observations, taken in practice,
frequently contain abnormally large residuals. This is not due
to a departure from the law of errors, but to a lack of sufficient
observations to fulfill the theoretical conditions. In such cases
there are not enough observations with normal residuals to balance
those with abnormally large ones. Consequently a closer approxi-
mation to the arithmetical mean that would have been obtained
with a more extended series of observations is obtained when the
abnormal observations are rejected from the series before taking
the mean.
Observations should not be rejected simply because they show
large residuals, unless it can be shown that the limit set by the
theory of errors, for the number of observations considered, is
exceeded. This can be judged approximately by comparing the
residuals of the given observations with the numbers given in the
first and last columns of the above table, but a more rigorous test
is obtained by applying Chauvenet's Criterion, as explained in the
following article.
70. Chauvenet's Criterion. The probability that the error
of a single observation, chosen at random, is less than SE is
expressed by P a in equation (106). Now, the taking of N inde-
pendent observations is equivalent to N selections at random from
the infinite number of possible accidental errors. Hence, by
equation (7), article twenty-three, the probability that each of
128 THE THEORY OF MEASUREMENTS [ART. 70
the N observations in any series is affected by an error less than
SE is equal to P N . Since all of the N errors must be either greater
or less than SE } the probability that at least one of them is greater
than this limit is equal to 1 P 8 N . Placing this probability
equal to one-half, we have
i - P." = i,
or
P. - (1 - (vii)
If the limit S is determined by this equation, there is an even
chance that at least one of the N observations is affected by an
error greater than SE.
Expanding the second member of (vii) by the Binomial Theorem
11 N -I I (N- l)(2N-l) 1
N 2 1-2-N 2 4 1-2- 3- N* 8
1-2-3 . . . K-N K
The terms of this series decrease very rapidly and all but the first
are negative. Consequently the sum of the terms beyond the
second is small in comparison with the other two; and, whatever
the value of N, (1 %) N is nearly equal to, but always slightly
less than, - ^-^ - . Since P 8 and S increase together, the limit
T determined by the relation
2N-1
2N
(108)
is slightly greater than the limit S determined by (vii). Hence,
if N independent direct observations have been made, the prob-
ability against the occurrence of a single error greater than
A r = TE (109)
is greater than the probability for its occurrence. Consequently,
if the given series contains a residual greater than A r , the prob-
able precision of the arithmetical mean is increased by excluding
the corresponding observation.
ART. 70] DISCUSSION OF COMPLETED OBSERVATIONS 129
Equations (108) and (109) express Chauvenet's Criterion for the
rejection of doubtful observations. In applying them, the prob-
able error E of a single observation is first computed from the
residuals of all of the observations by either equation (31) or the
first of equations (34) with the aid of Table XIV or XV. If any
of the residuals appear large in comparison with the computed
value of E, PT is determined from (108) by placing N equal to
the number of observations in the given series. T is then obtained
by interpolation from Table XII, and finally A r is computed by
(109). If one or more of the residuals are greater than the com-
puted A r , the observation corresponding to the largest of them is
excluded from the series and the process of applying the criterion is
repeated from the beginning. If one or more of the new residuals
are greater than the new value of A r , the observation correspond-
ing to the largest of them is rejected. This process is repeated
and observations rejected one at a time until a value of A r is ob-
tained that is greater than any of the residuals.
When more than one residual is greater than the computed
value of Ay, only the observation corresponding to the largest
of them should be rejected without further study. The rejection
of a single observation from the given series changes the arith-
metical mean, and hence all of the residuals and the value of E
computed from them. If r and r' are the residuals corresponding
to the same observation before and after the rejection of a more
faulty observation, and if A r and A r ' are the corresponding
limiting errors, it may happen that r' is less than A/, although r
is greater than Ay. Hence the second application of the criterion
may show that a given observation should be retained notwith-
standing the fact that its residual was greater than the limiting
error in the first application, provided an observation with a
larger residual was excluded on the first trial.
To facilitate the computation of Ay, the values of T corre-
sponding to a number of different values of N have been
interpolated from Table XII and entered in the second column
of Table XIII.
For the purpose of illustration, suppose that ten micrometer
settings have been made on the same mark and recorded, to the
nearest tenth of a division of the micrometer head, as in the first
column of the following table.
130
THE THEORY OF MEASUREMENTS [ART. 71
Obs.
r
r'
2.567
+0.0118
2.559
+0.0038
+0.0051
2.556
+0.0008
+0.0021
2.552
-0.0032
-0.0019
2.551
-0.0042
-0.0029
2.553
-0.0022
-0.0009
2.555
-0.0002
+0.0011
2.548
-0.0072
-0.0059
2.554
-0.0012
+0.0001
2.557
+0.0018
+0.0031
x =2.5552
[r] = 0.0364
[r>] = 0.0231
z'=2.5539
# = 0.0032
#' = 0.0023
IF = 2. 91
T' = 2.84
Ar = 0.0093
A/ = 0.0065
The residuals, computed from the mean x, are given under r.
The probable error E } computed from [r] by the first of equations
(34), with the aid of Table XV, is 0.0032. The value of T corre-
sponding to ten observations is 2.91 from Table XIII, and the
limiting error Ay is equal to 0.0093. Since this is less than the
residual 0.0118, the corresponding observation (2.567) should be
rejected from the series.
The mean of the retained observations, xi, is 2.5539, and the
corresponding residuals are given under r' in the third column of
the above table. The new value of the limiting error (A/), com-
puted by the same method as above, is 0.0065. Since none of
the new residuals are larger than this, the nine observations left
by the first application of the criterion should all be retained.
71. Precision of Direct Measurements. The first step in
the reduction of a series of direct observations is the correction
of all known systematic errors and the test of the completeness of
this process by the criteria of article sixty-eight. In general, the
systematic errors represent small variations of otherwise constant
errors; and, in making the preliminary corrections, it is best to
consider only this variable part, i.e., the corrections are so applied
that all of the corrected observations are left with exactly the
same constant errors. Thus, suppose that the temperature of a
scale is varying slowly during a series of observations, and is
never very near to the temperature at which the scale is standard.
It is better to correct each observation to the mean temperature
of the scale and leave the larger correction, from mean to standard
ART. 71] DISCUSSION OF COMPLETED OBSERVATIONS 131
temperature, until it can be applied to the arithmetical mean in
connection with the corrections for other strictly constant errors.
This is because the systematic variations in the length of the
scale are so small that the unavoidable errors in the observed
temperatures and the adopted coefficient of expansion of the scale
can produce no appreciable effect on the corrections to mean
temperature. The effect of these errors on the larger correction
from mean to standard temperature is more simply treated in
connection with the arithmetical mean than with the individual
observations.
Let 01, 02, . . . , o n represent a series of direct observations
corrected for all known systematic errors and satisfying the
criteria of accidental errors. We have seen that the most prob-
able value that we can assign to the numeric of the measured mag-
nitude, on the basis of such a series, is given by the relation
x = o m + c'+c"+ - +cfe>, (102)
where o m is the arithmetical mean of the o's, and the c's represent
corrections for strictly constant errors. If the c's could be deter-
mined with absolute accuracy, or even within limiting errors that
are negligible in comparison with the accidental errors of the o's,
the only uncertainty in the above expression for x would be that
due to the accidental error of o m . Hence, by equations (103), if
E x and E m are the probable errors of x and o m , respectively, we
should have
*. = *_ = 0.674 Vy '.' (HO)
. .
If we follow the usual practice and regard the probable error of a
quantity as a measure of the accidental errors of the observations
from which it is directly computed, equation (110) still holds
when the accidental errors of the c's are not negligible; but, as we
shall see, E x is no longer a complete measure of the precision of x
in such cases.
In practice each of the c's must be computed, on theoretical
grounds, from subsidiary observations with the aid of physical
constants that have been previously determined by direct or
indirect measurements. For the sake of brevity the quantities
on which the c's depend will be called correction factors. Since all
of them are subject to accidental errors, the computed c's are
affected by residual errors of indeterminate sign and magnitude.
132 THE THEORY OF MEASUREMENTS [ART. 71
When the probable errors of the correction factors are known the
probable errors of the c's may be computed by the laws of propa-
gation of errors with the aid of the correction formulae by which
the c's are determined.
Equation (102) gives x as a continuous sum of o m and the c's.
Consequently, if we represent the probable errors of the c's by
Ei t E 2 , . . . , E q , respectively, we have by equation (76), article
fifty-eight,
R x 2 = E m * + Ei* + +E q *, (111)
wnere R x is the resultant probable error of x due to the correspond-
ing errors of o m and the c's. To distinguish R x from the probable
error E X) which depends only on the accidental error of o m , we
shall call it the precision measure of x.
Although equation (111) is simple in form, the separate compu-
tation of the E'SJ from the errors of the correction factors on which
they depend, is frequently a tedious process. Moreover several
of the c's may depend on the same determining quantities. Con-
sequently the computation of x and R x is frequently facilitated by
bringing the correction factors into the equation for x explicitly,
rather than allowing them to remain implicit in the c's. Thus,
if a, )8, . . . , p represent the correction factors on which the c's
depend, equation (102) may be put in the form
x = F(o m ,a,0, . . . , P). (112)
Hence, by equation (81), article sixty,
where E a , Ep, etc., are the probable errors of a, ft, etc.
For example, suppose that o m represents the mean of a num-
ber of observations of the distance between two parallel lines
expressed in terms of the divisions of the scale used in making
the measurements. Let t\ represent the mean temperature of the
scale during the observations; L the mean length of the scale
divisions at the standard temperature U, in terms of the chosen
unit; a the coefficient of expansion of the scale; and ft the angle
between the scale and the normal to the lines. Then, if the
individual observations have been corrected to mean temperature
ti before computing the mean observation o m , the best approxima-
ART. 71] DISCUSSION OF COMPLETED OBSERVATIONS 133
tion that we can make to the true distance between the lines is
given by the expression
x = o m L\l]+a(ti - t ) I ,
in which the correction factors L, a, /?, fa, and to appear explicitly ,
as in the general equation (112). A more detailed discussion of
this example will be found in article seventy-three.
If we represent the separate effects of the errors E m , E a , . . . ,
E p on the error R x by D m , D a , D$, . . . , D PJ respectively, we
have
*-*/ D - - S E *-> :.:i ' D > * T P E < m >
and (113) becomes
R* 2 = D m * + D a 2 + Df + - - - + D P 2 . (115)
In some cases the fractional effects
_Drn, _D. . _D,
m ~ x ' a ~ x ' ' ' ' p ~ x
can be more easily computed numerically than the corresponding
D's. When this occurs, the fractional precision measure
is first computed and then R x is determined by the relation
R x = x-P x . (117)
While equations (112) to (117) are apparently more complicated
than (102) and (111), they generally lead to more simple numerical
computations. Moreover the probable errors of some of the
correction factors are frequently so small that they produce no
appreciable effect on R x . When either equation (115) or (116) is
used, such cases are easily recognized because the corresponding
D's or P's are negligible in comparison with D m or P m . Obvi-
ously the same condition applies to the E's in equation (111), but
the numerical computation of either the D's or the P's is generally
more simple than that of the E's in (111) because approximate
values of o m and the correction factors may be used in evaluat-
ing the differential coefficients in (114). The allowable degree of
approximation, the limit of negligibility of the D's, and some other
134 THE THEORY OF MEASUREMENTS [ART. 71
details of the computation will be discussed more extensively
in the next article.
If the true numeric of the measured magnitude is represented
by Xj the final result of a series of direct measurements may be
expressed in the form
X = xR x , (118)
where x is the most probable value that can be assigned to X on
the basis of the given observations, and R x is the precision measure
of x. In practice x may be computed by either equation (102)
or (112), or the arithmetical mean of the individually corrected
observations may be taken, and R x is given by equations (111),
(115), or (117), the choice of methods depending on the nature
of the given data and the preference of the computer.
The exact significance of equation (118) should be carefully
borne in mind, and it should be used only when the implied condi-
tions have been fulfilled. Briefly stated, these conditions are as
follows :
1st. The accidental errors of the observations on which x
depends follow the general law of such errors.
2nd. A careful study of the methods and apparatus used has
been made for the purpose of detecting all sources of constant
or systematic errors and applying the necessary corrections.
3rd. The given value of x is the most probable that can be
computed from the observations after all constant errors, system-
atic errors, and mistakes have been as completely removed as
possible.
4th. The resultant effect of all sources of error, whether acci-
dental errors of observation or residual errors left by the correc-
tions for constant errors, is as likely to be less than R x as greater
than R x .
The expressions in the form X = x E x , used in preceding
chapters, are not violations of the above principles because, in
those cases, we were discussing only the effects of accidental
errors and the observations were assumed to be free from all con-
stant errors and mistakes. Such ideal conditions never occur in
practice. Consequently R x should not be replaced by E x in
expressing the result of actual measurements in the form of equa-
tion (118), unless it can be shown by equation (115), and the given
data that the sum of the squares of the D's corresponding to all
of the correction factors is negligible in comparison with Z) m 2 .
ART. 72] DISCUSSION OF COMPLETED OBSERVATIONS 135
In the latter case E x and R x are identical as may be easily seen
by comparing equations (110), (111), and (115).
72. Precision of Derived Measurements. When a desired
numeric Z is connected with the numerics Xi, X 2 , . . . , X q
of a number of directly measured magnitudes by the relation
XQ = F (Xi, X%, . . . , X q ),
the most probable value that we can assign to X Q is given by the
expression
x = F(x 1 ,xt, . . . , x q ), (119)
where the x's are the most probable values of the X's with corre-
sponding subscripts. Each of the component x's, together with
its precision measure, can be computed by the methods of the pre-
ceding article. The precision measure of X Q may be computed
with the aid of equation (81), article sixty, by replacing the E's in
that equation by the R's with corresponding subscripts.
Sometimes the numerical computations are simplified and the
discussion is clarified by bringing the direct observations and the
correction factors explicitly into the expression for XQ. If o a ,
Ob, . . . , Op are the arithmetical means of the direct observa-
tions, after correction for systematic errors, on which Xi, x z , . . . ,
x q respectively depend, and a, /?, . . . , p are the correction
factors involved in the constant errors of the observations, equa-
tion (119) may be put in the form
x = d (o a , o b , . . . , o p , a, j8, . . . , p). (120)
The function 6 is always determinable when the function F in
(119) is given and the correction formulae for the constant errors
are known.
Representing the precision measure of XQ by R , and adopting
an obvious extension of the notation of the preceding article, we
have, by equation (81),
Introducing the separate effects of the E's,
*-*' ' ' ' = *=l^'
(121) becomes
*' ' ' ' ; '-*- (122)
. (123)
136 THE THEORY OF MEASUREMENTS [ART. 72
The fractional effects of the E's are
P _. . P =5*. P = ^. . P _A?
^ " XQ ' ' p x ' a Z ' p " X Q '
and the fractional precision measure of x is given by the relation
XQ
When the numerical computation of the P's is simpler than that
of the D's, PO is first computed by equation (124) and then RQ
is determined by the relation
#o = z Po. (125)
The expression of the final result of the observations and com-
putations in the form
XQ = XQ RQ
has exactly the same significance with respect to X Q , XQ, and R Q
that (118) has with respect to X, x, and R x . It should not be
used until all of the underlying conditions have been fulfilled as
pointed out in the preceding article. Confusion of the precision
measure R with the probable error E 0) and insufficient rigor in
eliminating constant errors have led many experimenters to an
entirely fictitious idea of the precision of their measurements.
When the correction factors are explicitly expressed in the
reduction formulae, as in equations (112) and (120), the only
difference between the expressions for direct and derived measure-
ments is seen to lie in the greater number of directly observed
quantities, o a , o&, etc., that appear in the latter equation. The
same methods of computation are available in both cases and the
following remarks apply equally well to either of them.
For practical purposes, the precision measure R is computed
to only two significant figures and the corresponding x is carried
out to the place occupied by the second significant figure in R.
The reasons underlying this rule have been fully discussed in
article forty-three, in connection with the probable error, and
need not be repeated here. In computing the numerical value
of the differential coefficients in equations (113), (114), (121), and
(122), the observed components, o m , o a , o&, etc., and the correc-
tion factors, a, , etc., are rounded to three significant figures,
and those that affect the result by less than one per cent are neg-
lected. This degree of approximation will always give R within
ART. 72] DISCUSSION OF COMPLETED OBSERVATIONS 137
one unit in the second significant figure and usually decreases the
labor of computation.
Generally the components o m , o a , o b , etc., represent the arith-
metical means of series of direct observations that have been
corrected for systematic errors. In such cases the corresponding
probable errors E mt E a , Eb, etc., can be computed, by equations
in the form of (110), from the residuals determined by equations
in the form of (103), with the aid of the observations on which
the o's depend. If the observations are sufficiently numerous,
the computation of the .27's.may be simplified by using formulae
depending on the average error in the form
E = 0.845 fl=> (34)
n Vn 1
where [f] is the sum of the residuals without regard to sign and n
is the number of observations. If the observations on which any
of the o's depend are not of equal weight, the general mean should
be used in place of the arithmetical mean and the corresponding
probable errors should be computed by equations (41), (42), or
(44), depending on the circumstances of the observations.
The o's in equation (120) are supposed to represent simultane-
ous values of the directly observed magnitudes. When any of
these quantities are continuous functions of the time, or of any
other independent variables, it frequently happens that only a
single observation can be made on them that is simultaneous
with the other components. In such cases this single observation
must be used in place of the corresponding o in (120), and its
probable error must be determined for use in equation (122).
For the latter purpose, it is sometimes possible to make an auxil-
iary series of observations under the same conditions that pre-
vailed during the simultaneous measurements except that the
independent variables are controlled. The required E may be
assumed to be equal to the probable error of a single observation
in the auxiliary series. Consequently it may be computed by
formulae in the form,
E = 0.674* /W
E = 0.845
n- I
or
[r]
138 THE THEORY OF MEASUREMENTS [ART. 72
where n is the number of auxiliary observations, and the r's are
the corresponding residuals. In some cases this simple expedient
is not available; and approximate values must be assigned to the
E's on theoretical grounds, depending on the nature of the meas-
urements; or more or less extensive experimental investigations
must be undertaken to determine their values more precisely.
Such investigations are so various in character and their utility
depends so much on the skill and ingenuity of the experimenter,
that a detailed general discussion of them would be impossible.
They may be illustrated by the following very common case.
Suppose that one of the components in equation (120) repre-
sents the gradually changing temperature of a bath. In com-
puting x Q we must use the thermometer reading o t taken at the
time the other components are observed. The errors of the fixed
points of the thermometer and its calibration errors enter the
equation among the correction factors a, /?, etc., and do not con-
cern us in the present discussion. In order to determine the
probable error of o t , the temperature of the bath may be caused
to rise uniformly, through a range that includes o t , by passing a
constant current through an electric heating coil, or the bath
may be allowed to cool off gradually by radiation. In either case
the rate of change of temperature should be nearly the same as
prevailed when o t was observed. A series of corresponding obser-
vations of the time T and the temperature t are made under
these conditions, and the empirical relation between T and t is
determined graphically or by the method of least squares. The
probable error of o t may be assumed to be equal to the probable
error of a single observation of t in this series, and may be com-
puted by equation (94), article sixty-four.
Some of the correction factors a, ft, etc., appearing as com-
ponents in equations (112) and (120), represent subsidiary obser-
vations, and some of them represent physical constants. The
subsidiary observations may be treated by the methods outlined
above. When the highest attainable precision is desired, the
physical constants, together with their probable errors, must be
determined by special investigation. In less exact work they
may be taken from tables of physical constants. Such tabular
values seldom correspond exactly to the conditions of the experi-
ments in hand and their probable errors are seldom given.
Generally a considerable range of values is given, and, unless
ART. 72] DISCUSSION OF COMPLETED OBSERVATIONS 139
there is definite reason in the experimental conditions for the
selection of a particular value, the mean of all of them should be
adopted and its probable error placed equal to one-half the range
of the tabular values. The deviations of the tabular values from
the mean are due more to differences in experimental conditions
and in the material treated than to accidental errors. Conse-
quently a probable error calculated from the deviations would
have no significance unless these differences could be taken into
account. The selection of suitable values from tables of physical
constants requires judgment and experience, and the general
statements above should not be blindly followed. In many cases
the original sources of the data must be consulted in order to
determine the values that most nearly satisfy the conditions of
the experiments in hand.
In good practice the conditions of the experiment are usually
so arranged that the D's, in equation (123), corresponding to the
direct observations o a , o&, etc., are all equal. None of the D's
corresponding to correction factors should be greater than this
limit, but it sometimes happens that some of them are much
smaller. Since R is to be computed to only two significant
figures, any single D which is less than one-tenth of the average
of the other D's may be neglected in the computation. If the
sum of any number of D's is less than one-tenth of the average
of the remaining D's they may all be neglected. A somewhat
more rigorous limit of rejection can be developed for use in plan-
ning proposed measurements, but it is scarcely worth while in
the present connection since the correction factors and all other
quantities must be taken as they occurred in the actual measure-
ments, and negligible D's are very easily distinguished by inspec-
tion after a little experience.
After #o has been determined, x may be computed by either
equation (119) or (120). If (119) is used the x's must first be
determined by (102) or (112). Sometimes the computation may
be facilitated by using a modification of (120), in which some of
the correction factors appear explicitly while others are allowed
to remain implicit in the z's to which they apply. Such cases
cannot be treated generally, but must be left to the ingenuity of
the computer. Whatever formula is used, the observed quanti-
ties and the correction factors should be expressed by sufficient
significant figures to give the computed X Q within a few units in
140 THE THEORY OF MEASUREMENTS [ART. 73
the place occupied by the second significant figure of R . Occa-
sionally the total effect of one or more of the correction factors is
less than this limit and may be neglected in the computation. For
f$ W 7?
a single factor, say a, this is the case when a is less than ~
73. Numerical Example. The following illustration repre-
sents a series of measurements taken for the purpose of cali-
brating the interval between the twenty-fifth and seventy-fifth
graduations on a steel scale supposed to be divided in centimeters.
The observations were made with a cathetometer provided with
a brass scale and a vernier reading to one one-thousandth of a
division. One division of the level on this instrument corre-
sponds to an angular deviation of 3 X 10~ 4 radians, and the ad-
justments were all well within this limit. The steel scale was
placed in a vertical position with the aid of a plumb-line, and,
since a deviation of one-half, millimeter per meter could have
been easily detected, the error of this adjustment did not exceed
5 X 10~ 4 radians. Consequently the angle between the two
scales was not greater than 8 X 10~ 4 radians, and it may have
been much smaller than this. The temperature of the scales was
determined by mercury in glass thermometers hanging in loose
contact with them. The probable error of these determinations
was estimated at five-tenths of a degree centigrade, due partly
to looseness of contact and partly to an imperfect knowledge of
the calibration errors of the thermometers.
Twenty independent observations, when tested by the last
two criteria of article sixty-eight, showed no evidence of the pres-
ence of systematic errors or mistakes. Consequently the mean
o m , in terms of cathetometer scale divisions, and its probable
error E m were computed before the removal of constant errors.
The following numerical data represents the results of the obser-
vations and the known calibration constants of the cathetometer.
Mean temperature of the steel scale, T 20 0.5 C.
Mean temperature of the brass scale, ti 21.3 =t 0.5 C.
Mean of twenty observations on the measured
interval in terms of brass scale divisions, o m . . 50.0051 db 0.0015 scale div.
Mean length, at standard temperature, of the
brass scale divisions in the interval used, S. . 0.999853 d= 0.000024 cm.
Standard temperature of brass scale, t 15.0 C.
Coefficient of linear expansion of brass scale, a. (182 12) X 10~ 7 .
Angle between two scales, /3, less than 8 X 10- 4 rad.
ART. 731 DISCUSSION OF COMPLETED OBSERVATIONS 141
The most probable value that can be assigned to the measured
interval is given by the expression
Since ft is a very small angle, -- - may be treated by the approxi-
COS p
mate formulae of Table VII, and the above expression becomes
where
t = fa-to.
The quantity S (1 -f- at) is very nearly equal to unity. Hence,
neglecting small quantities of the second and higher orders, the
correction due to the angle ft is
< 0.000016.
Since this is less than two per cent of the probable error of o m , it is
negligible in comparison with the accidental errors of observation.
Consequently the precision of x is not increased by retaining the
term involving ft, and we may put
x = OmS (1 + at). (a)
The probable error of t Q is zero, because the accidental errors of
the temperature observations, made during the calibration of the
brass scale, are included in the probable errors of S and a com-
puted by the method of article sixty-five. Consequently the
probable error of t is equal to that of fa, and we have
t = 6,3 0.5 C.
In the present case equation (115) is the most convenient for
computing the precision measure ,.R X of x. Only two significant
figures are to be retained in the separate effects computed by
equation (114). Consequently the factor (1 + at) may be taken
equal to unity, and the numerical values of o m and S may be
rounded to three significant figures for the purpose of this com-
putation. Thus, taking o m equal to 50.0, S equal to 1.00, and
the other data as given above, we have
142 THE THEORY OF MEASUREMENTS [ART. 73
D m = -E m = S(l+ at) E m = 1 X E m = 0.0015.
oo m
D,= ~ Q E t =o m (l + at) E,= 50 X E a = 0.0012.
do
D a =~E a = OmStE a = 50 X 6.3 X E a = 0.00038.
da
m =50 X 182 X 10~ 7 X E t = 0.00046.
ot
D m 2 = 225.0 X 10~ 8
A, 2 = 144.0 X 10~ 8
Z> 2 = 14.4 X 10~ 8
A 2 = 21.2 X 10~ 8
[D 2 ] = 404.6 X 10~ 8
Hence, by equation (115),
R x *= [D 2 ] = 404.6 X 10- 8 ,
JB X = V404.6 X 10- 8 = 0.0020.
For the purpose of computing x, it is convenient to put the
given data in the form
Om = 50 (1+0.000102),
S = 1- 0.000147,
at = 0.000115.
Then, by equation (a),
x = 50 (1 + 0.000102) (1 - 0.000147) (1 + 0.000115),
and by formula 7, Table VII,
x = 50 (1 + 0.000102 - 0.000147 + 0.000115)
= 50 (1 + 0.00007)
= 50.0035.
This method of computation, by the use of the approximate
formulae of Table VII, gives x within less than one unit in the last
place held, and is much less laborious than the use of logarithms.
Since the length S of the cathetometer scale divisions is given
in centimeters, the computed values of x and R x are also expressed
in centimeters and our uncertainty regarding the true distance L
between the twenty-fifth and the seventy-fifth graduations of the
steel scale is definitely stated by the expression
L = 50.0035 d= 0.0020 centimeters,
at the temperature
T r = 20.00.5C.
ART. 73] DISCUSSION OF COMPLETED OBSERVATIONS 143
The above discussion shows that the precision of the result
would not have been materially increased by a more accurate
determination of T, fa, and a, since the effects of the errors of
these quantities are small in comparison with that of the errors
of o m and S. The probable error of o m might have been reduced
by making a larger number of observations and taking care to
keep the instrument in adjustment within one-tenth of a level
division or less. But the given value of E m is of the same order
of magnitude as the least count of the vernier used, and, since
each observation represents the difference of two scale readings,
it would not be decreased in proportion to the increased labor of
observation. Moreover, the terms D m and D 8 in the above value
of R x are nearly equal in magnitude, and it would not be worth
while to devote time and labor to the reduction of one of them
unless the other could be reduced in like proportion.
CHAPTER XI.
DISCUSSION OF PROPOSED MEASUREMENTS.
74. Preliminary Considerations. The measurement of a
given quantity may generally be carried out by any one of several
different, and more or less independent, methods. The available
instruments usually differ in type and in functional efficiency. A
choice among methods and instruments should be determined by
the desired precision of the result and the time and labor that it is
worth while to devote to the observations and reductions.
Since the labor of observation and the cost of instruments in-
crease more rapidly than the inverse square of the precision
measure of the attained result, a considerable waste of time and
money is involved in any measurement that is executed with
greater precision than is demanded by the use to which the result
is to be put. On the other hand, if the precision attained is not
sufficient for the purpose in hand, the measurement must be
repeated by a more exact method. Consequently the labor and
expense of the first determination contributes very little to the
final result and the waste is quite as great as in the preceding
case. Sometimes the expense of a second determination is
avoided by using the inexact result of the first, but such a saving
is likely to prove disastrous unless the uncertainty of the adapted
data is duly considered.
In general the greatest economy is attained by so planning
and executing the measurement that the result is given with the
desired precision and neglecting all refinements of method and
apparatus that are not essential to this end. While these con-
siderations have greater weight in connection with measurements
carried out for practical purposes they should never be neglected
in planning investigations undertaken primarily for the advance-
ment of science. In the former case the cost of necessary measure-
ments may represent an appreciable fraction of the expense of
a proposed engineering enterprise and must be taken into account
in preparing estimates. In the latter case there is no excuse for
burdening the limited funds available for research with the expense
144
ART. 75] DISCUSSION OF PROPOSED MEASUREMENTS 145
of ill-contrived and haphazard measurements. The precision
requirements may be, and indeed usually are, quite different in
the two cases, 'but the same process of arriving at suitable methods
applies to both.
75. The General Problem. In its most general form the
problem may be stated as follows : Required the magnitude of a
quantity X within the limits R, X being a function of several
directly measured quantities X\, X 2 , etc. ; within what limits must
we determine the value of each of the components X\, X z , etc.?
In discussing this problem, all sources of error both constant and
accidental must be taken into account. For this purpose the
various methods available for the measurement of the several
components are considered with regard to the labor of execution
and the magnitude of the errors involved as well as with regard to
the facility and accuracy with which constant errors can be removed.
After such a study, certain definite methods are adopted pro-
visionally, and examined to determine whether or not the re-
quired precision in the final result can be attained by their use.
As the first step in this process, the function that gives the rela-
tion between X and the components, Xi, X 2 , etc., is written out
in its most complete form with all correction factors explicitly
represented. Thus, as in article seventy-two, the most probable
value of the quantity X may be expressed in the form
X Q = 0(o a ,o bj . . . , p ,a,/3, . . . , p), (120)
where the o's represent observed values of X\ t X 2 , etc., and a, /3,
. . . , p, represent the factors on which the corrections for con-
stant errors depend as pointed out in connection with equation
(112), article seventy-one.
The form of the function 0, and the nature and magnitude of
the correction factors appearing in it, will depend on the nature
of the proposed methods of measurement. Since all detectable
constant errors are explicitly represented by suitable correction
factors, all of the quantities appearing in the function may be
treated as directly measured components subject to accidental
errors only. Hence the problem reduces to the determination
of the probable errors within which each of the components must
be determined in order that the computed value of XQ may come
out with a precision measure equal to the given magnitude R Q .
If all of the components can be determined within the limits set
146 THE THEORY OF MEASUREMENTS [ART. 76
by the probable errors thus found, without exceeding the limits
of time and expense imposed by the preliminary considerations,
the provisionally adopted methods are adequate for the purpose
in hand and the measurements may be carried out with con-
fidence that the final result will be precise within the required
limits. When one or more of the components cannot be deter-
mined within the limits thus set without undue labor or expense,
the proposed methods must be modified in such a manner that the
necessary measurements will be feasible.
76. The Primary Condition. The present problem is, to
some extent, the inverse of that treated in articles seventy-one
and seventy-two. In the latter case the given data represented
the results of completed series of observations on the several
component quantities appearing in the function 0, together with
their respective probable errors. The purpose of the analysis was
the determination of the most probable value XQ that could be
assigned to the measured magnitude and the precision measure
of the result. In the present case approximate values of x and
the components in 6 are given, and the object of the analysis is
the determination of the probable errors within which each of the
components must be measured in order that the value of XQ,
computed from the completed observations, may come out with a
precision measure equal to a given magnitude R .
If D , Db, . . . , D p , D a) Dp, . . . , D p represent the separate
effects of the probable errors E a , Eb, . . . , E p , E a , Ep, . . . ,
E p of the components o aj o b , . . . , o p , a, /3, . . . , p, respec-
tively, we have, as in article seventy-two,
and the primary condition imposed on these quantities is given by
the relation
#o 2 = Da 2 + ZV + - + ZV + ZV + iy + - - . +D P 2 . (123)
The precision measure R and approximate values of the com-
ponents are given by the conditions of the problem and the pro-
posed methods of measurement. The E's, and hence also the
D's, are the unknown quantities to be determined. Conse-
quently there are as many unknowns in equation (123) as there
are different components in the function 0. Obviously the problem
is indeterminate unless some further conditions can be imposed
ART. 77] DISCUSSION OF PROPOSED MEASUREMENTS 147
on the D's; for otherwise it would be possible to assign an infinite
number of different values to each of the D's which, by proper
selection and combination, could be made to satisfy the primary
condition (123).
77. The Principle of Equal Effects. An ideal condition to
impose on the D's would specify that they should be so determined
that the required precision in the final result X Q would be attained
with the least possible expense for labor and apparatus. Un-
fortunately this condition cannot be put into exact mathematical
form since there is no exact general relation between the difficulty
and the precision of measurements. However, it is easy to see
that the condition is approximately fulfilled when the measure-
ments are so made that the D's are all equal to the same magnitude.
For, the probable error of any component is inversely proportional
to the square root of the number of observations on which it
depends and the expense of a measurement increases directly
with the number of observations. Consequently the expense
W a of the component o a is approximately proportional to 7^-5 or,
&a
n/j 1
since r is constant, to -^ 9 . Similar relations hold for the other
do a D a 2
components. Hence, as a first approximation, we may assume
that
A2 A2 A2 A2
where W is the total expense of the determination of x , and A is
a constant. By the usual method of finding the minimum value
of a function of conditioned quantities, the least value of W con-
sistent with equation (123) occurs when the D's satisfy (123) and
also fulfill the relations
_
dD a "* ^ dD a =
ML + ***?- o
dD b ^ * dD b -
=
SD * ^ dD
148 THE THEORY OF MEASUREMENTS [ART. 77
where K is a constant. Introducing the expressions for R<? and
W in terms of the D's, differentiating, and reducing, we have
and by equation (123)
where AT is the number of D's in (123) or the equal number of
components in the function 6. Consequently equation (123) is
fulfilled and the condition of minimum expense is approximately
satisfied when the components are so determined that the separate
effects of their probable errors satisfy the relation
D a = D b = - . - = D a = Dp = = -. (127)
Equation (127) is the mathematical expression of the principle
of equal effects. It does not always express an exact solution of
the problem, since A is seldom strictly constant; but it is the
best approximation that we can adopt for the preliminary com-
putation of the D's and E's. The results thus obtained will
usually require some adjustment among themselves before they
will satisfy both the preliminary considerations and the primary
condition (123). We shall see that the necessary adjustment is
never very great; and, in fact, that a marked departure from the
condition of equal effects is never possible when equation (123) is
satisfied.
Combining equations (122) and (127), we find
E ^ . ^ - E ^ . ^
VAT " de ' a VN ' de '
da
w Ro i .
* = ~~7= ' ~^7T >
VN <&' VN y,
do b 5/3
(128)
Hence, if the final measurements are so executed that the probable
errors of the several components are equal to the corresponding
values given by equations (128), the final result XQ, computed by
equation (120), will come out with a precision measure equal to
ART. 78] DISCUSSION OF PROPOSED MEASUREMENTS 149
the specified R Q , and the condition of equal effects (127) will be
fulfilled.
In computing the E's by equation (128), R Q is taken equal to
the given precision measure of X Q and N is placed equal to the
J/3
number of components in the function 0. The derivatives T
do a
etc., are evaluated with the aid of approximate values of the
components obtained by a preliminary trial of the proposed
methods or by computation, on theoretical grounds, from an
approximate value of XQ and a knowledge of the conditions under
which the measurements are to be made. Since only two sig-
nificant figures are required in any of the E's, the adopted values
of the components may be in error by several per cent, without
affecting the significance of the results. Moreover, any number
of components, whose combined effect on any derivative is less
than five per cent, may be entirely neglected in computing that
derivative. Consequently the function frequently may be sim-
plified very much for the purpose of computing the derivatives and
this simplification may take different forms in the case of differ-
ent derivatives. No more than three significant figures should be
retained at any step of the process and sometimes the required pre-
cision can be attained with the approximate formulae of Table VII.
Since equation (127) is an approximation, the E's derived from
equations (128) are to be regarded as provisional limits for the
corresponding components. If all of them are attainable, i.e., if
all of the components can be determined within the provisional
limits, without exceeding the limit of expense set by the prelim-
inary considerations, the solution of the problem is complete and
the proposed methods are suitable for the work in hand.
78. Adjusted Effects. Generally some of the E's given by
(128) will be unattainable in practice while others will be larger
than a limit that can be easily reached. In other words, it will
be found that the labor involved in determining some of the
components within the provisional limit is prohibitive while
other components can be determined with more than the pro-
visional precision without undue labor. In such a case the pro-
visional limits are modified by increasing the E's corresponding
to the more difficult determinations and decreasing the E's that
correspond to the more easily determinable components in such a
way that the combined effects satisfy the condition (123).
150 THE THEORY OF MEASUREMENTS [ART. 78
The maximum allowable increase in a single E is by the factor
. For, taking E a for illustration,
B0 a
and consequently
Hence (123) cannot be satisfied unless all of the rest of the D's
are negligibly small. For example, if there are nine components,
VN is equal to three. Consequently no one of the E's can be
increased to more than three times the value given by the condi-
tion of equal effects if (123) is to be satisfied. When, as is fre-
quently the case, the number of components is less than nine, or
when more than one of the E's is to be increased, the limit of
allowable adjustment is much less than the above. The extent
to which any number of E's may be increased is also limited
by the difficulty, or impossibility, of reducing the effects of the
remaining E's to the negligible limit.
If the probable errors given by equations (128) can be modified,
to such an extent that the corresponding measurements become
feasible, without violating the condition (123), the proposed
methods are suitable for the final determination of XQ. Other-
wise they must be so modified that they satisfy the conditions of
the problem or different methods may be adopted provisionally
and tested for availability as above.
Sometimes it will be found that the proposed methods are
capable of greater precision than is demanded by equations (128).
In such cases the expense of the measurements may be reduced
without exceeding the given precision measure of XQ by using less
precise methods. But such methods should never be finally
adopted until their feasibility has been tested by the process out-
lined above.
A discussion on the foregoing lines not only determines the
practicability of the proposed methods, but also serves as a guide
in determining the relative care with which the various parts of
the work should be carried out. For, if the final result is to come
out with a precision measure R Q , it is obvious that all adjustments
and measurements must be so executed that each of the com-
ART. 79] DISCUSSION OF PROPOSED MEASUREMENTS 151
ponents is determined within the limits set by equations (128),
or by the adjusted E's that satisfy (123).
79. Negligible Effects. In the preceding article it was
pointed out that the availableness of proposed methods of meas-
urement frequently depends on the possibility of so adjusting the
E's given by equations (128) that they are all attainable and
at the same time satisfy the primary condition (123). Generally
this cannot be accomplished unless some of the E's can be reduced
in magnitude to such an extent that their effect on the precision
measure R is negligible.
On account of the meaning of the precision measure, and the
fact that it is expressed by only two significant figures, it is obvi-
ous that any D is negligible when its contribution to the value of
73
#0 is less than y^. Thus, if Ri is the value of the right-hand
member of equation (123), when D a is omitted, D a is negligible
provided
or
0.
Squaring gives
0.81 Bo 2 < #i 2 ,
and by definition
R<? - RS = D*.
Consequently
0.81 #o 2 < #o 2 - D*,
and
Z> a 2 <0.19# 2 ,
or
D a < 0.436 #o.
Hence, if D a is less than 0.436 # , it will contribute lees than ten
per cent of the value of R Q . Since the true error of x is as likely
to be greater than R as it is to be less than R Q , a change of ten
per cent in the value of R Q can have no practical importance.
Consequently D a is negligible when it satisfies the above condi-
tion. However, the constant 0.436 is somewhat awkward to
handle, and if D a is very nearly equal to the limit 0.436 RQ, the
propriety of omitting it is doubtful. These difficulties may be
avoided by adopting the smaller and more easily calculated limit
of rejection given by the condition
D = R Q . (129)
152 THE THEORY OF MEASUREMENTS [ART. 79
This limit corresponds to a change of about six per cent in the
value of Ro given by equation (123), and is obviously safe for all
practical purposes. Since the above reasoning is independent of
the particular D chosen, the condition (129) is perfectly general
and applies to any one of the D's in equation (123).
When two or more of the D's satisfy (129) independently, any
one of them may be neglected, but all of them cannot be neg-
lected without further investigation for otherwise the change in
Ro might exceed ten per cent. This would always happen if all
T~)
of the D's considered were very nearly equal to the limit ~^-
o
However, by analogy with the above argument, it is obvious that
any q of the D's are simultaneously negligible when
+ D 2 2 + . . . + D 3 2 == Jflo, (130)
where the numerical subscripts 1, 2, . . . , q are used in place
of the literal subscripts occurring in equation (123) in order to
render the condition (130) entirely general. Thus DI may corre-
spond to any one of the D's in (123), D 2 to any other one, etc.
By applying the principle of equal effects, the condition (130)
may be reduced to the simple form
D, = D 2 = ... = D q = - ^ (131)
3 Vg
If some of the D's in (131) can be easily reduced below the limit
p
j=. , the others may exceed that limit somewhat without violating
3 V q
the condition (130). However, equation (131) generally gives the
best practical limit for the simultaneous rejection of a number of
D's, and all departures from it should be carefully checked by (130).
To illustrate the practical application of the foregoing discussion,
suppose that the practicability of certain proposed methods of
measurement is to be tested by the principle of equal effects
developed in article seventy-seven. Let there be N components
in the function 0, and suppose that q of them, represented by
ai, 2, . . . , a q , can be easily determined with greater precision
than is demanded by equations (128), while the measurement
of the remaining N q components within the^limits thus set
would be very difficult. Obviously some adjustment of the E's
given by (128) is desirable in order that the labor involved in the
various parts of the measurement may be more evenly balanced.
ART. 79] DISCUSSION OF PROPOSED MEASUREMENTS 153
The greatest possible increase in the E's corresponding to the
N q difficult components will be allowable when the E's of the
q easy components can be reduced to the negligible limit. To
determine the necessary limits, R is taken equal to the given
precision measure of XQ, and the negligible D's corresponding to
the q easy components are determined by equation (131). Then
by equations (122), the corresponding E's will be negligible when
E!=Z -^
3 Vq
1 1
If
dai
E 2 = -^L<
1
w
(132)
A r
J_^
6^
da q
If these limits can be attained with as little difficulty as the pre-
viously determined E's of the N q remaining components, the
corresponding D's may be omitted from equation (123) during
the further discussion of precision limits.
Since q of the D's have disappeared, the others may be some-
what increased and still satisfy the primary condition (123).
The corresponding new limits for the E's of the difficult components
may be obtained from equations (127) and (128) by replacing
N by N q. If these new limits together with the negligible
limits given by equations (132) can all be attained, without
exceeding the expense set by the preliminary considerations, the
proposed methods may be considered suitable for the final deter-
mination of XQ with the desired precision. Otherwise new methods
must be devised and investigated as above.
Equations (132) may also be used to determine the extent to
which mathematical constants should be carried out during the
computations. For this purpose the components i, 0% , ,
or part of them, represent the mathematical constants appearing
in the function 8. The corresponding E's, determined by equa-
tions (132), give the allowable limits of rejection in rounding the
numerical values of the constants for the purpose of simplifying
154
THE THEORY OF MEASUREMENTS [ART. 79
the computations. Thus, suppose that the volume of a right
circular cylinder of length L and radius a is to be computed
within one-tenth of one per cent, how many figures should be
retained in the constant TT? In this case
n / \ 17 9 T
(Oa , , , ) = y = *<,
RQ = 0.001 V = 0.001 7ra 2 L,
60 6V
= 0.00105.
0.001 7T
If TT is taken equal to 3.142 the error due to rounding is 0.00041 .
Since this is less than the negligible limit E r , four significant
figures in TT are sufficient for the purpose in hand.
It sometimes happens that the total effect of one or more of the
components in the function 0, on the computed value of x , is
negligible in comparison with RQ. This will obviously be the case
when
60 RQ
a^ a ^ IF'
for a single component a or when
KM \ 2 -L-/ de
z~~ a i) + (^~~
dai I \da2
da
for q components. Thus, on the principle of equal effects, the
components i, <* 2 , , <* 3 will be simultaneously negligible
when they satisfy the conditions
1 RQ 1
* 155 i
(133)
RQ 1
daz
7"> 1
\7^'~d0~
Such cases frequently arise in connection with the components
that represent correction factors.
ART. 80] DISCUSSION OF PROPOSED MEASUREMENTS 155
80. Treatment of Special Functions. During the foregoing
argument, it has been assumed that the function 6 in equation (120)
is expressed in the most general form consistent with the pro-
posed methods of measurement. Such an expression involves the
explicit representation of all directly measured quantities, and
all possible correction factors. Part of the latter class of com-
ponents represent departures of the proposed methods from the
theoretical conditions underlying them, and others depend upon
inaccuracies in the adjustment of instruments. In practice it
frequently happens that the general function is very compli-
cated, and consequently that the direct discussion of precision
as above is a very tedious process. Under these conditions it is
desirable to modify the form of the function in such a manner as
to facilitate the discussion.
Sometimes the general function 9 can be broken up into a series
of independent functions or expressed as a continuous product
of such functions. Thus, it may be possible to express 6 in the
form
XQ = 6 (o a , o b , . . ., a, |8, . . .)
= /i(ai,a 2 , . . . )/ 2 (&i,& 2 , . . . )/ 3 (ci,c 2 , . . .
or in the form
XQ = d (O a , O b) .
(134)
(135)
= /i(ai,a 2 , . . . ) X/2(&i,&2, . ) X/ 3 (ci,c 2 , . . .
X ... X / (mi, m 2) . . . ),
where the a's, &'s, . . . , and m's represent the same components,
o a , o b , . . . , a, 0, . . . , that appear in 6 by a new and more
general notation. The functions /i, / 2 , . . . , f n may take any
form consistent with the problem in hand, but the precision dis-
cussion will not be much facilitated unless they are independent
in the sense that no two of them contain the same or mutually
dependent variables. Sometimes the latter condition is imprac-
ticable and it becomes necessary to include the same component
in two or more of the functions. Under such conditions the expan-
sion has no advantage over the general expression for 0, unless
the effect of the errors of each of the common components can
be rendered negligible in all but one of the functions. It is
scarcely necessary to point out that equations (134) and (135)
represent different problems, and that if it were possible to expand
156 THE THEORY OF MEASUREMENTS [ART. 80
the same function in both ways, the component functions /i,
/2, , fn would be different in the two cases.
For the sake of convenience let
/I (Oi, 2, ) = 2
/2 (6l, 6 2 , . . . ) = ^2
jfn (Wi,m 2 ,. . . ) = 2
Then equation (134) may be written in the form
X = Zi 2 2! 3 . . . d= 2, (137)
and (135) may be put in the form
x = z l Xz z Xz 3 X . . . Xz n . (138)
First consider the case in which the function representing the
proposed methods of measurement has been put in the form of
(137). Since the precision measure follows the same laws of
propagation as the probable error, the discussion given in article
fifty-eight leads to the relation
# 2 = 7^2 + # 2 2 + R f + _ m + Rn 2 } ( 139)
where RQ is the precision measure of x , and each of the other R's
represents the precision measure of the z with corresponding sub-
script. Hence, by the principle of equal effects, provisional
values of the R's may be obtained from the relation
R, = R 2 = R, = . . . = R n = A . ( 140 )
The R's having been determined by (140), the corresponding
probable errors of the a's, 6's, etc., may be computed by the
methods of the preceding articles with the aid of equations (136).
If the provisional limits of precision thus found are not all attain-
able with approximately equal facility, the conditions of the
problem may be better satisfied by moderately adjusted relative
values of the probable errors as pointed out in article seventy-
eight. Obviously the adjusted values must satisfy equation (139)
if the value of x computed by (137) is to come out with a pre-
cision measure equal to the given R .
When the function representing the proposed methods can be
put in the form of (138) the computation is facilitated by intro-
ducing the fractional errors
P = ; Pl = ! ; P 2 = f2;...; P n = f" (141)
XQ Zi Zz Z n
ART. 81] DISCUSSION OF PROPOSED MEASUREMENTS 157
For, by the argument underlying equation (83), article sixty-two,
Po 2 = Pi 2 + P 2 2 + Pa 2 + . . . + P 2 , (142)
and, by the principle of equal effects, provisional values of the
P's are given by the relation
Pi = P 2 = P 3 = . . . = P = *=. (143)
Vn
Since RQ and approximate values of the components are given,
PO can be computed with sufficient accuracy with the aid of
(138) and the first of (141). Consequently provisional fractional
limits for the components can be determined by (143), and the
corresponding precision measures by the last n of equations (141).
Beyond this point the problem is identical with the preceding
case, except that the adjusted limits of precision must satisfy
(142) in place of (139).
The methods developed in the preceding articles are entirely
general and applicable to any form of the function 6, but they
frequently lead to complicated computations. In the present
article we have seen how the discussion can be simplified when the
function can be put in either of the particular forms represented
by (134) and (135). Many of the problems met with in practice
cannot be put in either of these special forms, but it frequently
happens that the treatment of the functions representing them
can be simplified by a suitable modification or combination of the
above general and particular methods. The general ideas under-
lying all discussions of the necessary precision of components
have been discussed above with sufficient fullness to show their
nature and significance. Their application to particular prob-
lems must be left to the ingenuity of the observer and computer.
81. Numerical Example. As an illustration of the fore-
going methods, suppose that the electromotive force of a battery
is to be determined, and that the precision measure of the result
is required to satisfy the condition
R = 0.0012 volts, (i)
T->
within the limits T?!>i- e -> #o must lie between 0.0011 and
=b 0.0013 volt. Preliminary considerations demand that the
expense of the work shall be as low as is consistent with the
required precision.
158
THE THEORY OF MEASUREMENTS [ART. 81
The given conditions are most likely to be fulfilled by some
form of potentiometer method. Suppose that the arrangement
of apparatus illustrated in Fig. 10 is adopted provisionally; and,
to simplify the discussion, suppose that the various parts of the
apparatus are so well insulated that leakage currents need not
be considered. The generality of the problem is not appreciably
affected by the latter assumption since the specified condition
can be easily satisfied in practice within negligible limits. With
what precision must the several components and correction
factors be determined in order that equation (i) may be satisfied?
-T&Z
FIG. 10.
L e t V = e.m.f. of tested battery BI,
Et = e.m.f. of Clark cell B 2 at time of observation,
t = temperature of Clark cell at time of observation,
Ri = resistance between 1 and 2,
Rz = resistance between 1 and 3,
/ = current in circuit 1, 2, 3, B 3 , 1 when the key K is open,
5i = algebraic sum of thermo e.m.f.'s in the circuit 1, 2, 6,
G, 1 when K is closed to 6,
2 = algebraic sum of thermo e.m.f. 's in the circuit 1, 3, a,
G, 1 when K is closed to a,
Ei5 e.m.f. of Clark cell at temperature 15 C.,
a. = mean temperature coefficient of Clark cell in the
neighborhood of 20 C.
ART. 81] DISCUSSION OF PROPOSED MEASUREMENTS 159
When the sliding contacts 2 and 3 are so adjusted that the
galvanometer G shows no deflection on closing the key K to
either a or 6,
RI RZ
Consequently
F = (^+6 2 )|- 1 -5 1 . (ii)
-fi/2
But
(in)
Hence
F = -B 16 !l-a-15)jf- 1 + 2 f- 1 -8 1 . (iv)
KZ n>z
The resistances RI and # 2 are functions of the temperature; but,
since they represent simultaneous adjustments with the cells BI
p
and Bz and are composed of the same coils, the ratio ~ is inde-
KZ
pendent of the temperature. Thus, if R t ' and R t " represent the
resistances of the used coils at t C., and ft is their temperature
coefficient,
RS Ri(l+ fit) Ri
whatever the temperature t at which the comparison is made.
This advantage is due to the particular method of connection and
adjustment adopted, and is by no means common to all forms of
the potentiometer method.
Under the conditions specified above, equation (iv) may be
adopted as the complete expression for the discussion of precision.
It corresponds to equation (120) in the general treatment of the
problem. Suppose that the following approximate values of the
components, which are sufficiently close for the determination of
the capabilities of the method, have been obtained from the
normal constants of the Clark cell and a preliminary adjustment
of the apparatus or by computation from a known approximate
value of V:
#15 = 1.434 volts; a = 0.00086;
t = 20 C.; Ri = 1000 ohms;
R 2 = 1310 ohms; V = 1.1 volts.
The thermoelectromotive forces 5i and 5 2 are to some extent
due to inhomogeneity of the wires used in the construction of
the instruments and connections. For the most part, however,
(v)
160 THE THEORY OF MEASUREMENTS [ART. 81
they arise from the junctions of dissimilar metals in the circuits
considered. Suppose that the resistances R\ and #2 are made of
manganin, the key K of brass, and that the copper used in the
galvanometer coil and the connecting wires is thermoelectrically
different. Both 5i and 5 2 would represent the resultant action
of at least six thermo-elements in series. While these effects can-
not be accurately specified in advance, their combined action
would not be likely to be greater than twenty-five microvolts per
degree difference in temperature between the various parts of the
apparatus, and it might be much less than this. Obviously 5i
and 6 2 are both equal to zero when the temperature of the appa-
ratus is uniform throughout.
By equations (133), article seventy-nine, the correction terms
depending on thermoelectric forces will be negligible in compar-
ison with the given precision measure R , when 5i and 62 satisfy
the conditions
. 1 #o 1 , - 1 flo 1
' l *3'vT5E ^s'vTE'
ddi dd 2
In the present case
Ro = 0.0012 volt; q = 2;
dV . dV R l
sE*-- 1 ' and srsr
Consequently the above conditions become
- 5^i? . _L _ 0.00028 volt = 280 microvolts,
3 v 2 1
_L - 0.00037 volt = 370 microvolts.
0.76
From the above discussion of the possible magnitude of the thermo-
electromotive forces in the circuits considered, it is obvious that
these limits correspond to temperature differences of approxi-
mately ten degrees between the various parts of the apparatus.
Since the temperature of the apparatus can be easily maintained
uniform within five degrees, the last two terms in equation (iv)
are negligible within the limits of precision set in the present
problem. Hence, for the determination of the required precision
of the remaining components, the functional relation (iv) may be
taken in the form
(vi)
ART. 81] DISCUSSION OF PROPOSED MEASUREMENTS 161
By equation (123), article seventy-six, the primary condition
for determining the necessary precision of the components is
R<? = 144 X 10~ 8 = Z>! 2 + > 2 2 + D 3 2 + > 4 2 + D<?, (vii)
where dV 67 67
67
(viii)
and EI, EZ, E 3 , E^ E$ are the required probable errors of EI$, a, t,
Ri, and Rz, respectively.
For the preliminary determination of the jE"s by the principle
of equal effects, equation (127), article seventy-seven, becomes
= 0.00054. (ix)
VN V5
Neglecting all factors that do not affect the differential coefficients
by more than one unit in the second significant figure and adopt-
ing the approximate values of the components given in (v),
67 R! 1000 n _
j- = p- = T^ = 0.76,
d-Cns /t2 lolU
=- E 15 a = - - 0.00094,
it/2
= Eu = 0.0011,
it 2
(x)
Hence, by combining (viii) and (ix), or directly from equations
(128), article seventy-seven,
, 0.00054
(xi)
E z
0.76
0.00054
ZEI V7.V7UV/I J. VWftVj
n nnnoQS
v*y
(b)
(c)
(d)
(e)
5.5
0.00054
= 0.57 C.
= =b 0.49 ohm
= db 0.65 ohm.
0.00094
0.00054
- o.oon
0.00054
0.00083
162 THE THEORY OF MEASUREMENTS [ART. 81
In practice the attainableness of these limits might be deter-
mined experimentally; but in the present case, as in most practical
problems, general considerations based on theory and previous
experience lead to equally trustworthy results. In the first place,
it is obvious that the temperature of the Clark cell can be easily
determined closer than 0.6 C. Consequently the limit (c) is easily
attainable and might possibly be reduced to a negligible quantity.
The constants of the normal Clark cell are known well within
the limits (a) and (b). But it requires very careful treatment of
the cell to keep Ei 6 constant within the limit (a), and new cells,
unless they are set up with great care and skill, are likely to vary
among themselves and from the normal cell by more than 0.0007
volt. Consequently the limit (a) is somewhat smaller than is
desirable in practical work of the precision considered in the
present problem. On the other hand, the limit (b) is very rarely
exceeded by either old or new cells unless they are very care-
lessly constructed and handled. Hence E 2 could probably be
reduced to the negligible limit.
With a suitable galvanometer, the nominal values of the resist-
ances Ri and R% can be easily adjusted within the limits (d) and
(e). But EI and E 5 must be considered practically as the pre-
cision measures of R i and R 2 . They include the calibration
errors of the resistances, the errors due to leakage between the
terminals of individual coils, and the errors due to nonuniformity
of temperature as well as the errors of setting of the contacts 2
and 3, Fig. 10. The resultant of these errors can be reduced
below the limits (d) and (e), but in the present case it would be
convenient to have somewhat larger limits in order to reduce the
expense of construction and calibration.
Hence, while all of the E's given by equations (xi) are within
attainable limits, the preliminary consideration of minimum
expense would be more likely to be fulfilled if the limits (a),
(d), and (e) were somewhat larger. Obviously the magnitude of
these limits can be increased without violating the primary con-
dition (vii) provided a corresponding decrease in the magnitudes
of the limits (b) and (c) is possible.
By equation (131), article seventy-nine, the separate effects D 2
and DZ will be simultaneously negligible if
n n 1 #o 1 0.0012
1/2 = DZ = 7^ = ~ ;= ^
3 Vq 3 V2
ABT.SI] DISCUSSION OF PROPOSED MEASUREMENTS 163
Hence, by equations (132), the errors of a and t will be negligible
when 0.00028
E 2 = ^~ =; 0.000051, (b')
and
00028
Ets mm ^o-3oc. (C ')
Since these limits can be reached with much greater ease than the
limits (a), (d), and (e), they may be adopted as final specifica-
tions and the corresponding Z)'s may be omitted during the deter-
mination of new limits for the components E 15) R 1} and R%.
Under these conditions, equation (ix) becomes
Hence the largest allowable limits for the errors of EM, Ri, and
RZ are OOOfiQ
~ = 0.00091 volt, (a')
= .63 ohm, (d')
While these limits cannot be quite so easily attained as (b') and
(c'), they cannot be increased without violating the primary con-
dition (vii). Consequently they satisfy the condition of minimum
expense, so far as the proposed method is concerned, and may be
adopted as final specifications.
The fractional errors corresponding to the specified precision
measure of V and the above limiting errors of the components
Po = Y = 0.0011 = 0.11%,
Pi = ~ = db 0,00063 = 0.063%,
P 2 = ^ = 0.059 = =t 5.9%,
P 3 = y = 0.015 = db 1.5%,
P 4 = ~ = =fc 0.00063 = d= 0.063%,
P 5 = f- 5 = 0.00063 = 0.063%.
164 THE THEORY OF MEASUREMENTS [ART. 81
Consequently in order to obtain a value of V that is exact within
0.11 per cent by the proposed method, a must be determined
within 5.9 per cent, t within 1.5 per cent, and E.-&, Ri, and R 2) each
within 0.063 per cent. These limits are all attainable in practice
under suitable conditions, as pointed out above. Hence the pro-
posed method is practicable.
If the final measurements are so devised and executed that the
above conditions are fulfilled, the precision of the result computed
from them will be within the specified limits and the expense of
the work will be reduced to the lowest limit compatible with the
proposed method. The desired result might be obtained at less
expense by some other method, but a decision on this point can
be reached only by comparing the precision requirements and
practicability of various methods with the aid of analyses similar
to the above.
CHAPTER XII.
BEST MAGNITUDES FOR COMPONENTS.
82. Statement of the Problem. The precision of a derived
quantity depends on the relative magnitudes and precision of the
components from which it is computed, as explained in Chapter
VIII. Thus, if the derived quantity X Q is given in terms of the
components x\, x^ . . . , x q by the expression
x = F (xi, x 2 , . . . , x g ), (144)
the probable error of X Q is given by the expression
E Q * = SSEJ + S 2 2 E 2 Z + + S q *E q 2 , (145)
where the E's represent the probable errors of the x's with corre-
sponding subscripts, and
AF AF AF
*-& *-&'< (146)
The error E, corresponding to any directly measured com-
ponent, is generally, but not always, independent of the absolute
magnitude of that component so long as the measurements are
made by the same method and apparatus. For example: the
probable error of a single measurement with a micrometer caliper,
graduated to 0.01 millimeter, is approximately equal to 0.004
millimeter, whatever the magnitude of the object measured so
long as it is within the range of the instrument. Hence, when
the methods and instruments to be used in measuring each of
the components are known in advance, the probable errors EI,
E 2 , etc., can be determined, at least approximately, by preliminary
measurements on quantities of the same kind as the components
but of any convenient magnitude. Under these conditions the
E's on the right-hand side of equation (145) may be treated as
known constants, and, since the S's are expressible in terms of
Xi, x z , etc., by equations (146), the value of E corresponding to
the given methods cannot be changed without a simultaneous
change in the relative or absolute magnitudes of the components.
165
166 THE THEORY OF MEASUREMENTS [ART. 82
Since equation (144) must always be fulfilled, and since the
value of XQ is usually fixed by the conditions of the problem, a
change in the magnitudes of the re's is not always possible. But
it frequently happens that the form of the function F is such that
the relative magnitudes of the components can be changed through
somewhat wide limits and still satisfy equation (144). Thus, if
a cylinder is to have a specified volume, it may be made long and
thin, or short and thick, and have the same volume in either case.
Consequently it is sometimes possible to select magnitudes for
the components that will give a minimum value of E and at the
same time satisfy equation (144).
The problem before us may be briefly stated as follows : Having
given definite methods and apparatus for the measurement of the
components of a derived quantity reo, what magnitudes of the
components will give a minimum value to the probable error EQ of
XQ and at the same time satisfy the functional relation (144)?
It can be easily seen that a practical solution of this problem
is not always possible. In the first place the form of the function
F may be such as to admit of but a single system of magnitudes
of the components, and consequently the value of EQ is definitely
fixed by equation (145). In some cases there are no real values
of the re's that will satisfy both (144) and the conditions for a
minimum of EQ. When values can be found that satisfy the
mathematical conditions they are not always attainable in prac-
tice. Finally the probable errors Ei, E 2 , etc., may not be inde-
pendent of the magnitudes of the corresponding components or
it may be impossible to determine them in advance of the final
measurements.
When the E's are not independent of the re's it sometimes
happens that the fractional errors
Pi = ? ; p * = ? '> p * = ? (147)
3/1 it/2 Xq
are constant and determinable in advance. In such cases the
problem may be solvable by putting (145) in the equivalent form
Ef = SfPfy? + SfPfxf +!>+ S q *P q *x q *, (148)
expressing the S's in terms of the components by equations (146),
and determining the values of the re's that will render (148) a
minimum subject to the condition (144).
ART. 83] BEST MAGNITUDES FOR COMPONENTS 167
When a practicable solution of the problem is possible, it is
obvious that the results thus obtained are the best magnitudes
that can be assigned to the components, and that they should
be adopted as nearly as possible in carrying out the final measure-
ments from which X Q is to be computed.
83. General Solutions. The general conditions for a mini-
mum or a maximum value of E Q 2 , when XQ is treated as a constant
and the variables are required to satisfy the relation (144), but
are otherwise independent, are
dF
^ A = U,
0)
where K is an arbitrary constant. By introducing the expressions
(145) and (146), transposing and dividing by two, equations (i)
become
Sl gtf 1 . + S ,g^ + ...
o O&1 ET 2 _j_ O 0O2 pi 2 i
1 dx 2 2 ^2
(149)
When the S's have been replaced by x's with the aid of equa-
tions (146), the q equations (149), together withj(144), are theoreti-
cally sufficient for the determination of all of the q + 1 unknown
quantities Xi, x 2 , . . . , x q , and K. However, in some cases a
practicable solution is not possible, and in others the components
or their ratios come out as the roots of equations of the second
or higher degree. The zero, infinite, and imaginary roots of these
equations have no practical significance in the present discussion
and need not be considered. Some of the real roots correspond to
a maximum, some to a minimum, and others to neither a maximum
nor a minimum value of E Z . In most cases the roots that corre-
spond to a minimum of E 2 can be selected by inspection with the
168
THE THEORY OF MEASUREMENTS [ART. 83
aid of equation (145), but it is sometimes necessary to apply the
well-known criteria of the calculus.
Dividing equation (145) by x Q 2 and putting
XQ dX 2 ' q XQ XQ dX q
XQ XQ dXi
gives the expression
PZ = EI
X 2
XQ
(150)
+ T*E* (151)
for the fractional error of XQ. Since XQ is a constant in any given
problem the maxima and minima of P 2 correspond to the same
values of the components as those of E Q 2 . Sometimes the form
of the function F is such that the expression (151), when expanded
in terms of the x's, is much simpler than (145). In such cases it
is much easier to determine the minima of P 2 than of E 2 . For
this purpose the equations of condition (i) may be put in the form
6X1
XQ dXi
KdF_
XQ 6X2
dx,
(152)
, q XQ dX q
and by substitution and transposition we have
dTi dT% dT g
1 dxi 2 dxi 2 q dxi
dT<
(153)
When the components are required to satisfy the condition (144)
and a given constant value is assigned to XQ, equations (153) lead
to exactly the same results as equations (149). In fact either of
these sets of equations can be derived from the other by purely
algebraic methods when the $'s and T's are expressed in terms of
the x's. In practice one or the other of the sets will be the simpler,
depending on the form of the function F; and the simpler form
ART. 83] BEST MAGNITUDES FOR COMPONENTS 169
can be more easily derived by direct methods as above than by
algebraic transformation.
In some problems the magnitude of one or more of the com-
ponents in the function F can be varied at will and determined
with such precision that their probable errors are negligible in
comparison with those of the other components. Variables that
fulfill these conditions will be called free components. Since any
convenient magnitude can be assigned to them, their values can
always be so chosen that the condition (144) will be fulfilled
whatever the values of the other components. Consequently the
latter components may be treated as independent variables in
determining the minima of E Q 2 or P Q 2 .
Under these conditions the E's corresponding to the free com-
ponents can be placed equal to zero, and either E 2 or P 2 can
sometimes be expressed as a function of independent variables
only by eliminating the free components from the S's or the T's
with the aid of equation (144). When this elimination can be
effected, the minimum conditions may be derived from equations
(149) or (153), as the case may be, by placing K equal to zero and
omitting the equations involving derivatives with respect to the
free components. This is evident because the remaining com-
ponents are entirely independent, and consequently the partial
derivatives of E Q 2 or P 2 with respect to each of them must vanish
when the values of the variables correspond to the maxima or
minima of these functions. When the elimination cannot be
accomplished, neither equations (149) nor (153) will lead to con-
sistent results and the problem is generally insolvable.
In practice it frequently happens that the free components are
factors of the function F, and are not included in any other way.
Under these conditions they do not occur in the T's corresponding
to the remaining components, since the form of equations (150)
is such that they are automatically eliminated. Consequently,
in this case, the conditions for a minimum are given at once by
equations (153) when K is taken equal to zero, since the derivatives
with respect to the free components all vanish and the correspond-
ing E's are negligible. It is scarcely necessary to point out that
the remarks in the paragraph following equations (149), except
for obvious changes in notation, apply with equal rigor to equa-
tions (153), whether K is zero or finite. The values of the x's
derived from these equations should never be assumed to corre-
spond to the minima of P 2 without further investigation.
170 THE THEORY OF MEASUREMENTS [ART. 84
84. Special Cases. Suppose that the relation between the
derived quantity XQ and the measured components xi, # 2 , and x s
is given in the form
XQ = ax?* + bxj 1 * + cxj 1 *, (ii)
where a, b, c, and the n's are constants. If the probable errors
Ei t E z , and E 3 of the x's with corresponding subscripts are known,
and independent of the magnitude of the components, what mag-
nitudes of the components will give the least possible value to the
probable error E of X Q ?
By equations (146),
Si = arnxi^-V; S 2 = bn&^'-V; S s = c/W^-D. (iii)
Consequently
dSi , i\ ( <>\ ^$2 rv ^$3 _.
-!(, -I)**-*; ._ =0; = 0,
Substituting these results in equations (149) and dividing the
first equation by Si, the second by $ 2 , and the third by S S) the
conditions for a minimum value of E Q 2 become
Efari! (m - 1) xi<*-*> = K,
Dividing the second and third of these equations by the first
and transposing the coefficients to the second member gives the
ratios of the components in the form
x 2 (n ^- 2) = EJani (ni-l)
T,(tti-2) ~~ EL2Jm n (n n - IV
(HI - 1)
~
(n s -
These two equations together with (ii) are theoretically sufficient
for the determination of the best magnitudes for the three com-
ponents xij Xzj and x$] but it can be easily seen, from the form of
the equations, that a solution is not practicable for all possible
values of the n's.
ART. 84] BEST MAGNITUDES FOR COMPONENTS 171
For example, if the n's are all equal to unity, the ratios of the
components given by (iv) are both indeterminate, each being
equal to ^- Consequently the problem has no solution in this
case. This conclusion might have been reached at once by
inspecting the value of E Q 2 given by equation (145), when the S's
are expressed in terms of the components. Thus, placing the n's
equal to unity in equations (iii) and substituting the results in
(145), we find
Since E<? is independent of the x's it can have no maxima or
minima with respect to the components.
When each of the n's equals two, equations (iv) are inde-
pendent of the x's, and consequently the problem is not solvable.
In this case (ii) becomes
XQ =
and (145) reduces to
E 2 = 4
Since these equations differ only in the values of the constant
coefficients of the x's, no magnitudes can be assigned to the com-
ponents that will give a minimum value to E Q 2 , and at the same
time satisfy the equation for XQ.
If each of the n's is placed equal to three, equation (ii) takes
the form
XQ = ax^ + bx 2 * + c#3 3 , (v)
and equations (iv) become
Xt~bEf'
(iv')
C# 3 2
In this case the problem can be easily solved when the numerical
values of the coefficients and the E's are known. As a very
simple illustration, suppose that
7 -f J 77T -TGI ~Ij1 XT'
a = o = c = 1, and J^i = & 2 MS &,
then, by (iv') and (v),
and, by (145) and (iii),
172 THE THEORY OF MEASUREMENTS [ART. 84
Since a decrease in the magnitude of one of the x's involves an
increase in that of one or both of the others, in order to satisfy
equation (v), and since the fourth power of a quantity varies
more rapidly than the third, it is obvious that the minimum
value of E 2 will occur when the x's are all equal. Consequently
the above solution corresponds to a minimum of E 2 .
It can be easily seen that there are many other cases in which
equations (ii) and (iv) can be solved, and also some others in
which no solution is possible. The extension of the problem to
functions in the same form as equation (ii), but containing any
number of similar terms, involves only the addition of one equa-
tion in the form of (iv) for each added component. Obviously
these equations hold for negative as well as positive values of the
coefficients and exponents of the x's.
As a second example, consider the functional relation
x = axi n i X xf*. (vi)
In this case the solution is more easily effected by the second
method given in the preceding article. By equations (150)
Consequently
and equations (153) reduce to the simple form
^ES=-K; %Ef = -K, ; (viii)
where EI and E 2 are the known constant probable errors of Xi and
#2. Eliminating K, we have
Consequently the problem is always solvable when n\ and n 2
have the same sign. When they have different signs the solu-
tion is imaginary. Hence there are no best magnitudes for the
components when the derived quantity is given as the ratio of
two measured quantities.
ART. 85] BEST MAGNITUDES FOR COMPONENTS 173
The extension of this solution to functions involving any num-
ber of factors is obvious. When the exponents of all of the
factors have the same sign the problem is always solvable but
the best magnitudes thus found may not be attainable in practice.
If part of the exponents are positive and others are negative the
solution is imaginary.
85. Practical Examples.
I.
In many experiments the desired result depends directly upon
the determination of the quantity of heat generated by an electric
current in passing through a resistance coil. Let I represent the
current intensity and E the fall of potential between the terminals
of the coil. Then the quantity of heat H developed in t seconds
may be computed by the relation
JH = TEt,
where J represents the mechanical equivalent of heat. If H is
measured in calories, I in amperes, E in volts, and t in seconds,
y is equal to 0.239 calorie per Joule and the above relation becomes
H = 0.239 lEt. (ix)
Suppose that the conditions of the problem in hand are such
that H should be made approximately equal to 1000 calories.
Since the resistance of the heating coil is not specified it can be so
chosen that 7 and E may have any convenient values that satisfy
the relation (ix) when H has the above value. Obviously t can
be varied at will, by changing the time of run, and (ix) will not
be violated if suitable values are assigned to / and E. If the
instruments available for measuring /, E, and t are an ammeter
graduated to tenths of an ampere, a voltmeter graduated to
tenths of a volt, and a common watch with a seconds hand, what
are the best magnitudes that can be assigned to the components,
i.e., what magnitudes of /, E, and t will give the computed H
with the least probable error?
By comparing equations (ix) and (vi), it is easy to see that
the present problem is an application of the second special case
worked out in the preceding article when a third variable factor
Z 3 n 3 is annexed to (vi). H corresponds "to x 0) I to Xi, E to x z , t to
# 3 /and all of the n's in (vi) are equal to unity. Consequently
174 THE THEORY OF MEASUREMENTS [ART. 85
the solution can be derived at once from three equations in the
form of (viii) if suitable values can be assigned to the probable
errors of the components.
With the available instruments, the probable errors E i} E e , and
E t of /, E, and t, respectively, will be practically independent of
the magnitude of the measured quantities so long as the range
of the instruments is not exceeded. Under the conditions that
usually prevail in such observations the following precision may
be attained with reasonable care:
E t = 0.05 ampere; E e = 0.05 volt; E t = 1 second.
The conditions for a minimum value of the probable error E
of H can be derived by exactly the same method that was used
in obtaining equations (viii), or these equations may be used at
once with proper substitutions as outlined above. Consequently
the best magnitudes for the components are given by the simul-
taneous solution of (ix) and the following three equations,
^ 2 _ K . ^_ E?
~P = ~ K > ~W~ ~ K > ~P =
Eliminating K and substituting the numerical values of the
probable errors we have
E_E e _. l_E t _
I ~ E<~ L > I~ Ei~
Consequently
E = I and t = 20 /. (x)
Substituting these results and the numerical value of H in (ix)
we have
1000 = 0.239 X 20 X / 3 ,
and hence
I = 5.94 amperes
is the best magnitude to assign to the current strength under the
given conditions. The corresponding magnitudes for the electro-
motive force and time found by (x) are
E = 5.94 volts and t = 119 seconds.
If the above values of the components and their probable errors
are substituted in equation (151), the fractional error of H comes
out
ART. 85] BEST MAGNITUDES FOR COMPONENTS 175
and the probable error of H is given by the relation
E Q = 1000 Po =15 calories.
If any other magnitudes for the components, that satisfy equa-
tion (ix), are used in place of the above in (151), the computed
value of E will be greater than fifteen calories. Consequently
the above solution corresponds to a minimum value of E Q .
In order to fulfill the above conditions the resistance of the
heating coil must be so chosen as to satisfy the relation
*-
Since our solution calls for numerically equal values of I and E,
the resistance R must be made equal to one ohm.
It can be easily seen that small variations in the values of the
components will produce no appreciable effect on the probable
error of H, ^ince the numerical value of E is never expressed by
more than two significant figures. Consequently the foregoing
discussion leads to the following practical suggestions regarding
the conduct of the experiment. The heating coil should be so
constructed that the heat developed in the leads is negligible in
comparison with that developed between the terminals of the
voltmeter. The resistance of the coil should be one ohm. The
current strength should be adjusted to approximately six amperes
and allowed to flow continuously for about two minutes. Under
these conditions the difference in potential between the terminals
of the coil will be about six volts. The conditions under which
7, E, and t are observed should be so chosen that the probable
errors specified above are not exceeded.
If the above suggestions are carried out in practice the value
of H computed from the observed values of /, E, and t by equa-
tion (ix) will be approximately 1000 calories, and its probable
error will be about fifteen calories. A more precise result than
this cannot be obtained with the given instruments unless the
probable errors of 7, E, and t can be materially decreased by
modifying the conditions and methods of observation.
II.
A partial discussion of the problem of finding the best magni-
tudes for the components involved in the measurement of the
strength of an electric current with a tangent galvanometer may
176 THE THEORY OF MEASUREMENTS [ART. 85
be found in many laboratory manuals and textbooks. Such dis-
cussions are usually confined to a consideration of the error in the
computed current strength due to a given error in the observed
deflection. On the assumption, tacit or expressed, that the effects
of the errors of all other components are negligible it is proved
that the effect of the deflection error is a minimum when the
deflection is about forty-five degrees. Although the tangent gal-
vanometer is now seldom used in practice it provides an instructive
example in the calculation of best magnitudes since the general
bearings of the problem are already familiar to most students.
In order to avoid unnecessary complications, consider a simple
form of instrument with a compass needle whose position is
observed directly on a circle graduated in degrees. Suppose that
the needle is pivoted at the center of a single coil of N turns of
wire, and R centimeters mean radius. Under these conditions the
current strength I is connected with the observed deflection (f> by
the relation
where H is the horizontal intensity of a uniform external magnetic
field parallel to the plane of the coil. In practice the plane of the
coil is usually placed parallel to the magnetic meridian and H
is taken equal to the horizontal component of the earth's mag-
netism.
N is an observed component but it can be so precisely deter-
mined by direct counting, during the construction of the coil,
that its error may be considered negligible in comparison with
those of the other components. Furthermore it can be given any
desired value when an instrument is designed to meet special
needs, and a choice among a number of different values is possi-
ble in most completed instruments. Consequently the quantity
x TT may be treated as a free component, represented by A, and
the expression for the current strength may be written in the
form 7 = A#.tan0. (xi)
Comparing this expression with the general equation (144) we
note that / corresponds to x 0) H to x\, R to x 2 , and to z 3 .
Since A is free, the components H, R, and </> are entirely inde-
pendent; and any convenient magnitudes can be made to satisfy
ART. 85] BEST MAGNITUDES FOR COMPONENTS 177
(xi) by suitably choosing the number of turns in the coil. Con-
sequently, as pointed out in article eighty-three with respect to
functions containing a free component as a factor, the conditions
for a minimum probable error of / are given by equations (153)
with K placed equal to zero. By making the above substitutions
for the x's in equations (150) and performing the differentiations
we have
I/' 7?' oi-r O ^ * V^^X
11 /L bill cp
Consequently
0/77 -i z\nn H^TI
ol i 1 . o J. 2 f\ OJ. 3 ~
dH = ~H~ 2 ' dH ; ~dH = '
*^/T7 *\ ATT "I fk T7
?l n ^ 2 _ _ L ^ 3 _ n.
dR ~ dR R 2 ' dR '
dTi ^ = n- dTz = 4cos2<?i>
d0 60 " 60 sin 2 2 '
and, if the probable errors of H, R, and are represented by E\ 9
EZ, and #3, respectively, equations (153) become
If EI and E 2 could be made negligible, as is tacitly assumed in
most discussions of the present problem, the first two of equations
(xiii) would be satisfied whatever the values of H and R. Conse-
quently these components would be free and would be the only
independent variable involved in equation (xi). Under these
conditions the minimum value of the probable error of 7 corre-
sponds to the value of derived from the third of equations (xiii).
The general solution of this equation is
0= (2n-l)|>
where n represents any integer. But, since values of greater
than I are not attainable in practice, n must be taken equal to
unity in the present case and consequently the best magnitude
for the deflection is forty-five degrees. It is obvious that (xi)
can always be satisfied when / has any given value, and is
equal to forty-five degrees by suitably choosing the values of the
free components 2V, H, and R.
178 THE THEORY OF MEASUREMENTS [ART. 85
If the fractional error of / is represented by P and the T's
given by equations (xii) are substituted in (151),
H 2 ' R 2 ' sin 2 20
Pi 2 + P 2 2 + Pa 2 ,
(xiv)
= Pi 2 + P 2 2 + P 3 2 ,
where
2
= : and
are the separate effects of the probable errors E\, EZ, and E 3)
respectively. If both ends of the needle are read with direct and
reversed current so that represents the mean of four observa-
tions, EZ should not exceed 0.025 or 0.00044 radians, and it might
be made less than this with sufficient care. Consequently, when
<j> is equal to forty-five degrees,
P 3 = 0.00088.
By an argument similar to that given in article seventy-nine it can
be proved that PI and P 2 will be simultaneously negligible when
they satisfy the condition
p l = P 2 = i A = 0.00021.
3V2
Hence, in order that the effects of E\ and E% may be negligible in
comparison with that of E 3 , H and R must be determined within
about two one-hundredths of one per cent.
With an instrument of the type considered it would seldom be
possible and never worth while to determine H and R with the
precision necessary to fulfill the above condition. In common
practice E\ and E 2 are generally far above the negligible limit
and it would be necessary to make both H and R equal to infinity
in order to satisfy the first two of the minimum conditions (xiii).
Hence there is no practically attainable minimum value of P .
This conclusion can also be derived directly by inspection of
equation (xiv). P 2 decreases uniformly as H and R are increased,
and becomes equal to Ps 2 when they reach infinity.
Although a minimum value of P is not attainable, the fore-
going discussion leads to some practical suggestions regarding
the design and use of the tangent galvanometer. For any given
values of E\, E 2 , and E 3 , the minimum value of PS occurs when <j>
is equal to forty-five degrees. Also PI and P% decrease as H and
R increase. Consequently the directive force H and the radius
ART. 85] BEST MAGNITUDES FOR COMPONENTS 179
of the coil R should be made as large as is consistent with the
conditions under which the instrument is to be used, and the
number of turns N in the coil should be so chosen that the observed
deflection will be about forty-five degrees.
The practical limit to the magnitude of R is generally set by a
consideration of the cost and convenient size of the instrument.
Moreover when R is increased N must be increased in like ratio
in order to satisfy the fundamental relation (xi) without altering
the observed deflection or decreasing the value of H. There
is an indefinite limit beyond which N cannot be increased with-
out introducing the chance of error in counting and greatly in-
creasing the difficulty of determining the exact magnitude of R.
Above this limit E 2 is approximately proportional to R, and, as
can be easily seen by equation (xiv), there is no advantage to
be gamed by a further increase in the magnitude of R.
H can be varied by suitably placed permanent magnets, but
it is difficult to maintain strong magnetic fields uniform and con-
stant within the required limits. Even under the most favorable
conditions, the exact determination of H is very tedious and
involves relatively large errors. Consequently Pi 2 is likely to be
the largest of the three terms on the right-hand side of equation
(xiv). Under suitable conditions it can be reduced in magnitude
by increasing H to the limit at which the value of EI begins to
increase. However, such a procedure involves an increased value
of N in order to satisfy equation (xi), and consequently it may
cause an increase in E 2 owing to the relation between N and R
pointed out in the preceding paragraph. In such a case the gain
in precision due to a decreased value of PI would be nearly bal-
anced by an increased value of P%.
In common practice the instrument is so adjusted that H is
equal to the horizontal component of the earth's magnetic field
at the time and place of observation. Unless H is very carefully
determined at the exact location of the instrument, EI is likely
to be as large as 0.005 ~5 and, since the order of magnitude
Cat,
of H is about 0.2 ^r , -Pi will be approximately equal to 0.025.
cm
Hence both P 2 and P 3 will be negligible in comparison with PI if
they satisfy the relation
P 2 = P 3 = - ^j= = 0.0059.
"3 V2
180 THE THEORY OF MEASUREMENTS [ART. 85
Under ordinary conditions R and < can be easily determined within
the above limit. Consequently, in the supposed case,
PO = PI = 2.5 per cent,
and it would be useless to attempt an improvement in precision
by adjusting the values of N, R } and <. With sufficient care in
determining H, PI can be reduced to such an extent that it be-
comes worth while to carry out the suggestions regarding the
design and use of the instrument given by the foregoing theory.
But when the value of H is assumed from measurements made in
a neighboring location or is taken from tables or charts the per-
centage error of / will be nearly equal to that of H regardless of
the adopted values of R and <. Under such conditions P Q can-
not be exactly determined but it will seldom be less than two or
three per cent of the measured magnitude of I.
The above problem has been discussed somewhat in detail in
order to illustrate the inconsistent results that are likely to be
obtained in determining best magnitudes when the effects of the
errors of some of the components are neglected. It is never
safe to assume that the error of a component is negligible until
its effect has been compared with that of the errors of the other
components.
III.
Figure eleven is a diagram of the apparatus and connections
commonly used in determining the internal resistance of a bat-
tery by the condenser method. G is a ballistic galvanometer,
C a condenser, R a known resistance, KI a charge and discharge
key, Kz a plug or mercury key, and B a battery to be tested.
Let Xi represent the ballistic throw of the galvanometer when
the condenser is charged and discharged with the key K 2 open,
and x z the corresponding throw when K 2 is closed. Then the
internal resistance R Q of the battery may be computed by the
relation
Ro = R ^L^l. ( XV )
Under ordinary conditions the probable errors of x\ and x^
cannot be made much less than one-half of one per cent of the
observed throws when a telescope, mirror, and scale are used. On
the other hand the probable error of R should not exceed one-tenth
of one per cent if a suitably calibrated resistance is used and the
ART. 85] BEST MAGNITUDES FOR COMPONENTS 181
connections are carefully made. When these conditions are ful-
filled, it can be easily proved that the effect of the error of R is
negligible in comparison with that of the errors of Zi and x 2 .
Furthermore any convenient value can be assigned to R, such
<T 2 R
" !L -A/WVW\AAAA/
B
FIG. 11.
that (xv) will be satisfied whatever the values of Xi and #2. Con-
sequently R may be treated as a free component and the throws
Xi and x z as independent variables.
For the purpose of determining the magnitudes of the com-
ponents R, xij and x z that correspond to a minimum value of the
fractional error P of RQ, we have by equations (150) and (xv)
Consequently
- X 2 )
(xvi)
Since x\ and x 2 are independent, K must be taken equal to zero
in the minimum conditions (153). Hence, dividing the first two
equations by T i} we have
1 xi 1
1
E, 2 -^
= o,
= 0,
(x,-xz) 2 x 2 x 2 2 (x l -xz) 2
where EI and E 2 are the probable errors of x\ and 2 , respectively.
182 THE THEORY OF MEASUREMENTS [ART. 85
Multiply each of these equations by -- 1 ^ 2 and they as-
sume the simple form
+- *
Since #i 2 and Ez 2 are always positive, it is obvious that there
are no values of Xi and x% that will satisfy both of these equations
at the same time. Hence, when Xi and x z can be varied inde-
pendently, they cannot be so chosen that the fractional error P
will be a minimum. However, if Xz is kept constant at any as-
signed value, PO will pass through a minimum when Xi satisfies
equation (a). On the other hand if any constant value is assigned
to Xi the minima and maxima of P will correspond to the roots
of equation (b).
In practice x\ is the throw of the galvanometer needle due to
the electromotive force of the battery when on open circuit; and
it is very nearly constant, during a series of observations, when
suitable precautions are taken to avoid the effects of polariza-
tion. Both Xi and Xz can be varied by changing the capacity
of the condenser or the sensitiveness of the galvanometer, but
their ratio depends only on the ratio of R to R. Consequently,
if any convenient magnitude is assigned to Xi, the root of equa-
tion (b) that corresponds to a minimum value of PO gives the
best magnitude for the component Xz.
Since x\ and x 2 are similar quantities, determined with the same
instruments and under the same conditions, E\ is generally equal
to EZ. Hence, if we replace the ratio -- by y, equation (b) be-
^-2^-1 = 0^ (b')
The only real root of this equation is
y = 2.2056.
By equations (151) and (xvi)
Putting E l = Ez = E and - = y,
Xz
Pl = y* + y*
E* x 2 -l 2 '
ART. 86] BEST MAGNITUDES FOR COMPONENTS 183
Since Xi is necessarily greater than x 2 , y cannot be less than unity.
P 2
Under this condition it can be easily proved by trial that -==
&
approaches a minimum as y approaches the value given above,
provided any constant value is assigned to x\.
Equation (xv) may be put in the form
R Q = R(y- 1),
and, by introducing the value of y given by the minimum condi-
tion (b')> we have
R = 0.83 R .
Consequently the greatest attainable precision in the determina-
tion of RQ will be obtained when R is made equal to about eighty
three per cent of RQ. If R is adjusted to this value Xi and x% will
satisfy equation (b), whatever the magnitude of the capacity used,
provided the observations are so made that E\ and E% are equal.
When the internal resistance of the battery is very low it is
sometimes impracticable to fulfill the above theoretical conditions
because the errors due to polarization are likely to more than off-
set the gain in precision corresponding to the theoretically best
magnitudes of the components. In such cases a high degree of
precision is not attainable, but it is generally advisable to make R
considerably larger than R Q in order to reduce polarization errors.
86. Sensitiveness of Methods and Instruments. The pre-
cision attainable in the determination of directly measured com-
ponents depends very largely on the sensitiveness of indicating
instruments and on the methods of adjustment and observation.
The design and construction of an instrument fixes its intrinsic
sensitiveness; but its effective sensitiveness, when used as an indi-
cating device, depends on the circumstances under which it is used
and is frequently a function of the magnitudes of measured quan-
tities and other determining factors. Thus; the intrinsic sensi-
tiveness of a galvanometer is determined by the number of
windings in the coils, the moment of the directive couple, and
various other factors that enter into its design and construction.
On the other hand its effective sensitiveness as an indicator in a
Wheatstone Bridge is a function of the resistances in the various
arms of the bridge and the electromotive force of the battery
used. An increase in the intrinsic sensitiveness of an instrument
may cause an increase or a decrease in its effective sensitiveness,
184 THE THEORY OF MEASUREMENTS [ART. 86
depending on the nature of the corresponding modification in
design and the circumstances under which the instrument is
used.
By a suitable choice of the magnitudes of observed components
and other determining factors it is sometimes possible to increase
the effective sensitiveness of indicating instruments and hence
also the precision of the measurements. On the other hand,
as pointed out in Chapter XI, the precision of measurements
should not be greater than that demanded by the use to which
they are to be put. In all cases the effective sensitiveness of
instruments and methods should be adjusted to give a result
definitely within the required precision limits determined as in
Chapter XI. Consequently the best magnitudes for the quan-
tities that determine the effective sensitiveness are those that
will give the required precision with the least labor and expense.
The methods by which such magnitudes can be determined depend
largely on the nature of the problem in hand, and a general treat-
ment of them is quite beyond the scope of the present treatise.
Each separate case demands a somewhat detailed discussion of
the theory and practice of the proposed measurements and only
a single example can be given here for the purpose of illustration.
Since the potentiometer method of comparing electromotive
forces has been quite fully discussed in article eighty-one, it will
be taken as a basis for the illustration and we will proceed to find
the relation between the effective sensitiveness of the galvanom-
eter and the various resistances and electromotive forces involved.
Since the directly observed components in this method are the
resistances R\ and R%, the effective sensitiveness is equal to the
galvanometer deflection corresponding to a unit fractional devia-
tion of Ri or R z from the condition of balance.
From the discussion given in article eighty-one it is evident that
the potentiometer method could be carried out with any conven-
ient values of the resistances R\ and R 2 provided they are so ad-
7-
justed that the ratio - satisfies equation (ii) in the cited article.
tiz
The absolute magnitudes of these resistances depend on the electro-
motive force of the battery J5 3 and the total resistance of the cir-
cuit 1, 2, 3, B 3 , 1 in Fig. 10. The effective sensitiveness of the
method, and hence the accuracy attainable in adjusting the con-
tacts 2 and 3 for the condition of balance, depends on the above
ABT.86] BEST MAGNITUDES FOR COMPONENTS 185
factors together with the resistance and intrinsic sensitiveness of
the galvanometer.
Since RI and R% are adjusted in the same way and under the
same conditions, the effective sensitiveness of the method is the
same for both. Consequently only one of them will be considered
in the present discussion, but the results obtained will apply with
equal rigor to either. The essential parts of the apparatus and
connections are illustrated in Fig. 12, which is the same as Fig. 10
with the battery B 2 and its connections omitted.
FIG. 12.
Let V = e.m.f. of battery BI,
E = e.m.f. of battery B 3 ,
R = resistance between 1 and 2,
W = total resistance of the circuit 1, 2, B s , 1,
G = total resistance of the branch 1, G, BI, 2,
I = current through B 3)
r = current through R,
g = current through BI and G.
When the contact 2 is adjusted to the balance position
Consequently
= 0, r = 7, and 7=^ = -^
(xvii)
This is the fundamental equation of the potentiometer and must
be fulfilled in every case of balance. Consequently E must be
186 THE THEORY OF MEASUREMENTS [ART. 86
chosen larger than V because R is a part of the resistance in the
circuit 1, 2, B z , 1, and hence is always less than W. Equation
(xvii) may then be satisfied by a suitable adjustment of R.
By applying Kirchhoff's laws to the circuits 1, G, BI, 2, 1, and
1, 2, B 3) 1, when the contact 2 is not in the balance position, we
have
Rr-Gg= V,
and Rr + (W - R) I = E.
But r = I - g.
Hence RI-(R + G)g = V,
and WI - Rg = E.
Eliminating I and solving for g we find
WV -RE
If D is the galvanometer deflection corresponding to the current
g and K is the constant of the instrument
g = KD.
Most galvanometers are, or can be, provided with interchange-
able coils. The winding space in such coils is usually constant,
but the number of windings, and hence the resistance, is variable.
Under these conditions the resistance of the galvanometer will be
approximately proportional to the square of the number of turns
of wire in the coils used. For the purpose of the present discussion,
this resistance may be assumed to be equal to G since the resist-
ance of the battery and connecting wires in branch 1, G, BI, 2,
can usually be made very small in comparison with that of the
galvanometer. The constant K is inversely proportional to the
number of windings in the coils used. Consequently, as a suffi-
ciently close approximation for our present purpose, we have
T
v
K = T=>
VG
where T is a constant determined by the dimensions of the coils,
the moment of the directive couple, and various other factors
depending on the type of galvanometer adopted. Hence, for any
given instrument,
ART. 86] BEST MAGNITUDES FOR COMPONENTS 187
VG
The quantity -jr is the intrinsic sensitiveness of the galvanometer.
It is equal to the deflection that would be produced by unit current
if the instrument followed the same law for all values of g.
By equation (xix) and (xviii)
VG WV-RE
T *R*-WR-WG'
The variation in D due to a change dR in R is
dD VG E(R*-WR-WG) + (WV-RE)(2R-W)
dR ' T ' (R*-WR-WGY
When the potentiometer is adjusted for a balance, D is equal to
zero and WV is equal to RE by equation (xvii). Hence, if d is the
galvanometer deflection produced when the resistance R is changed
from the balancing value by an amount dR, equation (xx) may
be put in the form
1 VVG
The fractional change in R corresponding to the total change dR
is
. I '-f : I
Consequently
1 VVO
~' '
is the galvanometer deflection corresponding to a fractional error
P r in the adjustment of R for balance. The coefficient of P r in
equation (xxi) is the effective sensitiveness of the method under
the given conditions. If this quantity is represented by S, equa-
tion (xxi) becomes
8 = SP r ,
8 I
All of the quantities appearing in the right-hand member of this
equation may be considered as independent variables since equa-
tion (xvii) can always be satisfied, and hence the potentiometer
188 THE THEORY OF MEASUREMENTS [ART. 86
can be balanced, when R, V, and E have any assigned values, if
the resistance W is suitably chosen.
If d' is the smallest galvanometer deflection that can be defi-
nitely recognized with the available means of observation, the frac-
tional error P/ of a single observation on R should not be greater
5'
than -~ Since the precision attainable in adj usting the potentiom-
o
eter for balance is inversely proportional to P/, it is directly pro-
portional to the effective sensitiveness S. By choosing suitable
magnitudes for the variables T, G, R, and E, it is usually possible
to adjust the value of S, and hence also of P/, to meet the re-
quirements of any problem.
From equation (xxii) it is evident that S will increase in magni-
tude continuously as the quantities T, R, and E decrease and that
it does not pass through a maximum value. The practicable in-
crease in S is limited by the following considerations: E must be
greater than V, for the reason pointed out above, and its variation
is limited by the nature of available batteries. Since E must
remain constant while the potentiometer is being balanced alter-
nately against V and the electromotive force of a standard cell,
as explained in article eighty-one, the battery B 3 must be capable
of generating a constant electromotive force during a considerable
period of time. In practice storage cells are commonly used for
this purpose and E may be varied by steps of about two volts by
connecting the required number of cells in series. Obviously E
should be made as nearly equal to V as local conditions permit.
When the potentiometer is balanced
V E
If R is reduced for the purpose of increasing the effective sensitive-
ness, W must also be reduced in like ratio, and, consequently, the
current 7 through the instrument will be increased. The prac-
tical limit to this adjustment is reached when the heating effect
of the current becomes sufficient to cause an appreciable change
in the resistances R and W. With ordinary resistance boxes this
limit is reached when 7 is equal to a few thousandths of an ampere.
Consequently, if E is about two volts, R should not be made much
less than one thousand ohms. Resistance coils made expressly
for use in a potentiometer can be designed to carry a much larger
ART. 86] BEST MAGNITUDES FOR COMPONENTS 189
current so that R may be made less than one hundred ohms with-
out introducing serious errors due to the heating effect of the
current.
The constant T depends on the type and design of the galva-
nometer. In the suspended magnet type it can be varied some-
what by changing the strength of the external magnetic field, and
in the D'Arsonval type the same result may be attained by chang-
ing the suspending wires of the movable coil. The effects of the
vibrations of the building in which the instrument is located and
of accidental changes in the external magnetic field become much
more troublesome as T is decreased, i.e., as the intrinsic sensitive-
ness is increased. Consequently the practical limit to the reduc-
tion of T is reached when the above effects become sufficient to
render the observation of small values of 6 uncertain. This limit
will depend largely on the location of the instrument and the care
that is taken in mounting it. Sometimes a considerable reduc-
tion in T can be effected by selecting a type of galvanometer
suited to the local conditions.
If the quantities T 7 , R, V, and E are kept constant, S passes
through a maximum value when G satisfies the condition
*?'
It can be easily proved by direct differentiation that this is the
case when
G =
Hence, after suitable values of the other variables have been de-
termined as outlined above, the best magnitude for G is given by
equation (xxiii). Generally this condition cannot be exactly ful-
filled in practice unless a galvanometer coil is specially wound for
the purpose; but, when several interchangeable coils are available,
the one should be chosen that most nearly fulfills the condition.
In some galvanometers T and G cannot be varied independently,
and in such cases suitable values can be determined only by trial.
Since the ease and rapidity with which the observations can be
made increase with T, it is usually advisable to adjust the other
variables to give the greatest practicable value to the second
factor in S, and then adjust T so that the effective sensitiveness
190 THE THEORY OF MEASUREMENTS [ART. 86
will be just sufficient to give the required precision in the deter-
mination of R.
As an illustration consider the numerical data given in article
eighty-one. It was proved that the specified precision require-
ments cannot be satisfied unless R is determined within a frac-
tional precision measure equal to 0.00063. Allowing one-half
of this to errors of calibration we have left for the allowable error
in adjusting the potentiometer
P r ' = 0.00031.
If a single storage cell is used at B$, E is approximately two volts,
and, with ordinary resistance boxes, R should be about one thou-
sand ohms, for the reason pointed out above. This condition is
fulfilled by the cited data; and, for our present purpose, it will be
sufficiently exact to take V equal to one volt. Hence, by equa-
tion (xxiii), the most advantageous magnitude for G is about
five hundred ohms; and, by equation (xxii), the largest practi-
cable value for the second factor in S is
ST = V Jf = 0.0224.
gf 1-41+0
With a mirror galvanometer of the D'Arsonval type, read by
telescope and scale, a deflection of one-half a millimeter can be
easily detected. Consequently, if we express the galvanometer
constant K in terms of amperes per centimeter deflection, we must
take 5' equal to 0.05 centimeter; and, in order to fulfill the specified
precision requirements, the effective sensitiveness must satisfy the
condition
S' 0.05
~P7~00003l~
Combining this result with the above maximum value of ST we
find that the intrinsic sensitiveness must be such that
0.0224 _
161
Hence the galvanometer should be so constructed and adjusted
that
G = 500 ohms,
and
T
K = = = 6.2 X lO" 6 amperes per centimeter deflection.
ART. 86] BEST MAGNITUDES FOR COMPONENTS 191
D'Arsonval galvanometers that satisfy the above specifications
can be very easily obtained and are much less expensive than
more sensitive instruments. They are so nearly dead-beat and
free from the effects of vibration that the adjustment of the poten-
tiometer for balance can be easily and rapidly carried out with
the necessary precision. Hence the use of such an instrument
reduces the expense of the measurements without increasing the
errors of observation beyond the specified limit.
CHAPTER XIII.
RESEARCH.
87. Fundamental Principles. The word research, as used
by men of science, signifies a detailed study of some natural
phenomenon for the purpose of determining the relation between
the variables involved or a comparative study of different phe-
nomena for the purpose of classification. The mere execution of
measurements, however precise they may be, is not research. On
the other hand, the development of suitable methods of measure-
ment and instruments for any specific purpose, the estimation of
unavoidable errors, and the determination of the attainable limit
of precision frequently demand rigorous and far-reaching research.
As an illustration, it is sufficient to cite Michelson's determination
of the length of the meter in terms of the wave length of light. A
repetition of this measurement by exactly the same method and
with the same instruments would involve no research, but the
original development of the method and apparatus was the result
of careful researches extending over many years.
The first and most essential prerequisite for research in any field
is an idea. The importance of research, as a factor in the advance-
ment of science, is directly proportional to the fecundity of the
underlying ideas.
A detailed discussion of the nature of ideas and of the conditions
necessary for their occurrence and development would lead us too
far into the field of psychology. They arise more or less vividly
in the mind in response to various and often apparently trivial
circumstances. Their inception is sometimes due to a flash of
intuition during a period of repose when the mind is free to respond
to feeble stimuli from the subconscious. Their development and
execution generally demand vigorous and sustained mental effort.
Probably they arise most frequently in response to suggestion or
as the result of careful, though tentative, observations.
A large majority of our ideas have been received, in more
or less fully developed form, through the spoken or written dis-
course of their authors or expositors. Such ideas are the common
192
ART. 88] RESEARCH 193
heritage of mankind, and it is one of the functions of research to
correct and amplify them. On the other hand, original ideas,
that may serve as a basis for effective research, frequently arise
from suggestions received during the study of generally accepted
notions or during the progress of other and sometimes quite differ-
ent investigations.
The originality and productiveness of our ideas are determined
by our previous mental training, by our habits of thought and
action, and by inherited tendencies. Without these attributes,
an idea has very little influence on the advancement of science.
Important researches may be, and sometimes are, carried out by
investigators who did not originate the underlying ideas. But,
however these ideas may have originated, they must be so thor-
oughly assimilated by the investigator that they supply the stim-
ulus and driving power necessary to overcome the obstacles that
inevitably arise during the prosecution of the work. The driving
power of an idea is due to the mental state that it produces in the
investigator whereby he is unable to rest content until the idea
has been thoroughly tested in all its bearings and definitely proved
to be true or false. It acts by sustaining an effective concentra-
tion of the mental and physical faculties that quickens his in-
genuity, broadens his insight, and increases his dexterity.
In order to become effective, an idea must furnish the incentive
for research, direct the development of suitable methods of pro-
cedure, and guide the interpretation of results. But it must
never be dogmatically applied to warp the facts of observation
into conformity with itself. The mind of the investigator must
be as ready to receive and give due weight to evidence against
his ideas as to that in their favor. The ultimate truth regarding
phenomena and their relations should be sought regardless of
the collapse of generally accepted or preconceived notions. From
this point of view, research is the process by which ideas are
tested in regard to their validity.
88. General Methods of Physical Research. Researches
that pertain to the physical sciences may be roughly classified
in two groups: one comprising determinations of the so-called
physical constants such as the atomic weights of the elements, the
velocity of light, the constant of gravitation, etc.; the other
containing investigations of physical relations such as that which
connects the mass, volume, .pressure, and temperature of a gas.
194 THE THEORY OF MEASUREMENTS [ART. 88
The researches in the first group ultimately reduce to a careful
execution of direct or indirect measurements and a determination
of the precision of the results obtained. The general principles
that should be followed in this part of the work have been suffi-
ciently discussed in preceding chapters. Their application to prac-
tical problems must be left to the ingenuity and insight of the
investigator. Some men, with large experience, make such appli-
cations almost intuitively. But most of us must depend on a
more or less detailed study of the relative capabilities of available
methods to guide us in the prosecution of investigations and in
the discussion of results.
In general, physical constants do not maintain exactly the same
numerical value under all circumstances, but vary somewhat with
changes in surrounding conditions or with lapse of time. Thus
the velocity of light is different in different media and in dispersive
media it is a function of the frequency of the vibrations on which
it depends. Consequently the determination of such constants
should be accompanied by a thorough study of all of the factors
that are likely to affect the values obtained and an exact specifica-
tion of the conditions under which the measurements are made.
Such a study frequently involves extensive investigations of the
phenomena on which the constants depend and it should be
carried out by very much the same methods that apply to the
determination of physical relations in general. On the other
hand, the exact expression of a physical relation generally involves
one or more constants that must be determined by direct or in-
direct measurements. Hence there is no sharp line of division
between the first and second groups specified above, many re-
searches belonging partly to one group and partly to the other.
The occurrence of any phenomenon is usually the result of the
coexistence of a number of more or less independent antecedents.
Its complete investigation requires an exact determination of the
relative effect of each of the contributary causes and the develop-
ment of the general relation by which their interaction is expressed.
A determination of the nature and mode of action of all of the
antecedents is the first step in this process. Since it is gen-
erally impossible to derive useful information by observing the
combined action of a number of different causal factors, it becomes
necessary to devise means by which the effects of the several
factors can be controlled in such manner that they can be studied
ART. 88] RESEARCH 195
separately. The success of researches of this type depends very
largely on the effectiveness of such means of control and the
accuracy with which departures from specified conditions can be
determined.
Suppose that an idea has occurred to us that a certain phenome-
non is due to the interaction of a number of different factors that
we will represent by A, B, C, . . . , P. This idea may involve
a more or less definite notion regarding the relative effects of the
several factors or it may comprehend only a notion that they are
connected by some functional relation. In either case we wish
to submit our idea to the test of careful research and to determine
the exact form of the functional relation if it exists.
The investigation is initiated by making a series of preliminary
observations of the phenomenon corresponding to as many vari-
ations in the values of the several factors as can be easily effected.
The nature of such observations and the precision with which they
should be made depend so much on the character of the problem
in hand that it would be impossible to give a useful general dis-
cussion of suitable methods of procedure. Sometimes roughly
quantitative, or even qualitative, observations are sufficient. In
other cases a considerable degree of precision is necessary before
definite information can be obtained. In all cases the observa-
tions should be sufficiently extensive and exact to reveal the gen-
eral nature and approximate relative magnitudes of the effects
produced by each of the factors. They should also serve to detect
the presence of factors not initially contemplated.
With the aid of the information derived from preliminary obser-
vations and from a study of such theoretical considerations as
they may suggest, means are devised for exactly controlling the
magnitude of each of the factors. Methods are then developed
for the precise measurement of these magnitudes under the con-
ditions imposed by the adopted means of control. This process
often involves a preliminary trial of several different methods
for the purpose of determining their relative availability and pre-
cision. The methods that are found to be most exact and con-
venient usually require some modification to adapt them to the
requirements of a particular problem. Sometimes it becomes
necessary to devise and test entirely new methods. During this
part of the investigation the discussions of the precision of meas-
urements given in the preceding chapters find constant applica-
196 THE THEORY OF MEASUREMENTS [ART. 88
tion and it is largely through them that the suitableness of
proposed methods is determined.
After definite methods of measurement and means of control
have been adopted and perfected to the required degree of pre-
cision, the final measurements on the factors, A, B, C, . . . , P,
are carried out under the conditions that are found to be most
advantageous. Usually two of the factors, say A and B, are
caused to vary through as large a range of values as conditions
will permit while the other factors are maintained constant at
definite observed values. At stated intervals the progress of the
variation is arrested and corresponding values of A and B are
measured while they are kept constant. From a sufficiently
extended series of such observations it is usually possible to make
an empirical determination of the form of the functional relation
A =/i(); C,Z>, . . . ,P. constant. (i)
On the other hand, if the form of the function /i is given as a
theoretical deduction from the idea underlying the investigation,
the observations serve to test the exactness of the idea and de-
termine the magnitudes of the constants involved in the given
function. By allowing different factors to vary and making
corresponding measurements, the relations
A =/ 2 (C); B,D, . . , P, constant,
A =/ n (P); ,C,Z>, ., constant,
(ii)
may be empirically determined or verified. As many functions of
this type as there are pairs of factors might be determined, but
usually it is not necessary to establish more than one relation for
each factor. Generally it is convenient to determine one of the
factors as a function of each of the others as illustrated above;
but it is not necessary to do so, and sometimes the determination
of a different set of relations facilitates the investigation.
During the establishment of the relation between any two
factors all of the others are supposed to remain rigorously con-
stant. Frequently this condition cannot be exactly fulfilled with
available means of control, but the variations thus introduced
can usually be made so small that their effects can be treated as
constant errors and removed with the aid of the relations after-
wards found to exist between the factors concerned, For this
ART. 88] RESEARCH 197
purpose frequent observations must be made on the factors that
are supposed to remain constant during the measurement of the
two principal variables. If the variations in these factors are not
very small all of the relations determined by the principal measure-
ments will be more or less in error and must be treated as first
approximations. Usually such errors can be eliminated and the
true relations established with sufficient precision, by a series of
successive approximations. However, the weight of the final
result increases very rapidly with the effectiveness of the means of
control and it is always worth while to exercise the care necessary
to make them adequate.
When the functions involved in equations (i) and (ii), or their
equivalents in terms of other combinations of factors, have been
determined with sufficient precision, they can usually be com-
bined into a single relation, in the form
or
A=F(B,C,D, . . . ,P),
F(A,B,C,D, ,P)=0,
(iii)
which expresses the general course of the investigated phenomenon
in response to variations of the factors within the limits of the
observations. Such generalizations may be purely empirical or
they may rest partly or entirely on theoretical deductions from
well-established principles. In either case the test of their validity
lies in the exactness with which they represent observed facts.
While an exact empirical formula finds many useful applications
in practical problems it should not be assumed to express the true
physical nature of the phenomenon it represents. In fact our
understanding of any phenomenon is but scanty until we can
represent its course by a formula that gives explicit or implicit
expression to the physical principles that underlie it. Conse-
quently a research ought not to be considered complete until the
investigated phenomenon has been classified and represented by a
function that exhibits the physical relations among its factors.
(i It is scat cely necessary to point out that a complete research
as outlined above is seldom carried out by one man and that the
underlying ideas very rarely originate at the same time or in the
same person. The preliminary relations in the form of equations
(i) and (ii) are frequently inspired by independent ideas and
worked out by different men. The exact determination of any
198 THE THEORY OF MEASUREMENTS [ART. 89
one of them constitutes a research that is complete so far as it
goes. The establishment of the general relation that compre-
hends all of the others and the interpretation of its physical signifi-
cance are generally the result of a process of gradual growth and
modification to which many men have contributed.
89. Graphical Methods of Reduction. After the necessary
measurements have been completed and corrected for all known
constant errors, the form of the functions appearing in equations
(i) and (ii), or other equations of similar type, and the numerical
value of the constants involved can sometimes be determined
easily and effectively by graphical methods. Such methods are
almost universally adopted for the discussion of preliminary obser-
vations and the determination of approximate values of the con-
stants. In some cases they are the only methods by which the
results of the measurements can be expressed. In some other
cases the constants can be more exactly determined by an appli-
cation of the method of least squares to be described later. Usu-
ally, however, the general form of the functions and approximate
values of the constants must first be determined by graphical
methods or otherwise.
Let x and y represent the simultaneous values of two variable
factors corresponding to specified constant values of the other
factors involved in the phenomenon under investigation. Suppose
that x has been varied by successive nearly equal steps through
as great a range as conditions permit and that the simultaneous
values x and y have been measured after each of these steps while
the factors that they represent were kept constant. If all other
factors have remained constant throughout these operations, the
above series of measurements on x and y may be applied at once
to the determination of the form and constants of the functional
relation
This expression is of the same type as equations (i.) and (ii).
Consequently the following discussion applies generally to all
cases in which there are only two variable factors. If the sup-
posedly constant factors are not strictly constant during the
measurements, the observations on x and y will not give the true
form of the function in (iv) until they have been corrected for
the effects of the variations thus introduced.
ART. 89] RESEARCH 199
As the first step in the graphical method of reduction, the
observations on x and y are laid off as abscissae and ordinates on
accurately squared paper, and the points determined by corre-
sponding coordinates are accurately located with a fine pointed
needle. The visibility of these points is usually increased by
drawing a small circle or other figure with its center exactly at
the indicated point. The scale of the plot should be so chosen
that the form of the curve determined by the located points is
easily recognized by eye. In order to bring out the desired rela-
tion, it is frequently necessary to adopt a different scale for ordi-
nates and abscissae. Usually it is advantageous to choose such
scales that the total variations of x and y will be represented by
approximately equal spaces. Thus, if the total variation of y is
numerically equal to about one-tenth of the corresponding vari-
ation of x, the i/'s should be plotted to a scale approximately ten
times as large as that adopted for the x's. In all cases the adopted
scales should be clearly indicated by suitable numbers placed at
equal intervals along the vertical and horizontal axes. Letters
or other abbreviations should be placed near the ends of the axes
to indicate the quantities represented.
The points thus located usually lie very nearly on a uniform
curve that represents the functional relation (iv). Consequently
the problem in hand may be solved by determining the equation
of this curve and the numerical value of the constants involved
in it. Sometimes it is impossible or inadvisable to carry out such
a determination in practice and in such cases the plotted curve
is the only available means of representing the relation between
the observed factors. In all cases the deviations of the located
points from the uniform curve represent the residuals of the
observations, and, consequently, indicate the precision of the
measurements on x and y.
The simplest case, and one that frequently occurs in practice, is
illustrated in Fig. 13. The plotted points lie very nearly on a
straight line. Consequently the functional relation (iv) takes the
linear form
y = Ax + B, (v)
where A is the tangent of the angle a between the line and the
positive direction of the x axis, and B is the intercept OP on
the y axis. For the determination of the numerical values of the
200
THE THEORY OF MEASUREMENTS [ART. 89
constants A and B, the line should be sharply drawn in such a
position that the plotted points deviate from it about equally in
opposite directions, i.e., the sum of the positive deviations should
be made as nearly as possible equal to the sum of the negative
deviations. If this has been carefully and accurately done, the
constant B may be determined by a direct measurement of the
intercept OP in terms of the scale used in plotting the y's-
0.10
05
25
FIG. 13.
50
75
The constant A may be computed from measurements of the
coordinates x\ and 2/1 of any point on the line, not one of the plotted
points, by the relation
If the position of the line is such that the point P does not fall
within the limits of the plotting sheet, the coordinates, Xi, y\ and
2, 2/2, of two points on the line are measured. Since they must
satisfy equation (v),
2/i = Axi + B,
and
2/2 = Ax 2 + B.
Consequently
A = and B
X 2
The points selected for this purpose should be as widely separated
as possible in order to reduce the effect of errors of plotting and
ART. 89] RESEARCH 201
measurement. The accuracy of these determinations is likely to
be greatest when the vertical and horizontal scales are so chosen
that the line makes an angle of approximately forty-five degrees
with the x axis. Space may sometimes be saved and the appear-
ance of the plot improved by subtracting a constant quantity,
nearly equal to B, from each of the y's before they are plotted.
Many physical relations are not linear in form. Perhaps none
of them are strictly linear when large ranges of variation are con-
sidered. Consequently the plotted points are more likely to lie
nearly on some regular curve than on a straight line. In such
cases the form of the functional relation (iv) is sometimes sug-
gested by theoretical considerations, but frequently it must be
determined by the method of trial and error or successive approxi-
mations. For this purpose the curve representing the observa-
tions is compared with a number of curves representing known
equations. The equation of the curve that comes nearest to the
desired form is modified by altering the numerical values of its
constants until it represents the given measurements within the
accidental errors of observation. Frequently several different
equations and a number of modifications of the constants must
be tried before satisfactory agreement is obtained.
When the desired relation does not contain more than two inde-
pendent constants, it can sometimes be reduced to a linear relation
between simple functions of x and y. Thus, the equation
y = Be~ Ax , . (vi)
represented by the curve in Fig. 14, is frequently met with in
physical investigations. By inverting (vi) and introducing ' log-
arithms, we obtain the relation
log* y = log* B - Ax.
Hence if the logarithms of the y's are laid off as ordinates against
the corresponding x's as abscissae, the located points will lie very
nearly on a straight line if the given observations satisfy the func-
tional relation (vi) . When this is the case, the constants A and
loge B may be determined by the methods developed during the
discussion of equation (v). If logarithms to the base ten are
used the above equation becomes
^|
log y = logio B - x,
202
THE THEORY OF MEASUREMENTS [ART. 89
where M is the modulus of the natural system of logarithms. In
^
this case the plot gives the values of logio B and -^ from which
the constants A and B can be easily computed. When the plotted
points do not lie nearer to a straight line than to any other curve,
y
10
\
\
0.5
1.0
1.5
FIG. 14.
equation (vi) does not represent the functional relation between
the observed factors and some other form must be tried. Many
of the commonly occurring forms may be treated by the above
method and the process is usually so simple that further illustra-
tion seems unnecessary.
The curve determined by plotting the x's and y's directly fre-
quently exhibits points of discontinuity or sharp bends as at p
and q in Fig. 15. Such irregularities are generally due to changes
in the state of the material under investigation. The nature- and
causes of such changes are frequently determined, or at least
suggested, by the location and character of such points. The
different branches of the curve may correspond to entirely differ-
ent equations or to equations in the same form but with different
constants. In either case the equation of each branch must be
determined separately.
The accuracy attainable by graphical methods depends very
largely on the skill of the draughtsman in choosing suitable scales
and executing the necessary operations. In many cases the errors
ART. 90]
RESEARCH
203
due to the plot are less than the errors of observation and it would
be useless to adopt a more precise method of reduction. When
the means of control are so well devised and effective that the
constant errors left in the measurements are less than the errors
of plotting it is probably worth while to make the reductions by
the method of least squares, as explained in the following article.
y
'FiG. 15.
90. Application of the Method of Least Squares. In the
case of linear relations, expressible in the form of equation (v),
the best values of the constants A and B can be very easily deter-
mined by applying the method of least squares in the manner
explained in article fifty-one. However, as pointed out in the
preceding article, very few physical relations are strictly linear
when large variations of the involved factors are considered.
Consequently a straight line, corresponding to constants deter-
mined as above, usually represents only a small part of the course
of the investigated phenomenon. Such a line is generally a short
chord of the curve that represents the true relation and conse-
quently its direction depends on the particular range covered by
the observations from which it is derived.
When the measurements are extended over a sufficiently wide
range, the points plotted from them usually deviate from a straight
line in an approximately regular manner, as illustrated in Fig. 16,
204
THE THEORY OF MEASUREMENTS [ART. 90
and lie very near to a continuous curve of slight curvature. Meas-
urements of this type can always be represented empirically by a
power series in the form
y = A + Bx + Cx* + . - - , (vii)
the number of terms and the signs of the constants depending on
the magnitude and sign of the curvature to be represented.
FIG. 16.
Since equation (vii) is linear with respect to the constants A, B,
C, etc., they might be computed directly from the observations
on x and y by the method of least squares. Usually, however,
the computations can be simplified by introducing approximate
values of the constants A and B. Thus, let A' and B' represent
two numerical quantities so chosen that the line
y' = A' + B'x
passes in the same general direction as the plotted points, in the
manner illustrated by the dotted line in pig. 16. The difference
between y and y' can be put in the form
y y' = (A A') + MI (B B'} -^ 4- M 2 C + . . . (viii)
MI M 2
where Afi, M 2 , etc., represent numerical constants so chosen that
*Y* s2
the quantities y - y', -=, etc., are nearly of the same order
ART. 90] RESEARCH 205
of magnitude. For the sake of convenience let
(ix)
and
The quantities s, 6, c, etc., may be derived from the observations,
with the aid of the assumed constants A', B', MI, M z , etc.; and xi,
x z , x S} etc., are the unknowns to be computed by the method of
least squares. After the above substitutions, equation (viii) takes
the simple form
xi + bx 2 + cx 3 + = s,
which is identical with that of the observation equations (53),
article forty-nine. As many equations of this type may be formed
as there are pairs of corresponding measurements on x andj y.
The normal equations (56) may be derived from the observation
equations thus established, by the methods explained in articles
fifty and fifty-three. Their final solution for the unknowns Xi, Xz,
xsj etc., may be effected by Gauss's method, developed in article
fifty-four and illustrated in article fifty-five, or by any other con-
venient method. The corresponding numerical values of the
constants A, B, C, etc., may then be computed by equations (ix).
These values, when substituted in (vii) , give the required empirical
relation between x and y.
If a sufficient number of terms have been included in equation
(vii), the relation thus established will represent the given measure-
ments within the accidental errors of observation. The residuals,
computed by equations (54), article forty-nine, and arranged in
the order of increasing values of y, should show approximately as
many sign changes as sign follows. When this is not the case
the observed y's deviate systematically from the values given by
equation (vii) for corresponding x's. In such cases the number of
terms employed is not sufficient for the exact representation of the
observed phenomenon, and a new relation in the same general
form as the one already tested but containing more independent
constants should be determined. This process must be repeated
until such a relation is established that systematically varying
differences between observed and computed y's no longer occur.
The observation equations used as a basis for the numerical
illustration given in article fifty-five were derived from the follow-
206
THE THEORY OF MEASUREMENTS [ART. 90
ing observations on the thermal expansion of petroleum by equa-
tions (viii) and (ix), taking
A' = 1000; B' = l; M l = 10; and M 2 = 1000.
X
temperature
volume
degrees
cc.
1000.24
20
1018.82
40
1038.47
60
1059 31
80
1081.20
100
1104.27
The computations carried out in the cited article resulted as
follows :
xi = 0.245; x 2 = - 1.0003; x 3 = 1.4022.
Hence, by equations (ix)
A = 1000.245; B = 0.89997; C = 0.0014022,
and the functional relation (vii) becomes
y = 1000.245 + 0.89997 x + 0.0014022 x\
The residuals corresponding to this relation, computed and tab-
ulated in article fifty-five, show five sign changes and no sign
follows. Such a distribution of signs sometimes indicates that the
observed factors deviate periodically from the assumed functional
relation. In the present case, however, the number of observa-
tions is so small that the apparent indications of the residuals are
probably fortuitous. Consequently it would not be worth while
to repeat the computations with a larger number of terms unless
it could be shown by independent means that the accidental errors
of the observations are less than the residuals corresponding to the
above relation.
Any continuous relation between two variables can usually be
represented empirically by an expression in the form of equation
(vii). However, it frequently happens that the physical signifi-
cance of the investigated phenomenon is not suggested by such
an expression but is represented explicitly by a function that is not
linear with respect to either the variable factors or the constants
involved. Such functions usually contain more than two inde-
pendent constants and sometimes include more than two variable
factors. They may be expressed by the general equation
y = F(A,B,C,. ,x,z,. . ), (154)
ART. 90] RESEARCH 207
where A, B, C, etc., represent constants to be determined and y t x,
z, etc., represent corresponding values of observed factors.
Sometimes the form of the function F is given by theoretical
considerations, but more frequently it must be determined, to-
gether with the numerical values of the constants, by the method
of successive approximations. In the latter case a definite form,
suggested by the graphical representation of the observations or
by analogy with similar phenomena, is assumed tentatively as a
first approximation. Then, by substituting a number of different
corresponding observations on y, x, z, etc., in (154), as many inde-
pendent equations as there are constants in the assumed function
are established. The simultaneous solution of these equations
gives first approximations to the values of the constants A, B, C,
etc. Sometimes the solution cannot be effected directly by means
of the ordinary algebraic methods, but it can usually be accom-
plished with sufficient accuracy either by trial and error or by
some other method of approximation.
Let A', B' ', C', etc., represent approximate values of the con-
stants and let 61, 5 2 , 5 3 , etc., represent their respective deviations
from the true values. Then
A=A' + 5 1 ; B = B' + d 2 ] C = C' + 5 3 , etc., (155)
and (154) may be put in the form
y-F\(A' + Sd, (B' + fc), (C" + .) ---- ,*,*, . - . | (x)
If the S's are so small that their squares and higher powers may
be neglected, expansion by Taylor's Theorem gives
y-F(A',B',C', . . . ,x,z, . .
dF dF , dF
,,. . .,,,..
By putting
y-F(A',B',C', . . . ,x,z, . . . ) = ;
(156)
and transposing, equation (xi) becomes
adi + 65 2 + c5 3 + . . . = s. (157)
As many independent equations of this type as there are sets of
corresponding observations on y, x, z, etc., can be formed. The
absolute term s and the coefficients a, 6, c, etc., in each equation
are computed from a single set of observations by the relations
208 THE THEORY OF MEASUREMENTS [ART. 90
(156) with the aid of the approximate values A', B f , C", etc. Since
the resulting equations are in the same form as the observation
equations (53), the normal equations (56) may be found and
solved by the methods described in Chapter VII. The values
of $1, 6 2 , 5 3 , etc., thus obtained, when substituted in (155), give
second approximations to the values of the constants A, B, C,
etc.
The accuracy of the second approximations will depend on the
assumed form of the function F and on the magnitude of the correc-
tions Si, 6 2 , 6 3 , etc. If these corrections are not small, the con-
ditions underlying equation (xi) are not fulfilled and the results
obtained by the above process may deviate widely from the correct
values of the constants; but, except in extreme cases, they are
more accurate than the first approximations A', B f , C', etc. Let
A", B", C", etc., represent the second approximations. The
corresponding residuals, n, r 2 , . . . , r n , may be computed by
substituting different sets of corresponding observations on y,
x, z, etc., successively in the equation
F(A",B",C", . . . ,x,z, . . . )-y = r, (xii)
where the function F has the same form that was used in comput-
ing the corrections 5i, ^2, 5 3 , etc. If these residuals are of the same
order of magnitude as the accidental errors of the observations
and distributed in accordance with the laws of such errors, the
functional relation
y = F(A",B",C", . . . ,x,z, . . . ) (158)
is the most probable result that can be derived from the given
observations.
Frequently the residuals corresponding to the second approxi-
mations do not atisfy the above conditions. This may be due
to the inadequacy of the assumed form of the function F, to
insufficient precision of the approximations A", B", C", etc., or
to both of these causes.
If the form of the function is faulty, the residuals usually show
systematic and easily recognizable deviations from the distribu-
tion characteristic of accidental errors. Generally the number of
sign follows greatly exceeds the number of sign changes, when the
residuals are arranged in the order of increasing y's, and opposite
signs do not occur with nearly the same frequency. Sometimes
the nature of the fault can be determined by inspecting the order
ART. 91] RESEARCH 209
of sequence of the residuals or by comparing the graph correspond-
ing to equation (158) with the plotted observations. After the
form of the function F has been rectified, by the above means or
otherwise, the computations must be repeated from the beginning
and the new form must be tested in the same manner as its prede-
cessor. This process should be continued until the residuals cor-
responding to the second approximations give no evidence that
the form of the function on which they depend is faulty.
When the residuals, computed by equation (xii), do not suggest
that the assumed form of the function F is inadequate, but are
large in comparison with the probable errors of the observations,
the second approximations are not sufficiently exact. In such
cases new equations in the form of (157) are derived by using A",
B" , C", etc., in place of A', B f , C', etc., in equations (156). The
solution of the equations thus formed, by the method of least
squares, gives the corrections 5/, 5 2 ', 5 3 ', etc., that must be applied
to A", B", C", etc., in order to obtain the third approximations
At tt A n I x / . T>itt ~Dir I <j / . r</n rut \ * t . 4.
= A -f- di ; > = n + 62 ; C = C + 03 ; etc.
These operations must be repeated until the residuals correspond-
ing to the last approximations are of the same order of magnitude
as the accidental errors of the observations.
Although an algebraic expression, that represents any given
series of observations with sufficient precision, can usually be de-
rived by the foregoing methods, such a procedure is by no means
advisable in all cases. In many investigations, a graphical repre-
sentation of the results leads to quite as definite and trustworthy
conclusions as the more tedious mathematical process. Conse-
quently the latter method is usually adopted only when the former
is inapplicable or fails to utilize the full precision of the observa-
tions. In all cases the choice of suitable methods and the estab-
lishment of rational conclusions is a matter of judgment and
experience.
91. Publication. Research does not become effective as a
factor in the advancement of science until its results have been
published, or otherwise reported, in intelligible and widely acces-
sible form. It is the duty as well as the privilege of the investiga-
tor to make such report as soon as he has arrived at definite
conclusions. But nothing could be more inadvisable or untimely
than the premature publication of observations that have not been
thoroughly discussed and correlated with fundamental principles.
210 THE THEORY OF MEASUREMENTS [ART. 91
Until an investigation has progressed to such a point that it makes
some definite addition to existing ideas, or gives some important
physical constant with increased precision, its publication is likely
to retard rather than stimulate the progress of science. On the
other hand, free discussion of methods and preliminary results is
an effective molder of ideas.
The form of a published report is scarcely less important than
the substance. The significance of the most brilliant ideas may
be entirely masked by faulty or inadequate expression. Hence
the investigator should strive to develop a lucid and concise style
that will present his ideas and the observations that support
them in logical sequence. Above all things he should remember
that the value of a scientific communication is measured by the
importance of the underlying ideas, not by its length.
The author's point of view, the problem he proposes to solve,
and the ideas that have guided his work should be clearly defined.
Theoretical considerations should be rigorously developed in so
far as they have direct bearing on the work in hand. But general
discussions that can be found in well-known treatises or in easily
accessible journals should be given by reference, and the formulae
derived therein assumed without further proof whenever their
rigor is not questioned. However, the author should always
explain his own interpretation of adopted formulae and point out
their significance with respect to his observations. Due weight
and credit should be given to the ideas and results of other workers
in the same or closely related fields, but lengthy descriptions of
their methods and apparatus should be avoided. Explicit refer-
ence to original sources is usually sufficient.
The methods and apparatus actually used in making the re-
ported observations, should be concisely described, with the aid
of schematic diagrams whenever possible. Well-known methods
and instruments should be described only in so far as they have
been modified to fulfill special purposes. Detailed discussion of
all of the methods and instruments that have been found to be
inadequate are generally superfluous, but the difficulties that have
been overcome should be briefly pointed out and explained. The
precautions adopted to avoid constant errors should be explicitly
stated and the processes by which unavoidable errors of this
type have been removed from the measurements should be clearly
described. The effects likely to arise from such errors should be
ART. 91] RESEARCH 211
considered briefly and the magnitude of applied corrections should
be stated.
Observations and the results derived from them should be
reported in such form that their significance is readily intelligible
and their precision easily ascertainable. In many cases graphical
methods of representation are the most suitable provided the
points determined by the observations are accurately located
and marked. The reproduction of a large mass of numerical data
is thus avoided without detracting from the comprehensiveness
of the report. When such methods do not exhibit the full pre-
cision of the observations or when they are inapplicable on account
of the nature of the problem in hand, the original data should be
reproduced with sufficient fullness to substantiate the conclusions
drawn from them. In such cases the significance of the obser-
vations and derived results can generally be most convincingly
brought out by a suitable tabulation of numerical data. An
estimate of the precision attained should be made whenever the
results of the investigation can -be expressed numerically.
Final conclusions should be logically drawn, explicitly stated,
and rigorously developed in their theoretical bearings. They
express a culmination of the author's ideas relative to the inves-
tigated phenomena and invite criticism of their exactness and
rationality. Unless they are amply substantiated by the obser-
vations and theoretical considerations brought forward in their
support, and constitute a real addition to scientific knowledge,
they are likely to receive scant recognition.
TABLES.
The following tables contain formulae and numerical data that
will be found useful to the student in applying the principles
developed in the preceding chapters. The four figure numerical
tables are amply sufficient for the computation of errors, but more
extensive tables should be used in computing indirectly measured
magnitudes whenever the precision of the observations warrants
the use of more than four significant figures.
The references placed under some of the tables indicate the
texts from which they were adapted.
TABLE I. DIMENSIONS OF UNITS.
Units.
Dimensions.
Fundamental.
Length, mass, time
Length, force, time.
Length
[L]
[M]
[T]
[LMT-*]
m
M
[L-W]
[Llr*\
[LT-i]
pNj
[LT-*\
[T- 2 ]
[LMT~ l ]
[L*M]
[LW7 7 - 1 ]
[L*MT-*]
[L-W7 1 - 2 ]
[LW7 7 - 2 ]
[Lwr- 8 ]
[L]
[L-iFT*]
[T]
[F]
[V]
[If]
[L-*FT*]
[LL-i]
[LT-i]
[T- 1 ]
[LT-*]
[T - 2]
[FT]
[LFT*]
[LFT]
[LF]
[L-*F]
[LF]
[LFT- 1 ]
Mass
Time
Force
Area
Volume
Density
Angle
Velocity, linear
Velocity, angular
Acceleration, linear
Acceleration, angular
Momentum
Moment of inertia
Moment of momentum
Torque
Pressure
Energy, work
Power
212
TABLES
213
TABLE II. CONVERSION FACTORS.
Length Units.
Logarithm.
1 centimeter (cm.) _ = 0. 393700 inch 1 . 5951654
" " = 0. 0328083 foot 2. 5159842
" = 0. 0109361 yard 2. 0388629
1 meter (m.) = 1000 millimeters 3. 0000000
" = 100 centimeters 2. 0000000
" = 10 decimeters. 1.0000000
1 kilometer (km.) = 1000 meters 3. 0000000
= 0. 621370 mile 1. 7933503
" = 3280. 83 feet 3. 5159842
1 inch (in.) = 2. 540005 centimeters 0. 4048346
1 foot (ft.) =12 inches 1.0791812
= 30. 4801 centimeters 1 . 4840158
1 yard (yd.) = 36 inches 1. 5563025
" =3 feet 0. 4771213
" = 91.4402 centimeters 1.9611371
1 mile (ml.) = 5280 feet 3 . 7226339
" = 1760 yards 3. 2455127
= 1609. 35 meters 3.2066497
= 0.868392 knot (U. S.) 1.9387157
Mass Units.
1 gram (g.) = 1000 milligrams 3. 0000000
" = 100 centigrams 2. 0000000
" = 10 decigrams 1. 0000000
= 0.0352740 ounce (av.) 2.5474542
" = 0. 00220462 pound (av.) 3. 3433342
= 0. 000068486 slugg 5. 8355997
1 kilogram (kg.) = 1000 grams 3. 0000000
1 ounce (oz.) (av.) = 28. 3495 grams 1. 4525458
= 0. 062500 pound (av.) 2. 7958800
" =0.0019415 slugg 3.2881455
1 pound (Ib.) (av.) = 16 ounces (av.) 1.2041200
" = 453. 5924277 grams 2. 6566658
= 0.0310646 slugg 2.4922655
1 slugg (sg.) = 32. 191 pounds (av.) 1. 5077345
= 515.06 ounces (av.) 2.7118545
= 14601. 6 grams 4. 1644003
1 short ton (tn.) = 2000 pounds (av.) 3. 3010300
= 907. 185 kilograms 2. 9576958
" =62. 129 sluggs 1 . 7932955
214
THE THEORY OF MEASUREMENTS
TABLE II. CONVERSION FACTORS (Concluded}.
Force Units.
The following gravitational units are expressed in terms of the earth's
attraction at London where the acceleration due to gravity is 32.191 ft. /sec. 2
or 981.19 cm./sec Logarithm.
1 dyne = 1 . 01917 milligram's wt 0. 0082469
" = 0. 00101917 gram's wt 3 . 0082469
" =2.2469 X 10- 6 pound's wt 6.3515811
1 gram's wt. = 981.19 dynes 2. 9917531
1 kilogram's wt. = 1000 gram's wt 3. 0000000
= 98. 119 X 10 4 dynes 5.9917531
= 2.20462 pound's wt 0. 3433342
1 pound's wt. =0. 45359 kilogram's wt 1 . 6566658
= 44.506 X 10 4 dynes 5.6484189
1 pound's wt. (local) = 0/32.191 pound's wt. at London.
g = local acceleration due to gravity in ft./secT 2 .
Mean Solar Time Units.
1 second (s.) = 0. 016667 minute 2. 2218487
" = 0. 00027778 hour 4. 4436975
= 0.000011574 day 5.0634863
1 minute (m.) = 60 seconds 1 . 7781513
" =0.016667 hour 2.2218487
= 0.00069444 day 4.8416375
1 hour (h.) = 3600 seconds 3. 5563025
= 60 minutes 1. 7781513
" = 0. 041667 day 2. 6197888
1 day (d.) = 86400 seconds 4. 9365137
= 1440 minutes 3. 1583625
" =24 hours 1.3802112
1 mean solar unit = 1 . 00273791 sidereal units 0. 0011874
Angle Units.
1 circumference = 360 degrees 2. 5563025
= 2 TT radians 0. 7981799
" = 6.28319 radians 0. 7981799
1 degree () = 0. 017453 radian 2. 2418774
= 60 minutes 1. 7781513
= 3600 seconds 3. 5563025
1 minute (') =2. 9089 X 10- 4 radians 4 . 4637261
= 0.016667 degree 2.2218487
= 60 seconds 1. 7781513
1 second (') = 4.8481 X KH 5 radians 6 . 6855749
= 2. 7778 X 10- 4 degrees 4. 4436975
= 0. 01667 minute 2. 2218487
1 radian = 57.29578 degrees 1. 7581226
= 3437.7468 minutes 3. 5362739
= 206264.8 seconds . . 5. 3144251
TABLES 215
TABLE III. TRIGONOMETRICAL RELATIONS.
a 3 . a 5 t <t\,
sma = a 777 +T? (!)
(2n-l)!
cos 2 a =
1 cos 2 a
2 cosec a
_ . ce a cos a tan a
= 2 sin ^ cos tr
2 2 cot a sec a
tan a 1
- = cos a tan a
Vl+tan 2 a VI + cot 2 a
= sin /3 cos (|8 a) cos /3 sin (/3 a)
= cos /3 sin (0 + a) sin /8 cos (/3 + a).
l/l
a =y
cos a
2~~
2 tan a
sin 2 a = 2 sin a cos a = .,
1 + tan 2 a.
sin 2 a = 1 cos 2 a = \ (cos 2 a 1).
sin (a j8) = sin a cos cos a sin 0.
sin a =fc sin /3 = 2 sin (a d= /8) cos |( =F 0).
sin 2 a + sin 2 /3 = 1 cos (a + /3) cos (a /8).
sin 2 a sin 2 = cos 2 /3 cos 2 a = sin (a + 0) sin (a /8).
V 1 + sin a = sin | a + cos a.
VI sin a = (sin | a cos \ a).
cos
cos 2 i a sin 2 | a
cot a
V 1 + tan 2 a V 1 + cot 2 a
sin a cot a 1
= sin a cot a
COS ^ a =
tan o; cosec a sec a
= cos cos (a + /8) + sin sin (a + 0)
= cos /? cos ((8 - a) + sin ft sin (0 a).
1 + cos a
216 THE THEORY OF MEASUREMENTS
TABLE III. TRIGONOMETRICAL RELATIONS (Continued).
cos 2 a = 2 cos 2 a 1 = 1 2 sin 2 a
1 - tan 2 a
= cos 2 a sm 2 a. = ^
1 + tan 2 a
cos 2 a. = 1 - sin 2 a = (cos 2 a + 1).
cos (a d= 0) = cos a cos T sin a sin 0.
cos a + cos = 2 cos 5 (a + 0) cos H 0)-
cos a cos = 2 sin (a -f 0) sin | (a 0) .
cos 2 a + cos 2 = 1 + cos (a + 0) cos (a - 0).
cos 2 a cos 2 = sin 2 sin 2 a = sin (a + 0) sin (a 0).
cos 2 a sin 2 = cos (a + 0) cos (a 0) = cos 2 sin 2 a.
sin a + cos a = V 1 + sin 2 a.
sin a cos a = Vi sin 2 a.
sin 2 a + cos 2 a. = 1.
sin 2 a cos 2 a: = cos 2 or.
tan a = a + | a 3 + - r 2 5 CK 5 + 3^5 a 7 + . . . w > a> TT
sin a. sin 2 a 1 cos 2
cos a 1 + cos 2 a sin 2 a
V'l cos 2 a _ 4 /
1+ cos 2 a " V
cos 2 a VI sin 2 a
= Vsec 2 a I
tan 2 a =
cosec a: Vcosec 2 a-l
= cot a 2 cot 2 a
cot a
sin (a + 0) + sin ( 0) _ cos (a 0) cos (a + 0)
cos (a + 0) + cos (a 0) sin (a + 0) - sin (a - 0)
2 tan a 2 cot a 2
1 tan 2 a cot 2 a 1 cot a tan a
tan f a. = - ; - = cosec a cot or.
1 + sec a
( . R\ tan a tan _ cos 2 cos 2 a
* W * 1 T tan a tan ~ sin 2 =F sin 2 a
sin (a 0)
tan a tan =
cos a cos
TABLES 217
TABLE III. TRIGONOMETRICAL RELATIONS (Concluded).
Ill 2
cot a. = -- - a j= a 3 ^r-= a 5 TT > a > IT
a: 3 45 olo
cos a _ sin 2 a _ 1 + cos 2 a
sin a ~~ 1 cos 2 a "~ sin 2 a
V/
1 + cos 2 o: _ cos a vl sin 2 a
1 cos 2 a Vl cos 2 a
= tan a. + 2 cot 2 a.
tan a.
_ 1 tan 2 a. _ cot 2 a 1 cot a tan a
" 2 tan a 2cota ~^~
cot - a = (1 + sec a) cot a
2<-. v j. | kjv/v; <-*. y vv/u c*. :
cosec a cot a
1 =F tan a tan /? cot cot =F 1
cot (a d= 0) =
tan a tan cot d= cot a
sin
TABLE IV. SERIES.
Taylor's Theorem.
/(*+&)=/(*) + AT (*) + ^/" (*)+;+ ^/W (x) +
f(x + h, y + k,
where u = f (x, y, z).
Maclaurin's Series.
/(0) + f /' (0) + !/" (0) + + fj/N (0).
218 THE THEORY OF MEASUREMENTS
TABLE IV. SERIES (Concluded}.
Binomial Theorem.
= xm + rn x ^ ly + m(n^ xm _^ +
. . . , * (m - 1) . . . (m - n + 1) ^- y>
when m is a positive integer, also when m is negative or fractional and
x > y. When x < y and m is fractional or negative the series must be
taken in the form
(x + y) m = y m + j y m -*x+ v ^ *' y *-'z +
m (m - 1) . . . (m - n + 1)
n!
Fourier's Series.
j- / \ It it ""E i t 2 7TX , 3 7TX .
/ (x) = - 6 + &i cos H &2 cos - + 6 3 cos H
C C C
. TTX . . 2irX . . STTX ,
+ 01 sin \- a 2 sin h a 3 sin f-
c c c
where
1 r + c ,/ v WTTX .
>m = ~ I / (*) COS - dx,
C / c t/
1 f +c r/ v . m-n-x ,
m = - \ f(x) sm dx,
C / c ^
2 / c , , , . WTTX ,
= - I / W sin - "
C /o C
provided / (x) is single valued, uniform, and continuous, and c > x >
c. For values of x lying between zero and c the function may be ex-
panded in the form
, / x . TTX . . 2-JTX . . 3 TTX ,
f (x) = 0,1 sin -- \-a-i sin -- H a 3 sin --- (- ,
where a
Also f(x) =^60 + 61 cosy 4-6 2 cos + 6 3 cos
2 r c - / x WTTX ,
where b m = - I / (x) cos - ax.
C JQ C
General Series.
xloga (x log a) 2 (x log a) 3 (x log a) n
~~ ~~ ~~ ~
.- :>}
TABLES
219
TABLE V. DERIVATIVES.
U, F, W any functions; a, 6, c constants.
dx
F 2
S : ***St^T?
axx
a , log a e.
_log a x= ,
dU
a . i at;
_ logaC7 . = __
V dx
= a x log a.
dx
d
dx (
a
dx
sm x = cos x.
ax
. r , .
sm aC7 = a cos ac7 ^ ,
ax ax
a l
tan x = r = sec 2 x:
ax cos 2 x
cos x = sm x.
ax
a -i
cot x = . , = cosec 2 x.
ax sin 2 x
sec x = tan x sec x;
oX
cosec x = cot x cosec x.
ax
log sinx = cotx;
log cos x = tan x.
ox
The following expressions for the derivatives of inverse functions hold
for angles in the first and third quadrants. For angles in the second and
fourth quadrants the signs should be reversed.
ax
tan- 1 x = =
ax i
.
T- cos- 1 x =
dx
i
220 THE THEORY OF MEASUREMENTS
TABLE VI. SOLUTION OF EQUATIONS.
The following algebraic expressions for the roots of equations of the
second, third, and fourth degrees are in the form given by Merriman.
(Merriman and Woodward, "Higher Mathematics"; Wiley and Sons,
1896.)
The Quadratic Equation.
Reduce to the form
x 2
Then the two roots are
x\ = a + a? 6; z 2 = a Va 2 b
The Cubic Equation.
Reduce to the form
= 0.
Compute the following auxiliary quantities :
B = - a 2 + 6; C = a 3 - f ab + c;
Then the three roots are
xi=-a + (si + s 2 ), _
x z =-a -Mi+s 2 ) +| V-_3( Sl -s 2 ),
x 3 = - a - HSI + s 2 ) - | V- 3 (si - s a ).
When B 3 + C 2 is negative the roots are all real but they cannot be de-
termined numerically by the above formulae owing to the complex nature
of si and s 2 . In such cases the numerical values of the roots can be deter-
mined only by some method of approximation.
The Quartic Equation.
Reduce to the form
z 4 + 4az 3 + 66z 2 + 4cz + d = 0.
Compute the following auxiliary quantities :
g = a*-b; h = 6 3 + c 2 -2abc + dg; fc = |ac - 6 2 - |d;
I = I (h + V^TF')* + 1 (h - VF+^)*;
u = g + l', v = 2g-l; w = 4u* + 3k - 12gl.
Then the four roots are
xi = a + ^u + Vy +
a u
in which the signs are to be used as written provided that 2 a 3 3 ab + c
is a negative number; but if this is positive all radicals except Vw are to
be changed in sign.
The above expressions are irreducible when h z + k* is a negative number.
In this case the given equation has either four imaginary roots or four real
roots that can be determined numerically only by some method of approxi-
mation.
TABLES 221
TABLE VII. APPROXIMATE FORMULA.
In the following formulae, a, /3, 5, etc., represent quantities so small that
their squares, higher powers, and products are negligible in comparison with
unity. The limit of negligibility depends on the particular problem in
hand. Most of the formulae give results within one part in one million
when the variables are equal to or less than 0.001.
1. (l+a) n =l+n; (1 -a) n = 1 - na.
4.
6 l = 1 --' , l = 1 +-
' Vl+ n' Vl -a n
7.
9. (x + a
When the angle a, expressed in radians, is small in comparison with unity
a first approximation gives
10. sin a = a', sin (x a) = sin x a cos x.
11. cos a = 1; cos (x a) = cos x =F a sin x.
12. tan a = a] tan (x d= a) = tana; ^
The second approximation gives
13. sin a = a -TT ; sin 2 a = a 2 1 ^r-
o \ o
a 2
14. cos a = 1 -5- ; COS 2 a = 1 a 2 .
3 / o \
15. tana = a + ^-| tan 2 a = a 2 ( 1 + ^ a 2 V
(Kohlrausch, "Praktische Physik.")
222 THE THEORY OF MEASUREMENTS
TABLE VIII. NUMERICAL CONSTANTS.
Logarithm .
Base of Naperian logarithms: e = 2. 7182818 ........ 0. 4342945
Modulus of Naperian log.: M = ^ = 2.30259 ........... 0.3622157
Modulus of common log.: = log e = 0. 4342945 ......... 1. 6377843
Circumference ,.. 1415 9265 . 0. 4971499
Diameter
2?r = 6.28318530 .............. 0.7981799
- =0.3183099 . 1.5028501
7T
Tr 2 = 9.8696044 . . ............. 0.9942998
V^ = 1.7724539 ............... 0.2485749
| = 0.7853982 ............... 1.8950899
5 =0.5235988 . 1.7189986
o
w = Precision constant; k = Unit error; A = Average error;
M = Mean error; E = Probable error.
4p = 0.31831 ................. 1.5028501
^ = 0.39894 ................. 1.6009101
^ = 0.26908 ................. 1.4298888
^ = 1.25331 ................. 0.0980600
A.
f = 0.84535 ................. 1.9270387
A.
= 0.67449 ................. 1.8289787
TABLES
223
TABLE IX. EXPONENTIAL FUNCTIONS.
X
logic (e*)
e*
e*
X
log 10 (O
e'
e~'
0.0
0.00000
1.0000
1.000000
5.0
2.17147
148.41
0.006738
0.1
0.04343
1.1052
0.904837
5.1
2.21490
164.02
0.006097
0.2
0.08686
1.2214
0.818731
5.2
2.25833
181.27
0.005517
0.3
0.13029
1.3499
0.740818
5.3
2.30176
200.34
0.004992
0.4
0.17372
1.4918
0.670320
5.4
2.34519
221.41
0.004517
0.5
0.21715
1.6487
0.606531
5.5
2.38862
244.69
0.004087
0.6
0.26058
1.8221
0.548812
5.6
2.43205
270.43
0.003698
0.7
0.30401
2.0138
0.496585
5.7
2.47548
298.87
0.003346
0.8
0.34744
2.2255
0.449329
5.8
2.51891
330.30
0.003028
0.9
0.39087
2.4596
0.406570
5.9
2.56234
365.04
0.002739
1.0
0.43429
2.7183
0.367879
6.0
2.60577
403.43
0.002479
1.1
0.47772
3.0042
0.332871
6.1
2.64920
445.86
0.002243
1.2
0.52115
3.3201
0.301194
6.2
2.69263
492.75
0.002029
1.3
0.56458
3.6693
0.272532
6.3
2.73606
544.57
0.001836
1.4
0.60801
4.0552
0.246597
6.4
2.77948
601.85
0.001662
.5
0.65144
4.4817
0.223130
6.5
2.82291
665.14
0.001503
.6
0.69487
4.9530
0.201897
6.6
2.86634
735.10
0.001360
.7
0.73830
5.4739
0.182684
6.7
2.90977
812.41
0.001231
.8
0.78173
6.0496
0.165299
6.8
2.95320
897.85
0.001114
.9
0.82516
6.6859
0.149569
6.9
2.99663
992.27
0.001008
2.0
0.86859
7.3891
0.135335
7.0
3.04006
1096.6
0.000912
2.1
0.91202
8.1662
0.122456
7.1
3.08349
1212.0
0.000825
2.2
0.95545
9.0250
0.110803
7.2
3.12692
1339.4
0.000747
2.3
0.99888
9.9742
0.100259
7.3
3.17035
1480.3
0.000676
2.4
1.04231
11.023
0.090718
7.4
3.21378
1636.0
0.000611
2.5
1.08574
12.182
0.082085
7.5
3.25721
1808.0
0.000553
2.6
1.12917
13.464
0.074274
7.6
3.30064
1998.2
0.000500
2.7
1 . 17260
14.880
0.067206
7.7
3.34407
2208.3
0.000453
2.8
1.21602
16.445
0.060810
7.8
3.38750
2440.6
0.000410
2.9
1.25945
18.174
0.055023
7.9
3.43093
2697.3
0.000371
3.0
1.30288
20.086
0.049787
8.0
3.47436
2981.0
0.000335
3.1
1.34631
22.198
0.045049
8.1
3.51779
3294.5
0.000304
3.2
1.38974
24.533
0.040762
8.2
3.56121
3641.0
0.000275
3.3
1.43317
27.113
0.036883
8.3
3.60464
4023.9
0.000249
3.4
1.47660
29.964
0.033373
8.4
3.64807
4447.1
0.000225
3.5
1.52003
33.115
0.030197
8.5
3.69150
4914.8
0.000203
3.6
1.56346
36.598
0.027324
8.6
3.73493
5431.7
0.000184
3.7
1.60689
40.447
0.024724
8.7
3.77836
6002.9
0.000167
3.8
1.65032
44.701
0.022371
8.8
3.82179
6634.2
0.000151
3.9
1.69375
49.402
0.020242
8.9
3.86522
7332.0
0.000136
4.0
1.73718
54.598
0.018316
9.0
3.90865
8103.1
0.000123
4.1
.78061
60.340
0.016573
9.1
3.95208
8955.3
0.000112
4.2
.82404
66.686
0.014996
9.2
3.99551
9897.1
0.000101
4.3
.86747
73.700
0.013569
9.3
4.03894
10938.
0.000091
4.4
.91090
81.451
0.012277
9.4
4.08237
12088.
0.000083
4.5
.95433
90.017
0.011109
9.5
4.12580
13360.
0.000075
4.6
.99775
99.484
0.010052
9.6
4.16923
14765.
0.000068
4.7
2.04118
109.95
0.009095
9.7
4.21266
16318.
0.000061
4.8
2.08461
121.51
0.008230
9.8
4.25609
18034.
0.000055
4.9
2.12804
134.29
0.007447
9.9
4.29952
19930.
0.000050
5.0
2.17147
148.41
0.006738
10.0
4.34294
22026.
0.000045
Taken from Glaisher's "Tables of the Exponential Function," Trans. Cambridge Phil. Soc.,
vol. xiii, 1883. This volume also contains a " Table of the Descending Exponential to Twelve
or Fourteen Places of Decimals," by F. W. Newman.
224
THE THEORY OF MEASUREMENTS
TABLE X. EXPONENTIAL FUNCTIONS.
Value of e x<t and er x<i and their logarithms.
X
<?
log e 2
e~* 2
log e'* z
0.1
1.0101
0.00434
0.99005
1.99566
0.2
1.0408
0.01737
0.96079
1.98263
0.3
1.0942
0.03909
0.91393
1.96091
0.4
.1735
0.06949
0.85214
.93051
0.5
.2840
0.10857
0.77880
.89143
0.6
.4333
0.15635
0.69768
.84365
0.7
.6323
0.21280
0.61263
.78720
0.8
.8965
0.27795
0.52729
.72205
0.9
2.2479
0.35178
0.44486
.64822
1.0
2.7183
0.43429
0.36788
.56571
1.1
3.3535
0.52550
0.29820
.47450
1.2
4.2207
0.62538
0.23693
.37462
1.3
5.4195
0.73396
0.18452
.26604
1.4
7.0993
0.85122
0.14086
.14878
1.5
9.4877
0.97716
0.10540
.02284
1.6
1.2936X10
1.11179
0. 77305 XlO- 1
2.88821
1.7
1.7993X10
1.25511
0. 55576 XlO- 1
2.74489
1.8
2.5534x10
1.40711
0. 39164 XlO- 1
2.59289
1.9
3.6966X10
1.56780
0.27052 XlO- 1
2.43220
2.0
5.4598X10
1.73718
0.18316 XlO- 1
2.26282
2.1
8.2269x10
1.91524
0.12155 XlO- 1
2.08476
22
1.2647X10 2
2.10199
0.79071 XlO- 2
3.89801
2.3
1.9834X10 2
2.29742
0.50417 XlO- 2
3.70258
2.4
3.1735X10 2
2.50154
0.31511 XlO- 2
3.49846
2.5
5.1801X10 2
2.71434
0.19305 XlO- 2
3.28566
2.6
8.6264X10 2
2.93583
0.1 1592 XlO- 2
3.06417
2.7
1.4656X10 3
3.16601
0.68232X10- 3
4.83399
2.8
2.5402X10 3
3.40487
0.39367X10- 3
4.59513
2.9
4.4918X10 3
3.65242
0.22263X10-3
4.34758
3.0
8.1031X10 3
3.90865
0.12341X10- 3
4.09135
3.1
1.4913X10 4
4.17357
0.67055x10-*
5.82643
3.2
2.8001X10 4
4.44718
0.35713 XlO- 4
5.55282
3.3
5.3637X10 4
4.72947
0.18644 XlO- 4
5.27053
3.4
1.0482X10 5
5.02044
0.95403 XlO- 5
6.97956
3.5
2.0898X10 5
5.32011
0.47851 XlO- 5
6.67989
3.6
4.2507X10 5
5.62846
0.23526 XlO- 5
6.37154
3.7
8.8204X10 5
5.94549
0.1 1337 XlO- 5
6.05451
3.8
1.8673X10 6
6.27121
0.53554 XlO- 6
7.72879
3.9
4.0329X10 6
6.60562
0.24796 XlO- 6
?. 39438
4.0
8.8861X10 6
6.94871
0.11254X10- 6
7.05129
41
1.9975X10 7
7.30049
0.50062 XlO- 7
.69951
4.2
4.5809X10 7
7.66095
0.21830 XlO- 7
S. 33905
4.3
1.0718X10 8
8.03011
0.93302 XlO- 8
9.96989
4.4
2.5582X10 8
8.40794
0.39089 XlO- 8
9.59206
4.5
6.2296X10 8
8.79446
0.16052X10- 8
9.20554
4.6
1.5476X10 9
9.18967
0.64614X10- 9
10.81033
4.7
3.9226X10 9
9.59357
0.25494X10- 9
10.40643
4.8
1.0143X10 10
10.00615
0.98594 XlO- 10
11.99385
4.9
2.6755X10 10
10.42741
0.37376 XlO- 10
11.57259
5.0
7.2005X10 10
10.85736
0.13888 XlO- 10
11.14264
TABLES
225
TABLE XI. VALUES OF THE PROBABILITY INTEGRAL.
t
P*
Diff.
t
^A
Diff.
1
^A
Diff
t
^A
Diff.
0.00
0.00000
1 IOC
0.50
0.52050
074
1.00
0.84270
A 1
1.50
0.96611
0.01
0.01128
1 I !
1 100
0.51
0.52924
o/ 1
O/3f>
1.01
0.84681
411
A AO
1.51
0.96728
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.02256
0.03384
0.04511
0.05637
0.06762
0.07886
0.09008
0.10128
liZo
1128
1127
1126
1125
1124
1122
1120
1 1 1C
0.52
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.53790
0.54646
0.55494
0.56332
0.57162
0.57982
0.58792
0.59594
ODD
856
848
838
830
820
810
802
7QO
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
0.85084
0.85478
0.85865
0.86244
0.86614
0.86977
0.87333
0.87680
403
394
387
379
370
363
356
347
O/M
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
0.96841
0.96952
0.97059
0.97162
0.97263
0.97360
0.97455
0.97546
113
111
107
103
101
97
95
91
OA
0.10
0.11
0.12
0.11246
0.12362
0.13476
1 1 J.O
1116
1114
1111
0.60
0.61
0.62
0.60386
0.61168
0.61941
/y^
782
773
7j
1.10
1.11
1.12
0.88021
0.88353
0.88679
O41
332
326
010
1.60
1.61
1.62
0.97635
0.97721
0.97804
89
86
83
Qf\
0.13
0.14
0.15
0.16
0.17
0.14587
0.15695
0.16800
0.17901
0.18999
1 1 i. i
1108
1105
1101
1098
1 AQ~
0.63
0.64
0.65
0.66
0.67
0.62705
0.63459
0.64203
0.64938
0.65663
< D^
754
744
735
725
71 t\
1.13
1.14
1.15
1.16
1.17
0.88997
0.89308
0.89612
0.89910
0.90200
OIo
311
304
298
290
oo/i
1.63
.64
.65
.66
.67
0.97884
0.97962
0.98038
0.98110
0.98181
oU
78
76
72
71
f*Q
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
0.30
0.31
0.32
0.20094
0.21184
0.22270
0.23352
0.24430
0.25502
0.26570
0.27633
0.28690
0.29742
0.30788
0.31828
0.32863
0.33891
0.34913
iuyo
1090
1086
1082
1078
1072
1068
1083
1057
1052
1046
1040
1035
1028
1022
1 A1 K.
0.68
0.69
0.70
0.71
0.72
0.73
0.74
0.75
0.76
0.77
0.78
0.79
0.80
0.81
0.82
0.66378
0.67084
0.67780
. 68467
0.69143
0.69810
0.70468
0.71116
0.71754
0.72382
0.73001
0.73610
0.74210
0.74800
0.75381
< 10
706
696
687
676
667
658
648
638
628
619
609
600
590
581
CT1
1.18
1.19
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
0.90484
0.90761
0.91031
0.91296
0.91553
0.91805
0.92051
0.92290
0.92524
0.92751
0.92973
0.93190
0.93401
0.93606
0.93807
Zo4
277
270
265
257
252
246
239
234
227
222
217
211
205
201
.68
.69
.70
.71
.72
.73
1.74
1.75
1.76
1.77
1.78
1.79
1.80
1.81
1.82
0.98249
0.98315
0.98379
0.98441
0.98500
0.98558
0.98613
0.98667
0.98719
0.98769
0.98817
0.98864
0.98909
0.98952
0.98994
Do
66
64
62
59
58
55
54
52
50
48
47
45
43
42
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.35928
0.36936
0.37938
0.38933
0.39921
0.40901
0.41874
lUio
1008
1002
995
988
980
973
Qf?r
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.75952
0.76514
0.77067
0.77610
0.78144
0.78669
0.79184
571
562
553
543
534
525
515
CA*7
.33
.34
.35
.36
.37
.38
.39
0.94002
0.94191
0.94376
0.94556
0.94731
0.94902
0.95067
195
189
185
180
175
171
165
1 JO
1.83
1.84
1.85
1.86
1.87
1.88
1.89
0.99035
0.99074
0.99111
0.99147
0.99182
0.99216
0.99248
41
39
37
36
35
34
32
01
0.40
0.41
0.42839
0.43797
yoo
958
nr:n
0.90
0.91
0.79691
0.80188
oU/
497
4P.Q
.40
.41
0.95229
0.95385
ItW
156
i ^.
1.90
1.91
0.99279
0.99309
ol
30
on
0.42
0.44747
you
n/io
0.92
0.80677
"oy
A>-r{\
1.42
0.95538
100
1 AO
1.92
3.99338
zy
oo
0.43
0.44
0.45689
0.46623
y4z
934
QOC
0.93
0.94
0.81156
0.81627
479
471
4fi9
1.43
1.44
0.95686
0.95830
148
144
140
1.93
1.94
3.99366
3.99392
28
26
9fi
0.45
0.47548
t/^O
QIC
0.95
0.82089
'\j
4CQ
1.45
0.95970
J.TAJ
IOC
1.95
3.99418
^O
OK
0.46
0.47
0.48
0.49
0.50
0.48466
0.49375
0.50275
0.51167
0.52050
7 j.o
909
900
892
883
0.96
0.97
0.98
0.99
1.00
0.82542
0.82987
0.83423
0.83851
0.84270
^too
445
436
428
419
1.46
1.47
1.48
1.49
1.50
0.96105
0.96237
0.96365
0.96490
0.96611
J.OO
132
128
125
121
1.96
1.97
1.98
1.99
2.00
3.99443
3.99466
3.99489
3.99511
3.99532
^O
23
23
22
21
oo
.00000
(Chauvenet, " Spherical and Practical Astronomy.")
226 THE THEORY OF MEASUREMENTS
TABLE XII. VALUES OF THE PROBABILITY INTEGRAL.
3
1
2
3
4
5
6
7
8
9
0.0
.00000
.00538
.01076
.01614
.02152
.02690
.03228
.03766
. 04303
.04840
0.1
.05378
.05914
.06451
.06987
.07523
.08059
.08594
.09129
.09663
. 10197
0.2
.10731
.11264
.11796
. 12328
. 12860
. 13391
. 13921
. 14451
. 14980
. 15508
0.3
.16035
. 16562
. 17088
. 17614
. 18138
. 18662
.19185
. 19707
.20229
.20749
0.4
.21268
.21787
.22304
.22821
.23336
.23851
.24364
.24876
.25388
.25898
0.5
.26407
.26915
.27421
.27927
.28431
.28934
.29436
.29936
.30435
.30933
0.6
.31430
.31925
.32419
.32911
.33402
.33892
.34380
.34866
.35352
.35835
0.7
.36317
.36798
.37277
.37755
.38231
.38705
.39178
.39649
.40118
.40586
0.8
.41052
.41517
.41979
.42440
. 42899
. 43357
. 43813
. 44267
.44719
.45169
0.9
.45618
.46064
.46509
.46952
.47393
. 47832
. 48270
. 48605
.49139
.49570
.0
.50000
.50428
.50853
.51277
.51699
.52119
.52537
.52952
.53366
.53778
.1
.54188
.54595
.55001
.55404
.55806
.56205
.56602
. 56998
.57391
.57782
.2
.58171
.58558
.58942
.59325
.59705
.60083
.60460
. 60833
.61205
.61575
.3
.61942
.62308
.62671
.63032
.63391
.63747
.64102
.64454
.64804
.65152
.4
.65498
.65841
.66182
.66521
.66858
.67193
.67526
.67856
.68184
.68510
.5
.68833
.69155
.69474
.69791
.70106
.70419
.70729
.71038
.71344
.71648
.6
.71949
.72249
.72546
.72841
.73134
.73425
.73714
.74000
.74285
.74567
.7
.74847
.75124
.75400
.75674
.75945
.76214
.76481
.76746
.77009
.77270
.8
.77528
.77785
.78039
.78291
.78542
.78790
.79036
.79280
.79522
.79761
.9
.79999
.80235
.80469
.80700
.80930
.81158
.81383
.81607
.81828
.82048
2.0
.82266
.82481
.82695
.82907
.83117
.83324
.83530
.83734
.83936
.84137
2.1
.84335
.84531
.84726
.84919
.85109
.85298
.85486
.85671
.85854
.86036
2.2
.86216
.86394
.86570
.86745
.86917
.87088
.87258
.87425
.87591
.87755
2.3
.87918
.88078
.88237
.88395
.88550
.88705
.88857
.89008
: 89157
.89304
2.4
.89450
.89595
.89738
.89879
.90019
.90157
.90293
.90428
.90562
.90694
25
.90825
.90954
.91082
.91208
.91332
.91456
.91578
.91698
.91817
.91935
2.6
.92051
.92166
.92280
.92392
.92503
.92613
.92721
.92828
.92934
.93038
2.7
.93141
.93243
.93344
.93443
.93541
.93638
.93734
.93828
.93922
.94014
2.8
.94105
.94195
.94284
.94371
.94458
.94543
.94627
.94711
.94793
.94874
2.9
.94954
.95033
.95111
.95187
.95263
.95338
.95412
.95485
.95557
.95628
3
.95698
.96346
96910
.97397
.97817
.98176
.98482
.98743
.98962
.99147
4
.99302
.99431
.99539
.99627
.99700
.99760
.99808
.99848
.99879
.99905
5
.99926
.99943
.99956
.99966
.99974
.99980
.99985
. 99988
.99991
.99993
TABLE XIII. CHAUVENET'S CRITERION.
N
T
N
r
AT
r
3
2.05
13
3.07
23
3.40
4
2.27
14
3.11
24
3.43
5
2.44
15
3.15
25
3.45
6
2.57
16
3.19
30
3.55
7
2.67
17
3.22
40
3.70
8
2.76
18
3.26
50
3.82
9
2.84
19
3.29
75
4.02
10
2.91
20
3.32
100
4.16
11
2.97
21
3.35
200
4.48
12
3.02
22
3.38
500
4.90
TABLES
227
TABLE XTV. FOR COMPUTING PROBABLE ERRORS BY FORMULA
(31) AND (32).
AT
0.6745
0.6745
AT
0.6745
0.6745
iV
VJv^T
VN(N-l)
iM
vim
v# (AT- i)
40
0.1080
0.0171
41
0.1066
0.0167
2
0.6745
0.4769
42
0.1053
0.0163
3
0.4769
0.2754
43
0.1041
0.0159
4
0.3894
0.1947
44
0.1029
0.0155
5
0.3372
0.1508
45
0.1017
0.0152
6
0.3016
0.1231
46
0.1005
0.0148
7
0.2754
0.1041
47
0.0994
0.0145
8
0.2549
0.0901
48
0.0984
0.0142
9
0.2385
0.0795
49
0.0974
0.0139
10
0.2248
0.0711
50
0.0964
0.0136
11
0.2133
0.0643
51
0.0954
0.0134
12
0.2029
0.0587
52
0.0944
0.0131
13
0.1947
0.0540
53
0.0935
0.0128
14
0.1871
0.0500
54
0.0926
0.0126
15
0.1803
0.0465
55
0.0918
0.0124
16
0.1742
0.0435
56
0.0909
0.0122
17
0.1686
0.0409
57
0.0901
0.0119
18
0.1636
0.0386
58
0.0893
0.0117
19
0.1590
0.0365
59
0.0886
0.0115
20
0.1547
0.0346
60
0.0878
0.0113
21
0.1508
0.0329
61
0.0871
0.0111
22
0.1472
0.0314
62
0.0864
0.0110
23
0.1438
0.0300
63
0.0857
0.0108
24
0.1406
0.0287
64
0.0850
0.0106
25
0.1377
0.0275
65
0.0843
0.0105
26
0.1349
0.0265
66
0.0837
0.0103
27
0.1323
0.0255
67
0.0830
0.0101
28
0.1298
0.0245
68
0.0824
0.0100
29
0.1275
0.0237
69
0.0818
0.0098
30
0.1252
0.0229
70
0.0812
0,0097
31
0.1231
0.0221
71
0.0806
0.0096
32
0.1211
0.0214
72
0.0800
0.0094
33
0.1192
0.0208
73
0.0795
0.0093
34
0.1174
0.0201
74
0.0789
0.0092
35
0.1157
0.0196
75
0.0784
0.0091
36
0.1140
0.0190
80
0.0759
0.0085
37
0.1124
0.0185
85
0.0736
0.0080
38
0.1109
0.0180
90
0.0713
0.0075
39
0.1094
0.0175
100
0.0678
0.0068
(Merriman, " Least Squares. ")
228
THE THEORY OF MEASUREMENTS
TABLE XV. FOR COMPUTING PROBABLE ERRORS BY FORMULAE (34).
N
0.8453
0.8453
N
0.8453
0.8453
^N(N - 1)
N^N-1
VN(N - 1)
N^W=1
40
0.0214
0.0034
41
0.0209
0.0033
2
0.5978
0.4227
42
0.0204
0.0031
3
0.3451
0.1993
43
0.0199
0.0030
4
0.2440
0.1220
44
0.0194
0.0029
5
0.1890
0.0845
45
0.0190
0.0028
6
0.1543
0.0630
46
0.0186
0.0027
7
0.1304
0.0493
47
0.0182
0.0027
8
0.1130
0.0399
48
0.0178
0.0026
9
0.0996
0.0332
49
0.0174
0.0025
10
0.0891
0.0282
50
0.0171
0.0024
11
0.0806
0.0243
51
0.0167
0.0023
12
0.0736
0.0212
52
0.0164
0.0023
13
0.0677
0.0188
53
0.0161
0.0022
14
0.0627
0.0167
54
0.0158
0.0022
15
0.0583
0.0151
55
0.0155
0.0021
16
0.0546
0.0136
56
0.0152
0.0020
17
0.0513
0.0124
57
0.0150
0.0020
18
0.0483
0.0114
58
0.0147
0.0019
19
0.0457
0.0105
59
0.0145
0.0019
20
0.0434
0.0097
60
0.0142
0.0018
21
0.0412
0.0090
61
0.0140
0.0018
22
0.0393
0.0084
62
0.0137
0.0017
23
0.0376
0.0078
63
0.0135
0.0017
24
0.0360
0.0073
64
0.0133
0.0017
25
0.0345
0.0069
65
0.0131
0.0016
26
0.0332
0.0065
66
0.0129
0.0016
27
0.0319
0.0061
67
0.0127
0.0016
28
0.0307
0.0058
68
0.0125
0.0015
29
0.0297
0.0055
69
0.0123
0.0015
30
0.0287
0.0052
70
0.0122
0.0015
31
0.0277
0.0050
71
0.0120
0.0014
32
0.0268
0.0047
72
0.0118
0.0014
33
0.0260
0.0045
73
0.0117
0.0014
34
0.0252
0.0043
74
0.0115
0.0013
35
0.0245
0.0041
75
0.0113
0.0013
36
0.0238
0.0040
80
0.0106
0.0012
37
0.0232
0.0038
85
0.0100
0.0011
38
0.0225
0.0037
90
0.0095
0.0010
39
0.0220
0.0035
100
0.0085
0.0008
(Merriman, "Least Squares.")
TABLES
229
TABLE XVI. SQUARES OP NUMBERS.
n
i
2
3
4
5
6
7
8
9
Diff.
1.0
1.000
1.020
1.040
1.061
1.082
1.103
1.124
1.145
1.166
1.188
22
1.1
1.210
1.232
1.254
1.277
1.300
1.323
1.346
1.369
1.392
1.416
24
1.2
1.440
1.464
1.488
1.513
1.538
1.563
1.588
1.613
1.638
1.664
26
1.3
1.690
1.716
1.742
1.769
1.796
1.823
1.850
1.877
1.904
1.932
28
1.4
1.960
1.988
2.016
2.045
2.074
2.103
2.132
2.161
2.190
2.220
30
1.5
2.250
2.280
2.310
2.341
2.372
2.403
2.434
2.465
2.496
2.528
32
1.6
2.560
2.592
2.624
2.657
2.690
2.723
2.756
2.789
2.822
2.856
34
1.7
2.890
2.924
2.958
2.993
3.028
3.063
3.098
3.133
3.168
3.204
36
1.8
3.240
3.276
3.312
3.349
3.386
3.423
3.460
3.497
3.534
3.572
38
1.9
3.610
3.648
3.686
3.725
3.764
3.803
3.842
3.881
3.920
3.960
40
2.0
4.000
4.040
4.080
4.121
4.162
4.203
4.244
4.285
4.326
4.368
42
2.1
4.410
4.452
4.494
4.537
4.580
4.623
4.666
4.709
4.752
4.796
44
2.2
4.840
4.884
4.928
4.973
5.018
5.063
5.108
5.153
5.198
5.244
46
23
5.290
5.336
5.382
5.429
5.476
5.523
5.570
5.617
5.664
5.712
48
2.4
5.760
5.808
5.856
5.905
5.954
6.003
6.052
6.101
6.150
6.200
50
25
6.250
6.300
6.350
6.401
6.452
6.503
6.554
6.605
6.656
6.708
52
2.6
6.760
6.812
6.864
6.917
6.970
7.023
7.076
7.129
7.182
7.236
54
27
7.290
7.344
7.398
7.453
7.508
7.563
7.618
7.673
7.728
7.784
56
2.8
7.840
7.896
7.952
8.009
8.066
8.123
8.180
8.237
8.294
8.352
58
2.9
8.410
8.468
8.526
8.585
8.644
8.703
8.762
8.821
8.880
8.940
60
3.0
9.000
9.060
9.120
9.181
9.242
9.303
9.364
9.425
9.486
9.548
62
3.1
9.610
9.672
9.734
9.797
9.860
9.923
9.986
10.05
10.11
10.18
6
3.2
10.24
10.30
10.37
10.43
10.50
10.56
10.63
10.69
10.76
10.82
7
3.3
10.89
10.96
11.02
11.09
11.16
11.22
11.29
11.36
11.42
11.49
7
3.4
11.56
11.63
11.70
11.76
11.83
11.90
11.97
12.04
12.11
12.18
7
3.5
12.25
12.32
12.39
12.46
12.53
12.60
12.67
12.74
12.82
12.89
7
3.6
12.96
13.03
13.10
13.18
13.25
13.32
13.40
13.47
13.54
14.62
7
3.7
13.69
13.76
13.84
13.91
13.99
14.06
14.14
14.21
14.29
14.36
8
3.8
14.44
14.52
14.59
14.67
14.75
14.82
14.90
14.98
15.05
15.13
8
3.9
15.21
15.29
15.37
15.44
15.52
15.60
15.68
15.76
15.84
15.92
8
4.0
16.00
16.08
16.16
16.24
16.32
16.40
16.48
16.56
16.65
16.73
8
4.1
16.81
16.89
16.97
17.06
17.14
17.22
17.31
17.39
17.47
17.65
8
4.2
17.64
17.72
17.81
17.89
17.98
18.06
18.15
18.23
18.32
18.40
9
4.3
18.49
18.58
18.66
18.75
18.84
18.92
19.01
19.10
19.18
19.27
9
4.4
19.36
19.45
19.54
19.62
19.71
19.80
19.89
19.98
20.07
20.16
9
4.5
20.25
20.34
20.43
20.52
20.61
20.70
20.79
20.88
20.98
21.07
9
4.6
21.16
21.25
21.34
21.44
21.53
21.62
21.72
21.81
21.90
22.00
9
4.7
22.09
22.18
22.28
22.37
22.47
22.56
22.66
22.75
22.85
22.94
10
4.8
23.04
23.14
23.23
23.33
23.43
23.52
23.62
23.72
23.81
23.91
10
4.9
24.01
24.11
24.21
24.30
24.40
24.50
24.60
24.70
24.80
24.90
10
5.0
25.00
25.10
25.20
25.30
25.40
25.50
25.60
25.70
25.81
25.91
10
5.1
26.01
26.11
26.21
26.32
26.42
26.52
26.63
26.73
26.83
26.94
10
5.2
27.04
27.14
27.25
27.35
27.46
27.56
27.67
27.77
27.88
27.98
11
5.3
28.09
28.20
28.30
28.41
28.52
28.62
28.73
28.84
28.94
29.05
11
5.4
29.16
29.27
29.38
29.48
29.59
29.70
29.81
29.92
30.03
30.14
11
n
1
2
3
4
5
6
7
8
9
Diff.
(Merriman, "Least Squares.")
230
THE THEORY OF MEASUREMENTS
TABLE XVI. SQUARES OF NUMBERS (Concluded).
n
i
2
3
4
5
6
7
8
9
Diff.
5.5
30.25
30.36
30.47
30.58
30.69
30.80
30.91
31.02
31.14
31.25
11
5.6
31.36
31.47
31.58
31.70
31.81
31.92
32.04
32.15
32.26
32.38
11
5.7
32.49
32.60
32 72
32.83
32.95
33.0633.18
33.29
33.41
33.52
12
5.8
33.64
33.76
33.87
33.99
34.11
34.2234.34
34.46
34.57
34.69
12
5.9
34.81
34.93
35.05
35.16
35.28
35.40
35.52
35.64
35.76
35.88
12
6.0
36.00
36.12
36.24
36.36
36.48
36.60
36.72
36.84
36.97
37.09
12
6.1
37.21
37.33
37.45
37.58
37.70
37.82
37.95
38.07
38.19
38.32
12
6.2
38.44
38.56
38.69
38.81
38.94
39.06
39.19
39.31
39.44
39.56
13
6.3
39.69
39.82
39.94
40.07
40.20
40.32
40.45
40.58
40.70
40.83
13
6.4
40.96
41.09
41.22
41.34
41.47
41.60
41.73
41.86
41.99
42.12
13
6.5
42.25
42.38
42.51
42.64
42.77
42.90
43.03
43.16
43.30
43.43
13
6.6
43.56
43.69
43.82
43.96
44.09
44.22
44.36
44.49
44.62
44.76
13
6.7
44.89
45.02
45.16
45.29
45.43
45.56
45.70
45.83
45.97
46.10
14
6.8
46.24
46.38
46.51
46.65
46.79
46.92
47.06
47.20
47.33
47.47
14
6.9
47.61
47.75
47.89
48.02
48.16
48.30
48.44
48.58
48.72
48.86
14
7.0
49.00
49.14
49.28
49.42
49.56
49.70
49.84
49.98
50.13
50.27
14
7.1
50.41
50.55
50.69
50.84
50.98
51.12
51.27
51.41
51.55
51.70
14
7.2
51.84
51.98
52.13
52.27
52.42
52.56
52.71
52.85
53.00
53.14
15
7.3
53.29
53.44
53.58
53.73
53.88
54.02
54.17
54.32
54.46
54.61
15
7.4
54.76
54.91
55.06
55.20
55.35
55.50
55.65
55.80
55.95
56.10
15
7.5
56.25
56.40
"56.55
56.70
56.85
57.00
57.15
57.30
57.46
57.61
15
7.6
57.76
57.91
58.06
58.22
58.37
58.52
58.68
58.83
58.98
59.14
15
7.7
59.29
59.44
59.60
59.75
59.91
60.06
60.22
60.37
60.53
60.68
16
7.8
60.84
61.00
61.15
61.31
61.47
61.62
61.78
61.94
62.09
62.25
16
7.9
62.41
62.57
62.73
62.88
63.04
63.20
63.36
63.52
63.68
63.84
16
8.0
64.00
64.16
64.32
64.48
64.64
64.80
64.96
65.12
65.29
65.45
16
8.1
65.61
65.77
65.93
66.10
66.26
66.42
66.59
66.75
66.91
67.08
16
8.2
67.24
67.40
67.57
67.73
67.90
68.06
68.23
68.39
68.56
68.72
17
8.3
68.89
69.06
69.22
69.39
69.56
69.72
69.89
70.06
70.22
70.39
17
8.4
70.56
70.73
70.90
71.06
71.23
71.40
71.57
71.74
71.91
72.08
17
8.5
72.25
72.42
72.59
72.76
72.93
73.10
73.27
73.44
73.62
73.79
17
8.6
73.96
74.13
74.30
74.48
74.65
74.82
75.00
75.17
75.34
75.52
17
8.7
75.69
75.86
76.04
76.21
76.39
76.56
76.74
76.91
77.09
77.26
18
8.8
77.44
77.62
77.79
77.97
78.15
78.32
78.50
78.68
78.85
79.03
18
8.9
79.21
79.39
79.57
79.74
79.92
80.10
80.28
80.46
80.64
80.82
18
9.0
81.00
81.18
81.36
81.54
81.72
81.90
82.08
82.26
82.45
82.63
18
9.1
82.81
82.99
83.17
83.36
83.54
83.72
83.91
84.09
84.27
84.46
18
9.2
84.64
84.82
85.01
85.19
85.38
85.56
85.75
85.93
86.12
86.30
19
9.3
86.49
86.68
86.86
87.05
87.24
87.42
87.61
87.80
87.98
88.17
19
9.4
88.36
88.55
88.74
88.92
89.11
89.30
89.49
89.68
89.87
90.06
19
9.5
90.25
90.44
90.63
90.82
91.01
91.20
91.39
91.58
91.78
91.97
19
9.6
92.16
92.35
92.54
92.74
92.93
93.12
93.32
93.51
93.70
93.90
19
9.7
94.09
94.28
94.48
94.67
94.87
95.06
95.26
95.45
95.65
95.84
20
9.8
96.04
96.24
96.43
96.63
96.83
97.02
97.22
97.42
97.61
97.81
20
9.9
98.01
98.21
98.41
98.60
98.80
99.00
99.20
99.40
99.60
99.80
20
n
l
2
3
4
5
6
7
8
9
Diff.
TABLES
TABLE XVII. LOGARITHMS; 1000 TO 1409.
231
1
2
3
4
5
6
7
8
9
100
0000
0004
0009
0013
0017
0022
0026
0030
0035
0039
101
0043
0048
0052
0056
0060
0065
0069
0073
0077
0082
102
0086
0090
0095
0099
0103
0107
0111
0116
0120
0124
103
0128
0133
0137
0141
0145
0149
0154
0158
0162
0166
104
0170
0175
0179
0183
0187
0191
0195
0199
0204
0208
105
0212
0216
0220
0224
0228
0233
0237
0241
0245
0249
106
0253
0257
0261
0265
0269
0273
0278
0282
0286
0290
107
0294
0298
0302
0306
0310
0314
0318
0322
0326
0330
108
0334
0338
0342
0346
0350
0354
0358
0362
0366
0370
109
0374
0378
0382
0386
0390
0394
0398
0402
0406
0410
110
0414
0418
0422
0426
0430
0434
0438
0441
0445
0449
111
0453
0457
0461
0465
0469
0473
0477
0481
0484
0488
112
0492
0496
0500
0504
0508
0512
0515
0519
0523
0527
113
0531
0535
0538
0542
0546
0550
0554
0558
0561
0565
114
0569
0573
0577
0580
0584
0588
0592
0596
0599
0603
115
0607
0611
0615
0618
0622
0626
0630
0633
0637
0641
116
0645
0648
0652
0656
0660
0663
0667
0671
0674
0678
117
0682
0686
0689
0693
0697
0700
0704
0708
0711
0715
118
0719
0722
0726
0730
0734
0737
0741
0745
0748
0752
119
0755
0759
0763
0766
0770
0774
0777
0781
0785
0788
120
0792
0795
0799
0803
0806
0810
0813
0817
0821
0824
121
0828
0831
0835
0839
0842
0846
0849
0853
0856
0860
122
0864
0867
0871
0874
0878
0881
0885
0888
0892
0896
123
0899
0903
0906
0910
0913
0917
0920
0924
0927
0931
124
0934
0938
0941
0945
0948
0952
0955
0959
0962
0966
125
0969
0973
0976
0980
0983
0986
0990
0993
0997
1000
126
1004
1007
1011
1014
1017
1021
1024
1028
1031
1035
127
1038
1041
1045
1048
1052
1055
1059
1062
1065
1069
128
1072
1075
1079
1082
1086
1089
1092
1096
1099
1103
129
1106
1109
1113
1116
1119
1123
1126
1129
1133
1136
130
1139
1143
1146
1149
1153
1156
1159
1163
1166
1169
131
1173
1176
1179
1183
1186
1189
1193
1196
1199
1202
132
1206
1209
1212
1216
1219
1222
1225
1229
1232
1235
133
1239
1242
1245
1248
1252
1255
1258
1261
1265
1268
134
1271
1274
1278
1281
1284
1287
1290
1294
1297
1300
135
1303
1307
1310
1313
1316
1319
1323
1326
1329
1332
136
1335
1339
1342
1345
1348
1351
1355
1358
1361
1364
137
1367
1370
1374
1377
1380
1383
1386
1389
1392
1396
138
1399
1402
1405
1408
1411
1414
1418
1421
1424
1427
139
1430
1433
1436
1440
1443
1446
1449
1452
1455
1458
140
1461
1464
1467
1471
1474
1477
1480
1483
1486
1489
(Bottomley, "Four Fig. Math. Tables.")
232
THE THEORY OF MEASUREMENTS
* TABLE XVIII. LOGARITHMS.
1
2
3
4
5
6
7
&
9
123
456
789
10
0000
0043
0086
0128
0170
O2I2
0253
0294
0334
0374
4812
17 21 2 5
29 33 37
11
12
13
0414
0792
"39
0453
0828
"73
0492
0864
1206
0531
0899
1239
0569
0934
1271
0607
9 6 9
I33
0645
100^
1335
0682
1038
1367
0719
1072
1399
0755
1106
1430
4811
3 7io
3 6 10
15 19 23
14 17 21
13 16 10
26 30 34
24 28 31
23 26 29
21 24 27
20 22 25
18 21 24
14
15
16
1461
1761
2041
1492
1790
2068
iffi
2095
1553
1847
2122
1584
1875
2148
i6i<:
1903
2175
164^:
1931
22OI
1673
1959
2227
1703
1987
2253
1732
2014
2279
3 6 9
36 8
3 5 8
12 15 18
ii 14 17
ii 13 16
17
18
19
2304
$1
2330
2577
2810
2355
2601
2833
2380
2625
2856
2405
2648
2878
2430
2672
2900
2455
2695
2923
2480
2718
2945
2504
2742
2967
2529
2765
2989
2 57
2 5 7
247
10 12 15
9 12 14
9 " I 2
17 2O 22
16 19 21
16 18 20
20
3010
3032
3054
375
3096
3"8
3139
3160
3181
3201
24 6
8 ii 13
15 17 19
21
22
23
3222
3424
3617
3243
3444
3636
3263
3464
3655
3284
3483
3674
3304
3502
3692
3324
3522
37"
3345
354i
3729
3365
3560
3747
3385
3579
3766
3404
3598
3784
2 4 6
24 6
2 4 6
8 10 12
8 10 12
7 9 ii
14 16 18
H 15 17
J 3 15 '7
24
25
26
3802
3979
415
3820
3997
4166
3838
4014
4183
3856
403 1
4200
3874
4048
4216
3892
4065
4232
3909
4082
4249
3927
4099
4265
3945
4116
4281
3962
4133
4298
245
235
235
7 9 ii
7 9 10
7 8 10
12 14 16
12 14 15
II 13 15
27
28
29
43H
4472
4624
4330
4487
4639
4346
4502
4654
4362
45 l8
4669
4378
4533
4683
4393
4548
4698
4409
45 6 4
47 J 3
4425
4579
4728
444
4594
4742
445 6
4609
4757
2 3 5
2 3 5
1 3 4
689
689
6 7 9
II 13 14
II 12 \i
10 12 13
30
477 1
4786
4800
4814
4829
4843
4857
4871
4886
4900
i 3 4
6 7 9
10 ii 13
31
32
33
4914
505i
5185
4928
5065
5198
4942
5079
5211
4955
5092
5224
4969
5105
5237
4983
5"9
5250
4997
5i|2
5263
5011
5*45
5276
5024
5159
5289
5 38
5172
5302
3 4
3 4
3 4
6 7 8
HI
10 II 12
9 II 12
9 10 12
34
35
36
5315
544i
55 6 3
5328
5453
5575
5340
5465
5587
5353
5478
5599
5366
5490
5611
5378
5502
5 6 23
5391
554
5 6 35
5403
5527
5647
54i6
5539
5658
5428
555i
5670
3 4
2 4
2 4
!.:;
5 6 7
9 10 ii
9 10 ii
8 10 ii
37
38
39
5682
5798
59"
5 6 94
5809
5922
5705
5821
5933
5717
5832
5944
5729
5843
5955
5740
58 II
5966
5977
5763
5877
5988
577 C
5999
5786
5899
6010
2 3
2 3
2 3
5 6 7
5 6 7
4 5 7
8 9 10
8 9 10
8 9 10
40
6021
6031
6042
6o53
6064
6075
6085
6096
6107
6117
2 3
4 5 6
8 9 10
41
42
43
6128
6232
6335
6138
6243
6345
6149
6253
6355
6160
6263
6365
6170
6274
6375
6180
6284
6385
6191
6294
6395
6201
6304
6405
6212
6314
6415
6222
6 3 2 5
6425
2 3
2 3
2 3
4 5 6
4 5 6
4 5 6
7 8 9
7 8 9
7 8 9
44
45
46
6435
6532
6628
6444
6542
6637
6454
^6
6464
6561
6656
6474
657 1
6665
6484
6580
6675
6 493
6590
6684
6 53
6599
6513
6609
6702
6522
6618
6712
2 3
I 2 3
I 2 3
4 5 6
4 5 6
456
7 8 9
7 8 9
7 7 8
47
48
49
6721
6812
6902
6730
6821
6911
6739
683O
6920
6749
6839
6928
6758
6848
6937
6767
6857
6946
6776
6866
6955
6785
?2 5
6964
6794
6884
6972
6803
6893
6981
I 2 3
I 2 3
I 2 3
4 5 5
4 4 5
445
6 7 8
678
678
50
6990
6998
7007
7016
7024
733
7042
7050
7059
7067
I 2 3
3 4 5
678
51
52
53
7076
7160
7243
7084
7168
7251
7093
7177
7259
7101
7185
7267
7110
7193
7275
7118
7202
7284
7126
7210
7292
7i35
7218
7300
7H3
7226
7308
7152
7235
73i6
I 2 3
122
I 2 2
3 4 5
3 4 5
345
678
6 7 7
667
54
7324
7332
7340
7348
735 6
7364
7372
7380
7388
7396
I 2 2
3 4 5
667
* From Bottomley's Four Figure Mathematical Tables, by courtesy of The Macmillan Company.
TABLES
233
TABLE XVIII. LOGARITHMS (Concluded).
1
2
3
4
5
6
7
8
9
1 23
456
789
55
7404
.7412
7490
566
^642
74i9
7427
7435
7443
745i
7459
7466
7474
122
3 4 5
5 6 7
56
57
58
7482
7559
7 6 34
7497
7574
7649
7505
7582
7657
75*3
7589
7664
7520
7597
7672
7528
7604
7679
7536
7612
7686
7543
7619
7694
755i
7627
7701
2 2
2 2
I 2
345
3 4 5
344
5 6 7
5 6 7
5 6 7
59
60
61
7709
7782
7853
7716
7789
7860
7723
7796
7868
773i
7803
7875
7738
7810
7882
7745
7818
7889
Ws
7896
7760
7832
7903
7767
7839
7910
7774
7846
7917
2
2
2
344
344
344
5 6 7
566
5 6 6
62
63
64
7924
7993
8062
7931
8000
8069
7938
8007
8075
7945
8014
8082
7952
8021
8089
79^Q
8096
7966
8035
8102
7973
8041
8109
7980
8048
8116
7987
8055
8122
2
2
2
334
334
334
566
5 5 6
5 5 6
65
8129
8136
8142
8149
8156
8162
8169
8176
8182
8189
2
334
5 5 6
66
67
68
.8195
8261
8325
8202
8267
833i
8209
8274
8338
8215
8280
8344
8222
8287
8351
8228
8293
8357
8235
8299
8363
8241
8306
8370
8248
8312
8376
8254
8319
8382
2
2
2
334
334
334
5 5 6
5 5 \
4 5 6
69
70
71
8388
8451
8513
8395
8457
8519
8401
8463
8525
8407
8470
8531
8414
8476
8537
8420
8482
8543
8426
8488
8549
8432
8494
8555
8439
8500
8561
8445
8506
8567
2
2
2
234
234
234
4 5 6
4 5 6
4 5 5
72
73
74
~75~
8573
8633
8692
8579
8639
8698
8585
8645
8704
8591
8651
8710
8597
8657
8716
8603
866 3
8722
8609
8669
8727
8615
8675
8733
8621
8681
8739
8627
8686
8745
2
2
2
234
234
234
455
455
4 5 5
875i
8756
8762
8768
8774
8779
8785
8791
8797
8802
2
233
4 5 5
76
77
78
8808
8865
8921
8814
8871
8927
8820
8876
8932
8825
8882
8938
8831
8887
8943
8837
8893
8949
8842
8899
8954
8848
8904
8960
8854
8910
8965
8859
8915
8971
2
2
2
233
233
233
4 5 5
4 4 5
4 4 5
445
4 4 5
445
79
80
81
8976
9031
9085
8982
9036
9090
8987
9042
9096
8993
9047
9101
8998
9053
9106
9004
9058
9112
9009
9063
9117
9015
9069
9122
9020
9074
9128
9025
9079
9U3
2
2
2
233
233
233
82
83
84
9138
9191
9243
9H3
9196
9248
9149
9201
9253
9154
9206
9258
9159
9212
9263
9165
9217
9269
9170
9222
9274
9175
9227
9279
9180
9232
9284
9186
9238
9289
2
2
2
233
233
233
4 4 5
445
445
85
9294
9299
9304
9309
93i5
9320
9325
9330
9335
9340
I 2
233
445
86
87
88
9345
9395
9445
935
9400
945
9355
9405
9455
9360
9410
9460
9365
94i5
9465
9370
9420
9469
9375
9425
9474
938o
943
9479
9385
9435
9484
9390
9440
9489
I 2
O
233
223
223
4 4 5
344
344
89
90
91
9494
9542
9590
9499
9547
9595
954
9552
9600
959
9557
9605
95*3
9562
9609
9518
95 66
9614
9523
957i
9619
9528
9576
9624
9533
9628
9538
9586
9633
O
O
223
223
223
344
344
344
92
93
94
~95~
9638
9685
973i
9643
9689
9736
9647
9694
974i
9652
9699
9745
9657
973
975
9661
9708
9754
9666
97 i 3
9759
9671
9717
9763
9675
9722
9768
9680
9727
9773
O
O
223
223
223
344
344
344
9777
9782
9786
9791
9795
9800
9805
9809
9814
9818
223
344
96
97
98
9823
9868
9912
9827
9872
9917
9832
9877
9921
9836
9881
9926
9841
9886
9930
9845
9890
9934
9850
9894
9939
9854
9899
9943
9859
9903
9948
9863
9908
9952
O
O
O
223
223
223
344
344
344
99
995 6
9961
9965
9969
9974
9978
9983
9987
9991
9996
I I
223
334
234
THE THEORY OF MEASUREMENTS
* TABLE XIX. NATURAL SINES.
0'
6'
12'
18'
24'
SO'
36'
42'
48'
54'
123
4 5
oooo
0017
oo35
0052
0070
0087
0105
OI22
0140
oi57
369
12 15
1
2
3
0175
0349
0523
0192
0366
0541
0209
0384
0558
0227
0401
0576
0244
0419
0593
0262
0436
0610
0279
0454
0628
0297
0471
0645
0314
0488
o663
0332
0506
0680
369
369
369
12 I 5
12 I 5
12 I 5
4
5
6
~7~
8
9
0698
0872
1045
7!5
0889
1063
0732
0906
1080
0750
0924
1097
0767
0941
"15
0785
0958
1132
0802
0976
1149
0819
0993
1167
0837
ion
1184
0854
1028
I2OI
369
369
369
12 I 5
12 14
12 14
1219
1392
1564
1236
1409
1582
1253
1426
1599
1271
1444
1616
1288
1461
1633
1305
1478
1650
J 323
1495
1668
1340
\&1
1357
1530
1702
!374
J 547
1719
369
369
369
12 14
12 14
12 14
10
1736
!754
1771
1788
1805
1822
1840
1857
1874
1891
369
12 14
11
12
13
1908
2079
2250
1925
2096
2267
1942
2113
2284
1959
2130
2300
1977
2147
2317
1994
2164
2334
2OII
2181
235 I
2028
2198
2368
2045
2215
2385
2062
2232
2402
369
369
368
II I 4
II 14
II I 4
14
15
16
TT
18
19
2419
2588
2756
2436
2605
2773
2453
2622
2790
2470
2639
2807
2487
2656
2823
2504
2672
2840
2521
2689
2857
2538
2706
2874
2554
2723
2890
257i
2740
2907
368
368
368
II 14
II I 4
II 14
2924
3090
3256
2940
3io7
3272
2957
3123
3289
2974
3 J 4
3305
2990
3156
3322
3007
3i73
3338
3024
3190
3355
3040
3206
337 1
3057
3223
3387
3074
3239
3404
3 6 8
368
3 5 8
II 14
II 14
II 14
20
3420
3437
3453
3469
3486
3502
35i8
3535
3551
35 6 7
3 5 8
II 14
21
22
23
~24~
25
26
3584
3746
3907
3600
3762
3923
3616
3778
3939
3633
3795
3955
3 6 49
3811
397 1
3665
3827
3987
3681
3843
4003
3697
3859
4019
37H
3875
4035
3730
3891
405 l
3 5 8
3 5 8
3 5 8
II 14
II 14
II 14
4067
4226
4384
4083
4242
4399
4099
4258
4415
4H5
4274
443i
4131
4289
4446
4147
435
4462
4163
432i
4478
4179
4337
4493
4195
4352
459
4210
4368
4524
3 5 8
3 5 8
3 5 8
II 13
II 13
10 13
27
28
29
4540
4695
4848
4555
4710
4863
457 i
4726
4879
4586
474i
4894
4602
475 6
4909
4617
4772
4924
4633
4787
4939
4648
4802
4955
4664
4818
497
4679
4833
4985
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
30
5000
5015
53o
545
5060
5075
5090
5 I0 5
5120
5135
3 5 8
10 13
31
32
33
5150
5299
5446
5*65
53H
546i
5180
5329
5476
5195
5344
5490
5210
5358
5505
5225
5373
5519
5240
5388
5534
5255
5402
5548
5270
5417
5563
5284
5432
5577
2 5 7
257
2 5 7
IO 12
10 12
IO 12
34
35
36
5592
5736
5878
5606
575
5892
5621
5764
5906
5635
5779
5920
5650
5793
5934
5664
5807
5948
5678
5821
5962
5693
5835
5976
577
5850
5990
572i
5864
6004
257
2 5 7
2 5 7
IO 12
IO 12
9 12
37
38
39
6018
6157
6293
6032
6170
6307
6046
6184
6320
6060
6198
6334
6074
6211
6347
6088
6225
6361
6101
6239
6374
6115
6252
6388
6129
6266
6401
6143
6280
6414
257
2 5 7
247
9 12
9 ii
9 ii
40
6428
6441
6 455
6468
6481
6494
6508
6521
6534
6 547
247
9 ii
41
42
43
6561
6820
6 574
6704
6833
6587
6717
6845
6600
6730
6858
6613
6743
6871
6626
6756
6884
6639
6769
6896
6652
6782
6909
6665
6794
6921
6678
6807
6934
247
2 4 6
246
9 ii
9 "
8 ii
44
6947
6959
6972
6984
6997
7009
7022
7034
7046
7059
246
8 10
* From Bottomley's Four Figure Mathematical Tables, by courtesy of The Macmillan Company.
TABLES
TABLE XIX. NATURAL SINES (Concluded).
235
0'
6'
12'
18'
24'
3O'
36'
42'
48'
54'
123
4 5
45
7071
7083
7096
7108
7120
7 J 33
7H5
7i57
7169
7181
246
8 10
46
47
48
7 J 93
73*4
743i
7206
7325
7443
7218
7337
7455
7230
7349
7466
7242
73 61
7478
7254
7373
7490
7266
7385
7501
7278
7396
75i3
7290
7408
7524
7302
7420
7536
246
246
246
8 10
8 10
8 10
49
50
51
7547
7660
7771
7558
7672
7782
757
7683
7793
758i
7694
7804
7593
7705
7815
7604
7716
7826
7 6l 5
7727
7837
7627
7738
7848
7638
7749
7859
7649
7760
7869
2 4 6
246
2 4 5
8 9
7 9
7 9
52
53
54
7880
7986
8090
7891
7997
8100
7902
8007
8111
7912
8018
8121
7923
8028
8131
7934
8039
8141
7944
8049
8151
7955
8059
8161
7965
8070
8171
7976
8080
8181
2 4 5
235
2 3 5
7 9
7 9
7 8
55
8192
8202
8211
8221
8231
8241
8251
8261
8271
8281
2 3 5
7 8
56
57
58
8290
8387
8480
8300
8396
8490
8310
8406
8499
8320
8415
8508
8329
8425
8517
8339
8434
8526
8348
8443
8536
8358
8453
8545
8368
8462
8554
8377
8471
8563
2 3 5
2 3 5
2 3 5
6 8
6 8
6 8
59
60
61
8572
8660
8746
8581
8669
8755
8590
8678
8763
8599
8686
8771
8607
8695
8780
8616
8704
8788
8625
8712
8796
8634
8721
8805
8643
8729
8813
8652
8738
8821
i 3 4
i 3 4
i 3 4
6 7
2 ?
62
63
64
8829
8910
8988
8838
8918
8996
8846
8926
9003
8854
8934
9011
8862
8942
9018
8870
8949
9026
8878
8957
9033
8886
8965
9041
8894
8973
9048
8902
8980
9056
i 3 4
i 3 4
i 3 4
1 I
5 6
65
9063
9070
9078
9085
9092
9100
9107
9114
9121
9128
I 2 4
5 6
66
67
68
9135
9205
9272
9M3
9212
9278
915
9219
9285
9157
9225
9291
9164
9232
9298
9171
9239
934
9178
9245
93"
9184
9252
9317
9191
9259
9323
9198
9265
9330
I 2 3
I 2 3
I 2 3
5 6
4 6
4 5
69
70
71
9336
9397
9455
9342
9403
9461
9348
9409
9466
9354
94i5
9472
936i
9421
9478
9367
9426
9483
9373
9432
9489
9379
9438
9494
9385
9444
9500
939i
9449
955
2 3
2 3
2 3
4 5
4 5
4 5
72
73
74
95 11
95 6 3
9613
95 l6
9568
9617
952i
9573
9622
95 2 7
9578
9627
9532
9583
9632
9537
9588
9636
9542
9593
9641
9548
9598
9646
9553
9603
9650
9558
9608
9655
2 3
2 2
2 2
4 4
3 4
3 4
75
9659
9664
9668
9673
9677
9681
9686
9690
9694
9699
I 2
3 4
76
77
78
9703
9744
9781
9707
9748
9785
9711
975i
9789
9715
9755
9792
9720
9759
9796
9724
97 6 3
9799
9728
9767
9803
9732
977
9806
9736
9774
9810
9740
9778
9813
2
2
2
3 3
3 3
2 3
79
80
81
9816
9848
9877
9820
9851
9880
9823
9854
9882
9826
9857
9885
9829
9860
9888
9833
9863
9890
9836
9866
9893
9839
9869
9895
9842
9871
9898
9845
9874
9900
I 2
O
O
2 3
2 2
2 2
82
83
84
9903
9925
9945
9905
9928
9947
9907
9930
9949
9910
9932
995i
9912
9934
995 2
9914
9936
9954
9917
9938
995 6
9919
9940
9957
992i
9942
9959
9923
9943
9960
O
O
2 2
I 2
I I
85
9962
9963
9965
9966
9968
9969
9971
9972
9973
9974
001
I I
86
87
88
9976
9986
9994
9977
9987
9995
9978
9988
9995
9979
9989
9996
9980
9990
9996
998i
9990
9997
9982
9991
9997
9983
9992
9997
9984
9993
9998
9985
9993
9998
I
O O O
O O O
I I
I I
O O
89
9998
9999
9999
9999
9999
I'OOO
nearly.
rooo
nearly.
rooo
nearly.
I'OOO
nearly.
I'OOO
nearly.
O O O
O O
236
THE THEORY OF MEASUREMENTS
* TABLE XX. NATURAL COSINES.
O'
&
12'
18'
24'
3O'
36'
42'
48'
54'
123
4 5
I '000
I'OOO
nearly.
rooo
nearly.
rooo
nearly.
rooo
nearly.
9999
9999
9999
9999
9999
o o o
1
2
3
9998
9994
9986
9998
999 8
9993
9984
9997
9992
9983
9997
9991
9982
9997
9990
9981
9996
9990
9980
9996
9989
9979
9995
9988
9978
9995
9987
9977
000
o o o
O O I
I I
I I
4
5
6
9976
9962
9945
9974
9960
9943
9973
9959
9942
9972
9957
9940
9971
995 6
9938
9969
9954
9936
9968
9952
9934
9966
9951
9932
9965
9949
9930
9963
9947
9928
o o
I
O I
I I
I 2
I 2
7
8
9
9925
9903
9877
9923
9900
9874
9921
9898
9871
9919
9895
9869
9917
9893
9866
9914
9890
9863
9912
9888
9860
9910
9885
9857
9907
9882
9854
9905
9880
9851
I
O I
I I
2 2
2 2
2 2
10
9848
9845
9842
9839
9836
9833
9829
9826
9823
9820
112
2 3
11
12
13
9816
9781
9744
9813
9778
9740
9810
9774
9736
9806
977
9732
9803
9767
9728
9799
9763
9724
9796
9759
9720
9792
9755
9715
9789
9751
9711
9785
9748
9707
112
I I 2
I I 2
2 3
3 3
3 3
14
15
16
973
9659
9613
9699
9655
9608
9694
9650
9603
9690
9646
9598
9686
9641
9593
9681
9636
9588
9677
9632
9583
9673
9627
9578
9668
9622
9573
9664
9617
9568
I I 2
122
122
3 4
3 4
3 4
17
18
19
95 6 3
95 11
9455
9558
955
9449
9553
9500
9444
9548
9494
9438
9542
9489
9432
9537
9483
9426
9532
9478
9421
9527
9472
94i5
95 21
9466
9409
95i6
9461
9403
I 2 3
i 2 3
I 2 3
4 4
4 5
4 5
20
9397
939i
9385
9379
9373
9367
9361
9354
9348
9342
I 2 3
4 5
21
22
23
9336
9272
9205
9330
9265
9198
9323
9259
9191
9317
9252
9184
93"
9245
9178
934
9239
9171
9298
9232
9164
9291
9225
9157
9285
9219
915
9278
9212
9H3
I 2 3
I 2 3
I 2 3
4 5
4 6
5 6
24
25
26
9135
9063
8988
9128
9056
8980
9121
9048
8973
9114
9041
8965
9107
9033
8957
9100
9026
8949
9092
9018
8942
9085
9011
8934
9078
9003
8926
9070
8996
8918
I 2 4
i 3 4
i 3 4
5 6
5 6
5 6
27
28
29
8910
8829
8746
8902
8821
8738
8894
8813
8729
8886
8805
8721
8878
8796
8712
8870
8788
8704
8862
8780
8695
8854
8771
8686
8846
8763
8678
8838
8755
8669
i 3 4
i 3 4
i 3 4
5 7
6 7
6 7
30
8660
8652
8643
8634
8625
8616
8607
8599
8590
8581
1 3 4
6 7
31
32
33
8572
8480
8387
8563
8471
8377
8462
8368
8545
8453
8358
8536
8443
8348
8526
8434
8339
8517
8425
8329
8508
8415
8320
8499
8406
8310
8490
8396
8300
2 3 5
2 3 5
235
6 8
6 8
6 8
34
35
36
8290
8192
8090
8281
8181
8080
8271
8171
8070
8261
8161
8059
8251
8151
8049
8241
8141
8039
8231
8131
8028
8221
8121
8018
8211
8111
8007
8202
8100
7997
2 3 5
2 3 5
235
7 8
7 8
7 9
37
38
39
7986
7880
7771
7976
7869
7760
7965
7859
7749
7955
7848
7738
7944
7837
7727
7934
7826
7716
7923
78i5
775
7912
7804
7694
7902
7793
7683
7891
7782
7672
245
245
246
7 9
7 9
7 9
40
7660
7649
7638
7627
7 6l 5
7604
7593
758i
757
7559
2 4 6
8 9
41
42
43
7547
7431
73H
7536
7420
7302
7524
7408
7290
7513
7396
7278
75 01
73 fl
7266
7490
7373
7254
7478
736i
7242
7466
7349
7230
7455
7337
7218
7443
7325
7206
246
246
2 4 6
8 10
8 10
8 10
44
7'93
7181
7169
7157
7H5
7133
7120
7108
7096
7083
2 4 6
8 10
N.B. Numbers in difference-columns to be subtracted, not added.
* From Bottomley'g Four Figure Mathematical Tables, by courtesy of The Macmillan Company.
TABLES
237
TABLE XX. NATURAL COSINES (Concluded).
O'
6'
12'
18'
24'
30'
36'
42'
48'
54'
123
4 5
45
7071
759
7046
734
7022
7009
6997
6984
6972
6959
246
8 10
46
47
48
6947
6820
6691
6934
6807
6678
6921
6794
6665
6909
6782
6652
6896
6769
6639
6884
6756
6626
6871
6 743
6613
6858
6730
6600
6845
6717
6587
6833
6704
6574
246
2 4 6
247
8 ii
9 u
9 ii
49
50
51
6561
6428
6293
6 547
6414
6280
6534
6401
6266
6521
6388
6252
6508
6374
6239
6494
6361
6225
6481
6347
6211
6468
6334
6198
6 455
6320
6184
6441
6307
6170
247
247
2 5 7
9 ii
9 ii
9 ii
52
53
54
6l 57
6018
5878
6i43
6004
5864
6129
5990
5850
6115
5976
5835
6101
5962
5821
6088
5948
58-07
6074
5934
5793
6060
5920
5779
6046
5906
57 6 4
6032
5892
5750
2 5 7
257
257
9 12
9 12
9 12
55
5736
572i
5707
5693
5678
5664
5650
5635
5621
5606
2 5 7
10 12
56
57
58
5592
5446
5299
5577
5432
5284
55 6 3
54i7
5270
5548
5402
5255
5534
5388
5240
55i9
5373
5225
5505
5358
5210
5490
5344
5 J 95
5476
5329
5180
546i
53H
5 l6 5
2 5 7
2 5 7
257
10 12
10 12
10 12
59
60
61
5 Z 5
5000
4848
5i35
4985
4833
5120
4970
4818
5105
4955
4802
5090
4939
4787
575
4924
4772
5060
4909
475 6
5045
4894
474i
5030
4879
4726
5i5
4863
4710
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
62
63
64
4695
4540
4384
4679
4524
4368
4664
459
4352
4648
4493
4337
4633
4478
4321
4617
4462
4305
4602
4446
4289
4586
443i
4274
457 1
4415
4258
4555
4399
4242
3 5 8
3 5 8
3 5 8
10 13
10 13
II 13
65
4226
4210
4195
4179
4163
4 J 47
4131
4"5
4099
4083
3 5 8
II 13
66
67
68
4067
3907
3746
405 I
3891
3730
4035
3875
37H
4019
3859
3697
4003
3843
3681
3987
3827
3665
397 1
3811
3 6 49
3955
3795
3633
3939
3778
3616
3923
3762
3600
3 5 8
3 5 8
3 5 8
II 14
II 14
II 14
69
70
71
3584
3420
3256
3567
3404
3239
355i
3387
3223
3535
337i
3206
35i8
3355
3190
3502
3338
3173
3486
3322
3156
3469
3305
3140
3453
3289
3123
3437
3272
3 J 07
3 5 8
3 5 8
3 6 8
II 14
II 14
II 14
72
73
74
3090
2924
2756
374
2907
2740
3057
2890
2723
3040
2874
2706
3024
2857
2689
3007
2840
2672
2990
2823
2656
2974
2807
2639
2957
2790
2622
2940
2773
2605
368
368
368
II 14
II 14
II 14
75
2588
257i
2554
2538
2521
2504
2487
2470
2453
2436
368
II 14
76
77
78
2419
2250
2079
2402
2233
2062
2385
2215
2045
2368
2198
2028
2351
2181
2OII
2334
2164
1994
2317
2147
1977
2300
2130
1959
2284
2113
1942
2267
2096
1925
368
369
3 6 9
II 14
II 14
II 14
79
80
81
1908
1736
i5 6 4
1891
1719
'547
1874
1702
1530
1857
1685
1513
1840
1668
1495
1822
1650
1478
1805
1633
1461
1788
1616
1444
1771
1599
1426
*754
1582
1409
3 6 9
3 6 9
369
12 14
12 14
12 14
82
83
84
1392
1219
1045
1374
I2OI
1028
1357
1184
IOII
1340
1167
0993
1323
1149
0976
1305
1132
0958
1288
i"5
0941
1271
1097
0924
1253
1080
0906
1236
1063
0889
369
369
369
12 I 4
12 I 4
12 14
85
0872
0854
0837
0819
0802
0785
0767
0750
0732
0715
3 6 9
12 I 5
86
87
88
0698
0523
0349
0680
0506
0332
o663
0488
03H
0645
0471
0297
0628
0454
0279
0610
0436
0262
0593
0419
0244
0576
0401
0227
0558
0384
0209
0541
0366
0192
369
369
369
12 15
12 15
12 15
89
oi75
0157
0140
0122
0105
0087
0070
0052
0035
0017
369
12 15
iV.B. Numbers in difference-columns to be subtracted, not added.
238
THE THEORY OF MEASUREMENTS
TABLE XXI. NATURAL TANGENTS.
O'
&
12'
18'
24'
3O'
36'
42'
48'
54'
123
4 5
oooo
0017
0035
0052
0070
0087
0105
OI22
0140
oi57
369
12 14
1
2
3
0175
0349
0524
0192
0367
0542
0209
0384
0559
0227
0402
577
0244
0419
0594
0262
0437
0612
0279
0454
0629
0297
0472
0647
0314
0489
0664
0332
0507
0682
369
369
369
12 I 5
12 15
12 I 5
4
5
6
0699
0875
1051
0717
0892
1069
0734
0910
1086
0752
0928
1104
0769
0945
1122
0787
0963
"39
0805
0981
"57
0822
99 8
"75
0840
1016
1192
0857
1033
I2IO
369
369
369
12 I 5
12 I 5
12 I 5
7
8
9
1228
1405
1584
1246
1423
1602
1263
1441
1620
1281
H59
1638
1299
H77
1655
1317
H95
1673
1334
1512
1691
1352
1530
1709
1370
1548
1727
1388
1566
1745
369
369
369
12 I 5
12 I 5
12 I 5
10
1763
1781
1799
1817
1835
1853
1871
1890
1908
1926
369
12 I 5
11
12
13
1944
2126
2309
1962
2144
2327
1980
2162
2345
1998
2180
2364
2016
2199
2382
2035
2217
2401
2053
2235
2419
2071
2254
2438
2089
2272
2456
2IO7
2290
2475
369
369
369
12 I 5
12 I 5
12 I 5
14
15
16
2493
2679
2867
2512
2698
2886
2 53
2717
2905
2549
2736
2924
2568
2754
2943
2586
2773
2962
2605
2792
2981
2623
2811
3000
2642
2830
3019
2661
2849
3038
369
369
369
12 l6
13 16
13 16
17
18
19
3057
3249
3443
3076
3269
3463
3096
3288
3482
3"5
3307
3502
3134
3327
3522
3i53
3346
354i
3172
3365
356i
3 J 9i
3385
358i
3211
3404
3600
3230
3424
3620
3 6 10
3 6 10
3 6 10
13 16
13 16
13 17
20
3640
3659
3679
3699
37 J 9
3739
3759
3779
3799
3819
3 7 I0
13 17
21
22
23
3839
4040
4245
3859
4061
4265
3879
4081
4286
3899
4101
4307
3919
4122
4327
3939
4142
4348
3959
4163
4369
3979
4183
4390
4000
4204
44"
4O2O
4224
4431
3 7 I0
3 7 I0
3 7 10
13 17
14 17
14 17
24
25
26
4452
4663
4877
4473
4684
4899
4494
4706
4921
45i5
4727
4942
4536
4748
4964
4557
477
4986
4578
479i
5008
4599
4813
5029
4621
4834
5051
4642
4856
573
4 7 10
4 7 ii
4 7 ii
14 18
14 18
15 18
27
28
29
5095
5317
'5543
5"7
5340
5566
5139
5362
5589
5161
5384
5612
5184
5407
5635
5206
5430
5658
5228
5452
5681
5250
5475
5704
5272
5498
5727
5295
5520
575
4 7 ii
4 8 ii
4 8 12
15 18
15 19
15 19
30
'5774
5797
5820
5844
5867
5890
59H
5938
596i
5985
4 8 12
16 20
31
32
33
6009
6249
6494
6032
6273
6519
6056
6297
6544
6080
6322
6569
6104
6346
6594
6128
6371
6619
6152
6395
6644
6176
6420
6669
6200
6445
6694
6224
6469
6720
4 8 12
4 8 12
4 8 13
16 20
16 20
17 21
34
35
36
' 6 745
7002
7265
6771
7028
7292
6796
7054
7319
6822
7080
7346
6847
7107
7373
6873
7 ! 33
7400
6899
7*59
7427
6924
7186
7454
6950
7212
748i
6976
7239
7508
4 9 13
4 9 13
5 9 H
17 21
18 22
18 23
37
38
39
7536
7813
8098
7563
7841
8127
7590
8156
7618
7898
8185
7646
7926
8214
7673
7954
8243
7701
7983
8273
7729
8012
8302
7757
8040
8332
7785
8069
8361
5 9 H
5 I0 M
5 10 15
18 23
19 24
20 24
40
8391
8421
8451
8481
8511
8541
857i
8601
8632
8662
5 1 '5
20 25
41
42
43
8693
9004
9325
8724
9036
9358
8754
9067
9391
8785
9099
9424
8816
9131
9457
8847
9163
9490
8878
9195
9523
8910
9228
9556
8941
9260
9590
8972
9293
9623
5 10 16
5 " 16
6 ii 17
21 26
21 27
22 28
44
9657
9691
97 2 5
9759
9793
9827
9861
9896
9930
9965
6 ii 17
23 29
* From Bottomley's Four Figure Mathematical Tables, by courtesy of The Macmillan Company.
TABLES
239
TABLE XXI. NATURAL TANGENTS (Concluded).
0'
6'
12'
18'
24'
3O'
36'
42'
48'
54'
123
4 5
45
I -0000
0035
0070
0105
0141
0176
O2I2
0247
0283
0319
6 12 18
24 30
46
47
48
1-0355
1-0724
1-1106
0392
0761
"45
0428
0799
1184
0464
0837
1224
0501
0875
1263
0538
0913
1303
0575
0951
1343
0612
0990
1383
0649
1028
1423
0686
1067
1463
6 12 18
6 13 19
7 13 20
25 3i
25 32
26 33
49
50
51
1504
1918
2349
1544
1960
2393
1585
2OO2
2437
1626
2045
2482
1667
2088
2527
1708
2131
2572
1750
2174
2617
1792
2218
2662
1833
2261
2708
1875
2305
2753
7 H 21
7 14 22
8 15 23
28 34
29 36
30 38
52
53
54
2799
3270
3764
2846
3319
3814
2892
3367
3865
2938
34i6
3916
2985
3465
3968
3032
35H
4019
3079
3564
4071
3127
3613
4124
3i75
3663
4176
3222
37i3
4229
8 16 23
8 16 25
9 17 26
3i 39
33 4i
34 43
55
4281
4335
4388
444 2
4496
4550
4605
4659
4715
4770
9 18 27
36 45
56
57
58
4826
5399
6003
4882
5458
6066
4938
5517
6128
4994
5577
6191
5051
5637
6255
5108
5697
6319
5166
mi
5224
5818
6447
5282
5880
6512
5340
594i
6577
10 19 29
10 20 30
II 21 32
38 48
40 50
43 53
59
60
61
6643
7321
8040
6709
739i
8115
6775
7461
8190
6842
7532
8265
6909
7603
8341
6977
7675
8418
7045
7747
8495
7113
7820
8572
7182
7893
8650
725 1
7966
8728
ii 23 34
12 24 36
13 26 38
45 5 6
48 60
5 1 6 4
62
63
64
1-8807
1-9626
2-0503
8887
9711
0594
8967
9797
0686
9047
9883
0778
9128
9970
0872
9210
0057
0965
9292
0145
1060
9375
0233
"55
9458
0323
1251
9542
041;
1348
14 27 41
15 29 44
16 31 47
55 68
58 73
63 78
65
2-1445
1543
1642
1742
1842
1943
2045
2148
2251
2355
1 7 34 5 1
68 85
66
67
68
2-2460
2-3559
2'475 i
2566
3673
4876
2673
3789
5002
2781
3906
5129
2889
4023
5257
2998
4142
5386
3109
4262
5517
3220
4383
5649
3332
454
5782
3445
4627
59i6
18 37 55
20 40 60
22 43 65
74 92
79 99
87 108
69
70
71
2-6051
27475
2-9042
6187
7625
9208
6325
7776
9375
6464
7929
9544
6605
8083
97H
6746
8239
9887
6889
8397
0061
734
8556
0237
7179
8716
0415
7326
8878
0595
24 47 7i
26 52 78
29 58 87
95 "8
104 130
"5 !44
72
73
74
3-0777
3-2709
3-4874
0961
2914
5 I0 5
1146
3122
5339
1334
3332
5576
1524
3544
5816
1716
3759
6059
1910
3977
6305
2106
4197
6554
'23 5
4420
6806
2506
4646
7062
32 64 96
36 72 108
41 82 122
129 161
144 180
162 203
75
3-732I
7583
7848
8118
8391
8667
8947
9232
9520
9812
46 94 139
i 86 232
76
77
78
4-0108
4-33I5
4-7046
0408
3662
7453
0713
4015
7867
IO22
4374
8288
1335
4737
8716
l6 53
5 I0 7
9152
1976
5483
9594
2303
5864
0045
2635
6252
0504
2972
6646
0970
53 107 i 60
62 124 186
73 146 219
214 267
248 310
292 365
79
80
81
5-I446
5-67I3
6-3138
1929
7297
3859
2422
7894
4596
2924
8502
5350
3435
9124
6122
3955
9758
6912
4486
0405
7920
5026
5578
6140
2432
0264
87 175 262
350 437
1066
8548
1742
9395
Difference-columns
cease to be useful, owing
to the rapidity with
which the value of the
tangent changes.
82
83
84
r"54
8-1443
9-5H4
2066
2636
9-677
3002
3863
9-845
3962
5126
IO-O2
4947
6427
10-20
5958
7769
10-39
6996
9152
10-58
8062
0579
10-78
9158
2052
10-99
0285
3572
11-20
85
n-43
11-66
11-91
12-16
12-43
12-71
13-00
13-30
13-62
I3-95
86
87
88
14-30
19-08
28-64
14-67
I9-74
30-14
15-06
20-45
31-82
I5-46
21-20
3J69
15-89
22-02
35-8o
16-35
22-90
38-19
16-83
23-86
40-92
I7-34
24-90
44-07
17-89
26-03
47-74
18-46
27-27
52-08
89
57'29
63-66
71-62
81-85
95-49
114-6
143-2
191-0
286-5
573-0
240
THE THEORY OF MEASUREMENTS
* TABLE XXII. NATURAL COTANGENTS.
O'
6'
12'
18'
24'
30'
36'
42'
48'
54'
Difference-columns
not useful here, owing
to the rapidity with
which the value of the
cotangent changes.
Inf.
573-o
286-5
191-0
143-2
114-6
95'49
81-85
71-62
63-66
1
2
3
57-29
28-64
19-08
52-08
27-27
18-46
4774
26-03
17-89
44-07
24-90
17-34
40-92
23-86
16-83
38-19
22-90
i6'35
35-80
22-02
15-89
33-69
2 1 -2O
31-82
20-45
15-06
19-74
14-67
4
5
6
14-30
ii'43
9-5I44
I3-95
II'2O
3572
13-62
10-99
2052
13-3
10-78
0579
13-00
10-58
9152
12-71
10-39
7769
12-43
10-20
6427
I2'l6
10-02
5126
11-91
9-845
3863
u-66
9-677
2636
7
8
9
8-1443
7'"54
6-3138
0285
0264
2432
9158
9395
1742
8062
8548
1066
6996
7920
0405
5958
6912
97S8
4947
6122
9124
3962
5350
8502
3002
4596
7894
2066
3859
7297
10
5-67I3
6140
5578
5026
4486
3955
3435
2924
2422
1929
123
4 5
11
12
13
4-7046
4-33I5
0970
6646
2972
0504
6252
2635
0045
5864
2303
9594
5483
1976
9152
5107
1653
8716
4737
1335
8288
4374
1022
7867
4015
0713
7453
3662
0408
74 148 222
63 125 i 88
53 107 160
296 370
252 314
214 267
14
15
16
4-0108
J4874
9812
7062
4646
9520
6806
4420
9232
6554
4197
8947
6305
3977
8667
6059
3759
5816
3544
8118
5576
3332
7848
5339
3122
7583
5105
29H
46 93 139
41 82 122
36 72 108
i 86 232
163 204
144 180
17
18
19
3-2709
3-0777
2-9042
2506
595
8878
2305
0415
8716
2106
0237
8556
1910
0061
8397
1716
9887
8239
5 2 4
9714
8083
1334
9544
7929
1146
9375
7776
0961
9208
7625
32 64 96
29 58 87
26 52 78
129 161
"5 *44
104 130
2*7475
7326
7179
734
6889
6746
6605
6464
6325
6187
24 47 7i
95 "8
21
22
23
2-6051
2-475 *
2-3559
5916
4627
3445
5782
454
3332
5649
4383
3220
5517
4262
3109
5386
4142
2998
5257
4023
2889
3906
2781
5002
3789
2673
4876
3673
2566
22 43 65
20 40 60
18 37 55
87 108
79 99
74 92
24
25
26
~27~
28
29
2-2460
2-1445
2-0503
2355
1348
0413
2251
1251
0323
2148
"55
0233
2045
1060
0145
1943
0965
0057
1842
0872
9970
1742
0778
988 3
1642
0686
9797
1543
0594
97"
17 34 5i
16 31 47
15 29 44
68 85
63 78
58 73
1-9626
1-8807
1-8040
9542
8728
7966
9458
8650
7893
9375
8572
7820
9292
8495
7747
9210
8418
7675
9128
8341
7603
9047
8265
753 2
8967
8190
7461
8887
8115
739i
14 27 41
i3 26 38
12 24 36
55 68
5 1 64
48 60
30
1-7321
7251
7182
7"3
745
6977
6909
6842
6775
6709
ii 23 34
45 56
31
32
33
1-6643
1-6003
1-5399
6577
5340
6512
5880
5282
6447
5818
5224
6383
mi
6319
5697
5108
6255
5637
5051
6191
5577
4994
6128
5517
4938
6066
5458
4882
II 21 32
10 20 30
10 19 29
43 53
40 5
38 48
34
35
36
1-4826
1-4281
1-3764
4770
4229
3713
4715
4176
3663
4659
4124
3613
4605
4071
3564
4550
4019
35H
4496
3968
3465
4442
3916
4388
3865
3367
4335
3814
3319
9 18 27
9 17 26
8 16 25
36 45
34 43
33 4i
37
38
39
1-3270
1-2799
1-2349
3222
2753
2305
2708
2261
3^27
2662
2218
3079
2617
2174
3032
2572
2131
2985
2527
2088
2938
2482
2045
2892
2437
2OO2
2846
2393
1960
8 16 23
8 15 23
7 14 22
3 1 39
30 38
29 36
40
1-1918
1875
1833
1792
!75o
1708
1667
1626
1585
1544
7 *4 21
28 34
41
42
43
1-1504
1-1106
1-0724
1463
1067
0686
1423
1028
0649
1383
0990
0612
1343
0951
0575
1303
0913
0538
1263
0875
0501
1224
0837
0464
1184
0799
0428
"45
0761
0392
7 13 20
6 13 19
6 12 18
26 33
25 32
25 31
44
1-0355
0319
0283
0247
O2I2
0176
0141
0105
0070
0035
6 12 18
24 30
N.B. Numbers in difference-columns to be subtracted, not added.
* From Bottomley's Four Figure Mathematical Tables, by courtesy of The Macmillan Company.
TABLES
241
TABLE XXII. NATURAL COTANGENTS (Concluded).
O'
6'
12
18'
24'
3O'
36'
42'
48'
54'
123
4 5
45
ro
0-9965
0-9930
0-9896
0-9861
0-9827
0-9793
'9759
0-9725
0-9691
6 ii 17
23 29
46
47
48
9657
9325
9004
9623
9293
8972
9590
9260
8941
955 6
9228
8910
9523
9i95
8878
9490
9163
8847
9457
9131
8816
9424
9099
8785
939i
9067
8754
9358
9036
8724
6 ii 17
5 ii 10
5 10 16
22 28
21 27
21 26
49
50
51
8693
8391
8098
8662
8361
8069
8632
8332
8040
8601
8302
8012
8571
8273
7983
8541
8243
7954
8511
8214
7926
8481
8185
7898
8451
8156
7869
8421
8127
7841
5 10 i5
5 10 15
5 I0 M
20 25
20 24
19 24
52
53
54
7813
7536
7265
7785
7508
7239
7757
748i
7212
7729
$3
7701
7427
7i59
7 6 73
7400
7133
7646
7373
7107
7618
7346
7080
7590
73i9
754
75 6 3
7292
7028
5 9 H
5 9 H
4 9 13
18 23
18 23
18 22
55
7002
6976
6950
6924
6899
6873
6847
6822
6796
6771
4 9 13
I 7 21
56
57
58
>6 745
6494
6249
6720
6469
6224
6694
6445
6200
6669
6420
6176
6644
6395
6152
6619
637 1
6128
6594
6346
6104
6569
6322
6080
6544
6297
6056
6519
6273
6032
4 8 13
4 8 12
4 8 12
17 21
16 20
16 20
59
60
61
6009
'5774
'5543
5985
5750
5520
596i
5727
5498
5938
574
5475
59H
5681
5452
5890
5658
5430
5867
5^35
5407
5844
5612
5384
5820
5589
5362
5797
5566
5340
4 8 12
4 8 12
4 8 ii
16 20
15 !9
15 !9
62
63
64
5317
595
4877
5295
573
4856
5272
505 1
4834
5250
5029
4813
5228
5008
479i
5206
4986
4770
5184
4964
4748
5161
4942
4727
5*39
4921
4706
5"7
4899
4684
4 7 ii
4 7 ii
4 7 ii
15 18
15 18
14 18
65
4663
4642
4621
4599
4578
4557
4536
4515
4494
4473
4 7 10
14 18
66
67
68
"445 2
4245
4040
443i
4224
4020
4411
4204
4000
4390
4183
3979
4369
4163
3959
4348
4142
3939
4327
4122
3919
4307
4101
3899
4286
4081
3879
4265
4061
3859
371
3 7 10
3 7 I0
14 17
14 17
13 17
69
70
71
3839
3640
'3443
3819
3620
3424
3799
3600
3404
3779
358i
3385
3759
356i
3365
3739
354i
3346
3719
3522
3327
3699
3502
3307
3679
3482
3288
3659
3463
3269
3 7 10
3 6 10
3 6 10
13 17
13 17
13 16
72
73
74
3249
3057
2867
3230
3038
2849
3211
3019
2830
3i9i
3000
2811
3172
2981
2792
3153
2962
2773
3134
2943
2754
3"5
2924
2736
3096
2905
2717
2698
3 6 10
369
369
13 16
13 16
13 16
75
2679
2661
2642
2623
2605
2586
2568
2549
2530
2512
369
12 16
76
77
78
2493
2309
2126
2475
2290
2107
2456
2272
2089
2438
2254
2071
2419
2235
2053
2401
2217
2035
2382
2199
2016
2364
2180
1998
2345
2162
1980
2327
2144
1962
369
3 6 9
369
12 15
12 15
12 I 5
79
80
81
1944
1763
1584
1926
'745
1566
1908
1727
1548
1890
1709
1530
1871
1691
1512
1853
1673
H95
1835
l6 55
H77
1817
1638
H59
1799
1620
1441
1781
1602
1423
369
369
369
12 I 5
12 I 5
12 I 5
82
83
84
1405
1228
1051
1388
I2IO
1033
1370
1192
1016
1352
"75
0998
1334
"57
0981
1317
"39
0963
1299
1122
0945
1281
1104
0928
1263
1086
0910
1246
1069
0892
369
369
3 6 9
12 15
12 15
12 15
85
0875
08 57
0840
0822
0805
0787
0769
0752
0734
0717
3 6 9
12 I 5
86
87
88
0699
0524
0349
0682
0507
0332
0664
0489
3i4
0647
0472
0297
0629
0454
0279
0612
0437
0262
0594
0419
0244
0577
0402
0227
0559
0384
0209
0542
0367
0192
369
3 6 9
369
12 15
12 15
12 I 5
89
oi75
0157
0140
0122
0105
0087
OC>7O
0052
0035
0017
369
12 14
N.B. Numbers in difference-columns to be subtracted, not added.
242
THE THEORY OF MEASUREMENTS
TABLE XXIII. RADIAN MEASURE.
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
123
4 5
0.0000
0017
0035
0052
0070
0087
0105
0122
0140
0157
369
12 15
1
0.0175
0192
0209
0227
0244
0262
0279
0297
0314
0332
369
12 15
2
0.0349
0367
0384
0401
0419
0436
0454
0471
0489
0506
369
12 15
3
0.0524
0541
0559
0576
0593
0611
0628
0646
0663
0681
369
12 15
4
0.0698
0716
0733
0750
0768J 0785
0803
0820
0838
0855
369
12 15
5
0.0873
0890
0908
0925
0942
0960
0977
0995
1012
1030
369
12 15
6
0.1047
1065
1082
1100
1117
1134
1152
1169
1187
1204
369
12 15
7
0.1222
1239
1257
1274
1292
1309
1326
1344
1361
1379
369
12 15
8
0.1396
1414
1431
1449
1466
1484
1501
1518
1536
1553
369
12 15
9
0.1571
1588
1606
1623
1641
1658
1676
1693
1710
1728
369
12 15
10
0.1745
1763
1780
1798
1815
1833
1850
1868
1885
1902
369
12 15
11
0.1920
1937
1955
1972
1990
2007
2025
2042
2059
2077
369
12 15
12
0.2094
2112
2129J2147
2164
2182
2199
2217
2234
2251
369
12 15
13
0.2269
2286
230412321
2339
2356
2374
2391
2409
2426
369
12 15
14
0.2443
2461
2478 2496
2513
2531
2548
2566
2583
2601
369
12 15
15
0.2618
2635
2653
2670
2688
2705
2723
2740
2758
2775
369
12 15
16
0.2793
2810
2827
2845
2862
2880
2897
2915
2932
2950
369
12 15
17
0.2967
2985
3002
3019
3037
3054
3072
3089
3107
3124
369
12 15
18
0.3142
3159
3176
3194
3211
3229
3246
3264
3281
3299
369
12 15
19
0.3316
3334
3351
3368
3386
3403
3421
3438
3456
3473
369
12 15
20
0.3491
3508
3526
3543
3560
3578
3595
3613
3630
3648
369
12 15
21
0.3665
3683
3700
3718
3735
3752
3770
3787
3805
3822
369
12 15
22
0.3840
3857
3875
3892
3910
3927
3944
3962
3979
3997
369
12 15
23
0.4014
4032
4049
4067
4084
4102
4119
4136
4154
4171
369
12 15
24
0.4189
4206
4224
4241
4259
4276
4294
4311
4328
4346
369
12 15
25
0.4363
4381
4398
4416
4433
4451
4468
4485
4503
4520
369
12 15
26
0.4538
4555
4573
4590
4608
4625
4643
4660
4677
4695
369
12 15
27
0.4712
4730
4747
4765
4782
4800
4817
4835
4852
4869
369
12 15
28
0.4887
4904
4922
4939
4957
4974
4992
5009
5027
5044
369
12 15
29
0.5061
5079
5096
5114
5131
5149
5166
5184
5201
5219
369
12 15
30
0.5236
5253
5271
5288
5306
5323
5341
5358
5376
5393
369
12 15
31
0.5411
5428
5445
5463
5480
5498
5515
5533
5550
5568
369
12 15
32
0.5585
5603
5620
5637
5655
5672
5690
5707
5725
5742
369
12 15
33
0.5760
5777
5794
5812
5829
5847
5864
5882
5899
5917
369
12 15
34
0.5934
5952
5969
5986
6004
6021
6039
6056
6074
6091
369
12 15
35
0.6109
6126
6144
6161
6178
6196
6213
6231
6248
6266
369
12 15
36
0.6283
6301
6318
6336
6353
6370
6388
6405
6423
6440
369
12 15
37
0.6458
6475
6493
6510
6528
6545
6562
6580
6597
6615
369
12 15
38
0.6632
6650
6667
6685
6702
6720
6737
6754
6772
6789
369
12 15
39
0.6807
6824
6842
6859
6877
6894
6912
6929
6946
6964
369
12 15
40
0.6981
6999
7016
7034
7051
7069
7086
7103
7121
7138
369
12 15
41
0.7156
7173
7191
7208
7226
7243
7261
7278
7295
7313
369
12 15
42
0.7330
7348
7365
7383
7400
7418
7435
7453
7470
7487
369
12 15
43
0.7505
7522
7540
7557
7575
7592
7610
7627
7645
7662
369
12 15
44
0.7679
7697
7714
7732
7749
7767
7784
7802
7819
7837
369
12 15
(Bottomley, " Four Fig. Math. Tables.")
TABLES
TABLE XXIII. RADIAN MEASURE (Concluded).
243
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
1 2 3
4 5
45
0.7854
7871
7889
7906
7924
7941
7959
7976
7994
8011
369
12 15
46
0.8029
8046
8063
8081
8098
8116
8133
8151
8168
8186
369
12 15
47
0.8203
8221
8238
8255
8273
8290
8308
8325
8343
8360
369
12 15
48
0.8378
8395
8412
8430
8447
8465
8482
8500
8517
8535
369
12 15
49
0.8552
8570
8587
8604
8622
8639
8657
8674
8692
8709
369
12 15
50
0.8727
8744
8762
8779
8796
8814
8831
8849
8866
8884
369
12 15
51
0.8901
8919
8936
8954
8971
8988
9006
9023
9041
9058
369
12 15
52
0.9076
9093
9111
9128
9146
9163
9180
9198
9215
9233
369
12 15
53
0.9250
9268
9285
9303
9320
9338
9355
9372
9390
9407
369
12 15
54
0.9425
9442
9460
9477
9495
9512
9529
9547
9564
9582
369
12 15
55
0.9599
9617
9634
9652
9669
9687
9704
9721
9739
9756
369
12 15
56
0.9774
9791
9809
9826
9844
9861
9879
9896
9913
9931
369
12 15
57
0.9948
9966
9983
0001
0018
0036
0053
0071
0088
0105
369
12 15
58
1.0123
0140
0158
0175
0193
0210
0228
0245
0263
0280
369
12 15
59
1.0297
0315
0332
0350
0367
0385
0402
0420
0437
0455
369
12 15
60
1.0472
0489
0507
0524
0542
0559
0577
0594
0612
0629
369
12 15
61
1.0647
0664
0681
0699
0716
0734
0751
0769
0786
0804
369
12 15
62
1.0821
0838
0856
0873
0891
0908
0926
0943
0961
0978
369
12 15
63
1.0996
1013
1030
1048
1065
1083
1100
1118
1135
1153
369
12 15
64
1.1170
1188
1205
1222
1240
1257
1275
1292
1310
1327
369
12 15
65
1.1345
1362
1380
1397
1414
1432
1449
1467
1484
1502
369
12 15
66
1.1519
1537
1554
1572
1589
1606
1624
1641
1659
1676
369
12 15
67
1.1694
1711
1729
1746
1764
1781
1798
1816
1833
1851
369
12 15
68
1.1868
1886
1903
1921
1938
1956
1973
1990
2008
2025
369
12 15
69
1.2043
2060
2078
2095
2113
2130
2147
2165
2182
2200
369.
12 15
70
1.2217
2235
2252
2270
2287
2305
2322
2339
2357
2374
369
12 15
71
1.2392
2409
2427
2444
2462
2479
2497
2514
2531
2549
369
12 15
72
1.2566
2584
2601
2619
2636
2654
2671
2689
2706
2723
369
12 15
73
1.2741
2758
2776
2793
2811
2828
2846
2863
2881
2898
369
12 15
74
1.2915
2933
2950
2968
2985
3003
3020
3038
3055
3073
369
12 15
75
1.3090
3107
3125
3142
3160
3177
3195
3212
3230
3247
369
12 15
76
1 . 3265
3282
3299
3317
3334
3352
3369
3387
$404
3422
369
12 15
77
1 3439
3456
3474
3491
3509
3526
3544
3561
3579
3596
369
12 15
78
1.3614
3631
3648
3666
3683
3701
3718
3736
3753
3771
369
12 15
79
1.3788
3806
3823
3840
385&
3875
3893
3910
3928
3945
369
12 15
80
1.3963
3980
3998
4015
4032
4050
4067
4085
4102
4120
369
12 15
81
1.4137
4155
4172
4190
4207
4224
4242
4259
4277
4294
369
12 15
82
1.4312
4329
4347
4364
4382
4399
4416
4434
4451
4469
369
12 15
83
1.4486
4504
4521
4539
4556
4573
4591
4608
4626
4643
369
12 15
84
1.4661
4678
4696
4713
4731
4748
4765
4783
4800
4818
369
12 15
85
1.4835
4853
4870
4888
4905
4923
4940
4957
4975
4992
369
12 15
86
1.5010
5027
5045
5062
5080
5097
5115
5132
5149
5167
369
12 15
87
1.5184
5202
5219
5237
5254
5272
5289
5307
5324
5341
369
12 15
88
1.5359
5376
5394
5411
5429
5446
5464
5481
5499
5516
369
12 15
89
1.5533
5551
5568
5586
5603
5621
5638
5656
5673
5691
369
12 15
INDEX.
A.
Absolute measurements, 5.
Accidental errors, axioms of, 29.
errors, criteria of, 121.
errors, definition of, 26
errors, law of, 29, 35.
Adjusted effects, 149.
Adjustment of the angles about a
point, 81.
of the angles of a plane triangle, 93.
of instruments, 15, 183.
of measurements, 21, 42, 63, 72.
Applications of the method of least
squares, 203.
Arithmetical mean, characteristic
errors of, 51.
mean, principle of, 29.
mean, properties of, 42.
Average error, defined, 44.
Axioms of accidental errors, 29.
B.
Best magnitudes for components,
fundamental principles, 165.
general solutions, 167.
practical examples, 173.
special cases, 170.
C.
Characteristic errors, defined, 44.
errors, computation of, 53, 57, 66,
71, 99, 101, 112, 114.
errors of the arithmetical mean, 51.
errors, relations between, 49.
Chauvenet's criterion, 127.
Computation checks for normal equa-
tions, 83.
Conditioned measurements, 17.
quantities, determination of, 92.
Constant errors, elimination of, 117.
errors, defined, 23.
Conversion factor, defined, 3.
factor, determination of, 8.
Correction factors, defined, 131.
Criteria of accidental errors, 121.
Criticism of published results, proper
basis for, 117.
Curves, use of, in reducing observa-
tions, 198.
D.
Dependent measurements, 17.
Derived measurements, defined, 12.
measurements, precision of, 135.
quantities, defined, 95.
quantities, errors of, 99.
units, 4.
Dimensions of units, 5.
Direct measurements, defined, 11.
measurements, precision of, 130.
Discussion of completed observa-
tions, 117.
of proposed measurements, general
problem, 145.
of proposed measurements, prelim-
inary considerations, 144.
of proposed measurements, primary
condition, 146.
E.
Effective sensitiveness of instru-
ments, 183.
Equal effects, principle of, 147.
Equations, observation, 74.
normal, 75.
Error, average, 44.
fractional, 101.
mean, 46.
probable, 47.
245
246
INDEX
Error, Continued.
unit, 31.
weighted, 67.
Errors, accidental, 26.
characteristic, 44.
constant, 23.
definition of, 18.
of adjusted measurements, 105.
of derived quantities, 99.
of multiples of a measured quan-
tity, 98.
of the algebraic sum of a number
of terms, 95.
of the product of a number of
factors, 102.
percentage, 104.
personal, 25.
propagation of, 95.
systematic, 118.
systems of, 33.
Examples, see Numerical examples.
F.
Fractional error, defined, 101.
error of the product of a number
of factors, 102.
Free components, 169.
Functional relations, determination
of, 15, 195, 198, 203.
Fundamental units, 4.
G.
Gauss's method for the solution of
normal equations, 84.
General mean, 63.
principles, 1.
Graphical methods of reduction, 198.
I.
Independent measurements, 17.
Indirect measurements, 11.
Intrinsic sensitiveness of instru-
ments, 183.
Law of accidental errors, 29, 35.
Laws of science, 2.
Least squares, method of, 72.
M.
Mathematical constants, use of, in
computations, 153.
Mean error, defined, 46.
Measurement, defined, 2.
Measurements, absolute, 5.
adjustment of, 21, 42, 63, 72.
derived, 12.
direct, 11.
discussion of, 117, 144.
independent, dependent, and con-
ditioned, 17.
indirect, 11.
precision of, 19, 130, 135.
weights of, 61.
Method of least squares, applica-
tions of, 203.
of least squares, fundamental prin-
ciples of, 72.
Mistakes, 26.
N.
Negligible components, 154.
effects, 151.
Normal equations, computation
checks for, 83.
equations, derivation of, 75.
equations, solution by determi-
nants, 114.
equations, solution by Gauss's
method, 84.
equations, solutions by indetermi-
nate multipliers, 105.
equations, solution with two in-
dependent variables, 78.
Numeric, defined, 2.
Numerical examples:
Adjustment of angles about a point,
81.
Adjustment of angles of a plane
triangle, 93.
Application of Chauvenet's crite-
rion, 129.
Best magnitudes for components,
173, 175, 180.
Characteristic errors of direct
measurements, 56, 70.
INDEX
247
Numerical examples Continued.
Coefficient of linear expansion, 78.
Discussion of proposed measure-
ment, 157.
Effective sensitiveness of potenti-
ometer, 190.
Errors of a derived quantity, 101.
Fractional errors, 101.
Precision of completed measure-
ment, 140.
Probable errors of adjusted meas-
urements, 113, 115.
Probable error of general mean, 69.
Propagation of errors, 101.
Solution of normal equations by
Gauss's method, 88.
Weighted direct measurement, 69.
O.
Observation, denned, 15.
equations, 74.
standard, 62.
Observations, record of, 16.
report of, 211.
representation of, by curves, 198.
P.
Percentage errors, 104.
Personal equation, 26.
errors, 25.
Physical tables, use of, 138.
Precision constant, 35.
Precision of derived measurements,
135.
of direct measurements, 130.
of measurement, denned, 19.
Precision measure, denned, 132.
Preliminary considerations for select-
ing methods of measurement,
144.
Primary condition, 146.
Principle of the arithmetical mean,
29.
of equal effects, 147.
Probability curve, 32.
function, 34.
Probability curve Continued.
function, comparison with experi-
ence, 40.
integral, 37.
of large residuals, 124.
of residuals, 30.
principles of, 28.
Probable error, denned, 47.
error of adjusted measurements,
111, 112, 116.
error of the arithmetical mean, 53.
error of direct measurements, com-
putation of, 54, 55, 57.
error of the general mean, 66, 68.
error of a single observation, 54,
68, 108.
error of a standard observation, 62.
Propagation of errors, 95.
Publication, 209.
R.
Research, fundamental principles,
192.
general methods, 193.
Residuals, defined, 27.
distribution of, 29.
probability of, 30, 124.
S.
Sensitiveness of methods and instru-
ments, 183.
Separate effects of errors, 133, 135.
Setting of instruments, 15.
Sign-changes, defined, 123.
Sign-follows, defined, 123.
Significant figures, use of, 19, 58.
Slugg, defined, 9.
Special functions, treatment of, 155.
Standard observation, defined, 62.
Systematic errors, defined, 118.
Systems of errors, 33.
of units, 7.
T.
Tables, list of, ix.
Transformation of units, 8.
Treatment of special functions, 155.
248 INDEX
U. W.
Unit error, 31. Weighted errors, 67.
Units, c.g.s. system, 7. mean, 63.
dimensions of, 5. Weights of adjusted measurements,
engineer's system, 7. 105, 112, 114.
fundamental and derived, 4. of direct measurements, 61.
systems in general use, 7.
transformation of, 8.
Use of physical tables, 138.
significant figures, 19, 58.
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