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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 

s^ 2 


a! se 






H H H 


ii H H n 






11111 


1 1 1? 






-2 J2 


^l^ 






1 








co d 








II II 








11 








bO 









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 

CO 10 




II II 


II II II II 


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 






-is 








bfi 






O Q 


g 1 g 


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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 = *&lt, 
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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