Mortin Duke
ISM Wa/nutStj Berkeley
PRACTICAL
LEAST SQUARES
PRACTICAL
LEAST SQUARES
BY
ORA MINER LELAND, B.S., C.E.
Dean of the College of Engineering and Architecture and the School of
Chemistry, University of Minnesota. Member of the American
Society of Civil Engineers, American Association of
Engineers, Society for the Promotion of Enqinecring
Education, American Astronomical Society, etc.
First Edition
Fourth Impression
McCx RAW-HILL BOOK CO^IPAXY, Ixc
N E W YO R K A X D L O X D O X
19 21
Copyright 1921, by the
McGraw-IIii.l ]?ook Compaxy, Inc.
PRIXTKD IX THE trX'ITED STATES OF AMERICA
TO
MY WIFE
whose i>oyal assistance
has contributed ix creat measure
to its prei'aration',
this book is aifkctionately
dedic:ated
EngjUa^S.
'J h
PREFACE
This book results from the author's experience in teaching
the subject of Least Squares and the Adjustment of Observations
to classes of civil engineering students at Cornell University.
As the time allotted to this work became more and more limited,
the available textbooks became less adaptable to the scope of
the course. To meet this condition, a series of chapters entitled
" Notes on the Adjustment of Observations " was prepared and
used as a text. With these notes as a basis, this book has been
written.
It is designed particularly for use in short courses of instruction
and In' engineers and scientists in connection with their private
practice. It will not replace the more elaborate treatises on the
subject but the author hopes that it will introduce the student
directly to the simpler methods of solving the ordinary problems
in adjustment.
The plan of the work is essentially practical. After a general
introduction devoted to a consideration of the character and
occm-n^nce of errors, the adjustment of direct, indirect, and
conditioned observations is taken up in detail and illustrated
by numerical applications to triangulation, leveling, astronomy,
and the derivation of emi:)irical formulas. Not until after this
practical treatment of the determination of the best values of
th(; unknown quantities is the precision of observations discussed,
togeth(M- witii the ('omputatif)n of the mean s(|uare and pi-oba])le
errors of the observations and results. Finally, the ])rinciples
of i)ro])ability and the analytical derivation of tlie Law of Error
are given in a]i})en(lices.
The utility of this arrangement should be o])vious. By far
the greater number of a]i]iHcations of Least Scjuares do not
require a consideration of the ])recision of thc^ i-esults nor a
kiunvknlge of the nunm sciuai'c or jiro])a])l(^ (>rrors. ^Moreover,
vii
viii PREFACE
the subject of the precision is usually the most troublesome part
of the work for the student or the beginner to understand.
Therefore, the practical methods of adjustment are explained
directly and fully, without regard to the probable errors or to
the theoretical derivation of the Law of Error. A special effoi't
has been made to explain the procedure in each case as com-
pleteh' as necessary for the beginner as well as the practitioner,
even at the risk of criticism for undue length. The usual
difficulties experienced by students seem to justify this effort.
In Appendix D there is given an outline of a short course of
instruction suitable for civil engineers. This plan was carried
out successfulh' by the author in sixteen lessons. While it is
not at all desirable to restrict the work so severely, if no more
time can be given to it the course is still very much worth while.
The author is indebted to many excellent works and has
endeavored to make specific acknowledgments throughout the
book wherever due. In the preparation of the original notes
and their application to class instruction, his thanks are due
to his former colleagues, Professors P. H. Underwood and L. A.
Lawrence, for their assistance and suggestions.
0. ^I. Lelaxd.
Minneapolis, Minn.
Sept:7nb('r, 1921.
CONTENTS
PAGE
Preface vii
CHAPTER I
INTRODUCTION
Art.
1. Discrepancies among Observations 1
3. Necessity for Adjustment 3
4. Errors of Observation 4
5. Systematic or Constant Errors 5
6. Theoretical Errors 5
7. Instrumental Errors 6
8. Personal Errors 6
9. Mistakes or Blunders 7
10. Accidental Errors of Observation 7
13. Accidental Errors, only, Considered in Adjustments 9
14. Assumption of the Arithmetic Mean 9
15. Residuals (v) 10
IG. Regularity in the Occurrence of Accidental Errors 10
17. Curve of Error 10
IS. Assumptions as to the Occurrence of Errors 11
19. Law of Error 12
20. Tests of the Law of Error 12
21. Method of Least Squares 13
22. Number of Observations 14
23. ^Fwo Uses of Least Squares 15
24. ( 'Uissification of Problems 15
CHAPTER II
DIIiECT OHSERAATIONS OF ONE (QUANTITY
25. Direct Observations: Readings 17
2i). 01)S('r\'ations Resulting from a Combination of Heaflings 17
27. The Mean IS
28. Comjnitation of the Mean 19
29. (^)iitrol or Cluvk ol ihe Ab^an 20
30. Weighted Observations 20
ix
X CONTENTS
Art. page
31. Definition of Weight {w) 20
32. Sources of Weights 21
33. The Weighted Mean 22
34. Principle of Least Squares for Weighted Observations 23
35. Control or Check of the Weighted Mean 24
36. Weighted Mean of Two Quantities 24
CHAPTER III
INDIRECT OBSERVATIONS, OF A FUNCTION OF THE
UNKNOW^N QUANTITIES
37. Indirect Observations 26
38. The General Function 26
39. The Linear Function 26
40. Observation Equations 27
41. Adjustment of Indirect Observations of Unequal Weight 27
42. Observations of Equal W^eight 29
43. Control or Check in the Formation of the Normal Equations. ... 29
44. Symmetry of the Normal Equations 30
45. Formation of the Normal Equations. Aids 30
47. Exam-pie of the Direct Formation of Normal Equations 32
48. Use of Assumed Approximate Values of the Unknowns 34
49. Adoption of New Unknowns to Equalize Coefficients 35
50. Example: Time by Star Transits 36
51. General Application of the Method 38
CHAPTER IV
SOLUTION OF NORMAL EQUATIONS
52. Methods of Elimination 40
53. The Gauss Method of Substitution 40
54. Requirements of a Good Method 40
55. Algebraic Elimination by Addition 41
56. Symmetry among the Derived Equations 43
57. Omission of Redundant Terms 44
58. The Series of Derived Equations 46
59. Control or Check in the Solution of the Normal Equation.s 46
60. Elimination by the Abridged Method; Example 47
62. Notes and Suggestions 49
63. ^'alues of the Unknowns 49
64. Final Check of the Unknowns 50
65. Refinement of the Computations 50
66. Mechanical Aids in the Solution 51
CONTENTS xi
CHAPTER V
OBSERVATIONS OF DEPENDENT QUANTITIES:
CONDITIONED OBSERVATIONS
Art. page
67. Dependent Quantities 53
68. The Observations 54
69. The Weights 54
70. Conditions 54
71. Number of Conditions 55
72. Statement of Conditions 57
73. Adjustment by the ISIethod of Correlates 59
74. Observations of Equal Weight 61
75. Controls or Checks upon the Computation 62
76. Tabular Forms for Computations 63
77. Example: Adjustment of Levels 64
78. Arrangement of Equations 70
79. Examjile: Local Adjustment of Angles bj- the Method of Correlates. 71
80. Special Case of One Condition Only 73
81. Adjustment by the Method of Indirect Observations 75
82. Example: Local Adjustment of Angles as Independent Quantities. . . 76
83. Comparison of the Two Methods 77
84. Adjustments not Rigid 78
(^HAPTER VI
ADJUSTMENT OF TRIANGULATION
85. Triangulation 80
86. Nature of the Conditions 81
87. Local Adjustment 82
88. Figure Adjustment. Notation 82
89. Classification of Figures 83
90. Angle Equations 84
91. Number of Angle Equations in a Figure 86
92. Side Ecjuations 87
9.'?. Side Equation of a Quadrilateral 88
94. Shorter Form of the Side Equation 91
95. Side Equation for a Central-point Figure 92
96. Mechanical Statement of Side Equations 93
97. Number of Side Equations in a Figure 95
98. Statement of All of the Conditions for a Figure .Adjustment 96
99. Exfinrple: Adju.stment of a (Quadrilateral; Method of Angles 99
100. Use of Directions Instead of Angles 105
101. Notation: Method of Directions 107
102. Lists of Directions 107
103. Statement of Conditions: Method of Direction.- 108
104. Example: Adjustment of a Quadrilateral; Method of F)ire('tions. . . Ill
xii CONTENTS
Art. page
105. Example. Adjustment of a Quadrilateral: Approximate Method .. , 118
106. Adjustment to Conform to Work Previously Adjusted or Fixed. ... 119
107. Two .Sides and the Included Angle Fixed 121
108. Quadrilateral with One Fixed Triangle 122
109. Fixed Triangle or Polygon with Central Point Unoccupied;
Exam-pie -. 123
110. Adjustment of a System between Points of Control 127
111. Adjustment of Trigonometric Leveling 130
112. Base Lines 130
CHAPTER VII
EMPIRICAL FORMULAS
113. Empirical Formulas 131
114. Their Uses 131
115. Nature of the Problem 132
116. The Form of the Equation 132
117. Straight Lines and Parabolic Arcs 133
118. Periodic Functions 134
119. Non-linear Forms 135
120. Exponential Functions 135
121. General Case of Reduction to Linear Form 137
122. Determination of the Constants 139
123. Test of Empirical Formula 139
124. Remarks 140
125. Example: Straight Line 141
126. Example: Parabola 143
127. Example: Exponential Curve 145
128. Example: Periodic Curve 148
129. References 150
CHAPTER Vni
PRECISION OF OBSERVATIONS AND RESULTS AND
COMBINATION OF COMPUTED QUANTITIES
131 . Precision 151
132. Precision and Accuracy 151
133. Index of the Precision 152
134. The Quantit}', h, in the Law of Error 152
135. The Mean Square Error f«) 153
136. The Probable Error (r) 155
137. The Average Error (t]) 157
138. Compari.son of the Indices of Precision 158
139. Precision of Direct Observations 160
140. Precision of a Single Observation Kil
141 . Precision of the Mean 163
CONTENTS xn\
Art. page
142. Example: Precision of the Mean 164
143. Precision of the Weighted Mean 165
144. Exam'ple: Precision of the Weighted Mean 167
145. Precision of Indirect Observations 167
146. Weights of the Unknowns 168
147. Precision of an Observation of Weight Unity 169
148. Example: Precision of Indirect Observations 170
149. Precision of Conditioned Observations 172
150. Examples: Differences of Elevation 174
151. Precision of Computed Quantities 177
152. Simple Propagation of Error 178
153. Example: Precision of the Mean 181
154. Compound Propagation of Error 181
155. Examples: Propagation of Error 182
Combination' of Computed Quantities
156. Weights from Mean Square or Probable Errors 187
157. Limitations 188
158. Example: Weighted Mean of Computed Quantities 189
159. Precision of the Adjusted Value 189
160. Example: Precision of the Adjusted Value 189
CHAPTER IX
CONCLUSION
161. Rejection of Observations 191
162. Criteria for Rejection of Observations 192
163. Methods of Observing 193
164. Precision Desired and Number of Observations 193
165. Ultimate Limit of Precision and Accuracy 194
166. Indication of Systematic Errors 195
167. Treatment of Discordant Observations 196
168. ArVjitrary Adjustments 196
169. Use and Abuse of Least Squares 197
170. Adju.<tments not Infallible 198
171. Other Laws of Error 198
172. Review; Outline of Methods of Adjustment 199
APPENDICES
A. HISTORY AND BIBLIOGRAPHY OF LEAST SQUARES
173. Historical Sketch 201
171. ( Irowth of the Literature 202
175. Hililio^a-aphy 202
xiv CONTENTS
B. PRINCIPLES OF PROBABILITY
Art. page
176. Definition 204
177. Two Sources of Probability 204
178. Simple Probability 20,5
179. Compound Probability. Independent Events 206
180. Compound Probability. Dependent Events 206
181. Number of Occurrences 208
C. DERIVATION OF THE LAW OF ERROR
182. The Law of Error 209
183. Assumptions. The Error Function 209
184. Derivation of the Law of Error 211
185. The Constant, C 213
186. Expansion of Law of Error in Series 215
187. Tables of the Law of Error 215
D. OUTLINE OF A SHORT COURSE OF INSTRUCITON
188. General Plan 217
189. List of Problems 218
E. TYPICAL CURVES FOR REFERENCE
Plate I. Straight Lines, Parabola. Hyperbola 221
II. Parabola 222
III. Hyperbola. Parabolas 223
IV. Parabolas 224
V. Parabolas 225
VI. Hyperbolas 226
VII. Exponential and Logarithmic Curves 227
VIII, Periodic Curves 228
F. TABLlvS
Table I. Probability of an Error Less than A: Argument is /,=//A. ... 229
II. Probability of an Error Less than A: Argument is A e 230
III. Probability of an Error Less than A; Argument is A/r 231
IV. Factors for Computing Probable Errors from Bcs-sel's
Formulas 232
PRACTICAL LEAST SQUARES
CHAPTER I
INTRODUCTION
1. Discrepancies among Observations. Measurements made
in the field, office, or laboratory directly depend upon readings of
scales, circles, micrometers, clocks, watches, etc. The readings
may be made to the nearest division, or graduation, or the space
between two adjacent graduations may be subdivided by estima-
tion, thus carrying the observation to a greater degree of refine-
ment.^ When successive settings or pointings of the measuring
apparatus are made, upon the same object, the corresponding
readings may be the same as the first if the graduations be coarse
and the nearest one, only, recorded. But if the divisions be very
fine and the readings made with the aid of a magnifier, or reading-
glass, and by estimation, there may be considerable variation
among them, especially in the last figure which is estimated.
For example, consider the following measurements of a line
made with a steel tape in a drizzling rain, using spring-balance,
hand-level, and plumb-bobs, the tape l)cing graduated to hun-
dredtlis.
899.754 ft. 899. 7():^ ft.
.7()1 .7r)()
.760 .7.')9
.7.")8 .7r)9
.7()2 .7(iO
If the readings had been made to the nearest hundrcxlth. all after
' It is oustomary to estimate to tenths, although an exiiericnccd observer
will sometimes record to five one-hundredths when the reading seems to lie
between two adjacent tenths, greater than the one and less than the other.
2 PRACTICAL LEAST SQUARES
the first would have been ahke and 899.76 ft.; if to the tenth only,
each reading would have been 899.8 ft., indicating that the care in
handling the apparatus would justify the use of a more precise
method of making the readings, or that some of the precautions
were unnecessary.^ Thus it will be seen that the observations
may be so rough or coarse as to show no variation whatever.
Their very agreement, in such a case, might be misleading, as
indicating a false precision.
Realizing the occurrence of these small discrepancies among
observations, when made with care, the observer makes a number
of readings, instead of a single one, and by some method of adjust-
ment adopts a certain value for the observed quantity as a result
of his series of observations. If they were made with equal care
and under the same conditions, he may consider them to be of
equal weight and that none is entitled to preference over the others,
in which case it will be reasonable to adopt the simple mean or
average of the set as the best value obtainable from these observa-
tions. In fact, this adoption of the mean is axiomatic.
2. It will be evident that absolute correctness in the observed
quantity is unobtainable as a result of the observations them-
selves. In the above example, it would be impossible to de-
termine the length of a line down to a millionth of a foot (the
sixth place of decimals), using this method of making the meas-
ures. Certainly, then, correctness to an infinite number of places
is beyond hope. Moreover, it is impossible to ascertain the
correct value of the next figure beyond the limit of our observa-
tions. Whatever value may be adopted as a result of adjust-
ment, it should be regarded as but an approximation to the true
one, that is, as the best available value within our knowledge.
The discrepancies among the observations of a (luantity, then,
show that these observations are not quite correct, — that the
work is not perfect, in other words, but is attended by errors of
observation. The differences between the readings are not the
errors themselves but serve to indicate thc^ existence of errors.
• Addiiif^ a zero after the last observed fisure, making the reading, in this
example, 899.80 instead of 899.8, is a habit of some beginners which should be
studiously avoided.
INTRODUCTION 3
If it were possible to ascertain the correct value of the observed
quantity the true error of each observation would be easily found
as the difference between the observation and the correct value. ^
But just as it is never possible to know the correct value, so the
true errors must be regarded as ideal and indeterminate.
3. Necessity for Adjustment. By making several observations
upon a quantity, in succession, two objects are attained, namely,
greater precision in the resulting mean than in a single observation,
and the check upon the work afforded by the agreement of the
various readings among themselves, within the limits of the small
discrepancies above described. The several observations having
been made, however, for the purpose of securing a better value of
the observed quantity than any one of the separate readings would
be Hkely to be, that is, a value presumably closer to the true or
correct value, it is necessary to arrive at, and adopt, some one
value of the quantity, for use in any computations which may
involve it, such use being the probable reason for making the
observations in the first place. This necessity arises from the fact
that if different values of the same quantity be used in the com-
putation, the results will fail to check.
Similarly, if two or more related quantities, resulting from
observations, be used in computations without having been
adjusted so as to satisfy the relation between them, the results
will be inconsistent and checks upon the computation will be sac-
rificed. For example, suppose the three horizontal angles of a
triangle have been measured in the field and their sum, as usual,
fails to equal the theoretical amount, namely, 180° plus the
spherical excess of the triangle. In order that the triangle may
be computed and the sides checked, the three angles must be
adjusted l)y the application of small corrections so as to satisfy
the theoretical sum. Also, if a scries of benchmarks be connected
' It is well to adopt the rule of su])tracting the incorrect or observed
ciuantity from the correct or adjusted one. algebraically, taking account of
the signs. The resulting dilference, with its sign, is then the correction to be
added algebraically to the observed (piantity to obtain the adjusted one.
Strictly, the error lias the opposite sign to the correction, but the latter is more
con\-enient in most cases, and the use of a fi.xed rule tends to avoid mistakes.
An old expression of this rule is, Suhtrnct llie fahc from the true.
4 PRACTICAL LEAST SQUARES
by lines of levels, some of which are check-lines forming with the
others complete circuits, it is necessary to adjust the differences
of elevation so that all of the circuits will close exactly, in order
that the difference of elevation between any two benchmarks will
be constant when computed through two or more series of lines,
that is, by two or more different routes.
Obviously, any computation could be carried out and checked
even though the original data were assumed and far from the
truth, provided they were not inconsistent. But it is not suf-
ficient that the data be consistent; they must be as near the truth
as our knowledge permits if the results are to be of the greatest
value. Observations are made for the purpose of securing infor-
mation with precision, and the results serve as a basis for accurate
computations. Therefore, it is important to so combine the obser-
vations as to give due consideration to each one and to obtain for
each quantity the best value which the given observations can
yield, that is, the value which they indicate to be nearest the
truth. However, the time and labor involved should not be
unreasonable or excessive in view of the objects to be secured.
The process of combining the various observations so as to
obtain the best values of the quantities concerned is called the
adjustment of the observations. The results are referred to as the
adopted, adjusted, or corrected values. The small quantities to be
added algebraically to the observations to obtain these adjusted
values arc known as the corrections.
4. Errors of Observation. Every observation made in the
process of measurement is likely to be in error from various causes,
that is, the actual reading is not the quantity really sought —
is not what it would be if conditions were ideal and perfection
attainable. Some of these causes are beyond tlie control of the
observer while others depend entirely upon his skill and i)erson-
ality. For example, the altitude of a star is measured with a
surveyor's transit. The star appears higher- than it really is,
owing to atmospheric refraction. The instrument is never in
perfect adjustment, so that wluni the star is seen on the hori-
zontal thread the vertical circle does not show the corre(;t altitude
of the line of sight. Moreover, the obsorvcM- liiius(4f may have tlie
INTRODUCTION 5
habit of noting the time when a star crosses a thread a fraction of a
second too late. Then he, or his recorder, may make a mistake
of a whole minute in taking the time from his watch. And finally,
he reads the vertical circle vernier to the nearest half-minute, per-
haps, with a possible error, therefore, of one-fourth of a minute.
It is customary to include the effects of all influences such as
those illustrated in the above example in the term, errors, and to
classify them as Systematic or Constant Errors, Mistakes or
Blunders, and Accidental Errors of Observation.
5. Systematic or Constant Errors occur in accordance with
fixed laws or are constant during a set of observations made under
unvarying conditions. Their effects are eliminated from observa-
tions, as far as our knowledge permits, in two ways: first, by
the application of corrections computed from the known laws of
the occurrence of the errors; and second, by making the observa-
tions according to a prearranged plan so that the conditions will
be reversed during half of the set, changing the signs of the cor-
responding systematic errors; these therefore neutralize those of
the other half-set when the observations of the whole set are com-
bined.^ Systematic errors are divided into three classes, namely,
Theoretical, Instrumental, and Personal Errors.
6. Theoretical Errors conform to certain laws from which
their effect upon observations made under given conditions may
be computed and corresponding corrections applied, as soon as
these laws are known. Refraction and aberration of light, expan-
sion of metals with rise of temperature, and dip of the horizon are
examples. The form of a law is usually determined theoretically
but its constants may result from observations. Theoretical errors
arc not errors in the sense of being accidents or inaccuracies, but
' This arrangement of a program for observing, so as to eliminate syste-
matic errors, is exceedingly important. Observers and computers should
always l)e on the lookout for new and unforeseen sources of these errors, as
the observations may not reveal them, and the results, apparently good, may
be erroneous to a suri)rising degree. The experience of the observer is inval-
uable in his study of the conditions under which his observing is done, with
this end in view. As our know'ledge of the sources of error increases, so does
our ability to bring the results of observations closer to the truth. (See Wright
and llayford: Adjustment of Observations, Art. 201.)
6 PRACTICAL LEAST SQUARES
rather are the effects of certain influences which operate to prevent
the observer's seeing or reading directly the quantity which he seeks
in his observations. They are included in the classification and
study of errors merely as a matter of convenience and as a result
of custom.
7. Instrumental Errors may be defined as imperfections in
the construction or adjustment of instruments, or the effects of
those imperfections upon observations made with the instruments.
Among these may be mentioned the graduation errors of scales
and circles, eccentricity of circles, inequality of pivots, collima-
tion error, and error of runs in a micrometer microscope. They
may be determined by measurement and the corresponding correc-
tions applied to the observations, or the observing plan may be
such as to eliminate their effects.
8. Personal Errors are generally referred to as Personal
Equation. They depend upon the habits of the observer and
his physical condition. They result, frequently, from the habit of
always setting the thread of a telescope slightly to one side of the
object sighted, or of always noting the time or giving a signal too
early or always too late. No one can hope to be free from such a
tendency, and some of the best observers the world has ever known
have had unusually large personal equations. Good, steady
observers in normal physical condition will have nearly constant
personal equations, whether large or small, and this steadiness
of habit is more important than that the error be small in amount.
If the observations be differential in character, the personal equa-
tion of the observer may have no effect, if it he constant and if all
the readings V)c made by him. This, for example, would be the
case in leveling, if the rod-target were alwa\'s placed too high oi'
always too low and by the same amount. Similarly, it may not
affect the measurement of angles in tiiaiigulation. But if dif-
ferent observers be involved, the results may be affected by the
sum or difference of their personal equations.
The effect of this error may be eliminated, in some cases, by an
exchange of observers, as in telegraphic longitude determinations;
or, its amount may be det(U'min(Hl by special experiments or
apparatus, for each observer, then assumed to be constant and
INTRODUCTION 7
applied as a correction to his subsequent observations of the same
kind when made under the same conditions, especially as regards
his personal comfort and health. However, the personal equation
of an observer must not be assumed as constant for any great
length of time, and there is always danger in assuming it constant
at all. It is safer to determine it at different times and to inter-
polate for its value between these results. Depending upon
personal peculiarities, it follows no law and is often the most
troublesome source of error to which observations are subject.
Fortunately, it is small in amount in most cases.
9. Mistakes or Blunders are irregular in their occurrence,
obeying no law, and are relative^ large in size. They result from
haste and carelessness, frequently, on the part of the observer,
during temporary lapses, perhaps, from his customary vigilance.
He may call out to his recorder one number while reading and
thinking another; he may read the wrong division of a circle or
scale; or he may read a clock wrong by a whole minute while he is
estimating tenths of a second. He may turn the wrong tangent-
screw while I'epoating angles, the rod-clamp may slip during level-
ing, or the wrong object may be sighted in triangulation or azimuth
work. The remedy lies in uninterrupted care on the part of the
observer to avoid these blunders, and watchfulness by the recorder
to detect them in any inconsistencies among the readings. Herein
lies one of th(^ chief virtues of a good recorder.
10. Accidental Errors of Observation, or simply Accidental
Errors, is a name given to a specific class of errors in connection
with the adjustment of observations by the Method of Least
Squares. They arc purely errors of observation and have no rela-
tion to systematic errors or the large mistakes already described.
They are small, for the most part, and their presence is indicated
by the discrepancies among a series of readings upon a fixed object
which have been made with the utmost care and precision, with an
instrument which can be read to a greater degi'ce of refinement
than the pointings can be made by the observer. These errors
are never known exactly because the true or correct value of the
quantity ol^served is never known, as has been explained in a
previous article. Thus it is staffed that they are indicated bv
8 PRACTICAL LEAST SQUARES
the discrepancies, not that they are the discrepancies them-
selves.
11. For example, suppose readings are made by a skillful
observer using a micrometer microscope, upon a graduation line or
scratch of a standard meter bar, the whole being enclosed in a
vault of constant temperature so that conditions are steady.
Further, suppose the observer to be able to set the parallel threads
so as to be equidistant from the scratch within 10 microns ^ and
that the micrometer reads directly to five microns and by esti-
mation to one half-micron, that is to 0.0005 millimeter. The
readings, then, might run as follows, the unit being one division
of the micrometer head (equal to 0.005 mm.) :
d d
46.4 45.9
45.6 45.3
46.0 46.1
46.1 45.8
45.9 45.8
46.6 45.2
46.7 46.1
45.4 46.8
46.5 45.1
45.9 46.3
By assumption, the conditions are very favorable for precise work
and the observer is skillful and is using great care in making the
readings; nevertheless, there are discrepancies and the readings
have a total range of 1.7 divisions. These are the discrepancies
which indicate the presence of accidental errors of observation.
They are so small as to be beyond the control of the observer, as he
is assumed to make each separate pointing as carefully as he can.
It may be noted, also, that most of the discrepancy is due to the
errors of pointing, that is, setting the threads on the mark, as
the estimation of tenths of a division of the head would seldom
be in error by a whole tenth.
12. In other examples, the discrepancies might be made up of
accidental errors of several different kinds, such as pointing the
^ A micron is one, one-tliovisaiulth of a milliinctor or one onc-Diillionth of
a meter. It is the unit used in verj' precise measurements of length by
means of micrometer microscopes.
INTRODUCTION 9
telescope upon a signal, setting the threads of the microscope
upon a division of the circle, and reading the micrometer head.
Other sources of errors which may have the nature of accidental
errors are the unsteadiness of the atmosphere and that of instru-
ment supports, and rapid changes of temperature. However, the
foregoing example of simple, direct readings is a clear illustration
of the occurrence of accidental errors without complication. If
the micrometer head had been graduated directly into one thou-
sand parts instead of one hundred, to be read with a magnifier,
the error of estimation in reading it would have disappeared and
the discrepancies might have been ascribed entirely to the acci-
dental errors of setting the threads upon the division on the
circle, — the simplest kind of a case.
13. Accidental Errors, only, Considered in Adjustments. It
has been shown that the effects of systematic errors arc eliminated
by corrections or by the observing program, as far as they arc
known to exist; and that the mistakes, or blunders, are avoided
by the exercise of care and vigilance, as much as possible. Of all
the kinds of errors, then, there remain the accidental ones, still
affecting the observations, and it is to minimize the effects of
these errors that adjustments are made. In all that follows in
this work, therefore, only this special class, the accidental errors,
will be considered, except as others may be specifically mentioned.
14. Assumption of the Arithmetic Mean.^ When eacli obser-
vation or reading has been made with the same care and under the
same conditions as all the others of a set made upon a certain
quantity, so that all arc of (Hjual value, or weight, there is no
reason for preferring any one to any other; the mean, or average,
of them all must be regarded, then, as th(> best value of the observed
(}uantity which can l)e ol)tained from the given set of observations.
The soundness of this i)rin('iple is so cn'idfMit that it is adopted as
the fundamental assumi)tioii in <levelo])ing the theory of the
adjustment of observations. The mean should b(^ regarded, not
as the true value of the ()])S(M-ve(I (juantity. but rather as the
^ I'ho word >nr<in. in tliis work, is uiiil(>rsloo(l to rcfiM' to the arithnulic
mean, or average, in every case. Tlie (ji auK Irn- nH\aii is the sijuare root of
the ])ro(hirt of two ([uantities.
10 PRACTICAL LEAST SQUARES
nearest approximation to it that the given observations will yield,
and subject to improvement if other or better observations should
become available.
15. Residuals (v). The difference between an observed value
of a quantity and the adopted one is known as the residual of that
observation. It should be taken in the sense, adopted minus
observed, for consistency in sign. If the adopted value, the mean,
for example, be the nearest approximation to the truth, then the
residuals obtained with that value would be the nearest approxi-
mations to the actual or true errors of the observations, to the
extent of our knowledge. The occurrence and behavior of the
residuals, then, will be our best indication as to the occurrence of
the true errors. In fact, we may reasonably assume that the
errors and the residuals conform to the same laws. In the inves-
tigation of such laws, therefore, it may be convenient, some-
times, to use the terms somewhat indiscriminately, — to use the
word error when residual is intended.
16. Regularity in the Occurrence of Accidental Errors. At
first thought, it may seem strange that there should beany
method at all in the occurrence of errors which are so small and so
evidently the result of accident or inaccuracy. However, it has
been found from a large number of investigations of observations
of almost ever}" conceivable sort, that these errors occur not only
with regularity but in conformity to a definite law, of which the
general form is the same for all kinds of observations. This law
of the occurrence of errors, or Law of Error, as it is called, is
expressed in the form of an equation which has been completely
derived, and tested, later, in a multitude of cases, with entire
satisfaction. In accordance with this law of error, the Method of
Least Squares has been devised and demonstrated for the adjust-
ment of observations.
17. Curve of Error. As an example, let us consider a large set
of direct observations, say 500 of them, such as the readings of
the micrometer microscope in the example of Art. 11, page 8.
By taking the mean of the entire series and subtracting from it
3ach separate reading, the rcsichials are obtained. Counting the
residuals of each size and sign, we find that there are 36 of +0.1,
INTRODUCTION
11
35 of -0.1, 33 of +0.2, 34 of -0.2, etc., the sum of all the numbers
being, of course, 500. These results are plotted as rectangular
coordinates, the magnitude of the error on the horizontal axis,
plus on the right and minus on the left of the origin, and the
corresponding number of errors of that size on the vertical axis,
upward. Thus one point is plotted for each size of error, and for
each sign. A smooth curve is then drawn so as to follow the
points as closely as possible, with the result shown in Fig. 1 :
Nuniler
of Errors
1 1 y^
aoN. 1 1
no \l !
10 1 \ I
; ^*>^..___ Size
-0.5 0 +0.5 +1.0
Fig. 1. Occurrence of Errors or Residuals
of Error
The form of the curve is typical of all those constructed from
observations in this manner, and it is called the Curve of Error,
or the Curve of Probability of Error, since an ordinate to the curve
may represent the probability ^ of the occurrence of an error as
well as the number of times that error occurs.
18. Assumptions as to the Occurrence of Errors. The error
curve has three properties which are evident from inspection:
first, it is symmetrical about the vertical axis; second, it has a
maximum point where it crosses that axis; and third, it approaches
the horizontal axis so gradually as to appear asymptotic. Gener-
alizing from these properties, the following assumptions, or axioms,
arc obtained, as to the occurrence of errors in any large set of
observations :
1. Positive and negative errors of the same magnitude
occur with equal frequency; they are equally probable.
2. Small errors occur more frequently, or are more prob-
able, than large ones.
^ The probability of an event is directly proportional to the number of
times it occurs. See Appendi.x B, for Princii^les of Probabilit}'.
12 PRACTICAL LEAST SQUARES
3. Very large errors seldom occur; they are likely to
belong in the class of mistakes rather than that of accidental
errors.
It should be remembered that the number of observations in a set
is assumed to be large. The smaller the set, the less closely will
the residuals conform to the ideal conditions, such as that of the
first of these assumptions, but even in a small set they will approx-
imate to their ideal occurrence. Obviously, the larger the number
of observations, the more closely should the mean approach the
true value of the quantity observed, in so far as the accidental
errors are concerned.
19. Law of Error. The general equation of the error curve, or
curve of probability, may be derived ^ from the assumptions of the
last article together with the principle of the mean (Art. 14, page 9).
The curve is seen to be continuous, and it is of special importance
to note that the number of errors, or the probability of an error,
is a function of the size of the error. The algebraic principles of
probability, also, are involved in the derivation. The resulting
Law of Probability of Error may be stated thus :
' ■ Vtt
in which p is the probability of an error A in a set of observations
for which }% is a computed constant; e is th(> base of natural
logarithms; and r = 3.1416+. The constant h lias the value
. in which e is a constant for each separate set of observations,-
and serves to change the general equation into a specific oik^ for
the particular set of ol)scrvations under consideration.
20. Tests of the Law of Error. The law may he tested by
applying it to many diffcM-ont kinds of observations so as to ascer-
tain whether the I'csiduals occur in conformity with it. Con-
versely, if the law be accepted as applicable to all observations,
1 This derivation may be found in Appendix C.
- € will be defined farther on as the mean square error of a single observa-
tion. Its \-alue, for a jiiA^cMi set of observations, depends upon their precision,
and i- determined from the resichials.
INTRODUCTION 13
the quality of a given set could be tested by the same method.
In general, then, it is a process of comparing theoretical results
with observed ones, or theory with practice. The method consists
of the comparison of the number of residuals, in the given series,
which lie between certain limits, with the number of errors which
ought to lie between those limits according to the Law of Error.
For example, e having been computed for the given observations,
the probability of an error between 0.00 and 0.30, say, is deter-
mined by integration and substitution in the equation (1).^
(It will always be less than unity, from the principles of prob-
ability.) Multiplying the total number of observations in the set
by this probability gives the number of errors which ought to lie
between the assumed limits according to the law. The residuals
which actually lie between those limits may then be counted
and their number compared with that obtained from the formula.
Crandall gives an example of a small set of 18 observations of
an angle, with the following results, e being 1.66".
Xvunber of Errors Less Than
1 1 • 1
O.o" 1" 1 2" 3" 1 4"
From theory
liy aftual count
I : 1 i
4.3 1 8.1 13.8 i It). 8 \ 17.8
6 8 14 ' 17 17
This agreement is very satisfactory but might be nuich closer in a
larger set of observations.
21. Method of Least Squares. The most prol)able value of the
observed quantity obtainable from a given set of observations will
be the one corresponding to the most probable set of errors or of
residuals. Consider a set of n observations of equal precision, of
which the most probal^le eri'ors are Ai, A^, X>,, . ■ . A„, respectively.
Siii('(> the probability of the siinultan(^ous occurrence of several
(n'cnis in a series is the product of theii' s(>pafate pro]jal)ilities,-
and tlie probability of an erroi'. A. is, fi'oin (1):
' The iis(> of the (equation is facilitalcnl by its transformation into series
from which tal)h^s have Ix'cn ('()mi)ute(l. See Appendices V and F.
- See Ap!)en(h\ W.
14 PRACTICAL LEAST SQUARES
„.-,„/.
it follows that the probability of the simultaneous occurrence of the
errors Ai, A2, A3, . . . A„, will be
■Vtt
that is
P:=[ _ ) g-'i2(Al2 + A22+ ... +An2) rg-v
But since these errors are to be the most probable ones, P must
have its maximum value. As h, t. n, and e are constant in a given
problem, and the exponent of e is always negative, the expression
will be a maximum when the exponent of e is a maximum, alge-
braically, that is, when the sum
Ai2+A2^+A3^+ . . . +^/ is a minimum (3)
Thus, the most probable value of the observed quantity, or the
best value, in other words, obtainable from the given set of obser-
vations, will be the one for which the sum of the squares of the
errors, or of the residuals, likewise, is a minimum. This is called
the Principle of Least Squares and the method which is based
upon it, for the adjustment of observations, is known as the
Method of Least Squares. It was first published by Legendre, in
1806, although used by Gauss as early as 1794. ^ In the general
case, involving the determination of several quantities, and obser-
vations of unequal weight, it provides that the most probable
values of the unknown quantities will be those for which the sum
of the weighted squares of the residuals is a minimum. This
form will be discussed later (Art. 34).
22. Number of Observations. In the development of the
]\Iethod of Least S(}uares, it is assumed that the number of obser-
vations is large. The assumptions as to the occurrence of errors
approach the truth more closely as the number of the errors
increases. However, if the method be applied to small sets of
^ See Appendix A,
INTRODUCTION 15
observations, although the results may be farther from the correct
values, still they may be regarded as the best ones obtainable
from the given observations, which is sufficient warrant for the
use of the method under such unfavorable conditions. It is
unreasonable to generalize too greatly from a very small set of
residuals, as to the precision of a result, but it is still permissible
to take the mean of even two observations, if they be the only
available data.
Regarding the number of observations, it must be remembered
that no adjustment is possible unless there are more observations
than unknown quantities. If the number be less, the unknowns
cannot be determined without additional information or assump-
tion. If the number be equal to that of the unknowns, there is
only one solution, namely, the rigid, algebraic one by means of
simultaneous equations.
23. Two Uses of Least Squares. The ^Method of Least Squares
is essentially a practical subject, being devoted to the solution of
numerical problems. Its applications may be divided into two
classes: first, the determination of the best values of the unknown
quantities obtainable from given observations, that is, the adjust-
ment of observations; and second, the investigation of the
precision of the observations and the results, and the influence of
errors upon them. Those two uses of the method are quite inde-
pendent; most problems require adjustment, but the precision
may not be investigated at all unless the results are to be compared
with those of other observations. In this treatment of the sub-
ject, therefore, immediate attention will be given to the adjust-
ment of the various kinds of oliscrvations, but the determination
of the precision of the results will be postpoiunl to a later chapter. ^
24. Classification of Problems. In the following chapters,
the typical problems are such as the engineer frequently encounters
in field work. A certain method of solution is adopted for each
type. The adjustment of the three great classes of observations
is taken up in th(^ usual order, namely:
1 Chapter VIII.
16 PRACTICAL LEAST SQUARES
Direct Observations of One Quantity,
Indirect Observations, of a Function of the Unknown Quan-
tities, and
Observations of Conditioned Quantities.
Following these, the investigation of precision and the propagation
of error will be explained. It is important that the student become
familiar with the characteristics of these classes of problems and
with the method of solution of each type. The special problem
of the derivation of empirical formulas and constants will be
treated in a separate chapter.^
' Chapter VII.
CHAPTER II
DIRECT OBSERVATIONS OF ONE QUANTITY
25. Direct Observations: Readings. In their simplest form,
direct observations consist of single readings made upon various
kinds of apparatus used in measurements, such as scales, circles,
micrometers, and timepieces. The example in Art. 11, of microm-
eter readings upon a fixed scale, is typical of this class. The
conditions under which the readings are made are assumed to be
constant or to vary according to a known law so that the discrep-
ancies among the readings maj" be reduced to the accidental errors
of pointing or setting the instrument and of reading.
Usually, however, the conditions are more complex and involve
several sources of error. In the example just cited, the tempera-
ture may vary, causing the position of the division line on the
scale to change. Then if the temperature be read from a mer-
curial thermometer simultaneously with the micrometer readings,
two corresponding sets of direct readings are obtained. Also,
when the altitude of a star is observed for time and azimuth,
each pointing on the star may be attended by readings of the
watch and the horizontal and vertical circles, so that three
simultaneous sets of direct readings result.
26. Observations Resulting from a Combination of
Readings. It fr(!(}uently happens, on the otlu^r hand, that
the so-called observed (juantity is the result of two or more
separate readings of the sanu^ kind. I'or cxanipk^, in tli(>
measurement of angles ])y ivpctition, a singl(> observation is
obtained by subti'acting the initial reading from the final one
and dividing tlu> difference by th(> nunibcM' of I'epelitions, in
the case of the direct uK^asure of tlu^ angl(> its(>lf and also, of
the n'V(M'S(Ml measure of its (>xplenieiit, the ni(>an of the two
17
18 PRACTICAL LEAST SQUARES
results being taken.^ In the measurement of a base line, each
observed length is the sum of several tape-lengths; the elemental
observations consisting of placing the rear scratch on the tape in
contact with a scratch on a marking-plate and of making a mark
on a plate opposite the scratch at the forward end of the tape.
Similarly, the observed difference of elevation between two
benchmarks consists of the algebraic sum of a series of fore- and
back-sight readings of the rod. It is customary, in all such
cases, to consider the result of a single complete measurement to
be the observed quantity, even though it consist of a combination
of separate readings. In its general sense, therefore, the term,
direct observation, may be taken to mean a single measurement
of the quantitj- desired.
27. The Mean. The adjustment of direct observations of a
single quantity consists in taking their mean as the best, or most
probable, value obtainable from the given observations, as ex-
plained in Art. 1-i. That this is in accordance with the principle
of least squares, ma}- be shown as follows :
Let JMi, 3/2 . . . j\In represent a series of observed values;
Xq, the best value of the observed quantity; and vi, vo, . . . I'n,
the corresponding residuals, n being the number of observations.
Then, for each observation there results an observation equation,
thus: ^ ^f _„
Xq — M 1 — I'l
.Tf) — Mo = V2
(4)
.ro — 3/)j = I'n
Squaring both members of each equation and adding the resulting
equations, we ()l)tain,-
(.ro--Vi)^' + (.ro-3/2)- + . . . +(xo-Mn)- = [r-] (5)
1 A simpler method is to subtrart from the reading on the right-liand
object the mean of the two readings on the left-hand object (first and last
readings) and divide the difference by the mimber of repetitions. 'J'he result
is the same.
-The square l)rackets, [ ], indicate^ the suiu of all such terms as arc in-
cluded by them. Thus, [r-] represents the sum of the sf]uares of all the /''s,
that is, rr + rj- + ':-+ ■ ■ +''«-• The symbol, ^, may be used to indicate
summation in the same manner.
DIRECT OBSERVATIONS OF ONE QUANTITY
19
According to the principle of least squares, the sum of the squares
of the residuals, that is, [y^], is to be a minimum. Therefore, we
differentiate the left-hand member and place the first derivative
equal to zero; whence, after dividing by two, we have:
{xo-M{) + {xo-M2)+ . . . +(a:o-Mn)=0 (6)
nxo-{Mx+M2+ . . . +ilf„)=0,
[M]
and
Xo=-
(7)
That is, the best value of the observed quantity — the one for
which the sum of the squares of the residuals is a minimum, is the
mean.
28. Computation of the Mean. Owing to the close agreement
of the observations of which the mean is to be taken, it is possible,
often, to abridge the numerical work by the assumption of an
approximate value of the mean. Suppose the mean of the follow-
ing 16 quantities to be desired
+ ''
— /'
+;•
-V
1403,49768
4
1463.49764
0
Assumed appro.x.
1463 . 49754
10
58
6
value, 60
1463 . 49763
1
66
2
1463 . 49765
1
63
1
1463 . 49759
5
65
1
Sum, +65
1463 , 49771
(
60
4
1463 . 49767
3
67
3
Mean, +4
1463.49765
1
70
()
Remainder, +1
IC) ' 16
11 12
Mean, 1463.19764
[r]
In the first phwo, it i.s cvidcMit from inspection that all but the last
two figures arc the same in all tlu^ quantities, so that there is no
need of writing them r('peat(Mlly. It is sufficient to write the first
nuni])(>r in full and thiM'cafter only the last two figures. Somc-
tiinc^s, in a long column of nunil:)ers, tlie constant part will change
suddcMily to anotli(>r on(\ in which cast^ it is well to write the last
as well as th(> first of each s(M'i(^s in full.
20 PRACTICAL LEAST SQUARES
Also, by inspection, it will be seen that the next to the last
figure in the mean will probably be 6. Thus we may take 60 as
an approximate value of the last two places, calling 54, for exam-
ple, — 6, and 71, +11. Then, adding mentally the figures in the
last place with 60 as a basis, we obtain the sum, +65, and the
mean, +4, so that the full mean is 1463.49760+4 in the last
place, or 1463.49764. This process may be simplified still more
by combining a 5 and a 7 in the next to the last place, as their
mean is 6, without modifying their last figures. Thus in the above
example, 59 and 71 would be added directly as 10 instead of —1
and +11.1
29. Control or Check of the Mean. Substituting equations
(4) in (6) of Art. 27,
Vi+V2-\-V3+ . . . -\-Vn = 0 or, [v] = 0 (8)
That is, the sum of the residuals should be zero, or the sum of the
positive residuals should be equal to that of the negative ones.
This check is very important and should l^e used whenever prac-
ticable. It will be satisfied rigidly unless there is a remainder
when the sum of the observations is divided bj' their number to
obtain the mean. In this case, the check fails by just the amount
of the remainder but with the opposite sign, so that the mean is
verified, nevertheless. In the example in the preceding article,
the sum of the residuals is — 1, and the remainder in taking the
mean is +1, so that the mean was correctly computed.
30. Weighted Observations. Thus far, wc have considered
only observations of equal fjuality or precision. In the general
case, however, one ()l)servation of a series may ])e better than
another, for some reason, and cntitk^l to have a greater influence
upon the result. WIkmi all of the (observations of a set are not of
the same quality or worth, tliey are called weighted observations,
or arc said to ho of une((ual weight.
31. Definition of Weight (w). By the weight of an observation
is meant its relative value among the others of a set. It is
' The l)e<j;iniicr will do ■well to learn to add mentally ])y combinations of
two or three fi^un\s at once, partitailarly those whose sum is 10, as 6 and 4,
7 and 3, or o, 2, and .'!, even though ancjther figure intervenes.
DIRECT OBSERVATIONS OF ONE QUANTITY 21
expressed as a number, and being strictly relative, may be mul-
tiplied by any factor so long as all the others in the set are mul-
tiplied by the same quantity. Thus, the weights may be integral
or fractional. If one observation has a weight of 3 and another
a weight of unity, the first may be considered as the mean of
three observations of the same size, each of which has the weight
unity. The weights could be stated as 6 and 2, as 1 and \, or as
0.153 and 0.051, as well as 3 and 1.
32. Sources of Weights. Either the observer or the com-
puter may assign the weights to the observations and it is largely a
matter of judgment. If the observer assigns them, during the
observing, he has the right to do it by estimation or arbitrarily.
For instance, in the measurement of angles in triangulation, the
atmosphere may be so unsteady during one observation that he
will give to that particular result a weight of one-half that of the
others. Or he may note in his record the fact that the atmosphere
was very unsteady at that time, and leave to the computer, in the
office or at headquarters, the dutj^ of assigning a low weight to
that observation, when making the adjustment. Of course, the
computer might give it a weight of 0.8 instead of 0.5, and thus
change the result somewhat. Or, an arbitrary rule might be
agreed upon so that both would assign the same weight under the
same circumstances. Similarly, two benchmarks may be con-
nected by two lines of levels giving discordant results. If one
run were made during a high wind or with a careless rodman, it
might be given a lower weight than the other.
In the second place, weights may be assigned upon the number of
observations, as a basis. If one measurement of an angle be made
with three repetitions and another with six, the second may be
given twice the weight of the first.
Finally,' the assignment of weights may ])e governed by //^cor//.
In the determination of time by transits of stars across the meridian,
the motion of a star near the ecjuator will be more rapid than that
of one of greater declination, and the rapidly moving one can be
observed more accurately than the other. Th(n-efore, a system
1 For the determination of weights from mean square errors, see Art. I.'i6,
Chapter \'1II, Combination of Computed (Quantities.
22
PRACTICAL LEAST SQUARES
of weights has been devised which depends upon the declinations
of the stars.
33. The Weighted Mean, The best value of the observed
quantity which is obtainable from a given series of weighted obser-
vations is known as the Weighted Mean. To determine it, each
observation is multiplied by its weight and the sum of these
products is divided by the sum of the weights. The analogy of
this process to the determination of the simple mean will be
evident from an example.
Let it be required to adjust the following set of four weighted
observations of an angle, the weights, w, being determined from
the number of repetitions and the notes as to weather conditions:
M
73° 18' 42,16"
41.96
41.70
42.23
Use 42.00 as
Approx. value
+0 . 72
11
+0.07
Mean, 42 07
11
2.16 or +.16
2.16
2.16
1.96
1.96
1.70
1.70
2 . 23
2.23
2.23
2.23
22.72
+ .16
+ .16
-.04
-.04
-.30
-.30
+ .23
+ .23
+ .23
+ .23
+ .72
+v
11
11
37
37
16
16
16
16
wM
+ .48
-.08
-.60
+ .92
-\-LCV
22
74
27
64
96
91
+ .72 9(i 91
By writing each observation a number of times equal to its
weight, and by using 42.00" as an assumed or approximate value
of the mean, the third column is obtained. According to the
definition in Art. 31, this reduces all the quantities in these
columns to the same, unit weight, and their number is the
sum of the weights. Therefore, their mean is the best value,
and by the methods of Art. 28, this is 42.07", with residuals
shown in the columns headed +-r and —v. The mean is checked
by its remainder, —5, against the sum of the residuals, +-5.
It is evident that instead of writing the first observation three
times in the third and fourth columns, it will be easier to multiply
DIRECT OBSERVATIONS OF ONE QUANTITY 23
it by three and write the product, and similarly with the other
observations and their weights; the sums would be unchanged.
Likewise, the residuals may be multiplied by the weights of the
corresponding observations and the products noted instead of
writing all the separate residuals. Thus, the last three columns
are obtained, giving the same results as the preceding ones. The
following rule, therefore, is given for the adjustment of direct
observations of unequal weight, whether the weights be integral
or fractional: Multiply each observation by its weight and divide
the sum of the products by the sum of the weights, to obtain the
weighted mean; and multiply each residual by the weight of the
corresponding observation, adding the products algebraically to
obtain the sum of the weighted residuals.
34. Principle of Least Squares for Weighted Observations.
Let Ml, M2, Ms, . . . Mn represent a set of n observations having
the respective weights, wi, w-z, W3, . • . Wn, and let xq be the best
value of the observed quantity, with vi, V2, V3, . . . Vn as the
corresponding residuals.^ Considering each observation of weight
w to be the mean of w equal observations of weight unity, the
residual of each of these latter observations would be the same
as that of the original one, but there would be w of them. As
stated in Art. 21, for the best value of the observed c^uantity, in
the case of equal weights, the sum of the squares of the residuals
will be a minimum. Therefore, to express this minimum for
weighted observations, each residual must be written a number of
times, w, equal to the weight of its observation. Thus,
{vr-\-vr'-\-vr-[- . . . iow\ terms) + (y2^ + ?-'2^+?^2^+ • . . iowo terms)
+ . . . +(/'«- + r«- + r„2+ ... to Wn terms) is to be a minimum;
that is,
u'V'r + wiv-r-i' ■ ■ ■ ■^-WnV,? must be a minimum (9)
or the sum of the weight od squares of the re-^idiials must hQ a min-
imum. Su])stituting for each v in (9) its value, Xo — M, with the
corrc^sponding subscripts,
IV \{xi)— M \)~ -\- ic-zixo— M -lY + ■ ■ ■ -^Wn(x() — Mn)'~ IS to be a mininunn.
1 Reference to the numerical cxauii)le of the preceding article will be of
assistance in following these steps.
24 PRACTICAL LEAST SQUARES
Differentiating this expression and placing the first derivative
equal to zero, for the minimum, we have, after canceling the fac-
tor 2:
WiiXo-Mi)-\-W2{Xo-M2)+ . . .WniXo-Mn)=0 (10)
Combining terms,
- {wiMi-\-W2M2-\-W3M3-{- . . . -\-WnMn)=0
and
[w]
that is, the best value of the observed quantity, for which the sum of
the weighted squares of the residuals is a minimum, is the weighted
mean, obtained by multiplying each observation by its weight
and dividing the sum of the products by the sum of the weights.
35. Control or Check of the Weighted Mean. If in (10),
above, v be substituted for xq — M, we have
WiVi+W2V2-\-lV3V3-h . . . -^-WnVn^O (12)
or, the sum of the weighted residuals should equal zero. As was the
case, however, in the control of the simple mean, the actual sum of
the weighted residuals should equal the remainder obtained with
the weighted mean but with the opposite sign. This is illustrated
in the example of Art. 33.
36. Weighted Mean of Two Quantities. The solution of this
special case is particularly convenient and instructive. With
the usual notation, let Mi and Mo be the two given quantities,
whose weights are u'l and u'2 respectively, and let .tq be their
weighted mean. Then from (11),
lV\Mi-\-lV2-^I'2 .,„s
.To- • (13)
lC\-+-tV2
Adding and subtracting W2M1 from the numerator,
ir\M\ -\-iV2^!^2^u'2M] — 2C2Mi
iri-^iC2
^ Ml (t/-i +K-2) ■i-W2(M2 - .1/1)
"•l+W'2
= Mi-h-^^{M2-Mi) (14)
Wi-{-W2
•To
DIRECT OBSERVATIONS OF ONE QUANTITY 25
Also, owing to the symmetry of (13), the subscripts may be inter-
changed, and therefore,
xo = M2-{- ^' {M1-M2)
Wi-\-W2
Thus, the weighted mean may be found by correcting one of the
quantities by an amount equal to the difference between the two
quantities multiplied by the weight of the other and divided by
the sum of the weights. Obviously, the mean lies between the
two quantities, so the sign of the correction will be evident. The
weighted mean divides the interval between the two quantities
in the inverse ratio of the weights of the adjacent quantities.
For example, the weighted mean of
6.784 Wt. 7
and 6.743 Wt. 2 is 6.784-|x41 = 6.784-9 = 6.775
the unit, for the correction, being conveniently taken in the last
decimal place. Similarly, the correction to the second quantity
would be +1^X41, with the same result.
CHAPTER III
INDIRECT OBSERVATIONS, OF A FUNCTION OF THE
UNKNOWN QUANTITIES
37. Indirect Observations are those in which the observed
quantity is related to the desired unknown quantities through a
known formula or function. The observed quantity is expressed
as an explicit function of the unknowns, which are usually two
or more in number, and is, therefore, the observed value of the
function. It may be that the unknowns cannot be separated so
as to be observed directly, and that they can only be determined
in combination. They are assumed to be mutually independent;
each may vary without causing a corresponding variation in the
others. Moreover, the number of observations must be greater
than that of the unknown quantities, as stated in Art. 22.
38. The General Function may be algebraic, logarithmic,
exponential, or trigonometric, and simple or complicated. How-
ever, it is always possible to reduce such a general function to the
linear form, that is, to the first degree, either by taking the loga-
rithm of each member or by developing the function by Taylor's
Theorem and neglecting the squares, products, and higher powers
of the small increments involved.^ Furthermore, the great ma-
jority of problems are concerned with the simplest form of func-
tion, namely, the algebraic one of the first degree. Therefore,
we shall here consider only this linear form.
39. The Linear Function between the unknowns, x, y, ?, etc.,
will have the following general form,
ax + by + cz-\- . . .+k (15)
in which a, h, c, etc., are known numerical coefficients or factors
and /.' is the constani t(M'in. As usual, the signs n^prosent algebraic
addition and the (quantities may be positive or n(^gative.
40. Observation Equations are the algebraic statements of
the separate; obscn'vations. Thus, if Mi, M-2, . . • Mn be the
i.See Arts. 119 121.
26
INDIRECT OBSERVATIONS 27
observed values of the function, with the respective weights,
wi, Wo, . . . Wn, the observation equations would be,
aix-\-hiy-{- . . . -\-ki = Mi Wt. wi
a2X+622/+ . . . -{-k2 = M2 W2
(16)
anX + bny+ . . . +A-„ = jT/„ Wn
Their number is the same as that of the observations, and each
subscript indicates its equation and observation.
If X, Y, Z, etc., be the best or most probable values of x, y, z,
etc., to be obtained from the given observations, the substitution
of these values in the above equations (16) would show a residual,
V, for each equation, inasmuch as the observations are subject to
error and no set of values of the unknowns would be likely to
satisfy exactly any one of the observation equations. The ideal
form of these equations, therefore, would be,
aiZ+6iF+ciZ+ . . . +ki = Mi-^vi \Yt. wi
a2X + b2Y-\-C2Z-\- . . . -\-k2 = M2 + V2 W2
(17)
Transposing the M of each ec^uation, and I'epresenting the differ-
ence, k — M, of the two constant (juantities by the constant term,
/, we have for the observation ecjuations,
aiX+6i}^ + ciZ+ . . . +/i = ri Wt. wi
a2X + 62r + C2Z+ . . . +/2 = r2 W2
(18)
anX + bn-Y-^-CnZ-]- . . . -\-In^Vn U'„
which are somotimes called Residual lu/uaiions.
41. Adjustment of Indirect Observations of Unequal Weight.
For th(^ l)est values of the unknown (|uaiilities, the sum of the
weighted scjuares of the rc-siduals is to be a iniiiinuuu. That is,
n:irr-\-W2V2~-\-u':',r:r-{- ■ ■ ■ +ii'nr,r must he a iniiiimuin (9)
Sinc(> .r, y, z, etc., ai'c iiidepeiideiit of ouv another, it follows that
\\\v first (l(M'ivative of the above expression with respect to each of
28 PRACTICAL LEAST SQUARES
them must separately equal zero, for the minimum. Differen-
tiating (9), therefore, and canceling the factor, 2, from each
equation, we have:
-"l , "«^2 I . avn f.
WlVi—--\-W2V2-—z+ . . . -hWnVn-r;^ = 0
dX dX dX
avi av2 . . dvn ^
dY dY dY
(19)
There will be one equation for each of the unknown quantities.
The differential coefficients in the first of these equations are the
coefficients of X in the successive equations (18), those in the
second are the coefficients of Y, etc. Substituting the value of
the v^s from (18), in the equations (19), then, we obtain
wiai{aiX+hiY+ . . . -\-h)+W2a2(a2X+b2Y-\- . . . +^2)
+ . . . +w)„a„(a„X+6„F+ . . . +U = 0 (20)
w;jbi(aiX+6iF+ . . . -\-h) +W2h2{a2X -{-boY + . . . +^2)
+ . . . -\-Wnhn{anX-\-bnY-i- . . . + ^=0
Whence carrying out the products indicated, and adding the similar
terms,
[iva^]X-{-[wab]Y+[wac]Z+ . . . -\-[wal] =0
[wba]X-h[wh'^]y-\-[wbc]Z+ . . . -\-[wbl] =0 (21)
[wca]X-i-[wcb]Y-i-[wc^]Z-\- . . . +[wd] =0
These are called the Normal Equations, as they are the same
in numljcr as the imknown quantities, and, therefore, may l)e
solved simultaneously to determine the latter. It will be seen
in the equations (20) that the first normal eciuation is formed by
nuiltipl\'ing the left-hand member of each observation equation
by its weight and the coeffici(Mit of A' in that ecjuation, and adding
all the resulting products. Likewise, the second normal equation
is formed by multiplying each observation equation by its weight
and the coefficient of F, and adding the products, and so on through
th(> series of unknown (}uantities.
INDIRECT OBSERVATIONS 29
The adjustment, then, consists in forming from the given
observation equations a set of normal equations, the same in num-
ber as the unknown quantities, the solution of which as simul-
taneous equations will give the best values of those quantities.
42. Observations of Equal Weight. This is a special case of
the foregoing, in which each weight may be replaced by unity so
that the tf's disappear from the normal equations (21), resulting,
therefore, in the following simpler form:
[a2]X+[a&]F + [ac]Z+ . . . +M] = 0
[hajXMb'^W-hlbcjZ-h . . . +[bl] = 0 (22)
[ca]X-\-[cb]Y+[c']Z+ . . . +[c/] =0
For purposes of illustration, it will be convenient to use these
equations (22) rather than the longer ones in which the weights
arc included.
43. Control or Check in the Formation of the Normal Equa-
tions. Referring to equations (18), let the sum of the numerical
coefficients and the constant term in each equation be represented
by s; thus, . , . , , ,
a2 + 62 + C2+ . . . -\-l2 = S2 (23)
On-\-hn + Cn-\- ■ . • +?» = S„
To form the first normal (xjuation, as shown in Art. 41, the terms
in the left-hand member of each of these equations are multiplied
by its weight and its first t(>rni or coefficient, namely, wioy, etc.,
and the resulting products ar(^ added, as in (21). P(>rfoi-niing
this operation at the same tim(> on the right-hand members above,
in (23), we have, using tlu^ first eciuation, only, as an illustration:
W'iar'+ii'ifli6i-f?/'iaici+ . . . +U!i«i/i = (<"i«i.s'i (24)
or, after addition,
[(/vr] + [(ra/;] + [»v/c]4- . . ■ +[?m/] = [(ra.s^] (25)
Thus, tlu> second inetiibcr of lliis (■(jualioii should ('(pial tlie sum
30 PRACTICAL LEAST SQUARES
of the numerical coefRcients and the constant term of the first
normal equation, which affords a check upon the numerical work
of computing these quantities and forming the normal equations.
This second member of (25) is therefore called the sum-term.
For the other normal equations, respectively, it has the form
[wbs], [wcs], etc. This check is very important and should always
be applied, except, perhaps, in the very shortest problems. Hav-
ing formed the sum, s, for each of the observation equations, it is
treated the same as the other quantities, a, b, c, etc., and w^hen a
normal equation is written, its sum-term should equal the alge-
braic sum of its other numerical quantities. It must be noted,
however, that the check may not hold exactly, in the last decimal
place, owing to discarded remainders, but this discrepancy will not
usually exceed one unit in that place.
44. S3nnmetry of the Normal Equations. Inspection of the
literal forms of the normal equations, in (21) and (22), reveals a
sj^mmetry which is useful as an aid to the memory, and which will
lessen the labor of computation both in forming the equations and
in their solution, as will be shown farther on. This symmetry
exists among the coefficients of the unknown quantities with refer-
ence to the diagonal line passing downward to the right. On this
diagonal will be found those terms which involve the squares of the
quantities, a, b, c, etc., as ^ [waa], [wbb], [voce], etc., or more simply,
without weights, [d^], \b^, [c~], etc. These terms being squares,
are always positive. Then the coefficients in any vertical cohnnn
occur in the same order as those in the corresponding horizontal
row. Thus, in the third coliunn and row the order is [ac], [be],
[cc], [dc], etc., c being the third of the original coefficients, and the
other factors having the original order, a, b, c, d, etc.
45. Formation of the Normal Equations. Aids. The com-
putation of the; necessary s(}uures and pixxlucts foi- the coeffi-
cients in tlie normal ecjuations is facilitated by the use of special
methods as well as tables and niectianical (Un'ices. Tlie choice
of the method or device will be governed, in general, by the size of
the numbers involved and the refinement of the computations.
' The Kcjuarcs of «, h, c, etc., arc oftcMi written ;is na, bb, cc, etc., to illustrate
this syTiiinetry as well as to avoid the us(> of exponents.
INDIRECT OBSERVATIONS 31
The tables used contain the logarithms, the squares, or the
products of numbers. Five-place logarithms are suitable in most
work, and four places are often sufficient. Hussey's five-place
tables are recommended as very convenient. Barlow's tables of
squares, cubes, roots and reciprocals of numbers up to 9,999 are
well known and satisfactory. Of the tables of products, Crelle's
Rechentafeln, giving the complete products of numbers of three
figures each, that is, up to 999 by 999, is probably the most useful,
although Zimmermann's and Peters' may more readily be used to
obtain products of larger numbers, as they give directly products
of numbers of four figures by those of two figures. In computing
the coefficients for normal equations by means of tables, the loga-
rithmic method is slowest, the use of squares is better, and the
tables of products are usually most satisfactory. In the absence of
these last, however, tables of squares may be used in either of two
ways for the computation of products, namely, by one of the fol-
lowing formulas:
a6 = i[(a+6)2-a2-62] (26)
and
a6 = i[(a+6)2-(a-6)2] (27)
The former requires but one new opening of the tables, as a^ and
b~ are separatel}' necessary' as coefficients.
The mechanical aids to computation consist of slide-rules and
computing machines. The ordinary 10-in. slide-rule is sufficient
for reading products to three significant figures. The Thatcher
slide-rule, however, reads directh' to four or sometimes five
figures and is excellent for solving normal equations as well as
forming them. Computing machines are of two types, for addi-
tion and foi- multiplication. We are concerned primarily with the
latter, although tlu> fornuM- may be used indirectly for multipli-
cation. Of the multiplying macliinos therc^ are two forms; the
Brunsviga type, in whicii one turn of the ci'ank multi])]ies !)y one
unit so that to multiply by 4.'), four turns would hv ixniuii'ed in
one position and three in tiie nexl : and th(^ Millioiuii' machine, in
which one tui'u of the crank nniltiplics by a whole digit, so that but
twt) turns would ])e recjuired to multiply by ()."). one for each digit.
If very lai'gc nunibcM's are to be inuhiphcd oi' (Hx'idcd, a computing
32 PRACTICAL LEAST SQUARES
machine is almost indispensable, but for ordinary work the tables
of products and the slide-rule are convenient and sufficient, espe-
cially since large numbers are avoided as much as possible.
46. The computation of the coefficients in the normal equations
is carried out conveniently in the form of a table in which each
quantity involved is shown, with its proper sign, first the given
ones and then the computed ones. Then it is highly important
that the multiplication of several quantities by the same factor
be performed in succession, as this plan in particular is adapted
to the use of slide-rules, multiplication tables, and computing
machines. Thus, for each observation equation, the factor, wa,
would be multiplied into a, h, c, . . . I, and s, in succession, and
the products entered in the proper columns of the table, so
that the sums of the columns would be the coefficients, [iwaa],
[wab], [wac], . . . [was], of the first normal equation. Next,
the factor, wb, would be multiplied into the same quantities,
beginning with 6, however, as the wab products are included in
the preceding series, and the column totals would be coefficients
for the second normal equation, and so on. As each normal
equation is completed, its coefficients should be tested with the
sum-term to assure the computer that the check is satisfied.
This would be indicated by a definite check-mark after the sum-
term if it checked exactly, or by the cancellation of its last figure
with the correct one written above so as to equal the sum of the
quantities in the equation.
In the simplest cases, when there are but few observations and
two unknown quantities, and when the coefficients are small
integers, it may not be worth while to carry out the tabulation for
the formation of the normal equations, but it is generally safer
to do so, cspeciall}' when the computer is subject to interruption
in his work. It is well, also, to write the algebraic signs for a
complete equation before forming and writing the numbers, always
writing all positive as well as negative signs.
47. Example of the Direct Formation of Normal Equations.
As an illustration of the preceding articles, the normal equations
will be formed directly from the following set of oljscrvation equa-
ti(Mis. For simplicity, the woiglits will be assunietl equal.
INDIRECT OBSERVATIONS
33
00
o
o
O
o
o
H
II
II
II
11
<
GO
CO
CO
o
00
GO
CO
r^
H
^
CO
'^
(M
1
1
+
+
Z
^
kH
>^
t^
o
OJ
o
00
H
'^
CO
LO
(>!
>
+
+
1
1
"^
H
'^
>^
W
ZD
^
>o
CO
+
+
1
1
O
o
p
>
o
cc
04
04
00
^
c^'
CO
i-H
lO
(^
-o
1^
t^
05
05
^
iO
a>
lO
1
1
+
+
+
o
o
00
00
04
oq
lO
'^
04
lO
,^^
lO
C4
04
t^
t^
^
CO
04
■*
1>
t^
04
1
1—1
1
04
1
1
CO
1
^
TjH
O
S
'^
o
04
CO
Tt<
-o
CO
O
1— 1
1>
lO
rO
r-(
l—{
CO
CO
+
+
+
+
+
>
00
ca
lo
04
t^
CC
o
ci
od
O
04
e
t-
CO
1
1
+
+
+
00
04
lO
00
CO
ca
CO
o
04
lO
■-0
to
o
?— 1
00
o
Cj
1
T— 1
1
04
1
1
00
1
o
QO
o
-*
04
-^
Tt<
C^
00
GO
CO
^
04
04
1>
+
+
+
+
+
o
o
lO
Ci
CO
ti!
ro
1-H
04
oo
a
+
+
+
+
+
00
CO
1^
^
04
^
04
7
04
1
7
CO
1
oc
CO
CO
CO
04
^^
X
CO
'f'
r-
CO
T
"\
+
Ol
+
•oi
1
Ol
CO
04
+
+
'i"
OJ
1
7
o
io
-1*
ir\
CO
OI
1 .
+
+
T
'
+
> >
+ +
'A CO
>.
+ +
+ +
34 PRACTICAL LEAST SQUARES
As there are no discarded remainders, the checks are exactly
satisfied.
(The solution of these equations by the methods of algebra
gives
Z- +11.52 and F= -0.25
as the best values of X and Y obtainable from the four given
observations.)
48. Use of Assumed Approximate Values of the Unknowns.
The constant term of the observation equations is sometimes large
as compared with the other numerical quantities, and to save labor
in the formation of the normal equations, recourse may be had
to a scheme similar to that used in Arts. 28 and 33 in the com-
putation of the mean, namely, the use of assumed, approximate
values of the unknowns, by which device the constant terms will
be reduced considerably in size. For each unknown in the obser-
vations, there is substituted its approximate value plus a small
correction, as,
X = Xo-\-x
Y=Yo-\-y, etc. (30)
where Xo and Yq represent the approximate values and x and y,
the small corrections. The approximate values may be obtained
by a trial solution of the necessary number of the observation
equations, namely, as many as there are unknown quantities.
Thus, in the example of the preceding articles, a solution of the
third and fourth of the observation equations results in
Z=+11.9 and F=-0.29
whence we may assume the approximate values,
A'o=+12.0 and 70= -0.3
Substituting for X and Y, therefore, in equations (28), the quan-
^^^^^"^ A = x+12.0 and F = /y- 0.3
we obtain for the first equation,
+ 6(a;+12.())+40(?/-0.3) -58.8 = 0 (31)
INDIRECT OBSERVATIONS 35
and for the entire set of observation equations, after simplifica-
tion,
+ 6a: +40t/+ 1.2 = 0
-t-4a:+ 327/ +0.1=0 (32)
-5a:-56?/+0.1 = 0
-3X-282/ =0
The constant terms have thus been diminished to very small
quantities and without the expenditure of much labor, so that the
formation of the normal equations will be considerably easier,
but in so far only, be it noted, as the terms involving I are con-
cerned. It is obvious that this scheme leaves the coefficients of
the unknowns entirely unaltered, the only changes being in the
constant terms.
49. Adoption of New Unknowns to Equalize Coefficients.
When the coefficients of any unknown in the observation equa-
tions are consistently large, they may be reduced in size by an
artifice similar to that of the preceding article, that is, by sub-
stituting for the unknown a new one obtained by multiplying the
former by a certain factor.
In the equations (32), for example, the coefficients of y are
much larger than those of x and would be easier to handle if they
were cUvided l^y, say, 20. Therefore, assume
y' = 20y or y = ^ (33)
SuV).stituting this value of y in the given equations, and writing
the coefficients in columns for simphcity, we have.
(34)
X
+6
y
+2.0
+ 1.2 =
= 0
+4
+ 1.0
+0.1
— 5
-2.8
+0.1
-3
-1.4
0
36 PRACTICAL LEAST SQUARES
which are much simpler than the original equations (28), both as
regards the formation of the normals and their solution. The
normal equations are
X y'
+86 +36.6 +7.1 = 0 (35)
+36.6 +16.36 +2.28
and their solution results in
a;=-0.48 and?/=+0.94 (36)
whence,
2/ = -^= +0.047
^20
Therefore,
X=+12.0+x= +12.0-0.48= +11.52
and (37)
F=- 0.3+2/=- 0.3+0.05=- 0.25
The advantages of reducing the size of the coefficients and con-
stants before forming and solving the normal equations, is less in
such a short problem as the one just solved than in the ones which
contain more unknown quantities and larger series of observa-
tions. It is generally advisable, however, even in the shorter
problems, to diminish the coefficients and constants to a size
which will be convenient for computation, and to equalize them to
some extent, at least, by using whole numbers for the approximate
values and the factors.
50. Example: Time by Star Transits. Let us consider, as
an example of indirect observations, the determination of time
by observed transits of stars on the mcri(Uan, using an astronomical
transit instrument. The times when each star is seen to cross
the successive threads are rocoixlcd by the observer, himself,
as he carries the lieats of the chronometer in his mind. The
mean of these times is taken as the time when the star crossed
the line of sight of the instrument. It is then corrected for diurnal
aberration, the rate of the chronometer, and the inclination of
the horizontal axis of the instrumcMit as detei'mined from the
readings of the stricHng level. The resulting time, d' , is subtracted
from the right ascension, a, of the star (which is the correct sidereal
time when the star crosses the incriiUdu) and the difference,
INDIRECT OBSERVATIONS
37
a—d', according to the usual notation, is therefore made up of
the chronometer correction, Ad, which is the quantity really desired
from the observations, and the corrections for azimuth, Aa, and
collimation, Cc, according to the formula,
Aa-\-Cc+Ae-{a-d') = 0 (38)
in which A and C are the known azimuth and collimation factors,
and a, c, and Ad are respectively the azimuth and collimation con-
stants and the chronometer correction, which are the three un-
knowns of the problem and which, therefore, will be represented
by X, y, and z. Each star thus furnishes an observation equation
of the form,
Ax + Cy-{-z-ia-d')^0 Wt. w (39)
the weight being determined from the star's declination, as stated
in Art. 32. The given data for the nine stars observed are:
,1
r
a-d'
IC
+0.15
+ 1.22
-9m. 06.81s.
0.9
+0.72
+ 1.00
06.98
1.0
- 1 . ()(•)
+3 . 30
07.69
0.2
-0 17
+ 1..V2
06.49
0.7
+0.09
- 1 . 2S
00.08
0.8
-0.18
- 1 . .-)3
06 . 32
0.7
+ 0.S0
- 1 02
06 . 27
1 0
+4.2.-,
+ 4 02
07 . 47
0.1
+ 0.C)0
^1 ()1
Of) . 00
1.0
(40)
As th(> valuers of a — 0' inv nearly ecjual, and the coefficients of z
are unity, it is (>vi(lent that the value of z will approximate to
a-d\ Therefore, let
2 = 2'-9ni.0(),00s. (41)
and the form of the typical (Hiuatioii Ix'comes
A.r + C// + / + / = () Wt. IV (42)
in which /= — Om.OG.OOs. — (a — 0')- The modified observation
equations, therefore, are
+-0,lo.r + 1.22?/ + / + 0.81-0 (Wt. 0.0), etc..
38 PRACTICAL LEAST SQUARES
or, arranged in tabular form,
X
y
z'
(l)
(Wt.)
+0.15
+ 1.22
+ 1
+0.81 = 0
0.9
+0.72
+ 1.00
+ 1
+0.98
1.0
-1.66
+3.30
+ 1
+ 1.69
0.2
-0.17
+ 1.52
+ 1
+0.49
0.7
+0.09
-1.28
+ 1
+0.08
0.8
-0.18
-1.53
+ 1
+0.32
0.7
+0.80
-1.02
+ 1
+0.27
1.0
+4.25
+4.92
+ 1
+ 1.47
0.1
+0.60
-1.01
+ 1
0
1.0
(43)
X
y
z'
(/)
(«)
+3.94
+0.38
+2.18
+ 0.38
+ 13.56
+ 0.19
+2.18
+0.19
+6.40
+ 1.00
+3.53
+3.09
= 0
= 0
= 0
+ 7.50
+ 17.66
+ 11.86
The quantities in the successive columns are the a, b, c, I, and w,
of the observation equations, so that it is unnecessar}' to tabulate
them again.
Forming the normal equations from the typical ones in (21)
by means of a tabulation similar to that of Art. 47, we have,
(44)
the solution of which, as algebraic simultaneous equations, gives
a:=+0.043s., ?/= -0.261s., z'= -0.491s.,
whence, from (41),
z = Ae = z'- 9m. 06.00s. = - 9m. 06.49s.,
so that a=+0.043s., c= -0.261s., and A^- -9m. 06.49s., rej)-
resent the best or most probable values of the thre(> unknowns
which can })e ol)tained from the given observations.
51. General Application of the Method. The process outlined
in this chapter for the adjustment of indirect observations may be
applied to any set of linear simultaneous ccjuations whose number
is gi'eater than that of the unknown quantities, although they may
not be observation ecjuations. Sometimes such a series of eciua-
INDIRECT OBSERVATIONS 39
tions may result from computations or from theoretical assump-
tions. Inasmuch, however, as the present method of adjustment
depends upon the assumptions as to the occurrence of error,
stated in Art. 18, the use of the method for the adjustment of
quantities other than those resulting from observations may be
justified only by the absence of a better scheme.
However, any other method is likely to be more laborious
than this one, if it takes into account all the given data. For
example, suppose the simplest case of three given equations involv-
ing two unknowns. If ignorant of the adjustment by means of
Least Squares, but desirous of utilizing all of the given equations
because there is no way of telling which one could be discarded
with least effect, the computer might reasonably select all possible
combinations of the three equations, two at a time, namely, three,
and solve each of the three pairs independently by algebraic
methods, thus obtaining three different values for each of the
unknowns, of which he would probably take the mean as the best
value within his knowledge. Certainly, the formation and
solution of two normal equations would be much easier than
such a process.
CHAPTER IV
SOLUTION OF NORMAL EQUATIONS
62. Methods of Elimination. As simple, simultaneous equa-
tions of the first degree, the normal equations may be solved by
any of the ordinary algebraic methods of elimination; by addition
or subtraction, by substitution, or by comparison. In fact, these
methods are satisfactory when there are but two equations to be
solved. But in larger sets, of three or more, it is possible to
shorten the numerical work by taking advantage of the peculiar
symmetry which all normal equations possess, as was pointed out
in Art. 44. It is much easier to solve a set of normal equations
than a set of ordinary, simultaneous equations of the same number
which do not have this symmetry.
53. The Gauss Method of Substitution has been for a century
the basis of the special methods for the solution of normal equa-
tions. Its notation is convenient, and in its general, literal form
it is given in nearly every work on Least Squares. However, it
has been modified and improved in various ways, particularly by
Mr. M. H. Doolittle, formerly a computer in the U. S. Coast
and Geodetic Survey, and, in the effort to confine ourselves to a
single method which shall be the most generally useful one for our
purposes, we shall omit the detailed explanation of the Gauss
process.
54. Requirements of a Good Method. It is important that
the method to be adopted be as universally useful as possible, in
both short and long problems, although modifications may ])e
convenient to adapt it to special or peculiar cases. The various
steps in the elimination should be identical, so that the work may
be performed mechanically, to a great extent, thereby avoiding
mistakes. The method should be as short as possible so as to
avoid unnecessary work. And finally, checks should be available
40
SOLUTION OF NORMAL EQUATIONS
41
at frequent intervals throughout the computation in order that
errors may be discovered and corrected without a great deal of
recomputation. All of these qualities should be borne in mind
and utilized as far as possible in every solution. It is believed
that the Abridged Method explained below fulfills these require-
ments and that it will be readily understood.^
55. Algebraic Elimination by Addition. Let us undertake
the solution of a simple set of normal equations in order that the
steps we shall take in the process may be clearly understood. The
method of elimination by addition will be used, although arranged
in a certain form to illustrate the shorter method which is to follow.
The given normal equations, with coefficients arranged in col-
umns, are:
(45)
For purposes of explanation, the equations are numbered at the
left, but for a reason which will appear later, the first is given the
Roman numeral (I). First, we eliminate x between the first and
second equations, by multiplying the first by such a quantity or
factor, as will make its first term equal to that of the second equa-
tion with the opposite sign, and then adding the two. This
factor will be the quotient of the first term of the second equation
with its sign changed, by the first term of the first equation, that is,
+2 6. Thus, indicating on the right the steps tak(ni, we write down
equation (2) and under it ihv first e(}uation multijilied by +2 G:
(40)
Published by M. II. Doolittle in (\ A: G. Survey Uejjort, 1S78, Ai)p, 8.
.r
,'/
-'
\li
(I)
+ 0
-2
+3
+2
= 0
(2)
-2
+3
-4
-3
= 0
(3)
+ 3
-4
+3
+ 1
= 0
(2)
(41
(III
■I- 11
z
(/)
— 2
+2.0
0
+3
-0.7
+2.3
-4
+ 1.0
-3.0
-3
+0.7
-2 3
= 0
= 0
= 0
(nx(+2 (n
(2) + !4i
42
PRACTICAL LEAST SQUARES
This equation, resulting from the elimination of the first unknown
is called a First Derived Equation, and is given the Roman numeral
(II) as marking the completion of a whole step in the process.
Next, X is eliminated in the same way between the first and
third equations, by multiplying the first by such a factor as will
make its first term equal to that of the third equation with its
sign changed, and adding the two equations. As before, the
factor will be the quotient of the latter first term with the reversed
sign by the former one, or — 3/6. Writing (3) first,
(3)
X
y
z
(l)
+3
-4
+3
+ 1
= 0
(5)
-3.0
+ 1.0
-1,5
-1,0
= 0
(I)X(-3,'6)
(6)
0
-3.0
+ 1.5
0
= 0
(3) + (5)
(47)
Equation (6) is like (II) in having no x-term, that is, it may
also be called a First Derived Equation. Therefore, y can be
eliminated from these two equations in the same manner as x
was eliminated above. Multiplying (II) by the first term of (6)
with its sign changed, divided by the first term of (II), and adding
the resulting equation to (6), we have, as before, writing the second
of the two equations first :
,'/
z
(I)
(iV)
(7)
aiii
-3.0
+3.0
0
+ 1.5 1
-3.9
-2.4
0
-3.0
-3.0
o c c
II II II
(II) X( +3. 0,2.3)
(i)) + (7)
(iS)
This conipk'tcs the s(H'ond step in the process, the second unknown
has been eliminated and this last ecjuation is therefore called a
Second Derived Erjuation and given the next Roman numeral (III).
The elimination is now complete^ and the last e(}nation may be
writt(n;i
-2.4.~-;5.() = 0
SOLUTION OF NORMAL EQUATIONS
43
giving the value of z directly, as 2= +3.0/ — 2.4= —1.25. Substi-
tuting this value in (II) gives
y=
+3.02+2.3 -3.75+2.3 -1.45
+2.3
+2.3
= -0.63
+2.3
Then, from (I),
+2y-?>z-2 -1.26+3.75-2 +0.49
(49)
x =
+6
+6
+6
+0.08
Some properties of this method of solution will now be con-
sidered.
56. Symmetry among the Derived Equations. The First
Derived Equations resulting from the elimination of the first
unknown, x, between the first normal equation and each of the
others in succession, will be one less, in number, than the unknowns;
therefore, in the above example they are two, namely.
y
2
(/)
(II)
(6)
+2.3
-3.0
-3.0
+1.5
— 2.3
0
= 0
= 0
(50)
They evidently are symmetrical about the diagonal in the same
respect as the normal equations, that is, the second coefficient in
the first row ( — 3.0) is the same as the second coefficient in the
first column ( — 3.0). This symmetry exists likewise in all sets of
first derived equations, whatever their ninnber, as may be proved
by carrying out a complete solution of the typical normal e(|ua-
tions with their literal coefficients. In this example, however, the
reason for the equality of these two coefficients may ])e seen by
indicating the operations through which they were obtained, using
certain symmetrical terms in the normal equations. Thus,
Second Coeff. of (II) =-4- (-2 6X+3) = -3.0
First Coeff. of (6) = -4- ( + 3 OX -2) = -3.0 (51)
The two — 4's are svnnnetrical in the normal eouations, as also
44
PRACTICAL LEAST SQUARES
are the two — 2's and the two +3's, while the +6 is the same in
both cases.
Likewise, the Second Derived Equations, resulting from the
elimination of 7j from the first derived equations, are symmetrical
among themselves, and so on with successive sets of derived
equations in the solution of a large number of normal equations
by this method of elimination.
57. Omission of Redundant Terms, (a) In each first derived
equation, x has been eliminated; that is, it has the coefficient zero.
Therefore, it is unnecessary to write the coefficients in the x-col-
umn at all during the elimination of x, as in (46) and (47), as we
know that they will add up to zero if the work is correct, and any-
way, there will be other and better checks on the correctness of
the work. Similarly, the y-column may be omitted during the
elimination of y, as in (48), and so on. However, the sum of the
remaining terms in each equation will not now equal zero, except
in the derived equations, where the omitted coefficients are always
zero. This will deprive us of the equation-checks except in the
derived equations, but these will still be sufficiently close together
to control the computation.
(6) By transposing all the terms of each equation into one
member, as was done in the above example, we maj^ omit the
symbols, " ="0," from each equation. As just stated, however,
these must be understood, in the cases of the derived equations,
as if written.
(c) Even the original normal equations may be simplified by
the omission of all the terms lying below the diagonal, these
being symmetrical to the ones above the diagonal. Thus, in the
normal equations, (4.5),
X
,v
^
(/)
(1)
+6
-2 1
+3
+2 !
^9
(2)
~2
+3
-4
-3 ;
F^0
CA)
+ 3
-4
+3
+ 1
?^o
(52)
Iho cancelod toi'ins would })e omitted. To read the original equa-
SOLUTION OF NORMAL EQUATIONS
45
tions, then, the omitted portion of each row must be replaced
by the symmetrical quantities in the corresponding column.
For example, the second equation is begun in the second
column and read downward to the diagonal and then hori-
zontally to the right along its own row, retaining, however, the
original order of the unknowns, as —2x-\-Sy — 4:Z — 3 = 0. Simi-
larly, the third equation is begun in the third column, read down-
ward to the diagonal and continued along the third row as usual:
-\-3x — 4:y-{-3z-{- 1.0 = 0. The simplified form of the equations,
then, would be:
X
y
1 ^
[l
(I)
+6
-2
+3
+2
(2)
+3
-4
-3
(3)
' +3
+ 1
(53)
It is evident that the original order of the equations must never be
changed if these abbreviations are to be used, as the symmetry
would then be destroyed.
(r/) In the second step of the elimination, it is unnccessar\' to
write equation (6) either in (47) or (48), but equation (7) may be
written directly below (5) in (47), the x- and ^/-columns being
omitted as above, and the three lines (3), (5), and (7) added, to
form (III). The factor for obtaining (7) from (II), namely,
— ( — 3.0 2.3), is the second term of (II) with its sign changed,
divided by the first term of (II), owing to the symmetry of the
derived equations as sliown in Art. 56. Thus, (47) and (48) may
be combined into:
'/
(3)
(n)
+ 3
- 1 . .")
-1,0
'r)Xf'-3 Gi
IT)
- :? . 0
-.so
(Hix'+i'S.o 2.
5)
fill)
-2.4
~:i 0
r.]) + <r)) + i7'-
(54)
46 PRACTICAL LEAST SQUARES
58. The Series of Derived Equations. Upon inspection of
(46) and (54)^ it will be seen that (II) is derived from (2) and (I),
and that (III) is derived from (3), (I), and (II). If there were a
fourth unknown and four normals, the derived equation (IV)
would be derived from the fourth normal equation and (I), (II),
and (III), and so on. Here, then, lies the reason for giving to the
first normal equation the Roman numeral (I); it is associated
with the derived equations in each step of the elimination. There-
fore, in writing a list of the derived equations, this equation is
written first, and is referred to as one of them. Such a list, in
order, has the property that each equation is complete and begins
with the second unknown of the preceding one, so that the series
is used for determining the successive unknowns in their reverse
order when the elimination has been completed, by substitution
back through them.
59. Control or Check in the Solution of the Normal Equations.
The check on the formation of the normal equations, explained
in Arts. 4.3 and 47, may be continued through the process of elimina-
tion so as to test the correctness of the computation at frequent
intervals. If the sum-term of each of the normal equations be
subjected to the same operations as its other terms, the resulting
modified sum-term will be equal to the sum of the corresponding
series of other terms. Aloreover, this relation will persist when
several equations have ))een added or sul^tracted, the sum of all
the sum-terms being ecjual to the sum of all the other terms.
Thus, the sum-terms which were used to check the formation of
the normal equations ma}' be used during their solution to test
the correctness of an ecjuation at any stage of the work. As was
pointed out in (a) of the last article, however, the omission of
redundant terms leaves the derived efjuations as the only com-
plet(> ones in the elimination. Therefore, this sum-check being
applicable to each derived equation as it is formed, should be
taken advantage of in every case.
8inc(! the check applies only to complete e(}uations, the coeffi-
cients of the normal equations must be i-ead down and to the right
as shown in the last article, when the simplified form is used.
SOLUTION OF NORMAL EQUATIONS
47
In the above example, then, the complete statement of the normal
equations in the simpler form, with their check-terms, is:
(55)
60. Elimination by the Abridged Method. This set of equa-
tions will now be solved in accordance with the devices explained
in the preceding articles for al)ridging the various operations as
much as possible. A comparison of this solution with the direct,
algebraic one given in Art. 55, will illustrate the different steps
and the saving of labor.
X
1
y
1
2
(0
is)
(I)
+6
-2 '
+3 '
+2
+9
(2)
+3
--1
-3
-6
(3)
+ 3
+ 1
+3
X
y
1 ^
(0
(•s)
(I)
(2)
(3)
(2)
^4)
(U)
(3;)
dill
+6
-2
+3
+2
+9 ^
^ Normal Equations
(r'ixf.+2 ())
+3
+3
-0.7
-4
+3
-4
: +1.0
-3
+ 1
-3
+0,7
-6
+3 .
-6
+3 . 0
+2.3
-3.0
-2.3
-3. On
' '2) + (4)
'I)X(-3 (i)
(II)X(+3.0 2.3)
' (3)+r5) + (()l
+3
-1.5
-3.9
+ 1
-1.0
-3.0
+3
-4 5
-3.9
-2.4
-3 0
-5.4^
Hic process may be outlined as follows: Wi'it(> (2); follow
the l(>ft-hand column of (2) u]i to (I) and find —2; chang(^ its
sign and divide^ by the first term of (1), giving +2 (>: multijily
this factor into the terms of (1). begimiing with The same left-hanc!
column of (2), and writing the results in line (4) under the corr(^
48 PRACTICAL LEAST SQUARES
spending ones in (2); add (2) and (4) to obtain (II). Next, write
(3); follow its left-hand column up to (I) and find +3; change
its sign and divide by the first term of (I), giving —3/6; multiply
this factor into the terms of (I), beginning with the left-hand col-
umn of (3), writing the products in their proper columns in line
(5), under (3). Again, follow the same left-hand column of (3)
up to (II) and find —3.0; change its sign and divide by the first
term of (II), giving +3.0/2.3; multiply this factor into the terms
of (II), beginning with the left-hand column of (3), writing the
results in line (6), below (5), and in their proper columns; add (3),
(5), and (6), to obtain (III), as the second and last step in the
elimination. If there were four normal equations, the fourth
would now be written, beginning with the fourth colunm; under
it would be written the products of the terms of (I), beginning
with the fourth, by a factor consisting of the fourth term of (I)
with its sign changed, divided by its first term; under this line
would be written the products of the terms of (II), beginning with
the third, by a factor consisting of this third term with changed
sign, divided by the first term of (II) ; and finally, under this line
would be written the products of the terms of (III), beginning with
the second, by a factor consisting of this second term with its sign
changed, divided by the first term of (III) ; whence the sum of the
four lines thus obtained would be (IV). This procedure could be
continued through any number of equations.
61. The mechanical character of this scheme of elimination
is apparent from the foregoing explanation. Each of the main
steps accomplishes the elimination of one unknown more than the
preceding step did, and results in the next derived equation. Each
step consists of the sum of its normal equation and as many others
as there are derived equations already formed, including (I);
so that the successive steps embrace the sums of two, three, four,
five, or more lines, up to the number of unknowns involved in the
problem. For each step, th(> numerators of the factors, with
opposite signs, are found in one column, namely, the one con-
taining the first term of the normal equation as written, that is,
the one corresponding to that equation, as third column for third
SOLUTION OF NORMAL EQUATIONS 49
equation, etc.; the denominators are the first terms of the corre-
sponding derived equations.
62. Notes and Suggestions. The arrangement of the work
in columns is essential to mechanical efficiency and " Data Sheets "
are convenient for this purpose. Ruled horizontal lines including
each derived equation make it prominent for quick reference.
By writing the algebraic signs of each line of products before
writing the numbers themselves, errors in sign may be avoided
to a great extent. In each line, all the signs will be the same as
those of the corresponding derived equation, or all opposite to
them. It will be noted that the ^7-5^ term in each of these lines of
products is always negative owing to changing the sign in the
numerator of each factor; this, also, affords a check on the signs.
Unavoidable discrepancies in the last figure of the check-term, due
to remainders, should be removed by arbitrarily correcting the
check-term before proceeding with the next step in the elimina-
tion; this is best done by drawing a line through the erroneous
figure and writing the correct one just above it. If the check is
exactly satisfied, it should be as carefully noted with a check-mark
in order to avoid uncertainty.
63. Values of the Unknowns. The process of elimination hav-
ing been completed and the derived equations checked as formed,
it remains to determine the last unknown from the last equation and
to substitute back in the preceding derived equations in reverse
order, to obtain the other unknowns; x being finally determined
from (I). If there ])e many equations, this process may be facil-
itatcnl by tal)ulating the products instead of indicating the work
as in (tO). A table is ])egun for each imknown by writing first,
with changcMl sign, the constant term of the derivcnl e(|uation from
which that unknown is to b(> obtained. Below this ar(> placed in
succ(>ssion tlu^ ])ro(lucts of the unknowns, as coniput(Ml, by their
respective co(>ffi('ients, with signs changed, in that ecjuation.
The sum of these (juantitic^s (Hvided ])y the fii'st coefficient in the
ecjuation gives the value of that unknown. The advantage lies
in the fact that each unknown, as conipuUMl, is niultiphed into
all of its coefiicicnit.s in tlie pr(M'e(Ung deiived (Hjuations, in sue-
50
PRACTICAL LEAST SQUARES
cession, rather than separately as needed for each case; thus, a
shde-rule, a multipHcation table, or a machine can be used with
profit. Applying this arrangement to the problem in Art. 60,
we have:
X
y
z
Constants
z-terms
,!/-terms
-2.0
+3.75
-1.26
+2.3
-3.75
-1.45
+2.3
-0.63 = ?/
+3.0
-2.4
-1.25 = 2
+0.49
+6.0
+0.08 = x
It is often convenient to write these little tables in the proper col-
umns just to the left of the computation of the derived equations,
that is, left of the elimination, as it is not necessary to write the
numbers of the various equations after the method has been
learned.
64. Final Check of the Unknowns. The use of the chock-
terms ends with the formation of the last derived equation. The
entire solution, however, including the values of the unknowns,
may be checked by substituting those values in the normal equa-
tions other than the first. If there were no neglected remainders
in the computation, all these equations should be exactly satis-
fied. Therefore, in any actual case, the discrepancies should be
very small.
When several etiuations are to l)e tested at once, the method of
tabulation explained in the last article may be used to advantage.
Obviously, the sum of each table should be very nearly zero if
no mistake has been made. This will be illustratetl in a later
chapter.
65. Refinement of the Computations. The number of decimal
places to 1)0 retained throughout the elimination will depend upon
the niunber desired in the n^sulting unknowns. T}u> labor of
solution inci-eases rapidly with thv size of the (juantities involved
SOLUTION OF NORMAL EQUATIONS 51
SO that it is very important to keep them as small as practicable.
The discrepancies due to neglected remainders during the elimina-
tion will seldom amount to more than one or two in the last place
and these will be revealed by the checks. Similar ones will
occur in the values of the unknowns, resulting in their failure to
check exactly when substituted in the original normal equations.
However, as the final values of the unknowns must be regarded
as but approximations to the correct values, which, of course, are
unattainable, it cannot be objectionable to alter the last figure
of an unknown arbitrarily, to make it check or to make it con-
sistent with the others, and this is sometimes necessary. There-
fore, it is unwise to carry the whole computation one or two places
farther merely to secure an exact check in a certain place without
forcing it.
As a general rule, it is well to carry the observations two
places beyond the last one which is regarded as known with cer-
tainty. For example, each reading will have its last figure the
result of estimation, to some extent, the preceding one being cer-
tain; then the mean of several readings would be carried one place
farther. This should determine the degree of refinement to which
the normal equations and the elimination should be carried,
the coefficients of the observation equations being modified as
shown in Arts. 48 and 49 so as to be consistent in size. The
unknowns may then be carried out one place farther, the last
figure to be retained or rounded-off to the preceding one as pre-
ferred. However, this is largely a matter of judgment derived
from experience. The beginn(>r is too apt to carry his work
farther than is justified by the precision of the observations.
He may ho guided l)y the rule to carry the computations one
place fartluM- than the given data; this is anipl(\
66. Mechanical Aids in the Solution. Wv have seiMi in Art. 45
how the formation of the normal ecjuations may l)e facilitated by
the us(^ of tal)k"s and nu'clianical (Un'ices. In \he solution of the
ecjuations, these articles are oxen moi'(> useful, perhaps, especially
the sli(l(>-rules. as th(\v admit of multiplying a series of mnnbers
by tlK> (juotient of two other numlxM-s at one st^tting of the rule.
The Thatcher, in jxirticular, is very coii\cni(Mit , and is good for
52 PRACTICAL LEAST SQUARES
four significant figures, but the ordinary 10-in. rule is sufficient
when but three figures are used and is commonly at hand. The
computing machines, while necessary for large numbers are less
advantageous for small ones, but they have the great advantage
over the slide-rules of causing little or no straining of the eyes.
CHAPTER V
OBSERVATIONS OF DEPENDENT QUANTITIES:
CONDITIONED OBSERVATIONS
67. Dependent Quantities. In the preceding chapters, the
quantities observed or determined from the observations have been
independent, that is, any one or more might vary without causing a
corresponding change in the others. Thus, in the determination
of time by star transits in Art. 50, the constants of the transit
instrument cannot be affected by any change in the chronometer
correction. Now, however, we shall consider a different class of
quantities, and one which is of particular importance to engineers,
inasmuch as it includes their most complex, but at the same time,
most useful, problems in the adjustment of observations. In
this second division of the subject, the observed quantities are not
independent of one another, but are inter-related by certain
theoretical requirements, called Conditions, which their adjusted
or adopted values must rigidly satisfy. The adjustment, then,
consists in determining the best set of values for the observed quan-
tities which shall exactly satisfy the prescribed conditions.
For example, if the three angles of a plane triangle be measured
with a protractor, they must be so adjusted that the sum will be
exactly 180°. Or, if the three angles of a triangle in the field be
measured with a transit or theodolite, they must be adjusted by
the application of a small correction to each, so that the sum of
the adjusted values will be 180° plus the spherical excess.^ Also,
^ The earth is api)rc)ximately si)heroidal but the figures in triangulation
are considered as spherical for convenience in eonii)Utation. The observed
horizontal angles, then, are those of .spherical triangles, since the plumb-lines
at the different stations are converg(>nt and the horizontal planes of the
angles are neither coincident nor par;dlel. Yvry small triangles, however,
may be considered plane, as the spherical excess is but 1" in a triangle con-
taining 75 square miles.
o3
54 PRACTICAL LEAST SQUARES
the horizontal angles completing the horizon at a station must be
adjusted so that the sum will be 360°; and the differences of ele-
vation in a closed circuit of levels must be adjusted so that their
algebraic sum will be zero, when proceeding continuously around
the figure, that is, clockwise or counterclockwise.
68. The observations to be adjusted will have been made
independently, as a rule, as in the case of a circuit of levels made
up of several differences of elevation between successive bench-
marks, each difference of elevation being determined independently
of the others. However, so-called " observations," entering into
an adjustment, may never have been actually observed but may
be the results of computation or of a previous adjustment of actual
observations. For example, an angle of a triangle may have been
determined by the addition or subtraction of two or more observed
angles, or from a local adjustment of the angles at that station. ^
Also, as stated in Art. 26, each observation may be the result of
several elemental observations or readings; in fact, this is usually
the case with dependent quantities. Generally, too, these obser-
vations are direct ones. In any event, however, the}' will be
adjusted as direct observations of dependent quantities, as this is
the most convenient and practical method.
69. The weights, in the general case, will be unequal, of course.
They are obtained as indicated in Art. 32, from the number of
observations, from theory, or bj' estimation. They ma}' be
determined from the nature of the observed quantities, inde-
pendently of the observations themselves, although subject to
modification in every case when the circumstances are unusual.
The basis of weights in observations of angles is usually the
number of observations; in leveling, the lengths of the lines, the
number of instrument stations, etc., may indicate the weights.
70. Conditions. The nature of the conditions which are to be
satisfied by the adjusted values of the observed quantities will
depend upon the character of the problem. The only limitations
1 It will be seen later on that it is often convenient to make two or more
small, partial adjustments instead of a single large one, so that it freqviently
happens that the given data to be adjusted are the results of a previous adjust-
ment.
OBSERVATIONS OF DEPENDENT QUANTITIES 55
upon them are: (1) that their number must be less than that of the
observations, as otherwise a sufficient number of them could be
solved as simultaneous equations so as to determine the unknowns
directly, without an adjustment; and (2) that they must be
independent of one another, that is, no condition may be included
twice in the same series. Furthermore, the correctness of the
conditions is not essential to the adjustment, itself, as this can be
carried out so as to force the unknowns to satisfy almost any arbi-
trary or unreasonable condition; but a correct adjustment requires
that the conditions be correctly stated. If an error be made in
the statement of a condition, and the proper method of adjust-
ment be used, the unknowns would satisfy the erroneous condi-
tion, and the error might not be discovered until, as a final check,
the adjusted values were tested by substitution in the original
conditions. Therefore, it is well to exercise great care in the
statement of each condition, and to be sure that all of the neces-
sary conditions, but no others, be included in an adjustment.
71. Number of Conditions. It is evident, in general, that a
certain numljcr of observations would be necessary for the deter-
mination of a certain number of quantities, if the observations
were strictly correct, — ideal. If extra observations are made,
beyond this necessar}' number, each of these would furnish a check
upon the work, that is, a condition to be satisfied. The rule, then,
could be stated that the number of independent conditions of a
certain kind, involved in a given series of observations, would be
equal to the number of extra observations of the corresponding kind,
that is, the excess over the necessary number of ideal, correct
observations.
Let Fig. 2 represent a system of levels connecting the bench-
marks, ^l, B, C, D, E, and F, the numbers in parentheses repre-
senting the Hues over which the differences of ek'vation are
observed. If the observations were absolutely correct, the dif-
ferences of elevation would be completely determined by the lines,
(1), (2), (8), (4), and (5). Then if (G) were addcxl, it would fur-
nish one chcM'k. and the condition that the whok^ outer circuit
should close to zero, if the signs of the separate lines were so
changed, if necessary, as to indicate running continuously around
56
PRACTICAL LEAST SQUARES
the figure. By adding tlie line (7), between C and F, another
check is obtained, with the corresponding condition that the
circuit A-B-C-F should close, or that the remainder of the figure,
namely, C-D-E-F, should close. Having taken the closure of
the whole figure as the first condition, onlv one of the two smaller
Fig. 2. System of Levels
circuits may be used as an independent condition, since the other
small circuit would then necessarily close, being the difference of
the other two circuits. Thus, as stated in the above rule, each
extra observation gives one independent condition. It is obvious
that any five lines connecting the six benchmarks could be con-
FiG. 'A. Horizontal Anglos at a Station
sidercd as the original, ne('(>ssaty obscM'vatioiis, and that cnty two
of the three circuits could l)e used for tlu^ two conditions.
As aiiotluM' example, let Fig. 3 r('i)rcsent a series of horizontal
angk^s around a station, coTniccting thc> {\yc signals, L-M XOF,
OBSERVATIONS OF DEPENDENT QUANTITIES 57
each angle observed being indicated by a number in parentheses
and a corresponding arc. Four angles would be sufficient to
connect the five signals, so that there are three extra observations
and, therefore, three independent conditions. If (1), (2), (5),
and (6), be regarded as the necessary angles, (j) would give the
condition that (l) + (2) + (3) — (6) should equal zero; (4) would
complete the horizon with the requirement that (4) + (5) + (6)
should equal 360°; and (7) would close the horizon, likewise, with
(6) — (1). Different combinations could be used, as well, for the
three conditions, such as (l) + (4) + (o) — (7) = 0, etc., and these
would be independent if each of the three extra observations were
used in one, and only one, condition. It is easily seen that the
Number of conditions =
(Number of angles observed) — (Number of signals) + 1.
72. Statement of Conditions. Although the conditions, as
functions of the observed quantities, might be very complex in
form, involving the higher powers, etc., still in the problems with
which the engineer is usually concerned, they are of the linear
form or easily reducible to th it form. Therefore, we shall con-
fine our attention to these simpler conditions and consider them
all as in the linear form.
The conditions express the relations which must be rigidly
satisfied l)y the final, adjusted values of the observed quantities.
Let these l)cst, adopted values be represented by T^, V2, V.i, ■ . . T'„;
the corresponding observations, hy Mi, Mo, Mi, . . . Mn', and the
small corrections to be added to the observations to obtain the
adjusted vahu^s by, v\, v-j, v-.i, . ■ . v„, in which /; is the number of
observations, which is also, in this case, the numl)er of ol)sorved
ciuantiti(^s. Then ]'\= Mi-\-v\, r2 = -l/2 + i'2, etc. The original
conditions will be stated in the following Condition Equations:
aiVi+02rL'+r/3r.s+ . . . +r/„T'„ + r/(, = ()
hiVi + h;V->^h,,V-,+ . . . +/;„r,+ /„ = 0 (56)
ClVl^C2V-2^C,.Vy,^ . . . +r„T'„+ro = 0
ill which th(> a's, //s, c':<. etc., ai'c known conslaiils. 'I'hcix^ will l)e
58 PRACTICAL LEAST SQUARES
as man}" of these equations, of course, as there are independent
conditions in the problem, and as many F's as observed quantities.
As the observed values approximate closely to the V's in all
observations which are carefully made, they will nearly satisfy
the conditions (56). Therefore, substituting M's for V's in (56)
will result in a small quantity, q, instead of zero, as the value of
each condition function, thus:
01.1/1+02^1/2+033/3+ . . . +o„-l/„+ao = 5i
6111/1 + 62.1/2 + 633/3+ . . . +6„3/„+6o = 92 (57)
Ciil/i + C23/2+ C33/3+ . . . + c„j/„+ Co = qs
q being the amount by which the observations fail to satisf}' a
condition equation, that is, it is the closure error of each condition
equation.
Xow substitute for V, in the equations (56), the value M-\-v,
and we have,
oiVi+a2r2+ . . . +a„r„+ (01.1/1+023/2+ - • • +o„.1/„+0()) = 0
6iri + 6,r2+ . . . +6„r„ + (6i-1/i + 623/2+ . . . +6J/„ + 6o)=0 (58)
Cii'i + C2r2+ • . • +c„r,.+ (cii1/i + C23/2+ . . . +c„,l/„ + Co) =0
in which the parenthetical expressions are the values of q in (57).
Therefore, the equations (58) take the form,
Oiri+02r2+ . . . +o„r„+ryi = 0
6iri + 62r2+ . . . +6„r„ + ry_. = 0 (59)
C\Vl+C-2V-2+ . . . +Cr.r„ + f/;s = 0
These arc the Reduced Condition Equations. They state the
rocjuired relation Ix'tween the corrections to the oljservations and
the closure errors of the ()i-i<2;inal conditions. These corrections
are the unknowns which are to be obtained as a result of Ihe adjust-
ment. The reduced conditions thus involve much smaller quan-
tities than the original ones, (56), and are more convenient to
handle.
OBSERVATIONS OF DEPENDENT QUANTITIES 59
Comparing the two sets of equations, (56) and (59), we note
that they are ahke in form but differ only in the substitution of
the small y's for the V's and the q's for the constant terms, oo,
bo, Co, etc. It is usually convenient, therefore, to write the con-
ditions in the reduced form in the first place, especially as the
constants, ao, bo, Co, etc., are zero in most of our problems. How-
ever, if the original equations be omitted, the sign of q should be
determined wdth great care. It should be the same as the error
of closure of that condition and opposite to that of the correc-
tions, in general. For example, if the sum of the angles closing
the horizon at a station be greater than 360°, q would be positive,
since the corrections to the angles, generally, would be negative
so as to reduce their sum to 360°. For the beginner, nevertheless,
it is safer to write the original conditions first, so as to avoid this
difficulty with the signs. It should be noted that if an adjust-
ment were carried out completely with the signs of all the q's
incorrect, it would result in a set of corrections having the wrong
signs throughout, which could be changed without altering the
adjustment computation in the least.
73. Adjustment by the Method of Correlates. The final,
adjusted values of the observed quantities must exactly satisfy
the prescribed condition-; of the problem, and must be, moreover,
the best, or most prol)a})lc, values, according to the Theory of
Least Squares, which will so satisfy them. Therefore, the sum
of the weighted scjuares of the corrections, which have the nature
of residuals, must be a minimum, as in Art. 34. That is,
[u'v~] = Wivr-}-V2i'2~^ii':iV:r^ . . . -i-WnVn'^^ii minimum (9)
which nuist be satisfied simultaneously with the conditions (59).
Multiplying th(^ condition eciuations of (50) in succession by
the factors, -2A, -2/i, -2r, (>tc., respectively,
-2ai.l/'|-2a_..l/'_>-2a:i.lr:i- . . . - 2^'„.l/'„-27ul = 0
-~2biIh'i-2h-2Br2-2h,,Bv:i- . . . -2h„Br„-~'2q:B = () (60)
-2ciCv,-2c-2Cv2-2c:,Cv:,- . . . ~2cJ-r„-2q:,C ^0
GO
PRACTICAL LEAST SQUARES
Adding these equations to (9) and collecting the coefficients
of the separate v's, we have the requirement that
wiVi2-2vi(aiA-\-b,B-\-ciC+ . . . ) +
-\-W2V2^-2v2ia2A+b2B-\-C2C+ . . . ) +
+WsVs^-2vs(a:iA-^bsB^C'sC+ . . . )+ (61)
+ +
-\-WnVn^-2Vn(anA-\-bnB + C„C+ . . . )
— 2 {qiA-\-q2B-{-q'sC-\- . . . )=a minimum
For the minimum, the derivative of this expression with respect to
each of the v's must be placed equal to zero. Therefore,
2wiVi-2iaiA-\-hiB+ciC^ . . .)=0
2w2V2-2{a2A-^b2B-\rC2C-\- . . . ) = 0 (62)
whence,
2w„Vn~2ia^ + b„B + c„C-\- . . . )=0
vi=—{aiA+biB-\-ciC+ . . .)
Wi
(63)
r„-~(a„A+6„7?+c„C+ .
Substituting these values of the r's in the eoncUtion equations (59),
and combining the coefficients of .4, B, C, etc., we obtain the
Normal Equations:
(64)
A +
-ab'
w
B +
'ac'^
IV
C'+.
.+71
= 0
~ah
IV
A +
1>K
w
/> +
be
IV
r'+ .
.+q2
= 0
~ ac
w
-4 +
be
w
/i +
cc
w
('+ .
. +r/.
= 0
Upon inspection of these (>(iuations, there is seen to exist among
them the same svmmetrv as shown, in Art. 44, amon"; those for
OBSERVATIONS OF DEPENDENT QUANTITIES 61
the adjustment of indirect observations. The diagonal coefficients,
down to the right, are sums of squares and, therefore, always posi-
tive, and the other coefficients are symmetrical about this diagonal.
However, the weights, w, occur here in the denominators of the
coefficients, instead of in their numerators as in the previous case,
and the constant terms, q, are the original closing errors of the
condition equations, in order. The number of the normal equa-
tions will always be the same as that of the conditions, so the
number of the g's will be the same.
The factors, A, B, C, etc., are obviously the same in number
as the conditions, and they correspond to the various condition
equations, in order. They are called Correlates or Correlatives,
and are the unknowns of the normal equations, from which they
are obtained by a solution according to the methods of the last
chapter.
Substituting the values of the correlates, resulting from the
solution of the normal equations, in the equations (63), we obtain
the desired corrections, vi, vo, Vz, etc., which, applied to the cor-
responding observations. Mi, AI2, M3, etc., give the best values,
Vi, V2, F.3, etc., of the observed quantities.
Thus, the process of adjustment may be stated in the rule:
Write the condition equations involving the unknown corrections
to the obscn'vations, and from them an equal number of normal
equations, the solution of which gives the values of the correlates,
from which the desired corrections to the observations are com-
puted to obtain the ])est values of the observed quantities. The
conditions (59) are fii'st written, then the normal equations (64)
are formed and solved, and lastly, the substitution of the cor-
relates in (63) gives the desired corrections.
74. Observations of Equal Weight. By placing the weights
equal to unity, in ccjuations (63) and (64), we have the simpler
forms:
Vi=a\A-j-hiB^ci('-\- . . .
r., = a^A +?>./^ + c,r'+ . . . (65)
r„ = nnA^h,Ji^CnC+ . . .
62
PRACTICAL LEAST SQUARES
and the normal equations,
[aa]A-\-[ab]B-{-[ac]C+ . . . +gi-0
[ab]A-\-[bb]B+[bc]C+ ... +92 = 0 (66)
[ac]A + [bc]B-\-[cc]C+ . . . +93 = 0
Here the coefficients are the same and occur in the same order as
those of the equations (22) in Art. 42, but the constant terms are
simpler as they may be taken directly from the conditions with-
out additional computation or combination.
75. Controls or Checks upon the Computation. The forma-
tion of the normal equations from the conditions is conveniently
checked by means of sum-terms similar to those explained in Art.
43 for indirect observations. In this case, however, the sum-
check does not include the constant terms, q, which are not formed
in the same manner as the coefficients. Therefore, the check-
equations have the form:
«1+&1+Cl+ . ,
which, multiplied bv — , becomes:
Wi
ttiOi ai6i aiCi
«'l Wi li'i
Adding all such ecjuations, wc have,
+
acf
ab-
, fflc
+
H
w
le
iw
Si
Wi
(67)
(68)
(69)
of which the left-hand member is the sum of the coefficients of the
first normal equation. A similar equation can be written for each
of the other normal ccjuations. Tlius, the sum-terms check the
formation of the coefficients of the normal ecjuations.
After the a])Ove checks have l^een verified, the constant term of
each normal equation may be added to its sum-term, so as to form
a check-terni which shall inchuk^ the constant, for use durinp; tlie
solution of the e(}uations, as ck^scribcMl in Art. 50.
The computation of the correlates is checked ])y suljstituting
them in the normal ecjuations otlun' than the first. The correc-
tions are checked by sul^stitution in the condition equations.
OBSERVATIONS OF DEPENDENT QUANTITIES
63
And finally, the resulting values of the observed quantities may
be substituted in the original conditions as a test of the correct-
ness of the entire adjustment. This is the ultimate test of the
work and should never be neglected. Beginners, in particular,
should make use of all available checks.
76. Tabular Forms for Computations. By arranging the given
data and the condition equations in the form of a table, the forma-
tion of the normal equations and the subsequent computation
of the unknown corrections will be greatly facilitated. As the
weights occur in the denominators of the coefficients, it is con-
venient to use their reciprocals throughout the computation.
The following form is recommended:
Form for Condition Equations
(v)
t'l [ r- Vs
Vi
Vo n
Vl
Const.
(1/uO
iM
lie.
; I w.
l/Wi
l/"'o
1, We
I/U'7
(9)
(«)
ai
(li
i 0-3
Ol
«0
Qe
a?
7i
(f>)
b,
hi
I ^3
h,
h
h
^7
Qi
(r)
Ci
Ci
; Cs
c.
Ci
(^6
C-,
'h
(«)
Si
Si
S3
Si
S5
Se
Sl
(70)
Th(^ parenthetical number at the left of each row is the symbol
Un- the quantities in that row. The reciprocals of the weights are
written in their proper columns just l)elow the corresponding
r's, while the sum of the coefficients is written at the foot of each
colunm. It frecpiently happens that most of these coefficients,
a, h, c, etc., are unity, so that the formation of the normal ecpiation
coefficients can be performed mentally, especially when the weights
are equal.
The solution of the normal ecjuations and th(^ substitution l)ack
throutih the (l(M-ive(l (Hiuations to obtain th(> correlates will be
carricMl out by the abridged method of tlie last chapiter in the form
there li-ivcii.
64
PRACTICAL LEAST SQUARES
The computation of i\w corrections, also, may be tabulated
conveniently, as follows :
Computation of the Corrections
t'l
t'2
V-i
I'i I'i re 1 Vj
1/w,
l/tt-2
1/Wi
l/Wi
1
l/"-7
(a A)
(bB)
(cC)
(iiA
hji
c,C
02.4.
62/i
C2C
a^A
b,B
cC
GaA
b,B
CiC
n^A
b,B
c,C
OeA
b,B
c,C
a,A
b:B
ciC
(Sum)
(Hum/w)
0')
Tl
''2
r,
Vi
!'-o /'6
V:
(71)
The first row is obtained from the first row of (70) b;.' multiplying
the a's in succession by the first correlate. A; the second row, by
multiplying the 6's of (70) by the second correlate, B, etc. ]\Iul-
tiphdng each sum by the reciprocal of its weight gives the prelim-
inary values of the f's, carried out, as were the correlates, one
decimal place farther than is required in the final corrections, which
are then wi'itten in the last row, taking the nearest figure in the
next to the last place. These r's would 1)0 tested in the condition
equations, and modified slightly, if necessary, so as to satisfy them
exactly, the modifications being shown by canceling the changed
figures and wiiting the adopted ones just to the right and ab()V(\
Finally, the adopted valuers of the observed cjuantiticvs would b;^
tested in the; original conditions, which they should rigidly
satisfy.
77. Example: Adjustment of Levels. The following o])served
differciices of elevation between the benchmai'ks, A, B, C, etc.,
will now be adjusted to the nearc^st hundredth, in accordance^ with
the foi'cgoing nielhod. In I'^ig. 4, each line of kn'els is numbered, in
pai-('iitlics(>s, nnd an arrow sliows tlu^ direction of running the levels
OBSERVATIONS OF DEPENDENT QUANTITIES
65
over it, or rather, the direction of stating it, as it may have been
run in either direction or both directions, but can be stated
with only one sign which must correspond to a certain direction,
plus or minus, according as the final benchmark is higher or lower
than the initial one. Lines like that between C and G, which are
parts of no complete circuit, do not enter into the adjustment in
any way. The observed differences of elevation are as follows:
(1) +2.18 (3) -3.47 (5) +4.70 (7) -6.86
(2) +5.06 (4) +1.32 (6) -9.82 (8) +3.46
Let the weights of the lines (7) and (8) be 2 each, and those of the
others, unity, or, for simplicity in the use of reciprocals, let the
former be unity and the others, one-half, giving 1 and 2 for the
reciprocals.
Fi<;. 4. System of Levels
As shown in Art. 71, each complete circuit furnishes the con-
dition that the sum of its adjusted differences of elevation shall
ec^ual zero when given the proper signs as if run continuously around
the circuit, clockwise or counter-clockwise. Also, the number of
independent conditions is the same as the number of extra observa-
tions above those necessaiy to connect the given benchmarks.
These neces.sary lines may be drawn, one at a time, starting at one
benchmai'k, as long as a new Ijenchmark is added for each line
drawn. Then each line adikxl t(j th(> figure, between two bench-
marks ali'eady sliown, gives one independent condition which
should always be written so as to inclwle that line. When the
complete figui'c has be(>n re})ro(luced on paper, in this manner,
omitting no hues, all of th(^ necessary, independent conditions for
66
PRACTICAL LEAST SQUARES
the adjustment will be indicated. Their number may be verified
by the rule that it is the same as the number of extra observations.
Also from the above construction,
Number of conditions = (Number of lines)
— (Number of bench-marks) + 1.
Assuming the lines (1), (2), (3), (4), and (5) to be those neces-
sary to connect all of the benchmarks, we write a condition for
each of the remaining lines, namely, (6), (7), and (8). It is
essential, of course, that all of the lines which form circuits should
appear in the conditions. Thus we obtain the original conditions,
+ Fi- 72 + ^3 + 74- F5-Fo = 0
+ Fi-F2 + F7-7o = 0 (72)
+ F5-F4-F8 = 0
The minus signs result from changing the directions of the arrows
so as to be continuous around each circuit. It is not necessary
that all the circuits be traversed in the same direction, however,
in a given problem. Substituting for each F its observed value,
we find the closure errors of these three circuits to be, respectively,
+ 0.09, +0.08, and -0.08. Therefore, the following table may
be formed directly, the coefficients being unity:
Condition Equations
('■)
Vi
t'2
?'3
t'4
rs
?'6
i'7
''8
Const.
a/ IV)
(a)
(b)
(c)
2
2
2
2
2
2
1
1
i'l)
+ 1
+ 1
-1
-1
+ 1
+ 1
-1
-1
+ 1
-1
-1
+ 1
-1
+0.09
+0.0S
-0.08
U)
+2
_2
+ 1
0
0
-2
+ 1
-1
(73)
The normal equations may be written by inspection, owing to
the simplicity of the condition equations. Each square or product
is formed in the propcM- column, above, and niultijilied l)y its 1, w;
the sum of all siniilai' i-(>sults IxMiig tlu^ coefficicMit foi- the normal
OBSERVATIONS OF DEPENDENT QUANTITIES
67
equation. Thus, there are six aa's, each being +1, three ab's, each
being +1, and two ac's, each of which is —1. Where no coeffi-
cient appears in the condition equation for a certain v, that coeffi-
cient is regarded as zero. Each aa/w will be 2X(-}-l) = +2, as
also, will be each ab/w. As there is no column in which both h
and c occur, the products, he, are zero. When all the coefficients
in any condition equation are unity, the sum of the squares, each
divided by its weight, is equal to the sum of the reciprocals of the
weights; thus, [66/w] = 2+2+2 + 1 = +7. Likewise, the product
terms may be written by inspection, but the signs must be care-
fully considered; thus, [ac/w]= — 2 — 2= — 4, etc. The sum-
terms are treated in the same way as the coefficients, to test
the correctness of the computations.
Normal Equations
(74)
It must be remembered that the sum includes all the coefficients
of an equation, whether written or not, so that, when the abridged
form is used, as above, the coefficients must be read down and to
the right as explained in Art. 57 (c).
Preparatory to solving the normal equations, the constants are
added to their respective sum-terms to form the check-terms for
use throughout the solution, in order that the operations performed
upon the constants may be included in the checks. In their form
for solution, therefore, these e(iuations are:
X()i{MAL Equations
(75)
A
B
C
(q)
Sum
+ 12
+6
-4
+0.09
+ 14
+7
0
+0.08
+ 13
+5
-0.08
+ 1
A
Jl
C
Const.
Check
+ 12
+0
+7
-4
0
+ 5
+0.09
+0.08
-0.08
+ 14.09
+ 13.08
+ 0.92
68
PRACTICAL LEAST SQUARES
These equations will now be solved by the Abridged Method as
explained in Art. 60, the separate operations being indicated.
(I)
(2)
(3)
(2)
(4)
(ID
(3)
(5)
(6)
(HI)
A
B
C
Const.
Check
+ 12
+6
-4
+0.09
+ 14.09
+7
0
+5
+0.08
-0.08
+ 13.08
+ 0.92
+7
-3
0
+2
+0.08
-0.04
+ 13.08
- 7.04
(I)X(-6/12)
(2) + (4)
(I)X(+4/12)
(IDX(-2/4)
(3) + (5) + (6)
+4
+2
+0.04
+ 6.04V
+5
-1.33
-1.00
-0.08
+0.03
-0.02
+ 0.92
+ 4.70
- 3.02
+2.67
-0.07
+ 2.60V
Correlates
A
B
C
Constants
-0.09
-0.04
+0.07
C-terms
+0.104
-0,052
+2.67
B-tcrms
+0.138
-0.092
+0.026 = r
+0.152
+4
+ 12
-0.023 = 7?
+0.013=. 4
(76)
Tests of C()1{UEi-.\tks
Equation (2) +0.078-0. 161+0.08= -0.003
Equation (3) -0.052+0. 130-0.08 = -0.002
These discrepancios would be reduced l)y caiTying out the corre-
OBSERVATIONS OF DEPENDENT QUANTITIES
69
--
-
d
1
CO
c<>
o
o
1
CO 'N
c o
d d
1 1
~
i-t
CO
o
d
1
CO
c
o
1
CO
o o
d d
1 1
s
(N
CO CO
r-H C^
o o
d d
1 +
o
o
o
+
o
!M CM
o o
o d
+ +
£
(M
CO CO
o §
o d
1 +
CO
r-H
o
d
+
CO
(N CO
o o
d d
+ +
^
«N
CO CD
O O
d d
+ 1
CO
I— 1
o
o
1
CO
(M CO
o o
d d
1 1
^
C^l
CO
o
o
CO
T-H
c
d
+
CO
CM CO
o o
o o
+ +
c'
M
CO CO
o o
o o
1 +
o
C-1 CM
o o
d d
+ +
~
1
CO CO
— 2J
+ T
c
T T
§
^ ^ vS-
'^
> >
8
8
o
o
II
o
II
o
II
o
00
o
o
o
+
o
+
o
1
§
u
I +
1 I
o
+
-:S -^3 T3
O
70 PRACTICAL LEAST SQUARES
lates to four places, instead of three. But these are satisfactory
when the corrections are desired to hundredths, only.
Upon testing the corrections by substituting them in the condi-
tion equations (73), it is found that the third condition fails by
+0.01. This discrepancy must be removed by arbitrarily altering
one or more of the corrections involved in it, as it is due to neglected
remainders. At the same time, the other two conditions, which
check exactly, must not be disturbed. Therefore, it is desirable
to find a correction which is used in the third condition only.
Such a one is vs which is seen to be too large by 0.004, and which,
moreover, belongs to one of the o]:)servations of greater weight so
that it would be expected to have a smaller correction. There is
reason, therefore, for reducing this correction by the necessary
0.01 in order to satisfy the condition. The change is made as
indicated so that the original figure remains. If there were no
single correction which could be modified without affecting other
conditions, it might be necessary to alter two or three corrections
in order to satisfy all the conditions by a given set of corrections.
The final test of the correctness of the adjustment consists in
substituting the adjusted values of the differences of elevation in
the original conditions, (72), or in the other conditions which were
not used because not independent of these. It is well to restate
the conditions, using the corrected differences of elevation, in
order to secure a check on the condition equations. Referring
to the diagram, Fig. 4, therefore, and appljdng to each observa-
tion the corresponding correction, we have:
Final Tests of the Adjusted Values
Circuit A-B~C-F: +2. 16-5. 08-0. 88+9, 80 = 0. OOv^
Circuit C-D-T^: -3.44-3.44 + 0.88 =0.()0\/ (79)
Civmit D-EF: +1.29-4.73+3.44 =O.O0V
These comprise all the elemental circuits, so that any coml)inati()ns
of these would also be satisfied.
78. Arrangement of Equations. Tlie larger the coefficients
OBSERVATIONS OF DEPENDENT QUANTITIES 71
of the normal equations, the greater will be the labor of solution,
generally speaking, so it is important, as was shown in the case of
indirect observations, to make them as small as practicable. The
methods of Arts. 48 and 49 do not apply directly to conditioned
observations, but it is possible to select the conditions and arrange
the condition equations in such a manner as to save some labor in
the solution of the normal equations.
Inspection of the equations (73) and (74) shows that the shorter
conditions, that is, those which involve fewer observations, will
produce smaller coefficients for the normal equations. Therefore,
it is important to select the shorter conditions, as far as practicable.
In the above example, for instance, the three small circuits might
have been used to advantage, although, in so short a problem the
advantage is less evident than in longer ones.
It is apparent, also, that by arranging the condition equations
in a certain order, with the shorter ones first, the larger coefficients
will occur later in the normal equations, instead of earlier, which is
an advantage especially in the abridged method of solution.
Then, too, it is possible to place those equations first which have
no terms in common, so that the product-terms, [ah/w], [ac/w],
etc., in the first normal equation, may be zero in some cases.
Each of such zero coefficients gives a zero elimination-factor which
saves writing a whole line in the elimination. In some problems
this is very important. In the above example, if the second and
third equations had been written as the first and second, respect-
ively, [ab/vi\ would have been zero, thus saving the second step
in the elimination, since the second normal equation would have
had no A-term and so would have been, itself, the first derived
equation, number (II). Sometimes, it is possible to save several
steps in the elimination in this manner.
79. Example : Local Adjustment of Angles by the Method of
Correlates. In triangulation, the, methods of measuring the
angles at a station may result in several extra angles being observed.
As shown in the latter part of Art. 71, each of these extra
observations yields one independent condition. To illustrate
the method of adjusting the angles so as to satisfy all these condi-
72
PRACTICAL LEAST SQUARES
tions, we shall consider the case shown in that article, Fig. 3,
assuming the weights to be equal.
p L
Fig. 3. Horizontal Angles at a Station
Observed Angles
Mi= 85° 14' 24.5" M5= 50° 23' 26.7"
M2= 83 45 32.0 71^6 = 210 35 17.5 (80)
Ms^ 41 35 24.0 ilf7-234 39 08.2
ilf4= 99 01 14.1
Adopting (1), (2), (5), and (6), as the necessary angles, the conditions
may be written,
Condition Equations
yi + F2+T^3-T>, =0
T^4 + r5 + Fc-360° = 0 (81)
-T'i + Fg + F7-3()0°-0
from which, by substituting for each V its value, M-{-v, we have.
Reduced Coxditiox Equatiox.s
(82)
(•i
r. , r.
!'4
'■.i ''ti . ''?
(n)
+ 1
+ 1
+ 1
-1
+;5.() = o
(h)
+1
+1
+1
-17
(<■)
-1
+1
i +1
+ 1.2
r.sO
0
+ 1
+ 1
+1
+1
+1
+1
OBSERVATIONS OF DEPENDENT QUANTITIES
Normal Equations
73
A
B C
Const.
Sum
Check
+4
-1
+3
-2
+ 1
+3
+3.0
-1.7
+ 1.2
+1V
+3^
+2V
+4.0
+ 1.3
+3.2
(83)
Solving these equations as in Art. 77, we obtain the following
values of the correlates:
A = -1.347 B=+0.G19 C =
whence the corrections to the observed angles are,
1.503
'•i 1 r.. Vs
'•-1 Vi
''6 ''7
+0.H)"
-i.3.r'
- 1 . 35"
+0.62"
+0.62"
+0.46"
- 1 . r,o"
(84)
which exactly satisfy the given conditions.
The adjusted values of the angles, therefore, are,
Vi = M,-^v,= 85° 14' 24.5"+0.16"= 85° 14' 24.66"
45 32.0 -1.35 = 83 45 30.65
35 24.0 -1.35 = 41 35 22.65
01 14.1 +0.62 = 99 01 14.72 (85)
23 26.7 +0.62 = 50 23 27.32
35 17.5 +0.46 =210 35 17.96
.39 08.2 -1.50 =234 39 06.70
As a final test of the adjustment, th(>se adjustcnl values ai-e sub-
stituted in th(> oi-iginal conditions, which th(\v arc found to satisfy.
80. Special Case of One Condition Only. Th(> gcncM-al con-
dition e([uation is the fir.-<t one of (59), nanu^ly,
airi+02r2+r/:ir:i+ . . . +anr„+r/ = 0 (86)
whence, the single normal ('([uation i.-^
\aa ;/•].! +ry = 0 (87)
Vo
= 83
F.i
= 41
F4
= 99
75
= 50
Fg
= 210
Vt
= 234
74 PRACTICAL LEAST SQUARES
and the corrections, from (63), are
vi=^{ai/wi)A; V2 = (a2/w2)A; etc. (88)
The case of special interest, however, is that in which the
coefficients of the condition equation are unity; thus,
Vi-\-V2+V3-\- . . . -\-Vn+q = 0 (89)
The normal equation, then, becomes,
[l/w]A + q^O (90)
so that
A = -q/[l/w] (91)
The corrections, with this value of A, are, therefore,
yi = -^rTT-i' ^2=-gY— "; etc. (92)
[1/w] [1/w]
Thus the corrections are proportional to the reciprocals of
the weights, and each correction is equal to the total closure cor-
rection divided by the algebraic sum of the reciprocals of the
weights and multiplied by the reciprocal of the corresponding
weight.
For example, suppose we have a single circuit of levels which
add to +0.24 instead of zero, and that the weights of the nine
lines are 2, 3, 1, 2, 3, 1, 1, 3, and 1. The least common multiple
of the weights is 6, and they may be written, 2/6, 3/6; 1/6, 2/'6,
3/6, 1/6, 1/6, 3/6, and 1/6, respectively, so that their reciprocals
are the following integers, in order, 3, 2, 6, 3, 2, 6, 6, 2, and 6,
whose sum is 36. The corrections, therefore, are obtained by
multiplying each of these reciprocals into the constant, —0.24/36,
resulting thus:
-0.020, -0.013, -0.040, -0.020, -0013,
-0.040, -0.040, -0.013, -0.040
Testing these corrections in (89), their sum is —0.239 instead of
— 0.24, so that it is ncK'cssary to add —0.001 to ()n(> of th(>m, prefer-
ably changing —0.013 to —0.014, in order to rigidly satisfy the
prescribed condition.
Th(^ important point is that 1h(^ corrections may be written
OBSERVATIONS OF DEPENDENT QUANTITIES 75
by inspection, in such cases, from the fact that they are propor-
tional to the reciprocals of the weights and that their sum must be
equal to — q. If any of the v's in the condition equation be nega-
tive, the signs of the corresponding corrections are changed. Thus,
if the condition equation were
Vi—V2 + V3 — V4-\-V5 — Vq-\-V7-'i-V8-^Vq+0.24: = 0
the corrections would be numerically the same as above, but the
signs of the second, fourth, and sixth would be plus instead of
minus. In testing the corrections in the condition equation, then,
these three would be multiplied by —1, so that the condition
would be satisfied as before.
This method of distributing the error of closure is somewhat
similar to that used in the special case of weighted mean of two
quantities, given in Art. 36.
81. Adjustment by the Method of Indirect Observations.
It is possible to adjust conditioned observations as if the quanti-
ties observed were independent, that is, by the method used in
Chap. Ill for indirect observations. Although this process is
generally longer and less satisfactory than the solution by the
method of correlates, it will be explained, briefly, in order that it
may be used when the circumstances are favorable, and that the
subject may be better understood.
In Art. 71 it was shown that a certain number of observations
would be necessary, in a given problem, for the determination of
the unknown quantities, on the assumption that those observa-
tions were correct, and that the remaining, extra, observations
would furnish one condition each, to be satisfied by the adjusted
quantities. Let those ol)servations which are selected as the
necessary ones be stated simply as observation equations, namely,
Vi=Mi; F2 = 3/2; V:i = M:i; etc. (93)
Tlien each condition, selected so as to involve ])ut one new (juan-
tity, may be expressed in terms of tlie other quantities only, so
that the total num])er of unknowns will not exceed those first,
necessary ones. From the entire set of these observation equa-
tions, the normal equations ai'e foi-nunl, as many as there are
76 PRACTICAL LEAST SQUARES
necessary (i.e., independent) unknowns, and their solution gives
the adjusted values of the quantities.
For example, in the local adjustment of angles at a station, in
Art. 79, and Fig. 3, the three conditions could be replaced by obser-
vation equations, as follows :
Conditions Observation Equations
F1 + F2 + F3-F6 =0 -F1-F2 + F6 =^Ms
F4 + F5 + Fo-360° = 0 -F5-Fg+360° = M4 (94)
- Vi + T^G + V7 - 360° - 0 + Fi - Fe +360° = M7
The entire seven observation equations, therefore, are,
+ Fi -M,
+ F2 -M2
-Vi -V2 +Vo -Ms
-F5 -Fg -M4 +360° =0 W4 (95)
+ F5 -Ms
+ Fg -Mg
+ Fi -Fg -Mj +360°
in which there are but the four unknowns, namely, the angles,
(1), (2), (5), and (6), and each M represents an observed value or
constant term. These equations correspond to (16). Forming
and solving the four normal equations ])y the methods of Chap.
Ill, the })est values of the angles are determined directly.
82. Example: Local Adjustment of Angles as Independent
Quantities. The solution of th(^ above example will be con-
tinued, to illustrate the method, Init with ecjual weights, for sim-
plicity. Let the observed angles be the same as those used in
Art. 79, as the compai'ison of the two methods will be useful.
These angles are given in (SO). Sul)stituting their values in (95),
and also for each T', the corresponding M^v, so as to reduce the
constant terms, which is ecjuivalent to assuming foi- thes(> T'"s
the corresponding M's as approximate values, as in Art. 48, we
have the simplified observation cfjuations:
= 0
wt.
= Wi
= 0
W2
= 0
Ws
= 0
W4
= 0
Wo
= 0
Wq
= 0
w-
OBSERVATIONS OF DEPENDENT QUANTITIES
77
;'i
''•2
''6
i-'e.
(0
is)
+1
+ 1
= 0
= 0
+ 1
+ 1
-1
-1
+ 1
-3.0"
= 0
-4.0
-1
-1
+ 1.7
= 0
-0.3
+ 1
+ 1
= 0
= 0
+ 1
+ 1
+ 1
-1
-1.2
= 0
-1.2
(96)
The normal equations are obtained by inspection ;
t'l
ih
Vb ?'o
Const.
Sum
+3
+ 1
+ 2
+2
-2
-1
+ 1
+4
+ 1.8
+3.0
-1.7
-3.5
II II II II
o o o o
+3.8 V
+5.0 V
+ 1.3 V
-1.5 V
(97)
Solving them, we obtain directly,
2,i = +0.17" y2=-1.36" y6=+0.61" and re = +0.48" (98)
whence,
Fi=8o° 14' 24.67" ¥5= 50° 23' 27.31'
72 = 83 45' 30.64 F6 = 210 35 17.98 (99)
by a combination of which, the angles V;i, V.i, and V7, are com-
puted. Thus,
Vs^Vq -Fi-72= 41° 35' 22.67"
T^4 = 860° -1^:5-1+,= 99° 01' 14.71" (100)
F- = ;36()°-T>, + ri = 234° 39' 06.69"
This completes Ihe adjustment. A comparison of th(>s(^ results
with those of (85) in Art. 79, shows them to be only slightly
diff(M'ent.
83. Comparison of the Two Methods. 'i1ic piincipal points
of differ(Mic(> between llu> nu^thods lie in \hv lal)oi' in\'olved and in
th(^ cliecks wliich ar(^ available. It may ha\'(^ \)cv\\ not(>d that in
tlie preceding article there is no ultimate ehcM-k upon th(> cor-
78 PRACTICAL LEAST SQUARES
rections or the adjusted angles. The corrections must satisfy
the normal equations, of course, as in any other adjustment, but
there is no check upon the observation equations (96) . The checks
afforded by the conditions, in the method of correlates, are for-
feited in the method of indirect observations, being used for the
determination of some of the unknowns, as F3, V^, and V7 in
the above problem. This is an evident disadvantage of the
latter method, inasmuch as the final check is very desirable and
important. The sum-checks controlling the formation and solu-
tion of the normal equations are present in both methods.
In the method of correlates, the number of normal equations is
equal to the number of conditions, which must be less than that of
the unknown quantities or observations. In the method of indi-
rect observations on the other hand, the number of normal
equations is that of the necessary, independent unknowns, and
therefore may be greater or less than in the former method. Usually,
however, the number of conditions is small as compared with the
number of independent unknowns, so that the method of correlates
is likely to be the shorter, although the determination of the cor-
rections from the correlates is a step which is not required in the
other method where the unknowns are obtained directly from the
solution of the normal equations, or at most, by a single addition
or multiplication. If the number of conditions happens to be
nearly as great as that of the independent unknowns, as in the
above example, the disadvantages of the method of indirect obser-
vations are less, and the simplicity of the normal equations, result-
ing from the considerable number of zeros in the observation
ec^uations, may give this method the advantage, even, although
this is seldom likely to be the case. ]\Ioreover, th(> alisencc of
the final check in the conditions is a serious defect, and gives to
the method of coiTolates the preference.
84. Adjustments not Rigid. The final, adjusted vahies of
the unknown quantities cannot be regarded as the correct ones, of
course, but are api)roxiiuations to th(un. As difTcrent nu^thods
may be used in the adjustment, and as diff(M'ent s(>ts of conditions
may ])e used in the same method, it is obvious that small dis-
crepancies ar(^ likely to exist Ix^tween the final values ol)lained
OBSERVATIONS OF DEPENDENT QUANTITIES 79
from different adjustments of the same data. Each of these sets
of results may satisfy all of the conditions as required and may
constitute an adjustment which is entirely satisfactory. Usually,
the discrepancies will be so small as to be negligible as com-
pared with the accidental errors of the observations.
CHAPTER VI
ADJUSTMENT OF TRIANGULATION
85. Triangulation. A system or network of triangulation
consists of a series of stations connected by lines in such a manner
as to form triangles having their vertices at the stations. The
length of one line, called the base-line, being determined by direct
measurement, usually with a tape, and the horizontal angles
between the lines at each station being measured with a transit
or theodolite, the lengths of all the lines become known by com-
putation from the base-line and the angles through successive
triangles. The differences of elevation between the stations are
obtained from observed vertical angles which determine the
elevations of the stations above sea-level when one of them has
been connected to sea-level by a line of precise, or geodetic, leveling.
The position of the system on the earth's surface is fixed by astro-
nomical observations for the latitude of one station, the longitude
of one station, and the azimuth of one line. The size of the system,
or net, depends, therefore, upon the length of the base; its shape,
or form, depends upon the horizontal angles; its position, upon the
astronomical obsc^rvations; and its elevations, upon the vei'tical
angles and the initial elevation. If the triangulation be based
u})on, or connected to, two stations of another system which
has been com{)lete]y determined and adopted in size, position, and
elevation, the line joining the; two stations may be used as the base-
line for the new work, and the azimuth and the latitude, longitude,
and elevation of one of the; stations will determine the position and
initial elevation of the new n(>t. However, if the new triangulation
be compk^te in itself in regard to one or more of these elements,
and in addition })e connected to previously adjusted and adopted
W(jrk, this connection affords checks upon the coi'responding
so
ADJUSTMENT OF TRI ANGULATION 81
elements, and therefore, from one to five conditions must be satis-
fied if all of the work is to be made consistent as to length, latitude,
longitude, azimuth, and elevation. The shape of the net, and
the differences of elevation, therefore, must be adjusted so as to
fulfill these requirements. Moreover, the horizontal angles must
be adjusted to conform to certain geometrical and trigonometrical
conditions which depend upon the arrangement of the lines and
stations and the angles observed.
The vertical angles are independent of the horizontal ones and
are adjusted by themselves in any case. The adjustment of a
system so as to close upon fixed, or adopted, work with regard to
any of the five elements of length, latitude, longitude, azimuth,
and elevation will be discussed farther on.^ There remains,
then, the adjustment of a system which is complete in itself.
In this, the length of the base and the initial latitude, longitude,
azimuth, and elevation are determined separately and inde-
pendently of the horizontal angles in the net, and so do not enter
into the adjustment as long as there is but one of each of these
elements. The adjustment of the horizontal angles, there-
fore, will now be considered.
86. Nature of the Conditions. The horizontal angles in
triangulation are subject to two classes of geometrical con-
ditions, namely, those which involve the angles at one station
only, and those which define relations between the angles at
two or more stations. The former are called local conditions
and the latter figure conditions, giving rise to local and figure
adjustments.
The local conditions express the nniuirement that the adjusted
values of the observed angles at a station shall satisfy the indi-
cated horizon-closures and algebraic sums.
The figure conditions aw of two kinds, known as angle equa-
tions and side eciuations. An angle e(}uation re(}uires that the
sum of the angles of a ti'iangle oi' polygon shall b(^ equal to the
nunibcM" of right angles pi'esci'ibcHl by <i'(M)inetry foi- a plane figure
plus \\\v sphcM'ieal excess. A sidi^ eciualion i'(>(iuires that if the
1 Art. IOC), ct s(Mi.
82 PRACTICAL LEAST SQUARES
length of a line in the figure be computed from another line through
two different series of triangles, that is, by two different routes,
the two results must be equal.
Since all of these conditions must be satisfied simultaneously,
they would enter into a single adjustment, ordinarily. As will be
explained later, however, it may be convenient to perform the local
adjustment separately, prior to the figure adjustment, the latter
being so arranged as not to disturb the former.
87. Local Adjustment. In modern field practice, simplicity is
sought for the sake of economy. Accordingly, observations are
arranged, as far as practicable, so as to lessen the office work
necessary for their reduction, but without a sacrifice of precision.
The angles at a station, therefore, are observed in such a manner
as to avoid combinations which introduce checks and conditions
requiring extensive local adjustment. It is customary to measure
one angle for each of the signals less one, and then a single
one to close the horizon, thus securing one check which involves
all of the observed angles. The local adjustment is thereby
reduced to one simple condition, with equal weights, also, in most
cases, so that it amounts to a mere distribution of the error of
closure, as explained in Art. 80. If extra observations have
been made, however, so that two or more conditions are to be
satisfied, the general method of adjustment must be used. This
has been demonstrated in Arts. 79 and 82, in the last chapter, in
which the number of the conditions was shown to be equal to the
number of extra observations. Thus, if S stations be observed,
S~l angles would be sufficient to connect th(nn, and if .V angles
be measured between them, the number of extra observations, and
therefore, the number of local conditions may be expressed in the
formula.
Number of Local Conditions at a Station^A"— aS + 1 (101)
88. Figure Adjustment. Notation. In order to distinguish
between stations occupied and unoccupied, and lines obsei-ved
in both directions or in one; direction only, lines shown in diagrams
of triangulation will be; broken at the (nuls fi'om which they are
ADJUSTMENT OF TRI ANGULATION
83
not observed, full lines indicating observation at both ends.
Stations which are sighted upon but not occupied with the instru-
ment will be recognized from the fact that all the lines at those
stations will be broken. Thus, in Fig. 5, the station Pan was
Ohrt
Bon
Arm
Dake
Dart
Fig. 5. Unobserved Lines and Unoccupied Station
not occupied, as no full lines radiate from it. Dake was occupied
and Pan and Bart were observed from it, but Ohrt was not observed
from it, although Dake was observed from Ohrt. The other
stations were occupied completely as shown by the lines being
unbroken at those ends.
89. Classification of Figures. Although the figures in tri-
angulation may be very complicated and the adjustment very
laborious, Ihe work in such a case loses its economic advantages of
covering a great area or distance at the minimum of cost con-
sistent with the accuracy desircxl. In th(^ best practice, there-
fore, simple figures are used, and special attention is given to
measuring each angl(> with the ref}uisite degree of precision.
These simple figures may hv. classified as triangles, quadrilaterals,
and central-point figures. A triangle consists of tlu'ee stations
connected by three lines. A cjuadrilateral has four stations con-
nected by six lines. A central-point figure is a jwlygon with a
station at each vertex and another station in the interior from which
lines radiate to tlie vertices; th(^ polygon usually has not more than
six sides. The lines in these figures may l)e full or partly broken,
as above. Fig. 5 represents a central-point figure. A typical
quadrilateral witli diagonals is shown in Fig. 6, while Fig. 7 is
84
PRACTICAL LEAST SQUARES
the simplest form of a central-point figure, which may be con-
sidered, also, as a quadrilateral. In Fig. 8 is shown a combina-
tion of a central-point figure with a polygon having diagonals;
Fig. 6. Quadrilateral
this is seen to increase the intricacy of the system, which would
have been a simple central-point figure had the diagonals KM
and MO been omitted.^
Fk;. 7. Central Point Fij^iire
C^ntral Point Figure with
p]xtra Diagonals
90. Angle Equations. The triangle is the unit figure in tri-
angulation. For each t]'iangl(» or other polygon of wliich all the
angles have been observed, an angle ('(juation may ])e written
expressing the conchtion that the sum of the adjusted angles shall
1 In the diagrams represent ing triangulation, it is assvnncd that there is
no station at the intersection of diagonals of a figure luiless there is an angle
at that i)oint in one of them. If, in llie r(>mote (•as(\ a station happcMied to
fail at this intersection, tlu^ diagram \voul(.l l)e slightly distorted so as to indi-
cate the fact without finest ion.
ADJUSTMENT OF TRIANGULATION 85
be the theoretical amount, namely, a certain number of right angles
plus the spherical excess, e, of the figure.^ Thus, referring to Fig.
6, in which the separate angles are numbered clockwise at each
station, and representing their adjusted values by F's, as usual,
the four triangles yield the following angle equations,
Triangle (a) ABC, Fi + F2 + F3 + F6-(180°+ea) =0
" {h) DAB, F2+F3 + F4+77-(180°+e,) = 0 (102)
'' (c) DBC, Fi + F5 + F6+F8-(180°+6c)=0
'' {(1) DAC, F4+F5 + F7 + F8-(180°+e.)=0
in which a, b, c, and d refer to the separate triangles as shown in
the figure. Since the spherical excess depends directly upon the
area of a figure,^ the excess for the entire quadrilateral should be
equal to the sum of the two excesses for the pair of triangles
formed by each diagonal. Therefore,
ea+ed=e6+ec (103)
which affords a check upon their computation. By inspection,
then, we find that from any three of the above angle equations it is
possible to derive the fourth by addition and subtraction. Also,
from the whole quadrilateral, we may write the condition,
F, + F2+F3 + F4+T;5 + T'G+F7 + Fs-(360°+e.+ 6.) = 0 (104)
and this eciuation is seen to Ijc the sum of the first and the fourth of
(101). Therefore, any thi'ce of the four triangl(>s may l)e sel(K'ted
from which to write the three independent conditions or angle
e(}uations. In other words, if two triangles foi'med by a single
diagonal satisfy thcnr ('oiiditions, the (>iitii-e figure nnist satisfy
its condition (KW); then if a third triangle condition, also. l)e
satisfied, the fourth one is sui'c to be, since the foui-th triangle is
e(iual to the whole figure minus the third one.
If w(> adopt the first three of tli(> e<|uations (102) as the inde-
pendent ones, and wi'ite foi' each F. in the usual manner, its value,
^ It is seldom that an aiifrlc ('(luatioii has to !)(> written for a fi^tire greater
than a triangle, as an (jpen ([iiadrilateral i without a diagonal) is not rigid and
should be avoided.
* From sphcTical trigonometry.
86
PRACTICAL LEAST SQUARES
M-i-v, in which M is the observed value of the angle and v is its
correction to be obtained from the adjustment, we have for this
quadrilateral the following set of final angle equations, corre-
sponding to (59):
V2 + V3 + V4-\-V7-\-q2 = 0 (105)
Vi+V5-hvQ^vs+q3 = 0
in which q is the error of closure of a triangle, positive when the
sum of the observed angles is too large. These closures may be
checked in the same manner as the spherical excesses, that is, the
closure for the whole figure must be the sum of the closures for
each pair of component triangles. Thus,
qa+qd = qb-\-qc (106)
91. Number of Angle Equations in a Figure. To determine
the number of independent angle equations in a given figure,
A-B-C-D-E, Fig. 11, we may proceed as follows: Start with
c
B ^^ / B
A
Fig. 9. Fig. 10 Fig. U.
Determination of the Number of Angle Equations in a Figure
two stations, A and B, connected by one line, as in Fig. 9. Add
the station C, with two lines to A and B, and one triangle is
obtained. Add station D and two lines, to A and B, and a second
triangle is formed, as in Fig. 10. Add the third line, from D to C,
and the quadrilateral is completed, making three independent
angle equations in all. If another station, E, be added, with
three lines to .4, C, and D, as in Fig. 11, two of these lines form a
new triangle, as before, and the third completes a (}iia(lrilatcral,
A-C-E-D, in which one triangle, A-C-D, formed a part of the
previous figure, A-B-C-D, and is therefore alread}' included in
the conditions. The third line from E thus adds but one new
ADJUSTMENT OF TRIANGULATION 87
condition, making a total of five angle equations. If the line
BE were added, it would be the second diagonal in the quad-
rilateral A-B-C-E, of which two triangles are already included
in the figure, so that the third angle equation, only, for that quad-
rilateral would be added by this line.
We may generalize from this procedure and write a formula
for the number of independent angle equations in any figure.
Starting with three lines and three stations in the form of a tri-
angle, we have one condition. If we add one station and one line
to it, no new conditions are introduced, but each additional line
to that station gives one new condition, and the same is true of
further additions of stations and lines until the entire figure has
been drawn. Therefore, each station added to the initial triangle
adds as manA' conditions as there are lines, less one, running to
that station, so that the total number of conditions would be the
aggregate of these conditions together with the one for the initial
triangle, that is, the whole number of lines minus the whole
number of stations, plus one. But if any one of these lines be
unobserved at one end, one angle of the corresponding triangle
would be missing and that line would not count for a condition.
Also, if one station were entirely unoccupied, as Pan, in Fig. 5,
page 83, it could enter into no complete triangle and would have
to be omitted from the stations counted in determining the number
of angle equations. Finally, then, we may write the following
formula for the number of independent angle equations in a given
figure :
Number of Angle E(}uations = L'-,S" + l (107)
in which L' is the numbcM' of /;/// lines in the figure and S' is the
number f)f occupied stdlions.
92. Side Equations. The ti-ianglcs of a figure may close
exactly to lS()° + e, and the angles still be inconsistcMit with regard
to th(> closure of th(> whole figure wIkmi the lengths are computed.
To illustrate this, sujipose the triangles of Fig. (>, page 84, to be
plotted in the following oidei-. the angles of each having been
adjusted to a closure and \\\v local conditions satisfied. Plot
triangle (a) to any convtMiicMit scale, as in Fig. 12. using the given
88 PRACTICAL LEAST SQUARES
angles. Upon the side AB, construct triangle (b) with vertex
falling at d'. Upon BC, construct triangle (c) and its vertex
might fall at (/". Then if triangle (d) were plotted upon AC as a
base, its vertex might fall at d'", and the angle at d'" would still
equal the sum of d' and d" as required by the local condition at
station D. Thus, the lengths of the lines running to this station
would fail to check because they did not intersect at a single
point. The side equation for this figure, therefore, would require
1^
1
— ^^^x/^'^'X
--
\.
\
;
?
\\^"
Fig. 12. Side Equation
that the three points, d', d" , and cV" , ])o coincident, which is
equivalent to the requirement that d" and d'" ho coincident, or
that the line Cd" in the triangle (r) ])e equal to the line Cd'" in the
triangle {d). In other words, the side equation requires that if one
line of a figure be computed from another line tlu'ough two series
of ti'iangles, or by two different routes, the resulting IcMigths shall
be eciual. Since the same initial lin(> is u.'^ed in ])oth cases, it can-
cels from the equation, and the angles, only, are concerned in the
condition.
93. Side Equation of a Quadrilateral. Starting witli the line
AB and computing CD through the ti'iaiigles (a) and (c), as indi-
cated by the upper dotted arrows, and using tlie final, adjusted
valu(>s of the angles, wliicli make the points d' , d" , and d'" coin-
cident, we have,
CD^BC'^^^^AB'^:^^ ^ (]0S)
sm Is sin \ 0 sm T g
ADJUSTMENT OF TRI ANGULATION 89
liikewise, computing through the triangles (6) and (d),
CD=^AD'^^^^ = Ab'^^^^^^^^ (109)
sm F5 sin Vi sin F5
Equating these expressions for CD and cancehng the factor AB,
sin F3 sin Fi _sin F2 sin F4 mm
sin Fe sin Fg sin F7 sin F5
Multiplying both members of (110) by the reciprocal of the second
member, we obtain a statement of the side equation in the form
sin V\ sin F3 sin F5 sin F7 _ n in
sin V2 sin F4 sin Fe sin Fg
in which the numerator contains the odd-numbered angles and the
denominator, the even ones, which happens as a result of our num-
bering the two angles at each station in clockwise order, and which
is a useful check on the formation of the side equation. To
reduce this equation to the first degree, we take the logarithm of
each member and equate them, whence,
logsin Fi+logsin Fs+logsin Fs+logsin F7
— logsin Fo — logsin F4 — logsin Fe — logsin Fg^O (112)
Equations (111) and (112) are original condition equations which
state the requirement which must be satisfied by the adjusted
values of the angles, and correspond to the form shown in (56).
It remains to derive the simpler reduced condition which expresses
the relation between the corrections to the observed angles, so that
it may be combined with the angle equations (105) into one adjust-
ment having the same unknowns, namely, the r's. That is, for
each T' must be substituted its value, J/+r, and the equation
reduced to the linear form to coi-respond to (59), page 58.
If a given angle, M , be altcu'cnl by a small conxH'tion, v, (expressed
in seconds, the logarithmic sini^ of the ang]t> would ])e changed
by a corresponding amount, naiuc^ly, the nuni})er of seconds in v
multiplied into the difference for ()iu> scH'ond in that particular
logarithmic sine, as taken from Ihc^ k)garithmic tables with the
proper algebraic sign, positive^ if th(^ angle lie in the first (juadrant,
in which the sine iiici-eascs with inci'cwsing angl(\ and negative
90 PRACTICAL LEAST SQUARES
if in the second quadrant, where the sine decreases with increase
of angle. That is,
logsin (3/+y)=logsin F = logsin M -j-v {dl") (113)
For example, if M = 76° 15' 14.5" and v=-4.1", logsin M =
9.9873797 with a difference for 1" of +5.1 in the seventh decimal
place. Then logsin (ilf +t;) = logsin 76° 15' 10.4" = 9.9873797 -
4.1(+5.1) =9.9873776. Substituting for logsin V, in (112), its
value given in (113), namely, logsin M-\-(d l")v, and collecting
the logarithms into one constant term, we have,
-(del")ve + {d-l")v7-{dsl")v8^(\ogsmMi
— logsin 3/2+logsin .^/a — logsin 3/4 + logsin il/s
— logsin J/6+ logsin M7 -logsin Mg) = 0 (114)
in which {dil") represents the difference for 1" in the logsine of
angle Mi, in the seventh place of decimals, assuming that seven-
place logarithms are used.^ Each of these differences for 1" is a
numerical coefficient for its v, and corresponds to a, b, c, etc., of
(59), page 58. Also, since the observations are carefully made,
the observed angles, M, will approximate closely to their adjusted
values, V, so that the algebraic sum of the logsines of the Jl/'s in
(114) will be, instead of zero as in (112), a sma.l error of closure, g,
expressed in units of the seventh decimal place, and equal to the
amount by which the sum of the positive logsines exceeds that of
the negative ones, in (114). The reduced form of the side equa-
tion becomes, therefore, if we assume it to be the fourth of the
condition equations so that its coefficients are f/'s (a, b, and c
being coefficients of the first three conditions respectively),
diVi — d2V2-\-d:',r-4 — d4V4:-\-d-)V-, — d(]VG-{-d7V7 — dsVs~^q4 = 0 (115)
in which each d is the numerical difference for I" in the logsine of
the corresponding angle. Thus we see that the side ecjuation
' It will be convenient, generally, to use seven-jilace logarithms but to
take the sixth {ilace unit for (dl") and q, thus moving the decimal ])oint one
])lace to the left in these (luantities. If the tal)les show a diilercnee for 10"
to be +51, therefore, ('/I") = +5.1 in the seventh place, or +0.51 in the sLxth
place.
ADJUSTMENT OF TRI ANGULATION
91
states that the corrections to the angles must be such that the
algebraic sum of the resulting corrections to their logsines will
equal —q and the original side equation (111) be satisfied. Since
the coefficients and the constant term of (115) are expressed in the
same unit, in the seventh place, this equation is consistent with the
angle equations (105), as stating a linear relation between the f's.
94. A Shorter Form of the Side Equation for the quadrilateral
may be obtained by computing one of the adjacent sides, as BC,
from AB, instead of the opposite one, CD, as above. In Fig. 13,
Fig 13. Side Equation; Quadrilateral.
the dotted arrows show the two routes of the computation, from
which the two resulting values for BC must be eciuated:
sin Vz
and
whence,
BC^AB
BC^AB
sin Fg
sin (r;5 + T'4) sin T's
sin ]': sin (I's + Vo)
sin F3 sin (T'n+F);) sin V~
= 1
(116)
(117)
sin (F3 + F-}) sin V(\ sin I's
and the rcHlucod side ecjuation Ix'coinos,
f/H?V, -'':',+ i(r:i + r4)+r/54-r,(r5 + '•,-,) -'/,;'■.! + '/7r7-'/sr8 + r/4 = 0 (118)
or, separating the various unknowns,
(f/3 — (/3 + 4)i':i — '/3+ 4?'.l +(/5+ (i''5 + (//o-r fi — 'A;)''0
+ (/7?'r-'/srs + ^y4 = 0 (119)
92
PRACTICAL LEAST SQUARES
in which ^3+4 represents the difference for 1" in the logsine of the
sum-angle, il/3+3/4, etc. Although this form is somewhat
shorter than (115) in that the angles at the station, B, lying be-
tween the sides AB and BC, do not appear, it is more troublesome
for the beginner. However, the fact that the angles at one sta-
tion are not concerned makes this the preferable form when one
of the stations of the figure was not completely occupied, in which
case the equation is expressed between the two exterior lines
adjacent to this station. Thus, in the above figure, station B
might have been unoccupied without affecting the form of equa-
tion (117).
95. Side Equation for a Central-point Figure. Let Fig. 14
c
Fig. 14. Side Equation; Contral-point Figure.
represent the general form of a central-point figure, and for the
sake of variety, suppose the central station to have been observed
from each of the others but not to have been occupied, as shown
by the lines l)eing })roken at that point, but that otluM'wise the
figure is (•omplet(\ The side equation will ho written between
two of the lines which meet at the central point, and the dotted
arrows show the two routes of computation from the line AO to
DO through two series of triangles.
T^^> ^/-^^i" ^1 ^"^ ^'i ^^^ ^^^ </->^"^ ^2 sin T'lo sin Tio
IJ(J = A(J 7 "T . -— -. — ., -A(J -. — ~ . — -. . — ~T-
sm I 4 sin I G sm I s sin \ n sm I (.> sm I 7
whence,
sin T'l sin T':{ sin T.-, sin V- sin 1'.. sin T'n
sin V2 sin 1^4 sin Vq sin Vs sin V\o sin T'l.)
= 1
(120)
(121)
ADJUSTMENT OF TRI ANGULATION
93
■fhe reduced side equation follows directly, as in Art. 93:
diVi— d2V2 + d3V3 — d4V4-\-d5V5 — d6VQ-\-d7V7
— d8V8-\-d9Vi) — dioVio^diiVn — di2Vi2-\-q = 0 (122)
It will be seen that the odd-numbered angles occur in the
numerator of (121) and the even ones in the denominator, which
results, as in Art. 93, from the clockwise numbering of the two
angles at each occupied station. Since the angles at the central
station were not observed, and do not occur in the side equation,
it is not necessary to number them. By comparison, also, with
(111) of Art. 93, it is evident that the side equation for a central-
point figure having four sides and the stations A, B, C, D, and 0,
would be identical with (111) written for a complete quadrilateral
with diagonals.
96. Mechanical Statement of Side Equations. The similarity
among the side equations (111), (117), and (121), in the occurrence
of the odd-numbered angles in the numerators and the even ones
in the denominators, would indicate the possibility of writing
these equations by inspection instead of using this property merely
as a check. This may be done b\' the following mechanical method
for all ordinary figures of the quadrilateral or central-point form.
Notation, (a) The pole for a side equation is some station,
or other point, to which a line rims from every (other) station of
the figure for which the equation is to be written. It may be at
the intersection of the diagonals of a quadrilateral, although there
is no station there. The point selected as the pole will be indicated
by a S7nnll circle (h'awn around it. as in the following figures:
Fig. 15. Fk;. 10. Fig. 17.
Location of Pole for Sidi^ Equaiion.
(h) At each station of the figui'e, there will l)e three lines, one
of which goes to the p()l(> and ina\- bo calliMl the pole line. Of the
94
PRACTICAL LEAST SQUARES
other two, one will be the left-hand line and the other, the right-
hand line, as we look into the figure, from the station towards the
pole, (c) At each station, the left-hand angle is the angle between
the left-hand line and the pole line, and the right-hand angle is the
one between the right-hand line and the pole line. This nota-
tion is illustrated in the following diagrams in which I and r indi-
cate the left-hand and right-hand lines and L and R, the left-hand
and right-hand angles, respectively:
Fig. 18.
Pole
Fig. 19.
Loft-hand and Right-hand Angles.
QPole
Fig. 20.
The side equation, then, is written by placing the product of
the sines of the left-hand angles equal to that of the sines of the
right-hand angles, or by placing the former in the numerator and
the latter in the denominator of a fraction which is placed equal to
unity. The reduced form of the equation, corresponding to (115),
(119), and (122), may be written as the sum of the dv's for the
left-hand angles minus the dv's for the right-hand angles plus q
equals zero, or,
[c/v](for left-hand angles) — [r/?;](for right-hand angles) -{-q = 0 (123)
in which d is the difference for 1" in the logsine and q is the sum of
the logsines of the left-hand angles minus the sum of the logsines
of the right-hand angles.
It will be noted that the ang^.cs at the pole do not enter into
the side equation at all, so that the pole is situated at the inter-
section of the two lines' between which the equation would be
written according to the analytical method of the preceding articles.
Thus, equation (111) would correspond to a pole at the intersection
of the diagonals of Fig. 12, page 88; in Fig. 1.3, page 91, the pole
would be at station B for equation (117); and in Fig. 14, page 92,
the pole would be at the central station, 0.
It is now apparent, as stated in Arts. 93 and 9o, that the odd-
ADJUSTMENT OF TRI ANGULATION 95
numbered angles occurred on one side of the side equations and the
even ones on the other because the two angles at each exterior
station of the figures had been numbered clockwise so that the odd
ones were on the left and the even ones on the right of the pole
lines.
The selection of the pole may be governed by the following
principles.^ In a central-point figure, it must be at the central
station. If a station was not occupied, or not completely occupied,
the pole should be at that station. Sum-angles may be avoided
by placing the pole of a quadrilateral at the intersection of the
diagonals, which is simpler for the beginner although it introduces
two additional angles and logsines. The pole should not be placed
where the smallest angles occur, as these angles should enter into
the side equations with their larger coefficients.^
97. Number of Side Equations in a Figure. In order that
there may be two routes, through two series of triangles, between
two lines in a figure, there must be at least three triangles in the
figure, and therefore, four stations with three lines to each station.
In other words, the quadrilateral is the simplest figure for which
a side equation may be written. Similarly, a central-point figure,
without diagonals, can have but one side equation since there are
but two series of triangles through which one side maj' be com-
puted from another. The quadrilateral and the central-point
figure, therefore, furnish one side equation each, and are the ele-
mental figures for these equations.
A complete central-point figure has as many outer lines and
the same numl)er of inner ones as there are exterior stations, so
that the total number of lines will equal twice the total number of
stations less one, which is true, also, in a quadrilateral. Since
each of these figures yields one side equation, the formula may be
written,
Number of Side Equations = L-20S- 1) + 1 =L-2,S+3 (124)
in which L is the total number of lines, full or ])roken, and S is the
' See Wright and Ilayford's Adjiistiiient of Oljservalions.
- The logsiiic of a small angle varies rapidly witl) change of angle, so that
the diilerence for 1" is large.
96
PRACTICAL LEAST SQUARES
total number of stations whether occupied or not. Adding to
either figure one station with three Hues from it to stations of the
figure, adds another quadrilateral or central-point figure, and
therefore, another side equation, which corresponds to an increase
of one {L — 2S) in the formula. Each additional line, without
increase of stations, makes possible the writing of one or more new
side equations, using that line, of which, however, only one can be
regarded as independent. Adding one station to a figure thus
adds as many side equations as there are lines from that station
to the figure, less two. For example, adding to Fig. 21 the station
A A
Fig. 21. Fig. 22.
Number of Side Equations in a Figure.
G with three lines to A, F, and E, adds one side equation which
could be written for the quadrilateral G-A-F-E or for the central-
point figure G-A-B-C-D-E, and if this latter had been the
original figure, the addition of the line AE would have formed the
quadrilateral A-F-E-G with its side equation. In each case the
new side equation must include; the added station or line.
98. Statement of All of the Conditions for a Figure Adjustment.
In the preceding articles, the three kinds of conditions, local,
angle, and side, which enter into the adjustment, have Ijcen
explained separatcl}'. The adjustment as a whole will now be con-
sidered.
Strictly, all of the conditions should ])e satisfied simultaneously
in one gc^ieral adjustment. The ln])()r of computation is greatly
reduced, how(>vei', by diminishing the number of conditions, and it
ADJUSTMENT OF TRIANGULATION 97
is especially convenient to perform the local adjustments sepa-
rately and in advance of the figure adjustment, inasmuch as it is
good practice to arrange the observations so as to have but one
local condition at a station, involving all of the angles, as explained
in Art. 87. Therefore, we shall assume that the necessary local
adjustments have been made, as in Arts. 79 and 82, preparatory to
the figure adjustment. However, if the angles at any station of
the figure complete the horizon, it will be necessary to include in
the figure adjustment a local condition providing that the sum of
these angles shall remain 360°, that is, that the algebraic sum of the
corrections to these angles must be zero. This is likely to be the
case at an interior station, such as F, in Fig. 22. Also, if a sum-
angle should be included among the conditions, as well as its
component angles, and with a separate number, a similar local
condition would be necessary to insure that the sum-angle would
remain equal to the sum of its components after adjustment;
but this may well be avoided by designating the sum-angle as the
sum of its components, as in Art. 94, instead of using a separate
symbol for it.^ In general, care must be taken that the prelim-
inary adjustment be not disturbed by the later one.
The selection of the angle and side equations for a given figure
or system must conform to the requirements that all the necessary
conditions be included, but no more, and that they be independent
of one another, so that no one of them could be obtained ])y com-
bining any of the others. If a dependent condition were included,
by mistake, it would be indicated during the solution of the normal
equations by a derived ociuation liaving all of its coefficients zero,
or nearly so, so that tlu^ corresponding correlate would be inde-
terminate. The necessaiy nuni])(>r of independent angle and side
ecjuations will \)e giv(>n by formulas (107) and (124), namely,
Number of Aii<;lc lMiuati()ns = L'-N' + 1 (107)
Number of Side l^cpiat ions =L — 2N +3 (124)
in wliicli L and N are the total luiinboi's of lines and stations, and
1 'riics(> local coiulitions an> avoided in th(^ fi^ur(^ adjust iiH-ut by using
dircclid/ts instead of (ni(jlc.<. as will he shown later on.
98 PRACTICAL LEAST SQUARES
U is the number of full lines and S' is the number of occupied sta-
tions. (For a station to be considered as occupied, at least two
lines must be unbroken at that station.) The best method of
writing the angle and side equations so as to be certain of their
independence as well as their number, is to draw a sketch of the
system or figure to be adjusted, adding one station at a time, with
its lines to the previous stations, and writing the equations intro-
duced by that station and those lines. For each station so added,
there will be as many angle equations as new full lines, less one,
and as many side equations as new lines, less two. As has been
stated, small angles should be used in the side equations where
practicable, although it is best to use each but once. In angle
equations, on the contrary, they should be avoided.
For example, the equations for Fig. 22, page 96, will be written.
In this case, L=13, L'=12, S = S' = 7, and there are six angle
and two side equations. The complete horizon at F, moreover,
requires a local condition. Beginning with the line AB, station
F, with the two lines to A and B, forms a triangle with one angle
equation (A), as shown below\ Adding C with two lines to B and
F, gives one angle equation, (B), and similarly, adding D with two
lines to C and F gives (C). Now, with E are added three linos
to A, D, and F, so that two angle equations, (D) and (E), are
formed and one side equation (//), for the w^hole figure
A^B-C-D-E-F, with pole at F. With G are added two full lines
and one broken line, giving one angle equation, (F), for the tri-
angle G-A-E, and one side equation (7) which might well be
written for the quadrilateral G-A-F-E, with pole at G since the
line FG is broken at G. Thus we have six angle and two side
ef^uations as required by the formulas above. The local equa-
tion (G) for the station F must be added. To facilitate the
formation and solution of the normal equations, these eonditioii
eciuations are so arranged as to place the simpler ones first and the
more complex ones with larger coefficients, last. (See Art. 78,
page 71.) The angle equations, therefore, will usually precede
the side equations. For a central point figure, also, several angle
equations may be written in succession having no angles in com-
mon, with the result that many of the coefficients in the first normal
ADJUSTMENT OF TRIANGULATION 99
equations will be zero, thus materially reducing the labor of solu-
tion. The above conditions are arranged as follows:
Angle: (A) F2 +Fii + Fi9-(180°+6a) = 0
(B) Fi +74 +7i4-(180°+66) = 0
(C) 73 +76 +7i5-(180°+6c)=0
(D) 75 +79 +7iG-(180°+6,) = 0
(E) Vs +7i2 + 7i7-(180°+e.)=0
(F) V7 +7io+7i3-(180°+e;)=:0 (125)
Local: (G) 7i4 + 7i5 + 7i6 + 7i7 + 7i8 + 7i9-360° = 0
sin Vi sin 73 sin Vo sin 78 sin Vn .
Side: (//)
sin V2 sin 74 sin 76 sin 79 sin 7i2
/yx sin V7 sin 7]7 sin (7i2 + 7i3)_..
sin (77 + 73) sin 7i3 sin 7i8
Substituting for each 7 its M-{-v, the M's being the observed values
of the angles, and computing the spherical excesses, the angle and
local equations are thrown into their reduced form as in (105), and
the reduced side equations are formed as in (115) and (119),
respectivel\^ The formation of the nine normal equations and
the remainder of the solution then follow as in the last chapter.
99. Adjustment of a Quadrilateral: Method of Angles.
To illustrate the foregoing principles, the following (juadrilateral
Beckuith
FiK. 23. Atljustmciit l)y Metliod of Angles.
will iK)w be adjusted in full. The final angles arc desired to hun-
(lr(Hltlis of a s(>('()nd. Weights are ecpial. Seven-place logarithms
will be used ])iit witli the unit tak(>n in \hc sixth placi^ foi' the side
100 PRACTICAL LEAST SQUARES
equation. The pole is taken at the intersection of the diagonals,
as indicated. The given angles are shown in the three triangles
which will be used in succession for the angle equations.
Given Angles
(A)
Beckwith (3) 26° 42' 51 .8"
NorthBase (1) 64 43 42.3
North Base (2) 43 44 02.0
SouthBase (8) 44 49 27.4
03.5
€a-0.05" 5„=+3.45"
(B)
Walter (6) 28° 17' 12.9"
NorthBase (1) 64 43 42.3
SouthBase (7) 42 09 40.3
SouthBase (8) 44 49 27.4
02.9
e, = 0.06" qh=+2.M"
(C)
Waher (5) 48° 03' 10.3
Walter (6) 28 17 12.9
Beckwith (4) (il 29 53.9
SouthBase (7) 42 09 40.3
57.4
e, = 0.08" (7.- -2.68"
ADJUSTMENT OF TRIANGULATION
101
o
Tfl
1— 1
o^
^
c^
^^
Oi
1— (
1— (
-M
1-H
CO
(M
+
+
+
+
^
00
lO
C3
C5
^
CO
X
1
^?
^
X
ori
o
t^
C5
t^
-^
X
t^
1
■■O
cc
o
T-H
CO
CO
1
c
CI
CO
i-O
00
r^
t^
CO
'*
t^
'^
o
o
S)
00
Oi
o
00
CO
CO
hj
03
05
03
05
Oi
C5
b
o
o
-t<
'M
^/-v
Ol
t^
o
0^
Vi
^
C:
t^
Ci
"~ ^
-*
Ol
'^
CO
,_t
00
-V
■^
o
Ol
-*
01
-^
-
X
C5
~
o
CO
s
o
,_,
Ci
CO
-^
-v
T— 4
01
+
+
+
+
^~,
OD
■s>
o
^
-o
+
C3
CO
■— H
O
^
l-~
CO
o
t^
^/^
t-
-^
X
CO
— ■
o
oa
-o
t^
1^
'■O
t^
"M
3
tt
C2
o
00
X
CO
>^
Ci
Ci
o
C5
o
p— -
CO
CO
CO
CO
01
^
3
o
-t
'■•'
'"^
^
^^
X-«
'^1
*o
C5
^^
-t>
-r
o
o
01
01
-r
-r
-H
CO
.0
t^
lO
^
X
o
•*
X
o
X
3
CO
+
oq
+
1
X
1
!M
Ol
+
+
1
o
1
CO
CO
CO
CO
i"
T-H
^
(M
Tt^
+
+
+
+
O
o
.—1
+
+
T
7
T
5^
O
o
o
Oi
05
H
<
^_|
,_l
01
13
C7
+
+
+
■^
X
~-
't
•<+
f-
r— 1
1-H
c
-
^
^
c
X
+
1
1
L/
Ci
■— '
>— '
^
+
-j-
+
05
o5
-■'
^H
01
,_,
+
1
i
S
01
+
+
+
-h
102
PRACTICAL LEAST SQUARES
K
a
3
+ 6.86
+ 5.29
+ 5.18
+50.83
(Factors)
-2/4= -0.50
0
-2/3= -0.667
-0.86/4= -0.215
+3. 14/3 = +1.047
-1.27/2.67= -0.476
0)
-a
O
+ 10.31
+ 8.13
+ 2.50
+42.03
+ 8.13
- 5.16
a
M
+
+ 2.50
0
- 1.99
>
1—1
i-O
d
+
+42.03
- 2.22
+ 3.12
- 0.24
S
01
+
s
+3.45
+2.84
-2.68
-8.80
+2.84
-1.72
+
-2.68
0
-0.75
CO
1
-8.80
-0.74
+ 1.17
+ 1.63
T
Q
+ 0.86
- 2.71
- 0.82
+53.50
- 2.71
- 0.43
CO
1
- 0.82
0
+ 2.09
+
+53 . 50
- 0.18
- 3.29
- 0.60
+
o
O C<I -* (MO
+ + +
M
+
CO
'f O ^
+ 1
-M
+
C<1 K*H T^ 1-1
+ + + 1
CO
+
-1
+
ssss s 5 g is i
ADJUSTMENT OF TRI ANGULATION
103
Correlates
+6.74
C^+3 43-0.17^±3^6^^^^^^^
2.67
2.67
-1.12+-0.43-2.44 -3.13
B= ^ =^- = -1.043
A =
-3.45-0.12+0+2.09
1.48
= -0.370
Test.s of Correl.\tes
Eq. (4)
Eq. (3)
Eq. (2)
Eq. (1)
A
B
C
D
(q)
Sum
-0.32
+2.83
-1.00
+7.28
-8.80
0.00
-2.09
+4.88
-0.11
-2.68
-0.74
-4.17
+2.44
-0.37
+2.84
-1.48
-2.09
0.00
+0.12
+3.45
0.00
—0.01
0.00
0.00
104
PRACTICAL LEAST SQUARES
^
-0.370
-1.043
-0.288
I— (
oo
77
-
CC r^ !>•
•rt< (M ^
CfM CC
^ ,-H C
1 + +
oo
++
~
^^ ^H C^
^ -M CC
O M Lt
7+?
cc cc
oo
1 1
i^
^ oc
I— 1 O
++
o
1^ oc
++
2
1—1 i^
(M i-H
.-id
+ 1
CO
o o
o o
1—1 1—1
++
~
T +
So
oo
++
T T T i
-
cc c; r- 1 Ol M
t7 +77
1 =^^^"^ ic
COt^ lO '*
o
,-1 c^ CO ic
CO
1> o t^ ^
X
lO
O X O 1--
CO
o
o CO L':' X
CO Tf i^ -*<
l^
s
d
X oi cc X
CO
1
11
o
05 0 0 05
OS
00
^
00
1
oi
CO CO >-- o
CO o "-0 t>.
c
3
,-i' -+ -M LO
s
[&
O LO -^ C<l
CO
"bb
^
+
c
'U~ ^ ?5' -^ ^
Ttl
^ — ^
1—1
0
0
t— 1
3
55
O
CO -^ X'*
+
«
-* -^ C<1 -*
o
o o o t^
H
OCO CO
<
Od Or-i
II II II +
x;
H
c^ Tf ox
"-^
LO -<t X ^
^
o
■* 00 o oc
a
u
CO(M C^ (N
'3
cc
(i.
++ 1 +
oo Ot-i
^
'/:
t^ t^ -^ C^l
.s
^'
LO -* t^ C^
X
h
.
o
h
X
X !M O LO
(M
rfi
1— t T— I O T— *
or^ CO o
X
^
1 1 + 1
O O iC Tj-
!M Tf CO X
:/2
CO t^ ^ X
O OQ ^ o
O LO t^ !>1
OOX X
CO
t^
o
CO
d o d d
+ + 1 +
3
CiOiOiOi
d
t~-. O X I>
Lh
o CO -^ -^
a;
ddr-i,-,'
^
C^ O X o
1 1 + +
m
o o r- i^
X X o r-
^
^ C<1 ^ o
(M !M O O)
H
I I +
ADJUSTMENT OF TRI ANGULATION 105
Computation of Triangles
The ultimate test of the adjustment occurs in the computation
of the lengths of the lines or sides of the triangles. If an error
were made in the original side equation, such as an erroneous
logsine or difference for \", or an error in adding the angles of a
triangle to obtain its error of closure, q, all of the subsequent
operations might check, to and including the tests of corrections.
The test of the side equation, using the adjusted angles and the
new logsines, checks the original logsines and differences for 1".
It remains to be seen in the computation of triangles whether or
not the adjusted angles " fill " each of the four triangles and at
the same time satisfy the side equation by giving the same results
for lengths which are computed in two triangles. The above
discrepancy of one in the last place of logarithms in the side equa-
tion test, would show, also, in the triangle computations, but is too
small to warrant further investigation. It would be corrected
arbitrarily so as to leave no inconsistency in the computed results.
(An example of the final triangle computations will be given at
the close of the adjustment of this quadrilateral by the ^Method of
Directions, which follows.)
100. Use of Directions instead of Angles. In the measure-
ment of angles with a direction instrument, as in primary triangu-
lation, the various signals are sighted independently and for each
pointing the horizontal circle is read, in a clockwise direction.
This is done in the direct and reversed positions of the instrument
and in various positions of the circle, and the mean of all of the
readings upon a certain signal is adopted as the direction to that
signal. The angle between any two signals is the direction of the
right-hand signal minus that of the left-hand one, and there is no
local adjustment. Even though the separate angles be measured
by reading directions in pairs, or by the method of repetitions, the
directions may be numbered, instead of the angles, and each angle
d(^<ignated by the difference of the two directions which hniit it,
the right-hand one minus the left. In Fig. 24, for example, angle
BAG would be designated by the symbols -(l) + (2), and CAE
would be represented by -(2) + (4). EAB would be -(4) + (l),
106 PRACTICAL LEAST SQUARES
and so always minus the left plus the right-hand direction, clock-
wise.
Fig. 24. Directions.
This method has certain advantages, especially in the adjust-
ment of the more complex systems, which render its use very
desirable, and it is deservedly popular among computers. One
of its strongest features lies in the fact that preliminary local
adjustments are not disturbed by later adjustments in which the
method of directions is used, so that no local condition for an
interior station would be necessary in a case such as that in Art.
97 and Fig. 22. Each direction is regarded as observed inde-
pendently, and the unknowns of the problem are the corrections
to the separate directions. The correction to an angle, therefore,
would be the correction to the right-hand direction minus that of
the left one, algebraically. There will be more directions, in a
given system, than angles, but this is not a serious objection
when the Method of Correlates is used. (In the Method of
Indirect Observations, any increase in the number of unknowns
produces a like increase in the normal equations but in the
Method of Correlates the number of normal equations is equal
to that of the conditions.)
The weights of the directions will be equal, in the general case,
but different weights may be assigned if certain signals were more
difficult to observe than others, owing, perhaps, to unsteady
atmospheric conditions or poor illumination. If it be desired to
use directions in the adjustment of angles of different weights, care
should be taken in giving weights to the corresponding directions
that the weights of adjacent angles be not seriously affected. If
ADJUSTMENT OF TRIANGULATION 107
two adjacent directions were assigned small weights, and thereby
received large corrections, the intervening angle might receive a
small correction (the difference of the two large ones) and so defeat
the purpose of the computer. If two adjacent angles have small
weight, the intervening direction might be given a smaller weight
and thus affect both angles. However, if the angles have different
weights, rather than the directions, it may be best to adjust by
the ^Method of Angles explained above. In using directions,
therefore, we shall assume that angle weights are equal; if separate
directions have different weights, they may be treated exactly as
in the Method of Angles.
If directions be used in local adjustments, it is advisable to
use the ^lethod of Indirect Observations, as in Art. 82, since the
local conditions would be identities of the form,
-(1) +(2) -(2) + (3) -(3) +(4) -(4) +(1) -360° = 0 (126)
Therefore, it will usually be preferable to use the Methods of Angles
and Correlates, illustrated in Art. 79, for the local adjustment.
101. Notation: Method of Directions. In numl^cring the
directions of a figure, one side may be regarded as the initial line,
as if fixed Ijy a previous adjustment, perhaps, and its numbers
omitted. In this case, it is well to place letters on the fixed line,
instead of numbers, to distinguish its directions, when writing the
equations, these lettered directions not to enter into the reduced
conditions, and to I'eceive no corrections. On the other hand, this
use of letters is not necessary, and numbers ma}^ be placed upon
all of the directions, if desired, without altering the method or
increasing the work to any considerable extent. Directions are
to be numbered clock\vis(>, invariably, at each station, so as to
avoid errors. Unobserved directions, shown by broken lines,
will not b(> numlxTcd.
102. Lists of Directions. Pi'cparatory to the adjustment, a
list of the dircH'tions at each station may be made from the given
data. The names of the observed signals are arranged in clock-
wise order, usually beginning with a prominent one which is given
the initial direction, 0° OO' 00.0", although any direction may be
taken as the zero liiu\ If angles wvvv (jbserved. and adjusted
108 PRACTICAL LEAST SQUARES
locally, if necessary, the resulting angles are added in the proper
order so as to obtain the angle from the assumed initial direction
to each of the other signals or stations, which will be its direction
in the list. The angle from one station to another, clockwise,
will then be the direction of the latter minus that of the former.
If only two or three angles were observed at each station, it may
not be worth while to form these lists of directions, but each angle
may be given the proper designation as the difference between
two directions, and two angles added or subtracted, when neces-
sary, to obtain a third.
C s
8
D
^\^
/
\
/.
\
11
\'.
15/ 16.
fU
A ^7 13 G
Fig. 25. Adjustment by Method of Directions.
103. Statement of Conditions: Method of Directions. To
illustrate the use of directions, the condition equations (125) for
Fig. 22, page 96, will be restated. The figure is reproduced in
Fig. 25, with the directions numbered. The line AB will be
regarded as fixed and its directions will be indicated by the letters
a and b, merely for convenient reference. Comparison of the fol-
lowing equations with (125) will make the process evident. As
stated above, the local condition is unnecessary when the Method of
Directions is used, as the closure of the horizon at F will not be
disturbed by applying corrections to the directions, since each
direction is common to two angles and its correction must increase
one by the same amount as it decreases the other, leaving the sum
unchanged.
ADJUSTMENT OF TRI ANGULATION
109
(N
o
CO
1
O
00
1—1
T
o
GO
\ — 1
1
o
00
T-H
T
O
00
T— 1
1
o
00
1— 1
T
TO
00
C5
o
(M
+
+
+
+
+
+
>-
^
+ + + + +
+ +
+ + +
+ + + + + + ^
-f
h-^
>^
+
+
>-
+
+
+
+ +
+
-^ cq
fe.
no PRACTICAL LEAST SQUARES
To state the reduced conditions, the direction letters, a and 6,
are omitted and the angle equations are written by replacing each
V by its V, and — (180°+e) by the closure error, q. The side
equations are more complicated owing to the combined subscripts.
For example,
logsin(-a+Fi5)=logsin( + Fi5)=logsin {-\-Mi5)-\-d+i5Vi5 (128)
and
logsin ( - Fs + F4) - logsin ( - F4 + F5)
= logsin (-ikr3+iVf4)- logsin (-il/4+M5)
+ C?-3 + 4(-l'3+t'4)-rf-4+5(-i'4 + l'5)
= logsin (-ilf3+ilf4)- logsin (-.¥4+3/5) -f/-3+4y3
+ (f/-3 + 4 + rf-4+5)t'4 — r/-4 + 5?'5
in which fi-3+4 is the difference for 1" in the logsine of the angle
( — Ms+Ah), etc. Applying these principles to the equations
(127), we obtain the reduced equations in the following form:
(A) —V2^Vi5 — V22 + V23+qa = 0
(B) —Vl+V2 — Vi^Vo — V23 + Vl8^gt> = 0
(C) —V3-\-Vi — V7^V8-Vl8 + VlC)^qc = 0
(D) — t'G + f? — y 1 1 + t'l 2 — i'l9 + 1'20 + gd = 0
(E) —Vio-\-Vn—Vi5-\-ViQ — V20-\-V22-\-qe = 0
(F) -VQ-\-vio-vi3 + vi4-vi(i-\-Vi7 + qj = 0 (129)
(G) -f/- 1 + 22^1 + (f/-] +2 + ^-2)^2 -f/-3 + 4?^3 + («'/-3 + 4+(/-4 + 5)?'4
— f/_4+5r5— f/_0 + 7?'0 + (f/-G + 7 + ^/-7 + 8)?'7
— f^-T + S^S — *^- 10+1 l^'lf) + (''/- 10+ 11+^/- 11 + 12)^11
— d- 11+ 12i'l2 + {d+ 15 +<"/- 1 5+ 1 g) ?'l 5 — f/- 1 5+ 1 G^'l G +^7 = 0
(//) (, — d-0+ 10 + f/- 9+ 1 1) t'9 + f/- 9+ lO^'lO — <"/- 0+ 1 1 1'l 1
— d- 20 + 2 1 i'20 + (''''- 20+ 2 1 + ''Z- 2 1 + 22) t'2 1 — ^/- 2 1 + 22?'22
— (/- 1 5+ 17'-'15 + (''/- 1 5+ I 7 — c/- 1 G+ 1 7) t'l 7 +''''- 1 G - 1 7^'] G +(?/! = 0
Inspection of eciuations (127) shows that there are more
unknowns than in (125), and what is more important, that adjacent
aiiji'lc equations have unknowns in common so that l<>ss of iho
ADJUSTMENT OF Till ANGULATION
111
product-coefficients in the normal equations would be zero in this
case than in the other. The diagonal coefficients (squares) are
larger, also, owing to the greater number of unknowns. These
are disadvantages which may offset the omission of the local
condition, in a central-point figure, so that the Method of Angles
might actually involve less work than the Method of Directions,
in such a case. A rearrangement of the above equations, how-
ever, would simplify the normal equations, to some extent, by
collecting the zero coefficients near the beginning. The following
order might be used: {B), (L»), {F), (E), (C), (A), (H), (G).
104. Adjustment of a Quadrilateral: Method of Directions.
Beckuith
Walter
Fig. 2G. AdjiistmcMit of Quadrilateral; Method of Directions.
As an example of tlio use of directions, the (juadrilatcral of Art.
99 will be adjusted. A comparison of the two methods of solving
the same problem will b(! instructive. The* figin-e is shown in
Fig. 26 with the new notation. The number of angles being small,
it will not l)e necessary to write lists of directions, btit the separate
angles of the triangles will be represented by the proper directions,
instc^ad, and other angles may be obtained from them by addi-
tion or subtraction, the symbols being subjected to the same
operations. Thus, adding the two angles, — (3) + (4) and
-- (d )-!-(•")), w(^ ol)(a!n th(Mr sum as — (3)4-('"))- The pole for
the sid(» (Hjuation is tak(Mi at Sotith Base. The right-hand
angles happen to have Ixhmi written on the left side of the
('(juation, and vi('(> versa, which is equival(Mit to changing
all the signs in th(> (^luation without affcH'ting th(> i-esults.
The ('()nii)utati()n of the triangles is added in order to com-
plete tlie solut ioTi.
112 PRACTICAL LEAST SQUARES
(A)
Beckwith - (3) + (4) 26° 42' 51 . 8'
N. Base -(a) + (2) 108 27 44.3
S. Base -(10) + (6) 44 49 27.4
€„ = 0.05" ga=+3.45"
Walter -(7) + (8)
N. Base -(«) + (!)
S.Base -(9) + (6)
66 = 0.06"
(C)
Walter -(6) + (8) 76°
Beckwith -(4) + (5) 61
S.Base -(9) + (10) 42
e. = 0.08" qc=-2J
03.5
28° 17'
12.9'
64 43
42.3
86 59
07.7
02.9
g,= +2.84"
20'
23.2'
29
53.9
09
40.3
57.4
ADJUSTMENT OF TRIANGULATION
113
00
Oi
1—1
^
,_,
05
lO
-TS
+
+
+
1
>o
OT
1— (
lO CO
o
a>
00
05 00
■■ — ^
J>
o
CO
1— 1 1—1
1
ta
t^
CO
iC
O CO
(N
CO
t^
CO CO
kO
lO
00
03 C5
11
,S
<r>
05
03
lO >o
W)
ai
d
03
t^ l>
§
:i
1 +
00
00
(M
,_!
(M
CO
lO
Tt<
c^
.^— s
O
(M
CO
o
bC
TtH
^
(N
a
Q
<^
o
Tt<
CO
-C5
c^
o
t^
c3
^
A
^— V
,. V
.. s
t(— >
^
y—i
00
q;
v£^
+
+
+
'ro'
e
CO
1
1
1
lO
--;
I— 1
^
,^
r^
Oi
1-H
CO
+
1
+
+
L'J
o
1 CO
r-^
c^
LQ
X
CI
L-O
t^
30
^
CO
CO
1^
LQ
CO
w
-t^
t^
t^
C3
c
Ci
CI
CO
lO
do
Ci
C2
O
t^
h3
+
^
C3
05
iQ
^
^H
X
'^
C^
t^
t^
!:X
-M
C^J
<;
, ,
X
X
— *
w
; ;
— '
J^
1
St
1 \r-^
■M
X
■ ""
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~^
+
+
+
1 "*
^
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1
'
1
0^
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CC
o
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«
o
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H
, ^
<!
r;^
&
^
a
y
>^
o
t^
a
^
n
fl
o
s
+3.45
+2.84
-2.68
-0.90
o
t— 1 T— 1
1 + ^
^
1 1 1
^
o o
^ --H CO lO
+ + + +
_r-
1— 1 ,— 1
Oti C5
r-l CO •'fl
1 1 1
~
1— 1 O)
.-H O O
1 + 1
i'
LO lO
1-1 ^ Ol
+ + +
^
+ 7 '? 'T
_ro
X X
1 + +
~
t~ ?0
+ T +
~
CI — '
C3 c:
+ T +
■ '? ^ i ? s
114
PRACTICAL LEAST SQUARES
s
- 8,21
+ 12,32
+ 15.37
+81.26
(Factors)
0
+2/4
-2/4
+ 10.21/4
- 6.32/4
- 1.11/4
y
U
- 4.76
+ 15.16
+ 12,69
+80,36
CO
+ ^
CO
lO
+
C5 X 00
CO CO »o
CM CM t^
+ 1 1
CM
+
+80.36
-12.15
-23.95
- 0.75
>
+
-
lO ^ X o
TfH X CO O
CO CM CM O
+ + 1 1
X
CM O
+
X
CM
+
-2.68
+ 1.72
- 1 , 42
X
Ol
1
-0,90
+8,81
-4,49
+0,66
X
c
+
^
-10.21
+ 6,32
+ 9.37
+75.78
05
+ ^
C^l
CO
CO
+
+ 9.37
- 5.10
- 3,16
+
+75.78
-26.06
- 9,99
- 0,30
CO
CO
+
01 M -o;
1 + +
Ol 3
+
+
+ 77
+
+
+ ^
+
+
! ? S S S
?I '?
1 '" ' I "^ 1 '
^
ADJUSTMENT OF TRI ANGULATION
Correlates
-4.08
115
-1.244+0.654-2.84
B = : = -0.858
^ 0+1.244-1.056-3.45
A= : =-0.815
Te.sts ov Correi.ati:.-;
Eq. (I)
Eq. (2)
Eq. (3)
Eq. (4)
+8.32
-5.42
+5.83
-7.83
-0.90
A
B
C
D
(q)
-3.26
0
-1.24
+ 1.06
+3.45
0
-3.43
+ 1.24
-0.66
+2.84
+ 1.63
-1.72
+3.73
-0.96
-2.68
+0.01
-0.01
0
0
116
PRACTICAL LEAST SQUARES
c o
+ +
00 (M
00 CO
o o
+ I
OO C^ <M
iC C<I "O
00 CO CO
o c o
I + I
o
+
oo
oo
I I
CO
+ 1
o do
+ I +
+
++
o o
I +
CO
CO-*
o c
++
CO
I I
++
o c
I I
o o
++
o c
++
o t^ t^
o o
I I
o o
I I
d
I
I I
5 o o
O »-, ^: ^ o
P o o 't CO
R CO -HO
3+1 + 1
o II II
-r CO 'M O C
^ ++ I +
+
'^
c; X cj
o
1
?+ +
00 CO o c^
CO 00 Lo LI
c
1
— CO
1 ++
ci X, lO
coco
1 1 ++
>
O 05 x
CD O CO
t^co >o
(M CO l>
lO >^ X
CO 03 c
05 Ol 03 I t^
CO i^ ^
■* -* <N
o ■* ^
C<) CO «>
2 +++
<; CO c3 CO
S M T
5- ,_! t^ LO
- CO "M 1^
. CI LO CO
^ X O CO
J- CO t-- LO
H c c CO
C5 C5 O I t^
+++
TTT
ADJUSTMENT OF TRI ANGULATION
117
^
^
^
+
+
I
Tt< 00 t^ C5
.-105
Tt^ ^ ^ O
'^^ C3
aa^.T^t-
1.0 05
•* '^ X lO
/*r« —->
t^ CO (M lO
t^ CO
t^c^ C5 00
oc 00
CO rt CO :o
C^ X
X ^ C<l I^
t>- c^
O CO iC-^
CO ■<t<
ic CO c o
X Ci
^ CO o :o
t^ C3
X i-it^ CO
CO t^
Tf CMO^
xr^
■^ CO CO CO
t^^ X X
cc
—< -N cc Tf
X o
■<* r^ t^ 00
O 00
rt< ^ vC cr.
X LO
X c^ coo
X o
X o o --^
Ci X
•^ 't 1^ tT
CO o
^ C^l i-O C2
o c^l
:c — -*C^
O-M
OOCOt^
LO CO o ao
1> 00
LO CO c; CI
X X
X oox
t^ X
XC X X
t^ t^
coo c;c5
coco
CO c C-. o
CO CO
CO O O d
CO CO
CO O O C5
coco
C C CO "O
>-0 Tf (M
+
X CO
-* o
CO o
o
• T-H Tfl lO
X X
CO CO
+
o o
C; o
CO :C ^
O X !M
X C5 (M
lO 1— I CM
^^ = -
CM (M -jti CO
■ ^ ^ o
t^ X if rf
CO c;
^ c ^
I I I
X
CO
-r
I^
-r 01
*-)
01
2
01
X
■4
I I I
CM CO CO X
CM CO CM t^
+ + +
+ + +
l^O CO
1.0 CM O
X
I -
y.'
1 I I
V. X —
I J
/< :C X: x £ X I X; ::; X: xJ ?^ x i x ?^ — x ::^ xJ i x. ::c X. ;
+ + +
X 10 --
++ +
I ! 1
1.0 01 I^
+++
118 PRACTICAL LEAST SQUARES
In this standard form of computation of the triangle sides, the
given side is written first, followed by the opposite station and the
other two in clockwise order around the triangle. The correc-
tions are applied to the given angles to obtain the adopted (spher-
ical) ones, from which the spherical excesses are deducted and the
plane angles (to be used in the logarithmic computation) are found.
The sum of these plane angles, of course, should be exactly 180°.
The cologsine of the first angle is written below the log distance,
followed by the logsines of the other two angles. Covering the
fourth logarithm with a pencil or strip of paper, the first three are
added to obtain the sixth, and the fifth is then obtained as the sum
of the first, second, and fourth, by covering the third. In order
that the computed lengths may be consistent throughout, a certain
value is adopted for each distance and logarithm, and the neces-
sary modifications are made by the application of small correc-
tions as shown. It is a good plan to arrange the triangles in the
above form before beginning the adjustment of the figure. Then
the symbols and the observed angles (after local adjustment, if
any) are in convenient form for use, together with the spherical
excesses. After the adjustment, the corrections are inserted and
the form completed.
105. Adjustment of a Quadrilateral: Approximate Method.^
The angles of a quadrilateral may be made to satisfy the angle
equations exactly and the side equation very nearly, by an approx-
imate adjustment which, although not rigorous, may be sufficient
for subordinate triangulation or detached figures in which great
precision is not required. The weights of the angles are assumed
to be equal.
The two triangles formed by one diagonal are first closed by
correcting the four angles of each b}^ one-fourth of the closure-error
for the triangle. One of the other triangles is then closed by cor-
recting each of its four new angles by one-fourth of its closure-
error, which correction is also applied to the remaining four angles
(of the fourth triangle), with the opposite sign, so that all four
triangles are thus satisficnl exactly. Taking the pole for the side
1 Due to Prof. T. W. Wright.
ADJUSTMENT OF TRIANGULATION 119
equation at the intersection of the diagonals, each of the eight new
angles is corrected by one-eighth of the error of closure of the log-
sines divided by the algebraic mean of the eight differences for
1", the angles on the right being corrected with the opposite sign
to those on the left, so as to bring the sums of their logsines closer
together. If the eight angles were equal, the side equation, also,
would be exactly satisfied by this method ; in this case the figure
would be a square.
For example, let us adjust the quadrilateral in Fig. 23, page 99,
with the data and notation there given. (See next page.)
106. Adjustment to Conform to Work Previously Adjusted or
Fixed. Triangulation of a subordinate character is frequently
carried on in connection with a main scheme or net in order that a
number of points may be located from the main stations without
reoccupying them expressly for this purpose. In primary tri-
angulation, for instance, it is customary to read directions from the
stations upon prominent objects such as church-spires, which may
be used later by local surveyors for obtaining initial positions and
azimuths. Such points do not enter into the adjustment of the
main figures but are adjusted subsequently and usually separately,
upon the previously adjusted work as a basis. Also, secondary
or tertiary figures may be connected to or based upon primary
ones so as to require separate adjustment which will not disturb
the previous work. If the connection be to one fixed line^ only,
that line would be used as a base-line, and no condition woukl 1)0
introduced. But if a triangle be fixed, or two sides and the in-
cluded angle, the new conditions must be so written as not to dis-
turb the previous adjustment. The Alethod of Directions is
particularly convenient when fixed lines are involved, as the
dii'cctions may l)e omitted from those lines and they will not be
affected by the adjustment. The angles are assumed to have been
adjusted locally, in advance. The use of letters on the fixed lines,
instead of numbers, serves to identify them without giving them
the character of unknown directions, although an (>xp(M'i(>nced
computer usually omits the letters as well as the nunil)ers on the
fixed lines. If th(> Method of An<z;k>s were used, local conditions
120
PRACTICAL LEAST SQUARES
!3
a-
^
^
V
^
>
"3
^ O 05 C^
00
CD 05 OOC^
lO
c
GO 02 lo r^
o
>o ^ CO 05
o
fa
Tt< ^ (N o
d
lO <— 1 >— 1 >-H
o
lO --H --H ^
o
(M Tt< O lO
o
fl G>
^
TtH o -H ro
rH Ol 05 rO
1-1 05(M .-H
00 LO
r- CO
1 «t
'T*
T-l rH CO C<)
(M <M ^
+ + + +
+ + + +
- ^S
W
m o
a> 4J
b£
O (M
05 (N
CO CO
gSg
CO lO
lO (M
CDO
C5 l>
^ t-
00 1^
^ f^
^ ^
oo o
i-H CO
CO CO
> O
,
CO lO
00 Oi
t^
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1
TfH t^
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o
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■^-^
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00 00
CO
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t--co
ooo
CO --<
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CO
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o
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CO CO
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t^
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CO t^
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+
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CD (M
r^
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t^ (N
iO to
o
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00 00
05 CD
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C3 Ci
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^
>
>
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GO
CO (M >0 >0
lO
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d
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t^ Ol Ol r-H
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C2 CO -f 01
o ^
^
Ol o ^ o
lO
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C3
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^
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(^
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00
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u
o
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CO --< CM CO
c3
S
>
rtHcD t^i-H
1-H (N CO lO
t^ 05 1>- ■<#
CD 00 CD I— I
O CO lO 00
CO ■* t^ TfH
CO Ol CD 00
o
CO
05 05 C3 05
Ol
00 1— 1 02 CD
CO GO lO lO
i-i •* CM lO
O lO 1-H CM
Tf O^ t^ 05
■* CM .-HTt<
CO i-H 00 ■*
^ CD C<1 Tt<
o
^^S~oo
(n 1"^ T-H o O
H 00 CM t^ lO
L^ C t^ CO CD
f^ CO t^ Tt< 00
CD CM -H CD
iO »0 t^ CM
02 CD 00 00
<y> o^ c^ a
25 <M CD CM
""* OS 03 1^
T-H CO "Q l>.
ADJUSTMENT OF TRIANGULATION 121
would have to be added. The following simple cases will be
considered. From the condition equations the solution proceeds
in the usual manner.
107. Two Sides and the Included Angle Fixed. Fig. 27. The
adjacent sides, A and B, are fixed in length and the angle between
them, also, must not be altered. If the missing diagonal had been
Fig. 27. Two Sides and Included Angle Fixed
observed, it would have to be considered as fixed, since the sides,
A and B, and the included angle, determine the triangle completely.
That case will be discussed later. The only new lines, then, are
the three w^hich run to C, forming the two triangles. According
to our rules, there are two angle equations, one for each triangle,
and no side equation. However, the fact that two lines are fixed
in length renders a condition necessary, which shall require the
angles to be so adjusted that when one fixed line is computed
from the other, the result will l^e equal to its known lengtli.
This condition is called a length equation. It has the same natun^
as a side equation, but involves two known lengths. The three
('([uations are:
(.4) -(6) + Fi-F2 + (r/)-F5-fFr,-(180°+eJ = ()
(/i) -ri + (r)-(a) + T^3-74+F5-(180°+e,)=0 (1:^0)
sin i-a-^V-,) sin (-F5 + F6) _
^^ sin (-r4+r5) sin(-r2+r/)
II is ol)vi()us that the angle (—a+ !':{), for cxainplc, may l)e desig-
122
PRACTICAL LEAST SQUARES
nated by (+F3) since there is to be no correction to the direction
(a). The equations (130) may therefore be written,
(A) +7i-F2-F5 + F6-(180°+6a) = 0
(B) -Fi + F3-F4+F5-(180°+6,)=0
Asin(+F3)sin(-F5 + F6)
(131)
(C)
5 sin (-F4 + F5) sin (-F2)
= 1.
To obtain the error of closure, q, for the length equation, the log-
arithm of the length A must be added and that of B subtracted, in
the series of logsines. As these lengths are fixed, they do not
appear in the reduced conditions, which have the form,
(A) +fl— ^'2 — t'S + t'O+^a^O
(B) -vi + V3-V4 + V5+qt. = 0 (132)
(C) d-2V2-\-d+3V3-\-d-4 + 5V4:—(d-4^ + 5-hfJ-5 + Q)V5 + d-5 + 6VQ+qc = 0
108. Quadrilateral with One Fixed Triangle. Fig. 28. The
quadrilateral being complete would have one side and three angle
equations. The angle equation for the fixed triangle is satisfied
in advance, however, so that there remain but two angle equations,
and the side equation as independent conditions. Using the tri-
Fic. 2S. One Triangle Fixed
angles D-A-B and D-B (', and placing the pok^ at D for tiie side
o(iuation, we write the three conditions:
ADJUSTMENT OF TRI ANGULATION
123
{A) -(a) + 7i-y2+(^/)-F4+F5-(180°+6a)=0
{B) -(c) + F2-F3 + (/)-F5-^Fg-(180°+6,)=0
sin ( -g+Fi) sin (-C+F2) sin (-Fs+e) _
(133)
sin {-h + Vx) sin (-F2+^) sin (-F3+/)
After obtaining the constants, q, the lettered directions, a, h, c, d,
e, and/, would be omitted, as usual, in forming the reduced equa-
tions, although it is convenient to use them in the subscripts of
the side equation to distinguish between those angles which differ
only by the fixed angle at A, B, or C. Thus the reduced side
equation would be,
(C) {(l-a+l—d-b+\)Vi-\-{d-c+2+d-2 + d)v2
-{d-s+e-d-3+f)v3-\-qc = 0 (134)
109. Fixed Triangle or Polygon with Central Point Unoccupied.
Figs. 29 and 30. In this case there are no new triangles and,
therefore, no angle equations whatever. The pole for the single
side equation is placed at the central, unoccupied (or concluded^
station. If this station be outside of the triangle, the side equation
would be the same as (C) in (133) and (134), above, but if it be
inside the figure, the side equation has a characteristic synunetrj'
Fk;. 29. Fic. 30.
Fixed Tri;uiRl(^ or Polyfjon with Concluded Station
in that every numl)er('d dii'CM'tion occiu's in botli numerator and
denominator, and the algel)raic signs are positive in th(> ntmierator
and negative in the (kMiominator, or vice vei'sa. Thus, foi- I'^ig. 29.
tlie side (H^uation would l)e as follows, omitting the lettered direc-
tions which ai-e mmecessarv in this very simple case,
sin ( + ]',) sin ( + ]\') sin i+V:,]
sin f— T'l) sin (— F2) sin (— ]':i)
= 1
(135)
124 PRACTICAL LEAST SQUARES
Similarly, for Fig. ,30, the side equation would be,
sin ( + 7i) sin ( + F2) sin ( + F3) sin ( + 74) sin ( + "^'5)_
sin ( - Fi) sin ( - V2) sin ( - F3) sin ( - V^) ^In ( - F5) ^ '
This case, especially Fig. 29, occurs so frequently in the loca-
tion of subordinate stations that it may well receive special atten-
tion here. In its reduced form, equation (135) may be written,
(d+i+d-i>i + ((i+2+C?-2)y2 + (^+3+C^-3)y3 + g-0 (137)
in which rf+i is the difference for l'' in the logsine of angle {-\-M\),
(l-i is that difference for angle { — Mi), etc. This equation has
the same form as (86) of Art. 80, page 73, so that ai = (d+i-\-d-i),
a2 = {d\.2-\-d-2), etc. Assuming equal weights, which will usuall}^
be the case, the correlate for the single equation will be, from (87),
A = ~ (138)
[aa\
and the corrections will follow from (88),
— Q — Q
vi = aiA = ai-f — r; V2 = a2A = a2-r — r; etc. (139)
[aa] [aa]
It is easy, then, to arrange the logsines in positive and negative
columns, and to take their algebraic sum as q. The algebraic sum
of the differences for 1" corresponding to the directions (those in
the negative column having their signs changed) will be the a's,
and the sum of the squares of these a's is the denominator of the
factor, —q/[aa], in (139). Each v is computed by multiplying its
a into this factor.
For example, let us adjust the following observed angles for
Fig. 29, the weights being (>(|ual. Each angle is followed l:)y its
logsine and the difference for 1" in that logsine, the left-hand
angles, in the left-hand colunm, being considered })ositivc. The
sphericid excess for the fixed triangle is 0.30".
ADJUSTMENT OF TRIANGULATION
125
+ + +
00
6b
I— 1
1— 1
Cl
a>
o
o
C5
o
+
^ ^
w
c
o
o
-f
o^
Ol
-S
t^
o
o
GO
-1-J
^e ^"
CO
iC
<M
72
lO
^
-*
O
.s
ll
s
o
^
T— 1
^
^.
Ci
t^
+ + +
+ + +
o
00
or
oa
TO
^
r/^
+
+
+
i—<
CD
^/^
o
L-O
-^
+ + +
~
iC
~
-f
'T
^
§
^
r^
.— •
QC
-^
X
l-H
^-~.
—
I-
1-
^
+
C:
C:"
c^
a-.
+ + +
+ + +
126
PRACTICAL LEAST SQUARES
00
+
IN
+
cS
+
+
:0
o o O
+ + +
^ ±
O
O
+
o
o
o
+
t^ t^ o
+ + +
o
o
t^ ^ t^
I I I
>
>
CM
r^ C:
+
+ +
ADJUSTMENT OF TRIANGULATION 127
The side equation is satisfied, since the sums of the positive
and negative logsines are equal, namely, 9.0918108. Also, the
sum of all the angles remains unchanged, and each of the three
angles of the fixed triangle is the same as before the adjust-
ment, since each correction was applied both positively and
negatively. An ordinary slide-rule is sufficient for the arith-
metical work, and after the sum of the aa's is obtained, each v is
found at one setting of the rule. The above illustration of the
process is given in greater detail than is necessary when the
method is understood.
110. Adjustment of a System between Points of Control.
Large systems of triangulation, such as the primary work of the
U. S. Coast and Geodetic Survey, may extend over strips of coun-
try for hundreds of miles. In such great distances, errors of
various kinds are likely to have a cumulative effect which becomes
too great to be tolerated. It is necessary, therefore, to control,
or check, the triangulation at intervals which will depend upon
the precision of the observations, the points of control being
farthest apart in first-class, or primary systems. The lengths
are controlled by measured base-lines, the positions, by astro-
nomical observations for azimuth, latitude, and longitude, and
the elevations, by precise spirit leveling, although the astro-
nomical observations may really control all three elements
of a system, that is, its size, shape, and position. (See Art. 85,
page 80.)
In general, the controlling points for these different purjwscs
will not be coincident. The observations for azimuth may not
be made at the same stations as those for latitude or longitude,
Of at the base-line stations. To illustrate the character of the
coiiti'ol, how('\'('i-. it will 1)(> assuin(>(l foi- exainpl(> that a given sys-
tem, or net, starts at a certain line, ,1/i. Fig. 31, whose length
and azinuUh ar(^ known as w(>ll a.s the latitude, longitude, and ele-
^■ati()n of on(> of its (muIs, and that it extends to anotlier line, CD,
which is fixed in the sain(> maniuM', in length, dii'ection, ])osition,
and elevation. This lin(\ CI), may hav(> \)vvn i'lxvd by original
o1)S(M-vations and nieasurcMuents, as if it were a detached or isolatcnl
line, or it nia\- be a line in a previously adjusted triangulation
128
PRACTICAL LEAST SQUARES
system which is so precise or so strong that it is not subject to
modification by the subsequent work.
Fig. 31. Triangulation System with Control
If the separate elemental figures, such as quadrilaterals, are
adjusted in advance, with local, side, and angle equations, and
the lengths and positions are computed from the initial side, AB,
through the system, the final line might fall at CD' instead of CD.
If, then, all the lines of the sj'stem were flexible except AB, and
CD' were picked up and forced to coincide with CD, it is easily
seen that all of the lines and angles would probably be distorted.
The adjustment, therefore, affects all of the angles in the net.
The Base-line, or Leyigth, Equation provides that the length of CD',
computed from AB, shall be equal to the fixed length, CD. This
condition is similar to the length equation (C) of Art. 107, page 121,
but must extend through the whole net. The Azimuth Equation
re(}uircs that CD' shall be parallel to CD. The Latitude Equation
states that the latitude of a point such as C shall be equal to the
fixed latitude of the corresponding point, C, and the Longitude
Equation expresses the same requirement for their longitudes.
It is evident that these conditions are independent — that any
one or more of them could ])e satisfied without forcing the others
to be fulfilled. For example, the line CD' might have the same
l(>n^th as CD, and C might coincide with C, and still the azimuths
mi.^lit be diff(>rent. Of course, the amount of the discrepancy is
exaggerated in the figure.
In a I'igid adjustment, all of these conditions woukl be com-
bined with the local, side, and angle conditions and satisfied sinml-
taiieously. It is usually sufficient, and much more convenient,
ADJUSTMENT OF TRIANGULATION 129
however, to perform the figure adjustments and then modify
them so as to effect the closure upon the controlling hne through
the above four conditions. In primary triangulation of the highest
grade, the rigid, complete adjustment may be required.
When an extensive system contains several points of control,
such as base-lines or astronomical stations, it is customary to
regard the net as subdivided at these points into sections, and to
adjust each section independently. This method has practical
advantages which outweigh its divergence from the ideal adjust-
ment of an entire system as a single problem.^
The special case sometimes occurs in which a detached net,
having its own base-line, but only approximate astronomical
position, is connected, after its figure adjustment, to a fixed
system through a single figure, such as a quadrilateral or central-
point figure. If the discrepancy in length between the two sys-
tems be small, it may be thrown entirely into the intervening figure,
which would have, therefore, two fines fixed in length, as shown in
Figs. 32 and 33. In addition to the usual angle and side equa-
tions, the figure would have a length equation such as (C) of Art.
107, page 121. It is assumed that the geographic positions for the
detached system are to be obtained, through this connection, from
the fixed one.-
Fig. 32. I.oii'rth iMiuation
1 For a thorougli treatment of the adjustment of large systems, and for
sperial methods apphcable to trianguhition in general, see Wright and Hay-
ford's Adjustment of Observations; Sjiccial Publication No. L'S of the V . S.
Coast and (leodetic Survey, by O. S. Adams; and .Jordan's \"eiinessungs-
kunde, Band L
-A method for the adjustment of triangulation by eorrectiiig tlie ])relim-
inary latitudes and longitudes of the stations is presented by Mr. .\darns ni
Special l'ul)lication Xo. 28 of the U. S. Coast and Ceodctic Su'Acy.
130 PRACTICAL LEAST SQUARES
111. Adjustment of Trigonometric Leveling. The adjust-
ment of the horizontal angles in triangulation is generally inde-
pendent of the vertical angles, which will be used to compute the
difference of elevation between the various stations. Although
Fig. 33. Length Equation
these vertical angles may be adjusted directly, it is usually easier
and at the same time satisfactory, to adjust the computed differ-
ences of elevation, and this is done by the method illustrated in
Art. 77, page 64. A long net may be divided into sections to facili-
tate the adjustment, and if a control point becomes available in
the form of a station whose elevation has been determined directly
through a line of precise levels, the entire net intervening between
the initial elevation and this final one may be adjusted to conform
to this total difference of elevation by a slight modification of the
partial adjustments without disturbing their conditions, as the
proportionate discrepancies will be very small in carefully executed
work.
112. Base-lines. The measurement of a base-line is carried
out in sections, and the total length is the sum of the sections. It is
customary to measure each section two or more times, in both
directions and under different conditions. The separate measures
of a section are then adjusted as direct observations, by taking
their mean or weighted mean. It is seldom that weights are
required, however, since additional measures are made if there is
too much discrepancy between vhe first two, and doubtful results
are subject to rejection in the field.
CHAPTER VII
EMPIRICAL FORMULAS
113. Empirical Formulas. Experimental investigations fre-
quently comprise the determination of the values of a certain
function corresponding to known, assigned, or observed values of
its independent variable. It is often desirable to express the
relation thus determined, between the function and the variable,
in the form of an equation. Should the observations be the same
in number as the unknown constants or coefficients of the equation,
a rigid solution of the problem would result, as explained in Art. 22-
But, as it is customary to make a larger number of observations in
order to obtain increased precision, the problem becomes one of
determining the equation which will best represent the entire
group of observations, thus involving an adjustment by the
Method of Least Squares. Such an expression, depending upon
experimental data, is known as an Empirical Formula.
114. Their Uses. Empirical formulas are sometimes called
interpolation formulas from the fact that one of their principal
uses is to facilitate the interpolation of values of the function
among the observations. The curve which represents the formula
is smooth and continuous and avoids the disci'epancies among the
various observations, so that interpolation is usually safe and
reasonable. However, th(M-e is generally a teridency to use the
formula beyond th(^ limits of the obsei-vations, that is, to extra-
polate along an extension of the curve. While this yields, in
many cases, very useful and inteivsting results, care must l)e taken
that such results l)e Tiot considered trustworthy except within
reasonable limits.
Sometimes it seems impossibU^ to deiive a th(H)i-(Mical relation
b(^twe(Mi two variables, while it is c^vident from the observations
that some conncH'tion does c^xisl. Here the em{)iri('al formula
may Ix^ the only oiu* avail;\bl(\
It is not essential that the inflation expi-(^ss(Hl by the function
131
132 PRACTICAL LEAST SQUARES
have any foundation in theory. It may be purely accidental, as
is the case in many statistical investigations. A formula may be
stated between the death-rate of a city and the time or season,
or between the depth of a pond and the distance from the shore.
However, the existence of a close relationship, such as cause and
effect, is sometimes indicated by an empirical formula, resulting
in the subsequent development of the rigid formula by theoretical
analysis. In this manner some well-known laws have been dis-
covered.
115. Nature of the Problem. Equations may be partially or
wholly empirical. For example, the form of the expression may be
developed theoretically and regarded as known, leaving only the
numerical constants and coefficients to be obtained empirically.
Or, nothing whatever may be known concerning the formula, in
which case it is necessary to assume a form for the equation and
then determine the constants by an adjustment of the observa-
tions. In any event, the problem is, to ascertain those constants
which will make the given expression, whether of previously known
or assumed form, represent the observations as nearly as possible.
Should there be uncertainty as between different forms which
could be assumed, or should the residuals resulting from a solution
be unsatisfactorily large, one or more other forms may be assumed
and the constants be determined for each of them, that one being
finally adopted for which the sum of the squares of the residuals
is the least.
116. The Form of the Equation may be known from theoretical
considerations, as when it is a special case of a group of expressions
the nature of which is known. But in the great majority of cases,
it must be obtained from the observations themselves. This is
conveniently done by plotting them as rectangular coordinates,
representing the values of the function, y, as ordinates, and those
of the independent variable, x, as abscissas, each point thus plotted
corresponding to one observation. A smooth curve is then
sketched so as to follow the plotted points as nearly as prac-
ticable. An inspection of this curve will generally throw it into
one of throe classes, namel.y: (1) a portion of a conic section,
such as a straight line or a parabola; (2) a periodic or wave-like
EMPIRICAL FORMULAS 133
curve; or (3) a curve which is non-hnear with respect to the
unknown coefficients, that is, one which involves their products,
squares, or higher powers, or functions.^
To assist in the selection of a suitable form, a number of curves,
with their equations, are shown in Appendix E. Apparent prop-
erties of the desired curve should be carefully noted, as positions
of axes, asymptotes, points of inflection, points of crossing of
axes, maxima and minima, regular or irregular periodicity, etc.,
so that the equation selected may be capable of representing these
features. In general, however, it is convenient to utilize an expres-
sion in the form of a series which can embrace all the curves in a
certain group. This is particularly useful in the first two of the
above classes of curves, and will now be illustrated.
117. Straight Lines and Parabolic Arcs. The simpler curves
vary from the straight line, through the forms which appear uni-
formly curved, to those in which the sharpness of curvature
increases or decreases continuously in one direction. It is possible
to represent any of these by a series of the form,
y = a-\-bx + cx^-\-dx^ + ex-^-\- . . . (140)
The character of the curve will determine the number of terms to
be used in this equation. Thus, if a straight line be desired,
the first two terms would suffice, giving,
y = a^bx (141)
If the curvature is slight, or if the curve straightens towards one
end, the parabolic form may be assumed,
y = a + hx^cx^ (142)
Oi-, if it be desired to represent the ])l()tted points still nioic closeh',
one or more terms may ho added, the principle Ixniig that the
greater the number of terms used, the more nearly will the ivsult-
ing formula fit the observations. If an unnect^ssarily large num-
l)cr (jf terms is used, the coefficients of those which miglit be omitted
' It must be remembered that in the derivation of empirical formulas, the
variables, .r ;ind //, are not the unknowns as they are in the Adjustment of
Indirect ()l)servat ions. Chapter III. Here, the variables are the observed
((Uantities and the eoeffieients are the imknowns which are to be determined.
As will appear later, the methods of solution are analogous.
134 PRACTICAL LEAST SQUARES
will come out quite small or negligible, and a re-solution with the
simpler form may be advisable.
It should be noted that the straight line is a special case, and
that although the plotted points seem to lie very close to such a
line it is usually best to use the formula for a parabola and obtain
a curve which approximates closely to the straight line. This
parabolic form is of very general application for empirical formulas
because of its convenience and adaptability.
118. Periodic Functions. If the curve is composed of similar
elements which repeat themselves as x increases, the function is
evidently periodic, that is, the values of y corresponding to increas-
ing values of x will pass through similar cycles or periods. The
curve in many cases will have a wave-like form, and it may be
simple or very complex. The general formula to be used is a
Fourier series,
. 360° 360°
y = a-^o sm x-\-c cos x
m m
, . 360° 360° , , ,
-\-d sm 2a; + e cos 2.r+ . . . (143)
m m
in which a, h, c, etc., are the constants to be determined.^ By
using a sufficient number of terms, this equation may represent
any curve whatever, for finite values of the variables, but in
the case of periodic functions it is particularly useful. If the ele-
mentary parts of the curve are alike and not complicated, the
first three terms will be sufficient; otherwise, succeeding pairs
of terms should be added, involving the nmltiples of x. Unless a
complex formula is expected, it is well to sketch each wave in the
curv(^ so it will !)(> symmetrical about its middle ordinate. If the
total angk^ corix'sj^onding to a cycle should be LS0° instead of
360°, this nuinh(M- should l)e substitut(>(l for the latter in the for-
mula.
Th(^ constant (iuantit>-, ni, is the nuinb(>r of units of x in one
cycl(> or period, and is assumed from an inspection of the curve
and the observations. For example, sui)pose the brightness of a
' For ail interesting ai)i)li('ati()n of harmonic analysis to this jiroblem, see
Brunt's Combination of Observations, Chai)ter XI
EMPIRICAL FORMULAS 135
variable star to be observed from day to day, and when plotted as
a function of the time to seem to have a period of about nine days.
Here, x would be the number of days elapsed since an assumed
epoch (such as the date of the first observation) and m would be
assumed as 9. Thus, x/m is an abstract number; 360°a:/m is a
number of degrees; and 360 °/m is a constant coefficient of x in
any single problem. Different values of m may be assumed, and
the problem solved for each, if deemed worth while, that one being
adopted for which the sum of the squares of the residuals is the
least. In determining the period, m, from the curve, it is well to
measure it at several places, if possible, and take the mean.
AVhen the empirical formula of this periodic type has been
determined, it may be transformed into a more convenient expres-
sion in the following manner: Let hq, ni, n^, Ni, N-z, etc., be
auxiliary quantities determined from the assumptions,
no = a ; /; i sin A'l = 6 ; ^2 sin A"^ = '/ ;
??icosA^i = c; n2CosA^2 = c; etc. (l-l-i)
Substituting in (143), and combining, we have
/360° ^ \ /360° ^ \ , ,
7/ = /(() + /( 1 cos .r — A 1 )+/;■> cos 2.r — A2)+ • • • (145)
\ m / \ m J
which is shorter than (143). From (144),
h ,, h
n\^-: — ^-, tanAi=-, etc.
sm A 1 c
119. Non-linear Forms. As stated in Art. 38, equations of
higher dogrco can be reduced to linear form, in general, by Taylor's
Theorem, and in the case of exponential equations by the use of
logarithms. Thus, it is not necessary to treat these higher degi-ee
expressions (^xcept by reducing to the linear form and then applying
th(> usual methods. These j^i'ocesses of i-(Hlu('tion will now be
(explained. They are ajjplieable, of (■ours(\ to e(iuatioiis wliicli
ai'e non-linear as to the independent variable as well as to those
which are non-linear as to the eoc^ffieients.
120. Exponential Functions. I'(iuations in which the unknown
constant occurs as an exponent constitute a special ca^e tor I'ecluc-
tion to linear form which, owing to its simplicity-, will he discussed
136 PRACTICAL LEAST SQUARES
first. In brief, the method is to throw the equation into the
logarithmic form, by taking the logarithm of each member, and
the resulting function will be linear with respect to the desired
coefficient. Suppose the function to be of the form
2/ = arc^ (145)
in which a and b are to be determined so as to fit all of the observa-
tions as well as possible. Taking the logarithm of each member,
log y = \og a-\-h log X (146)
which has the linear form
y' = A-]-bx' (147)
where A and b are the unknown constants.
By plotting log x and log y as coordinates, or by using loga-
rithmic cross-section paper for plotting x and y, the above exponen-
tial formula would be represented by a straight line. Thus the
assumption of this form of equation can be easily checked.
Special attention must be given to the weights in this case of
exponential functions, for the weights of the reduced, linear equa-
tions will not be the same as before reduction to the linear form,
even though they were then equal.^ If the weights of the original
observations of yi, y2, ys, etc., are wi, W2, ws, etc., the correspond-
ing weights of the functions, log yi, log y2, log yn, etc., will be yi^wi,
y-i^wo, yii^ws, etc.- Or, if the original weights are equal, the
reduced equations will be weighted directly as the squares of the
corresponding observed values of y. If the empirical formula
1 This matter was first brought to the attention of the author several years
ago, bj- Mr. C. K. \'an Orstrand.
2 It will be shown in the next chapter (Art. 143) that the weights are in-
versely as the squares of the mean square errors, and that (Art. 152) the mean
square error of a function of ?/ is equal to the mean square error of y multiplied
by the derivative of the function with respect to //. Thus,
(I (log i/) 1
€iog !/ = «;/ 7"~" = f2/- (l-47a)
dy y
and
"'iogj/ = "Vy' (1476)
the mean sfpiare errors being represented by e.
EMPIRICAL FORMULAS 137
follows the observations very closely, however, as is usually the
case, these weights will not have much effect. In fact, the errors
of observation may warrant neglecting them in most cases.
121. General Case of Reduction to Linear Form. A simple
example of an equation of the non-linear form with respect to
the coefficients would be the following:
2y = «2+53a;+c.T2+r/'V+ • • . (148)
Thus, each observation equation would be a function of a, b, c, d,
etc., since x and y would be the observed numerical quantities,
so that if the observed values of the function, y, be represented
as usual by Mi, M2, Mz, . . . Mn, the observation equations
would have the form,
fi(a, h, c, .
. . )=Mi
hia, h, c, .
. . )-M2
h{a, h, c, .
. . )=Mz
(149)
fn{a, b, C, . . . )=Mn
The functions /i, /2, fs, etc., on the left-hand side of these equa-
tions are different owing to their having cHfforont nuuKM-ical
values of x. Now let the ])est or most probable values of a, b, c,
etc., namely, those which will result from this solution, l)e A, B, C,
etc., and let Ao, Bo, Co, etc., represent approximate values of
A, B, C, etc., determined bj- the solution of some of the ()])sei-va-
tion equations as simultaneous equations. Then let
.l=.4o + a'
B^Bo + b' (!.-)())
C=Co + c'
in which a', //, c' , . . . are small correclions lo \\\v assumed
138 PRACTICAL LEAST SQUARES
approximate values, to be determined by this solution. The
observation equations may now be written,
/i(Ao+a', 5o+fo', Co+c', . . . )=Mi+yi
f2{Ao+a', Bo+h', Co+c', . . . )=M2+V2 (151)
/3(Ao+a', 5o + &', Co + c', . . . )=M;+Vi
the residuals being represented by fi, y2, vs, . . .
These functions will now be expanded by Taylor's Theorem.
The unknown corrections, a' , h', c', . . . being small, it is per-
missible to neglect the terms involving their products and higher
powers. The constant terms, /i(Ao, -Bo, Co, . . .),f2(Ao, Bo, Co, . . .),
etc., will be combined with the corresponding Mi, M2, etc., and
represented hy l\, h, etc., thus,
MAo,Bo,Co, . . . )-Mi = h (152)
The observation equations will then become,
cIAq aBo clCo
1 .^-^^ df2 df2 ,
t2 + 7T-a'+;T^6'+-^77-c'+ . . . =V2 (lo3)
dAo dBo dCo
which are linear with regard to a', 1/ , c' , . . . The differential
coefficients arc obtained by differentiating the left-hand members
of (149) with respect to a, b, c, etc., and then substituting for
these quantities their approximate values, Aq, Bo, Co, etc. If now
we let the differential coefficients be represented by ai, hi, Ci,
etc., with the subscripts of the corresponding equations, we obtain,
aia' + 6i6' + cic'+ . . . +/i=t'i
a2a'^b2l/ + C2c'-\- . . . +?2 = ?^'2 (154)
an(i' + hJ)'-\-Cnc'-^ . . . +/„ = /'„
which arc similar to (18), page 27. Normal ec}uations having boon
formed as in (21) or (22), tlioir solution in tlie usual manner results
in the dc^sired corrections, a', h' , c' , etc., which applied to the
approximate values, yio, Bo, Co, etc., as in (150), give the most
EMPIRICAL FORMULAS 139
probable values, A, B, C, etc. From these, the desired non-
linear coefficients of the original equation are computed directly,
giving finally the empirical formula sought.
If the observations are of different weight, the general form of
normal equations, (21), would be used as in Indirect Observa-
tions, Chapter III.
122. Determination of the Constants. The plotted observa-
tions having been investigated and a suitable form selected for the
eciuation, reduced, if necessary, to the linear form as just explained,
it remains to form the observation equations and from them the
normal equations, the solution of which is to give the desired
constants for the empirical formula. In general, it is similar to
the case of Indirect Observations, and the methods of Chapter III
arc applicable. The function will be stated in the explicit form,
y^J{x), although, of course, these quantities may be reversed, if
desired, to fit the conditions, into x—f{y), which form may some-
times be simpler than if fractional exponents were used.
The observation equations are formed, one for each observa-
tion, by substituting for x and y their observed values. The
processes of Arts. 48 and 49 may be utilized for the simplification
of the equations, and the normal eciuations will take the form
of (22) or (21) according as the weights are equal or unequal.
The solution of the normal et^uations will be carried out by the;
usual methods, and the resulting values of the unknowns, modified
as necessary, will furnish the constant term and coefficients of
the empirical formula.
123. Test of Empirical Formula. There are two methods of
determining how closely the formula corresponds to the observa-
tions, namely, ])y plotting the curve of the formula and by com-
})uting the residuals.
The residuals arc^ f()fin(Ml by substituting the observed values
of the varia1)le, .r, in tli(^ euipirical formula and computing the cor-
responding values of y. Subtracting from tliesc^ the observed
values of ?/, we obtain the residuals with the signs of corrections
to the observations. The sum of the sc^uares of th(>se i'(\si(luals
is the quantity which should be a mininunn if the empirical
formiila is the most probable one.
Having ploftcd the values of y, coinpu1(Ml as above from the
140 PRACTICAL LEAST SQUARES
formula, the need of other, intermediate values in order to accu-
rately define the curve may be seen at once and such values com-
puted and plotted and the curve drawn by means of a French
curve. If this be done on the sheet showing the original observa-
tions, the value of each residual is shown to scale by the vertical
distance from the corresponding observation up or down to the
curve, according as the residual is plus or minus, measured on its
ordinate. Inspection of these graphical residuals will determine
whether or not another form of curve should be assumed and the
work repeated in order to find a closer approximation to the obser-
vations. If this should be done, the sums of the squares of the
residuals in the two cases would be compared and that formula
adopted for which this sum is the smaller. In a case of great
importance, especially one that involves a large number of observa-
tions, several trials of this kind might be made in order to obtain
the best formula.
124. Remarks. The above method of deriving empirical
formulas is evidently closely analogous to the adjustment of
Indirect Observations, that is, observations of a function of several
quantities, and it must be borne in mind that in this method the
errors of observation are assumed to lie in the values of the func-
tion, y, and not in those of the variable, x. At least, the errors in x
are assumed to be negligible in comparison with those of tj}
A final word of caution must be added with regard to the use
of the empirical formula. In general, it is safe to use it within
the range of the observations, that is, in interpolation; but
only in very exceptional cases should it be depended upon for
extrapolation, outside of these limits. Duncan ^ cites the example
of the stress-strain curve, which is practically a straight line until
the clastic limit is reached, but which, at that point, suddenly
breaks into a sharp curve. An extrapolation from the straight
line would be greatly in error.
Again, it nuist be emphasizcnl that the form of the empirical
cfiuation is assumed at the outset and from considerations outside
' For an investigation of the case when x and y arc equally subject to error,
see Report of C. & G. Survey, 1890, page 687, or ^\'right's Adjustment of
Observations, Art. 106.
2 Practical Curve Tracing l)y R. II. Duncan.
EMPIRICAL FORMULAS
141
of the Method of Least Squares. From that point as a beginning,
this method determines the best values of the coefficients for
that form of equation, but unless a suitable form has been selected
the resulting empirical formula may be no better than a rough
guess. Therefore, great care should be exercised in choosing the
form of the equation.
When the observed data are few and widely scattered, it is
scarcely worth while to go to the trouble of a Least Squares adjust-
ment to establish an empirical formula. In such a case, it is
usually sufficient to sketch a smooth curve through the plotted
observations and to determine the constants of the curve by
scaling various elements from it, in connection with its known
properties. In particular is this method applicable to straight
lines and to those hyperbolic forms which appear as straight lines
when plotted on logarithmic paper.
125. Example : Straight Line. Let it be required to derive
a formula which shall fit the following observations as nearly as
possible, preference being given to a straight lino, if reasonable.
.r
y
X
y
-1.0
+ 14.0
+ 14.0
+5.0
+ 1.0
13.0
17.0
2.9
5.0
10.7
20.0
1.0
9.0
8.0
LTpon plotting these observations, as in Fig. 34, it is seen that
they fall nearly in a straight line, so we shall assume the form
Y
0 o 19 JJ ZO
Fig. 34. Straight Line and Parabola
142
PRACTICAL LEAST SQUARES
5.0 = 0
2.9 = 0
1.0 = 0
of the equation to be, y = A-\-Bx, A and B to be determined.
Substituting the observed data in this form and reversing the order
we obtain the equations,
- 5+^-14.0 = 0
+ 5+^-13.0 = 0
+ 5S+A-10.7 = 0
+ 9B-\-A- 8.0 = 0 (155)
+ 14B+A-
+ 175+A-
+20jB+^
Considering these to have the form, aiA + 6i5+/i =yi, to cor-
respond to equations (18), the normal equations take the form of
(22), and become
+9935+65A -263.8 = 0
+ 65B+ 7 A- 54.6 = 0 (156)
the solution of which gives A = +13.62, and 5= —0.63, so that
the required empirical formula is,
^=13. 62 -0.63a: (157)
Substituting the values of A and B in (155), with the original
values of X, the computed values, ?/', are obtained, and subtracting
from these the corresponding observed values, ij, we find the resid-
uals V, which, for further reference, will be squared and added.
■ -^ "/
.1-
y
y'
V
r-
-1.0
+ 14.0
+ 14.25
+ .2o
. 0()2.-,
+ 10
1.3 . 0
12.99
-.01
1
5 . 0
10.7
10.47
-.23
529
9.0
SO
7.9.-)
- . 0.")
25
14.0
Ti.O
4.80
-.20
400
17.0
2.9
2.91
+ .01
1
20.0
1.0
1.02
+ .02
4
= .1.5S5
0.0
+ 13.62
+21.0
0.0
EMPIRICAL FORMULAS 143
The line is easily plotted from the points where it crosses the
axes, that is, where x = 0 and where y = 0, which have been added
to the above table. It is shown in Fig. 34. The residuals are
indicated as the vertical distances of the observations from the
plotted line.
126. Example: Parabola. From the observations in the
preceding article, let us determine a curve instead of a straight
line, using the parabolic form,
y^A^Bx+Cx^ (158)
Substituting the observed values of x and y, and reversing the
order, we obtain the observation equations,
C- 5+^-14.0 = 0
C+ 5+^-13.0 = 0
25C+ 55+^-10.7 = 0
81C+ 95+ A- 8.0 = 0 (159)
196C+145++- 5.0 = 0
289C+175+A- 2.9 = 0
400C + 205+.4- 1.0 = 0
In order to reduce the coefficients of the first two unknowns, we
l(>t C = \QOC, and B' = 105, as in Art. 49. Then we have,
.OIC- .15' + .4- 14.0 = 0
.01C'+ .15'+.4-13.()
.25(:"+ .55' + .! -10. 7
.Sir+ .95' + .!- S.O (160)
1.9()r'+I.45' + .l- 5.0
2.S9(" + 1.75' + .4- 2.9
4.()()r" + 2.()/r + .l- 1.0
144 PRACTICAL LEAST SQUARES
The resulting normal equations are,
c
ir
A
+28.91
+ 16.50
+ 9.93
+ 16.50
+ 9.93
+ 6.50
+9.93
+6.50
+7.00
-31.61
-26.38
-54.00
= 0
(161)
and their solution yields the values, .4. = +13.52, B'=— 5.63,
and C'=-0.34. Then B = 0.15'= -0.56, and C = 0.01C' =
— 0.0034. The empirical formula is, therefore.
y= 13.52- 0.56a; - 0.0034^2
(162)
The similarity of the first two terms of the second member to
those of (157), as well as the very small coefficient of x', indicates
that the curve approximates closely to the straight line of the
previous article. However, we shall investigate the residuals to
see how closely the observations are followed. Computing the
values of y and designating them by y' as before, we find:
X
y
y'
V
r2
- 1.0
+ 14.0
+ 14.08
+0.08
.0064
+ 1.0
13.0
12.96
- 4
16
5.0
10.7
10.64
- 6
36
9,0
8.0
8.20
+ 20
400
14.0
5.0
5.01
+ 1
1
17.0
2.9
3.02
+ 12
144
20.0
1.0
0.96
- 4
16
,()()77
Evidently, this curve is much closer to the observations than is
the straight line. The residuals are smaller and the sum of their
scjuares is smaller bj" more than half. The plotted curve is
shown, also, in Fig. 34, wIkm'c its advantages are appai'ent.
EMPIRICAL FORMULAS
145
127. Example: Exponential Curve. The following observa-
tions are plotted in Fig. 35, and an exponential curve seems reason-
able to assume. In order to investigate the equation more in
X
y
log X [ c
[iff.
log y
diff.
y-
0,2
4.6
9.301
0.663
21
0.6
5.8
9.778
48
0.763
.10
34
1.2
7.6
0.079
30
0.881
.12
58
1.6
9.6
0.204
12
0.982
.10
92
2.0
11.5
0.301
10
1.061
.08
132
2.4
14.4
0.380
08
1.158
.10
207
2.8
17.5
0.447
07
1.243
.08
306
3.0
20.0
0.477
03
1.301
.06
400
detail, the common logarithms of x and y are tabulated, also, with
their successive differences.
20
r
a/
i 1
/
1 '
!
1
/
/
!
/!
1
10
/t
i
y\ log y=0.G3-h o.nx
>^ 1
1
^^^
0
! ! i
X
'J
•
J
1,
J
Fk;. 35. l-"-x]K)n(nitial Function; Siini)l(' Plotting
It i.-- evid('nt froiu uu insi)ection of these (liff(>rciic('.^ tliat there is
no .-^traight-line relation between log x and log //, and plotting
146
PRACTICAL LEAST SQUARES
these values as coordinates shows a distinct curve, in Fig. 36.
However, the differences in log y are seen to correspond quite
Log X
Fig. 36. Exponential Function; Logarithmic Plotting
closely with those in x itself, and this is verified b}' plotting x
and log y, in Fig. 37. Therefore, the desired equation will have
Log
1.5
Y
J^
^
^^
^
^
1.0
^^
log y
= 0.63
t- 0.22
t
0.5
0
1
0
^
0
3
0
Fig. 37. Exponential Function; Semi-logarithmic Plotting
the form,
\ogy = A+Bx (163)
or, if A =log A',
?/-^l'(10^^) (164)
Writing the former equation in the ustial order for observation
ec^uations,
Bx+A-\ugy = 0 (165)
Substituiing the values (jf x and log y from the tabl(>, abov(>,
EMPIRICAL FORMULAS
147
and carrying the numerical work to two places, only, we have,
Wt.
0.25+A-0.66 = 0 0.2
0.65+A-0.76 0.3
1.25+^-0.88 0.6
1.6B+A-0.98 0.9 (166)
2.05+A-1.06 1.3
2.45+^1-1.16 2.1
2.85+A-1.24 3.1
3.05+A-1.30 4.0
The weights of the original observations are assumed equal.
Those of log y, and the observation equations, will then be directly
as the squares of the ?/'s. In the table these have been divided by
100 to lessen numerical labor.
The normal equations, formed in accordance with (21), are,
+80 . 885+30 . 70A - 37 . 18 = 0
+30.705+12.50^-14.63 = 0 (167)
and from their solution, 5= +0.22 and A = +0.63, so that the
empirical formula is,
log y = 0. 63+0. 22x (108)
or,
a; = 4. 55 log ?/- 2. 86 (169)
or,
7/ = 4. 27(10"-"^) (170)
Computing the values of log y' and from them those of y\
corresponding to the successive values of x, and forming the
residuals, we obtain tlie following table:
.(•
li'S //'
1"« //
'■i
//'
i //
V-l
0.2
.07
. 00
+ .01
4.7
4.0
+0.1
O.G
.70
.70
0
") . 8
.') . 8
0
1.2
.89
,88
1
7.8
7.0
2
l.f)
.98
.98
0
9.0
9 . 0
0
2.0
1.07
1.00
1
11.8
11,.")
3
2 4
1.10
■ 1.10
0
14.4
14 4
0
2 . S
1 . 25
1.24
+ .01
17,8
17.5
+0.3
:> 0
1.29
1 :]()
- . 01
19 .-)
20 . 0
-0.5
148
PRACTICAL LEAST SQUARES
The curve is plotted in Fig. 35, and the straight Une, using log y,
in Fig. 37. The residuals of log y, in the column headed vi, are
practically negligible; those of y, called vo, are somewhat larger,
and increase numerically with x. This may seem surprising in
view of the increasing weight used, and in order to illustrate this
effect, the normal equations were formed a second time without
considering weights at all, and solved with the following equation
as a result :
log t/= +0.61+0. 23a; (171)
The residuals of log y are about the same as before, but those of
y are, respectively, —1, 0, 0, 0, +.3, 0, +.7, and 0, indicating
the diminishing weights. However, the curve follows the obser-
vations so closely that the weights have little effect upon the empir-
ical formula.
128. Example : Periodic Curve. The following set of observa-
tions is given for the purpose of determining the equation which
will best represent them. They are of equal weight.
X
y
X
y
+0.1
+8.0
+ 1.8
+ 1.0
0.5
+6.8
2.1
-2.0
0.9
0.0
2.-4
+5.0
1.2
+0.5
2.7
+9.5
1 ..",
+4.5
'.\:2
+ 1.0
These data are plotted in Fig. 38 and from a curve sketched through
the points it is evident that the function is a periodic one. Also
Y
lO.C
y=.3.00-2.2', siti ir>n.r^t.:.', ms imr°+3..93 .^iti 30nx°+n.0D rns SOOx'
Yic. :^)S. Compound Periodic Curve
EM PI RICA L FORM ULA S
149
the waves occur in pairs, one large and the next smaller. The
cycle or period is completed in approximately 2.4 units of x, and
this value will be assumed for m in (143). Owing to the fact that
the waves in the curve are not equal, the first five terms of (143)
will be used, namely,
. 360° ^ 360°
y = A-^B sm x-\-C cos x
. 360° ^ 360°
-\-D sm 2x-\-E cos 2x
m m
(172)
which becomes, upon inserting the above value of m,
y^A^B sin 150.t° + C cos lbOx°
+D sin 300.r°+7^ cos 300a; ° (173)
Substituting the various values of x and y, and looking up the
natural sines and cosines to two places, we obtain the observation
equations, which will be written, for convenience, in the reverse
order:
+0.87£'+0.50Z)+0.97C+0.26fi+A-8.0 = 0
-0.87/i'+0.50i) + 0.26C + 0.97i?+A-0.8 = 0
-1.00Z)-0.71C+0.715+A =0
+ 1.00^ -l.OOC +.4-0.5 = 0
+ 1.00Z)-0.71C-0.715+A-4.5 = 0
-l.OOE -1.00B + -4- 1.0 = 0 (174)
-1.00/:) + 0.7ir-0.71/i+A+2.0 = 0
+ 1 . OOE + 1 . OOC + .4 - 5 .0 = 0
+ 1.00/) + 0.71C+0.717?+A-9.5 = 0
- 0 . oOE-0 . 87/)- 0 . .5()f ' + 0 . S7/i+.4 -1.0 = 0
The normal (xiuatioiis. forincd in tlie usunl mainier, are,
/•;
1) 1 <■
y.' .1
-f-4.77
+0.44
+ 0.81)
-0,0.")
+ 0..")0
- .").04 =
- 0
+ 5. 26
+ 1.0.")
-0 1.")
+ 0.13
-22..")3 =
= 0
+ 5.20
+ 0.00
+ 0.73
-1.") ()4 =
= 0
+ 4,77
+ 1.10
^10,00
-13 .")1 =
-31 :;() :
17.5)
150 PRACTICAL LEAST SQUARES
the sub-diagonal terms being omitted for the abridged solution.
Solving these equations, the following values of the unknowns are
obtained :
A = +3.00, B=-\-2.24, C=+1.73, Z)=+3.93, ^=+0.09.
The empirical formula, therefore, will be,
^ = 3.00+2.24 sin 150a;° + 1.73 cos loOa;°
+3.93 sin 300.t°+0.09 cos 300.t° (17G)
or, expressed in the form mentioned at the close of Art. 118,
t/ = 3.00 + 2.83 cos (150a;°-o2° 19')
+3.93 cos (300a;°-88° 41') (177)
The curve is plotted in Fig. 38. For this purpose, a numljcr of
extra values of y were computed so as to determine the curve with
greater precision. It is evident that a larger number of obser-
vations would be desirable in the case of an equation as complicated
as this one. The curve conforms to the observations fairly well,
and it is doubtful that a recomputation with a different value of m
for the period would be worth while. It is useful to note in con-
nection with the plotting that the same value of y will correspond
to values of x which differ by multiples of m. Thus, for .r = 0.1
and 2.5, we have the same value of ij, namely, +7.32.
129. References. The reader is referred to the following works
in which useful information and methods concerning empirical
formulas will be found. The collections of examples given ])y
Weld and Bartlctt are worthy of note.
\Vuight: Adju.stinent of Observations.
Comstock: Method of Least Squares.
Meiiuim.\x: Method of Least Squares.
AVkld: Thc'ory of Errors and Least Squares.
B.\utlktt: Method of Least Squares.
Helmeiit: Auspcleic'hungsrechnmig,
Duxc.w: Practical Curve Tracing.
BiirxT: Combination of Observations.
LiPK.\: Graphical and Mechanical Oomi)utation.
CHAPTER VIII
PRECISION OF OBSERVATIONS AND RESULTS AND
COMBINATION OF COMPUTED QUANTITIES
130. Having considered the determination of the best values
of the unknown quantities to be obtained from given observations,
it remains to investigate the degree of confidence which may be
placed in the observations and the computed results, so that they
may be compared with the results of other observations of the
same quantities.
131. Precision. If two sets of direct observations of the same
kind be compared, and in the first the component quantities are
scattered over a wider range or are more discordant than in the
second, it is natural to conclude that the observations of the
first set were mads with less care or under less favorable cir-
cumstances than those of the second set. The latter are more
consistent and evidently more precise; their differences or dis-
ci'cpancios are smaller. Furthermore, even though the number
of oljservations in the two sets were equal, the mean of the second
set would be regarded as of greater reliability or weight than the
mean of the first set, merely l^ecause of the greater consistency, i.e.,
smaller discrepancies, among its original observations. Since
these smaller discrepancies correspond to smallci- I'csiduals from
the mean, it is evident that the precision of the mean is indicated
by the size of its ]-esiduals, being grc^ater as the residuals are smaller,
and ^■i(•(■ vci'sa.
132. Precision and Accuracy.' This ])reci>ion must not be
coufu.-cd witli the accurac\', that is. the correctness, of tlie results.
The latter is affected by systematic errors (\v\. .")). Thus, a
sei'ies of observations may be very closely grouped, showing a
' .'<c(' .John-nn. I'heory (if llrrors ami Mctluid of Lca>t .SfjUarcs, (Jhaj). \'II,
for ail cxtcndcii trcatineiiT of tlii< -ulijcct.
l.jl
152 PRACTICAL LEAST SQUARES
high degree of precision, but each separate observation, and there-
fore, the mean, may be in error by a large amount due to some
influence which is unknown or not taken into account. Precision
has reference to the accidental errors of observations made under
constant conditions and indicates the care exercised by the observer,
the closeness with which the instrumental readings are made, and
the suitabilit}" of the method used. Discordant observations are
not precise; but precise determinations may or may not be
accurate.
133. Index of the Precision. It is easy to obtain an idea as
to the precision of the observations from an inspection of them
or of the residuals of their mean. But in the comparison of the
results of diiTerent sets of observations of the same quantities,
it is very convenient to have a numerical index from which the
precision of each set may be determined without actually inspecting
the observations themselves. Since this precision is indicated, in
general, by the size of the residuals, it is evident that the desired
index would logically be some function of these residuals. The
precision of a result will depend, also, upon the numljcr of obser-
vations from which it is obtained. Obviously, the larger the
series of observations, the greater should be the precision of their
mean as well as that of the typical single observation. Thus, we
might use the mean of the residuals, without regard to signs, or
the square root of the sum of their squares, and either of these
would give us some idea of the consistency of the observations,
this hypothetical I'csidual being smaller in the case of greater
precision.
From the very inception of the ]\Iethod of Least Squares, the
investigation of tlie precision was regarded as of considei-a])l('
iniportajice. Sevei'al ([uanlities have been iis(h1 to indicate it.
( lauss designated the (}uantity, h, in the Law of luTor, as a " meas-
ure of prcM'ision.'" However, other indices have hcon more gen-
erally used, namely, certain selectcnl eri'ors, theoi'etically defined, as
the Mean Square Error, the ProJxihle Error, and the Average Error.
These will now be considered in oi'(l(>r.
134. The Quantity, h. in the Law of Error. If we consider two
sets of observations of the same (-luantit;', in;;(l-^ in th(> same man-
PRECISION OF OBSERVATIONS AND RESULTS
153
ner, to be represented by the curves in Fig. 39, the area between
each curve and the axis of A will be unity, that is, the probability
of an error between the limits— oo and +00. Then, the taller
the curve, i.e., the greater the p-intercept, the larger will be
the portion of the area immediately adjacent to the p-axis and the
more numerous the smaller errors will be in comparison with the
entire group; in other words, the greater will be the precision.
By inspection of the Law of Error,
h
V-
-h2A2
V71
h . h
it is seen that when A = 0, p^ — =, so that the p-intercept is — =
Vtt Vtt'
Therefore, Vtt being a constant, h may be regarded as an index of
the precision, with which it varies directly.
Fig. 39. Curves of Probability
135. The Mean Square Error (e) of an observation is defmed
as the square root of the mean of the squares of the errors in a
given series of observations.^ It will be represented by e or
m. s. e. To determine its relation to h of the previous article, we
proceed as follows:
According to thc> Law of Error, the probability of the occiu'renco
of an error. A, in a given set of ol^servations, is,
V-K^)=^e->>'-' (178)
1 The moan square error is sometimes referred to as the uiean error. This
introduces an ambiguity with the average error, or mean of the errors, and is
an unfortunate use of the term. Clerman writers call it der mittlere Feliler
but this involves no ambiguity as they designate the average error as der
(hireli.-ielinittliehe Feldcr.
154 PRACTICAL LEAST SQUARES
and the probability of an error between the hmits A and A-\-dA is,
-^e-^'^'dA (179)
Vtt
The number of these errors will be equal to their probability times
the total number of observations (or errors) in the set/ that is,
nh ,,,,
—^e-^'^^'dA (180)
Vx
and the sum of their squares will be,
^e-'''^'A^dA (181)
then the sum of the squares of all of the errors, between the limits
— 00 and + cc , ^vill be,
nh r
and the mean of their squares, equal to e- by definition, is
nh r+ '^-
e2 = --^ e-""-^'A^dA (183)
^-^f'\-'"^'A^dA (184)
Vtt J- '^
Substitutmg in (184) the value of the definite integral,^
1 Sec Appendix C.
2 This integral maj* be evaluated in the following manner (Bartlett):
The probability of an error between the limits — co and + oo is unity (cer-
tainty), that is,
-^„-. I c-"'^'Ma = 1 (184o)
V 71
,- , e-'-^^A2</A (182)
or,
e-"'^>/A=--^ (184&)
h
Differentiating both members with respect to h, we obtain,
A-dAdh= '~dh (lS4c)
h-
hence,
For another solution, sec .Idi-dan. Ilamlbuch dor VtTmessungskunde, I, .ItU.
PRECISION OF OBSERVATIONS AND RESULTS 155
(185)
' V^V2/iV"2/i2
Hence,
(186)
cr.
hV2
h = — ;=, as stated in Art. 19.
eV2
The geometrical interpretation of the mean square error is
that it corresponds to the abscissa of the point of inflexion of
the Eri'or Curve. Differentiating (178) and placing the second
derivative equal to zero, we have,
— 2h^A
f'{A)=—y-e-''^' (187)
Vtt
/"(A) =^c-^^^^+ -^e-"=^^ (188)
Vtt Vtt
2h^
= -— e- "'^\2h^A^ - 1) = 0 (189)
Vtt
Therefore, for the point of inflexion,
2/rA--l = 0 or, 2/;2a2=1 (190)
and
A = T'-^e, from (ISO) (191)
hV2 ^ ^
which shows that the point of infl(>xion corresponds to tlie mean
scjuare error of an oljservation.
136. The Probable Error (r) of an o])servation in a given
series is the middk^ one of all the errors when they are an-anged
in numerical order, each Ix^ing written as inany tinu^s as it occurs.
As many of the errors are gn^iter than it as nvv less, and so the
])n)l)ability of an error greater than the j)i-()babl(^ error is e([ual to
that of an (M-ror less than it , namely, ()..">, since the total ]ii-()b;ibiiity
is unity. It is an even chanei^ that an ei'ror taken at I'andom
from the series will ])e gi-(>at(>r or k'ss than the pi'obable ei'ror.
This is not the most pmbnhle error in the series, for that would
];)e zero, to correspond to the maxinunn ordinate^ to the Error
Curve, and it is unforiunate tliat the nanu^ has com(^ to be quite
156 PRACTICAL LEAST SQUARES
generally used in this country. It is simply a quantity from which
the precision of the observations can be estimated or deter-
mined, in comparison with similar quantities referring to other
observations. A better name for it would be the middle error.
It is represented by the letter r.
The probability that the error of an observation will be numer-
ically less than the probable error is, by definition, ^. Then
from the law of Error,
h r+' 1
or changing the lower limit,
^j'.-«VA = i (193)
It is not feasible to determine the value of r in terms of h directly
from this equation, so we make use of the following process :
In the Law of Error, let
t = hA, whence dA = ~.
h
h r^ 2 n
^\ e-"'''\l\ = --^\ e-'\lt (194:
■Kja V TT Jo
Then we have for the proba])ility of an error less than A,
Vti
This expression is evaluated for various values of t, by expansio:i
into a series,^ and the results are tabulated with t as an argument.-
By interpolation in this table with the value of the probability
0.5, the corresponding value of t is found to be 0.4769, which i.
the value of t = li\ when A is the probable error, r. Thus,
/ir = 0.4769 (195,)
and
0.4769
r = — (196)
Since the probability that an error will lie between certain
limits is represented by the area bounded by thc^ Error Curve,
the horizontal axis, and the ordinates at those limits; and since
1 Sec Appendix C, page 215.
2 See Table I, page 229.
PRECISION OF OBSERVATIONS AND RESULTS 157
the entire area between the curve and the horizontal axis repre-
sents the probabiHty of an error between — oo and + ^ , that is,
unity (certainty); it follows that the ordinate of the probable
error divides the area on either side of the vertical axis into two
equal parts corresponding to the probability, |.
137. The Average Error (7?) is the mean of all the errors in a
set without regard to signs. Since positive and negative errors
are equally likely to occur, the probability of a positive error
between A and A+dA will be one-half of that of any error between
those limits, that is, it will be equal to
h r^""
—V I e-^'-^V/A (197)
2V7rJ-x
The number of the positive errors will be their probability times
the total numlx^r of errors, n, namely,
^^ I e-'^^^^rfA (198)
2V7rJ-=c
and their sum is,
nh r+"
-^ e-^^'^''^d^. (199)
But the sum of the negative errors is numerically equal to that
of the positive ones, so that the total sum will he twice the above,
that is,
*'- ,"". -^=^=Ar/A, (200)
V7
nh n
or,
^'"" ' -^^'^'AdX (201)
V
and the average of all of the errors is therefore,
<() that.
^f\^-'''^'Adl (202)
Vtt Jo
-^ r c'-"'^'(-2/i2A)r/A (203)
:— . J r---^ (204)
— - (205)
158 PRACTICAL LEAST SQUARES
The ordinate of the average error passes through the center
of gravity of the area between the curve of error and the horizontal
axis on either side of the vertical axis. For, if Ao represent the
abscissa of the center of gravity, by considering vertical strips of
width dA and length
Vtt
and taking moments about the origin, we have.
~ re-'^^'dA^-^ Ce-"'^'AdA
ttJo 'Vtt. Jo
Ao4= I e-'"^'dA^-^\ e-'"^'AdA (206)
Vti
But since the total probability area is equal to unity, the area
on one side of the vertical axis is 1/2, that is
(207)
Hence,
2h C"^
Ao = -iL( e-'-'^'Af/A =77, from (202). (208)
138. Comparison of the Indices of Precision. From (186),
(196), and (205), we obtain directly,
eV2 '" tjVtt
- = 1 . 4 142 6 = 2 . 09()6r = 1 . 7726r? (2 1 0)
e=1.4826r=1.2533r7 ]
r = 0.07456 = 0.84587? \ (211)
r? = 0.7979e=1.1829r J
Thus it is seen that the mean square error, the probable error,
and the average error are related b}- constant factoi-.'^. Therefore,
they may be used interchangeably in various formulas and math-
ematical investigations b,y simply providing for the numerical
factors.
In Fig. 40, these quantities are shown in their correct relative
positions and magnitudes. The abscissa' rcpi'o.sent the erroi-s
PRECISION OF OBSERVATIONS AND RESULTS
159
and the ordinates their probabihties. It will be recalled that the
intercept on the vertical axis is -— =.
Vtt
The quantity, h, is directly proportional to the precision.
However, it is inconvenient in practice and is not generally used.
The three representative errors, e, r, and rj, on the other hand,
are inversely proportional to the precision; the smaller these
errors, the more precise and consistent are the observations.
They are sometimes said to indicate the uncertainty, therefore,
instead of the precision. Each of the three errors occurs in a
Fig. 40. Relations between the Varioas Indices of Precision
certain relativi; position when all the errors in a set of obsei'vations
are arranged in the order of their numerical magnitude, as, for
example, the probable error occupies the middle of the series.
This feature is what one would naturally expect in an index of
the precision (Art. b38).
The average^ error, also, is not used in practice as an index,
although it would be a satisfactory one. It may be used, how-
ever, in the process of determining e and r.
Tlie mean scjuare eiTor and tlic pr()l)abl(> error are in common
use as indices of the precision. The former has l)een almost uni-
v(M'sally used by writers in German and othei- foixMgn languages, as
w(>ll as by some Americans, notably in the classic Adjustment of
( )t)S('rvati()ns, by Pi-ofessoi' T. W. Wi'ight, and by ( 'hauvenet,
160 PRACTICAL LEAST SQUARES
Newcomb, and Crandall. Its principal advantages lie in the facil-
ity of its theoretical derivation; in its priority (it was used by
Gauss); in its use by the Germans and French, who have made
the most numerous contributions to the subject of Least Squares ;i
and in its avoidance of the misnomer of the probable error which is
frequently a stumbling-block to the beginner.
The probable error is used by most American and British
writers. Its name is its greatest enemy, but there may be some
advantage in its mere reference to probability. The person who
does- not clearly understand its significance is apt to take it at its
face value and so interpret it. However, it is hoped that such
persons will learn what it means or leave it alone. It should be
understood simply as an index of the precision.
Whichever index is used, it is written after the quantity to
which it refers and separated from it by the sign, ±. This is
merely a convention and the sign is never to be used algebraically.
There is never any reason for increasing or diminishing a quantity
by the amount of its mean square error or probable error. A
better method of designating it would be to use instead of the sign,
±, the symbol for the mean square error (e or m. s. e.), or that
for the probable error (r or p. e.), as 7653.28 (€ = 0.02). But
the use of the ± sign is well established as is also the term probable
error.
139. Precision of Direct Observations. We have seen how
the precision in a set of observations may be indicated by the mean
square error, the probable error, etc., and it is evident that if
we could know the true value of the observed quantity, and
therefore, the true errors. A, we could ascertain the numerical
value of the index of the precision dircctl}^ from those ei'rors, by
definition, as
e^ = , r] = — , and r = the middle error. ^
n n
But as these true errors are unknown, it remains to determine the
precision index from the given observations or n^sidiuils. Know-
' Sec Appendix A.
^ The syuiVjol fur the .sum without reg;ir(l to signs is [
PRECISION OF OBSERVATIONS AND RESULTS 161
ing the relations between the three indices as stated in (211), it
will suffice to determine the mean square error in each case and
from it to express the probable error and the average error.
140. Precision of a Single Observation. Each observation
has its own individual error and when we refer to a " single obser-
vation " in this connection, we mean an observation such as those
in the set which is being discussed, not any single one of them,
but a hypothetical one which is never evaluated, but which is
typical of the entire set in so far as precision is concerned.
Using the notation of Chapter I, let M represent an observa-
tion of a directly observed quantity; v, its residual from the
arithmetic mean, xq; X, the true value of the observed quantity;
A, the true error of an observation; Ao, the true error of the
arithmetic mean; and n, the number of observations in the
series. Then, using subscripts to indicate the separate obser-
vations,
X = .To+Ao = il/i+Ai=ilf2+A2 . . . (212)
Vi^xo — Mi, V2^xo — M2 . . . (213)
Ai-xo+Ao-i¥i, A2 = a:,)+Ao-M2, . . . (214)
Ai=i'i+Ao, A2 = y2+Ao, . . . (215)
Squaring both members and adding the n resulting equations,
[A->[i'2]+2AoH+nAo2 (216)
From (8), [y] = 0; and the unknown true error, Ao, of the mean is
assumed, for this demonstration, to be equal to the mean square
error of the moan, the value of which is determined in Art. 153
to bo (see Art. 141),
6
(217)
(218)
(219)
60= .
Thoroforo,
divic:
ling
(216) by
11
n n
whoiico,
,2_ V-\
n-\
and
102 PRACTICAL LEAST SQUARES
Then, from (211),
r = 0.6745
7? = 0. 7979a P^^ (221)
\n— 1
These three formulas are known as Bessel's Formulas and the
first two are in general use. In long series of observations,
however, it is more convenient to use Peters' Formulas, which
involve the sum of the residuals without regard to signs, [v,
instead of the sum of their squares. They may be derived as
follows :
From (217) and (218),
[A2] [,2]
n n—1
(222)
k1 = "--[A2] (223)
n
and,
v,=J''-^Ai, V2 = J'^^A2, . . . (224)
\ n y n
Adding these n equations, neglecting the signs of v and A, wc have,
since, by definition, rj = [A/n,
whence,
!^[A = ,J^, (225)
and from (211),
, = -^^=^ (226)
vn(n— 1)
W
e = 1 . 2533— =i z:^ (227)
V7i(n— 1)
r = 0 . 84o3-y J:L= (228)
V ??,(?!— 1)
An ayjiroximate value of the probable error of a single o})serva-
tion in a series of from 20 to 30 observations may l)c determined
by taking one-sixth of the rangc^ of the set, or one-third of the
largest residual. Fi'om the table of values of the I^aw of Error,
PRECISION OF OBSERVATIONS AND RESULTS 163
it is found that the probability of an error three times as great
as the probable error is about 0.04, or 1 in 25.^ That is, in a
series of 25 observations, the maximum error is likely to be about
three times the probable error of a single observation. And,
since there are as many positive as negative errors, the total range
of the observations in an ordinary set of, say, from 20 to 30 obser-
vations, is likely to be about six times the probable error, or about
four times the mean square error of a single observation. Con-
versely, knowing the precision index, and the approximate number
of observations in the set, we can estimate the range. Frequently
this fact affords the most tangible idea as to the consistency of the
observations, especialh^ to the beginner, since, by doubling the
mean square error of a single observation he obtains an approximate
value of the maximum residual.
141. Precision of the Mean. The arithmetic mean being the
best value of the observed quantity obtainable from the given
direct observations (Arts. 14, 27), it is obvious that the precision
of the mean wall be greater than that of a single observation, and
also that the precision will increase with the nvmibcr of observa-
tions in the set. In Art. 153 it is shown that if e be the mean
square error of a single observation, and eo that of the mean of the
set of n observations,
60 = -^ (229)
Vn
which expresses the very important relation that the precision of
the mean increases directly as the srpiare root of the number of obser-
vations. In other words, to double the precision, that is, to divide eo
by two, it is necessary to make four times as many observations.^
'See Ai)i)en(li\ 1'', pajjo 231.
The probability of an error less tlian three times the probable error is
0.957, eorrespoiidiiifi to A r = 'A.(); th(>n the probability of an error greater
than this would l)e 1-0.9.57 =0.01.3.
- It must not be assumed that by increasing the number of observations
without limit, the precision can be indefinitely increased. There are always
infhuMices which make it extremely didicult, if not ((uite impossible, to ap-
proach c(>rtainty beyond a definite limit. In this connection, the reader is
rcf(>rre<l to th(> admirable tn^itment of tliis matter in ^^'right and Hayford's
Adjustment of Oliservations. Arts. 3S to 4l).
164
PRACTICAL LEAST SQUARES
This principle is used in determining the most economical or advis-
able number of observations to make in a certain program.
From (229) and the formulas of the preceding article, wc obtain
the following expressions for the three precision indices of the
mean, by dividing by y/n in each case;
Bessel's Formulas:
(230)
(231)
(232)
1 = 0.7979
Peters' Formulas:
n(n—l)
V
Vo
nv n— 1
€0 = 1.2533-
ro = 0 . 8453
nVn—1
[v
(233)
(234)
(235)
wvn— 1
The values of the factors of [v~] and [v in these formulas are
tabulated for various values of n to facilitate; computation. Such
a table for (231) will be found in Appendix F, Table IV.
142. Example: Precision of the Mean. Let us consider the
prol)lein in Art. 28, consisting of 16 ol)servations. Here, w=16,
[v = 5r), and [r-] = 305, the unit being in the fifth place of decimals.
The results are as follows :
e
!
I'cters
4.5
4.5
1.1
1.1
:i . 0
3.0
0.8
0.7
'I'he raiig(^ of tlie observations is 16; one-sixth of this gives 3 as
the approximate value of the probable error of a single observa-
tion. Tsing mean s(}uar(^ errors, then, the best value from the set
of 16 observations would be (from Art. 28),
1463. 49764 ±0.00001
PRECISION OF OBSERVATIONS AND RESULTS 165
143. Precision of the Weighted Mean. Since the weights are
merely relative quantities, as explained in Art. 31, we shall consider
them as reduced to integers. The weight of any observation will
then be regarded as the number of elemental observations of weight
unity of which that observation is the mean. The mean square
error, ei, of an observation of weight wi, then, will be that of an
observation of unit weight, nameW, e', divided by \/wi, from (229) :
e
Vwi Vw2
(236)
and
whence.
SiVwi= €2^^^= e3Vw3= ... = e' (237)
.2
(238)
er _W2
which states the fundamental principle that the weights are inversely
as the squares of the mean square (or probable) errors. Also, since
the weight of the weighted mean is, by the definition of weights,
[w], from (237) we have.
e
eo = — = (239)
V[w]
which corresponds to (229).
To find the expression for e', the mean square error of a single
observation of weight unity, we proceed as in the case of equal
weights, Art. 140. Beginning with equations (215) and using the
first one only, to illustrate the process,
Ai = ri+Ao, A2 = V2+Ao, ... (215)
Squaring,
Ai2 = yi2+2Ao^'i+Ao2 (240)
]\Iultiplying each ecjuation by its weight,
ivilr=wirr^2\)WiVr}-WiAo^ (241)
Since iri represents iho number of elemental observations of unit
weight which make up th<^ first actual observation of weight wi,
it will also be the num])er of the errors A], so that i/']Ai^' would be
the sum of the squai-cs of these elemental errors; also, by defini-
tion, this sum is equal to the num])er, ir\, times the corresponding
mean square error sciuanvl, and tlierefore,
wilr = wier=e'- from (236) (242)
166 PRACTICAL LEAST SQUARES
in which ei is the mean square error of an observation of weight wi.
Hence,
e''^ = wiVi^+2AoWiVi-\-wiAo^ (243)
Adding the n equations of this kind, and assuming as in Art. 140
that Ao = eo,
n e'2 = [^^y2j _^ 2Ao[wv] + [w] eo^ (244)
But, from (12), [wv] = 0,
Therefore,
= [wv^] + [w]eo^
(245)
^[wv^]^e'^ by (239)
(246)
._[wv^]
r947^
Whence,
^ /2
and the mean square error of an observation of weight unity is,
(248)
, j[wv']
Then from (239),
'"^;fe=V[#:fT) (249)
and using the relations stated in (211), wc have.
r' = 0^6745 J I'"-"'!
> ?? — 1
(250)
r ()Cu]-^J f''"''^
(251)
(252)
\[w](n-l)
,' = 0.7979./^""^'^)
V // — 1
,v n 7070-c'' f""''"^
(253)
^ [m'J(^; — 1)
If the weights are eciual, a' = l, [w] = n, and tlie last six forniular-
rr)rrospond to thos(> of Arls. ]40 and 141.
PRECISION OF OBSERVATIONS AND RESULTS 167
By analogy, the Peters' Formulas for weighted observations
may be written. They are,
[V^
ivv
VO-
Vn(n— 1)
[Vwv
V[w]n{n — 1)
(254)
(255)
e'=1.2533-i±^ (256)
Vn{n-1)
eo = 1 . 2533-— LJ^= (257)
V[iv]n(n-1)
/ ^0.8453 L^!^ (258)
Vn(n— 1)
ro = 0.8453— ^ ^^''' — • (259)
V[w]n(n-1)
144. Example: Precision of the Weighted Mean. In the
problcMii of Art. 33, [ir] = ll, /)=4, [tt-r-] =4247, and the mean
square error of the weighted mean is, eo = 0.11". The complete
result is,
a;o = 73° 18' 42.07"±0.11"
145. Precision of Indirect Observations. The process of find-
ing the mean square errors of th(^ best values of the unknowns from
indirect o})servations is nuich more involved than in the case of
direct observations. Also, the precision is required in compar-
atively few cases in which engineers are concerned. The method
will b(> outlined, however, without developing the complete theory,
for which the reader is referred to the works ])y Jordan, and
Wright and Ilayford.
The determination of tli(^ ])i'ecision of the results from in-
direct observations is divided into two parts, namely, (a) the
computation of the relative weights of the adjusted values of
the unknowns. X. Y, Z. etc., and (h) the determination of the
mean s(^uai'f> (m-i-oi-, e', of an observation of weidit unity. Then
168
PRACTICAL LEAST SQUARES
the mean square errors of these unknowns are obtained from
the relation (237) :
Vwr:=
or,
eJ^Wx = ey^Wy =
146. "Weights of the Unknowns. There are three methods of
determining the weights of the unknowns. We shall use the fol-
lowing one, which utilizes the principle of undetermined coeffi-
cients. In the normal equations (22), page 29, to find the weight
of X, replace the constant term of the X (i.e., first) equation, [al],
by — 1, and the other constant terms by zeros. The solution of the
equations thus modified will give as the value of X, the reciprocal
of its weight, that is, 1/wx- Similarly, substituting —1 {or [hi] in
the second equation, and zeros for the other constant terms, and
solving the set of equations for F, we obtain l/wy. Thus, —1 is
substituted for each constant term in succession, the others being
replaced by zeros, and the equations are solved, in each case, for
the corresponding unknown, the resulting value of which is its 1 'w.
This process is tedious at best, but it can be simplified as follows.
It is evident that as the constant terms, only, are altered, the
preceding columns of the elimination in the solution of the normal
equations will be unchanged. Therefore, referring to the cciua-
tions (55) page 47, we may add as many columns as there are un-
knowns, between {I) and {sj, designating them as {x\), iy'z), (23),
etc., in which to write the new constant terms. These would be
included in the check-terms (.s') and carried through the elimina-
tion the same as other coefficients or constants. Then the weight
of each unknown would be determined by substituting ])ack in
the proper column until that unknown was ck^terminod, and
taking the reciprocal of its value. Of course it would be unneces-
sary to substitute farther in that particular column, as but o\w
weight is ol)taincd from (vich colunm. This arrangement of the
eciuations (00) would Ix',
(2()0)
X
,'/
il)
(.'■0 (■//■i)
(-:.) 1 (*•')
(2)
+ 6
_2
+ '■'>
-4
+2
— .'5
+ 1
-1 0
0 -1
0 0
0 +s
0 i -7
-1 • +2
PRECISION OF OBSERVATIONS AND RESULTS 169
Since the last equation in the ehmination will have the quantity
— 1 as its absolute term in its added column, it follows that the
coefficient of the last unknown, in that equation, will always be its
own weight. The last added column, (23), in the above example,
may therefore be omitted.
If the original observations are of unequal weight, the same
process is followed, using (21) instead of (22) as the form for the
normal equations, and replacing the terms [wal], [wbl], etc., by
— 1, and zeros, as above.
147. Precision of an Observation of Weight Unity. Let the
number of unknowns be represented by m, the number of obser-
vations being n, as usual. Then the formulas for the mean
square and probable errors of an observation of unit weight, are,
\wv\
n — in y n — m
r' = 0.6745x/^'^ or 0. 6745a M^ (261)
^ n — m y n — m
Liiroth's Formulas arc,
e' = 1 . 2533-J^^„ or 1 . 2.533-J=
V ??. {n — m ) V ?? (?i — 7)i)
r^ = 0.8453 L^^^''' or 0.8453— JL= (262)
V7}{n — ))i) 'Vn(n — m)
If there is but one unknown, m= 1 and these formulas ])0('()nic those
of Bcssel and Peters for direct observations (Art. 140).
The usual method of determining the residuals is to substitute
the adjusted values of the unknowns ])aek into the obscM'vation
equations and obtain a residual for each ccjuation. IIo\v('\-(m-, the
sum of the squares of the resicUials or of the \vei,ulit(Ml ic.-^ichials,
that is, [v^] or [ivv^], may b(> obtaiiuMl more (visily, in most cases,
in the following manner, along with the solution of tlie normal
C(iuations. Form the term at the foot of th(> diagonal, namely,
[f\ or [ivP], and perform a cori'esponding step in the (elimination
as if there were more terms following it. The resulting sum in the
170
PRACTICAL LEAST SQUARES
Z-column will then be [v^] or [wv^], as the case may be. Also, it may
be obtained from the relation
and from,
[wv^] = [wal]x-\-[whl]y -\-[wcl]z-\-
[wv^] = [wvl]
-\-[wf] (263)
(264)
which latter requires that the separate v's be known.
148. Example: Precision of Indirect Observations. To illus-
trate the foregoing articles, the modified observation equations
(43), page 38, will be solved to determine the best values of the
unknowns and their mean square errors. To the normal equations
(44) are added the term [wf] and the columns (xi), {y2), and (23),
and the check terms are modified to include all of these.
.r
U
z'
(/)
(.r.)
(y.)
(23)
(,s-')
I
II
III
IV
+3.94
+ 0.38
+ 13.56
+2.18
+0.19
+6.40
+ 1.00
+3.53
+3.09
+2.66
-1
0
0
0
-1
0
0
0
-1
+ 6.50
+ 16.66
+ 10.86
+ 10.28
+ 13.56
- 0.04
+0.19
-0.21
+3 , 53
-0.10
0
+0.10
-1
0
0
0
+ 16.66
- 0.63
+ 13.52
-0.02
+3.43
+0.10
-1.00
0
+ 16.03^/
+6.40
-1.21
0
+3 . 09
-0.55
+0.01
0
+0.55
0
0
0
0
-1
0
0
+ 10.86
- 3.59
+ 0.02
+ 5.19
+2.55
+0.55
0
- 1 . 00
+ 7.29v'
["■'- =
+2.66
-0.25
-0.87
-1.25
+0.25
-0.03
-0.27
0
+0.25
0
0
0
+0.49
+ 10.28
- 1.65
- 4.07
- 3.5S
+ 0.29
-0.05
+0.25
+0,49
+ O.OSy
0 , 29
i 1
PRECISION OF OBSERVATIONS AND RESULTS 171
Unknowns
5.19
-3.43-0.10 -3.53
« = - ■ = = — 0.2ol
^ 13.52 13.52
-1.00 + 1.07+0.10 +0.17 , ^ ^.o
X = = — = +0 . 043
3.94 3.94
Residuals
Substituting in the observation equations (43) and determining
the i''s, we find directly, [wv-]^0.2S and [wvl]= 0.26, while the
evaluation of (263) gives [wv^] = 0.2Q. From the above elimina-
tion, the first term of IV is [wv^] = 0.29. The average value,
is therefore, .027.
Weights
23 = ^^—7 and M.v = 5.2
o. 19
5.19
i/2 = and ii'v = 13 . 5
^ 13.52 "
5.19
-010
yi= = -0.007
^ 13.52
+ 1.00 + 0.23 1 , .^ „
Xi= = and Wx = o.2
3.94 3.2
^NIe.v.x Sqiake Errors
Average value of [»7'''] = 0.27; /i=9; //; = 3.
e' = .. f lirl ^ ^ /().()475 = 0.22
e' 0 22
6:, = -4=: =---^^-^ = 0.12
Vic. 1.8
0 22
e, = --^ = O.OG
0 22
ey = ^^-^ = 0.10
2.3
172 PRACTICAL LEAST SQUARES
Results
X =+0.043d=0.12
y =-0.261±0.06
/--0.491±0.10
149. Precision of Conditioned Observations. In general, it is
necessary, as in the preceding case, to determine the precision of
an observation of weight unity and also the weight of each un-
known, from which the precision of the unknowns is obtained from
the usual relation that the mean square errors are inversely pro-
portional to the square roots of the weights, that is, from,
e/wx = ey~Wy = . . . = e'2
If the conditioned observations are adjusted as indirect obser-
vations by the method stated in Art. 81, the precision of those
unknowns which are involved in the normal equations may be
determined by the methods just explained in Arts. 145 to 148.
Then by a second solution, ehminating a different set of unknowns,
the normal equations may be made to involve those which were
not included in the previous set, and their precision may be found
in the same manner. Obviously, this is a tedious method except
in cases of a few observations.
Since the number of unknowns which may thus be made inde-
pendent is the total number, m, minus the number of conditions,
m', the formula for the mean square error of a single observation
of weight unity may be derived direct^ from (261) and (262)
by substituting for m, m — m'. Thus,
: ^ f ) J- ^
y n—m-\-7n > /;
//; + ///
/ = 0.6745\/ — ''^~^— or 0.6745x/ ^"^ — , (265)
^ n — 7n-\-m ^ n — ni-\-ni
But in most of the cases with which we are concerned each unknown
is directly observed so that n = m, when the above formulas become.
ivi'-
r' = 0.6745a/' -^ or 0.6745\r (266)
PRECISION OF OBSERVATIONS AND RESULTS 173
in which m' represents the number of conditions. Liiroth's
Formulas (262) may be similarly modified by substituting m — m'
for m; and when n = m, the denominators become \^nin' .
The residuals, v, are the corrections, v, to the observations, as
determined in the adjustment (Art. 73). As a check upon the
direct computation of [wv^], however, we may use the formula,
[^i;2^=-Aq,-Bq2-Cqz . . . (267)
A, B, C, . . . being the correlates, and qi, 92, ^3, . . . being the
absolute terms of the reduced condition equations (59) or the
normal equations (64). Or, in the solution of the normal equa-
tions, a step may be taken similar to the one described in Art. 147
for indirect observations. Here, however, zero is written for the
last term in the constant (q) column. The elimination process is
continued to include this term, and the resulting sum will be
— [wv"^].
The method of correlates, however, will generally be used in
the adjustment. The weights of the adjusted values are not
determined directly, in this case, but the weight of a function
of these values is determined, and this function may be merely
unity times one of them. Examples of the functions of the
adjusted values for which the precision may be desii-ed are: A
side of a triangle or an unobserved line in a system of triangula-
tion, when computed from the adjusted angles; and a computed
difference of elevation in a lev(4 net, det(M-min(Hl from adjusted
values of observed difference's. The fmiction must not involve
more of the unknowns than can be made indc^pcndent by (elimina-
tion with the conditions, that is, not more than tii-^)/)'. The
method is as follows:^
Since any function can be I'cduced to the liii(\ai' form, this one
will be assumed to have that lorni.
in which 1^, \-2, 1':;. • ■ • ;ii"e the adjusted values of ihe unknown-
(Art. 72), namely, Vi= }f]+r\. etc. If any of the t(>i-!n-; in (2()8)
are missing from the desired function, give to the cori'espoiKhng
' See Jordan. Handbuch tier W'rmo.-sunjrskundc, I5d. I, jiar. lii. or Wrigiit's
.\djustmont of Observations, page 229.
174
PRACTICAL LEAST SQUARES
coefficients, /, the value zero. Now, referring to the condition
equations (56) or (59) for the notation, and representing the
original weights of the observations by Wi, W2, Wz, . . ., we form
the terms, m' + l in number.
«/
Y
Lw\
iwl
-1' \-\ -m
WA IwA iwA
(269)
Writing these terms in order in an additional column, between the
constant and check in the normal equations, or in place of the con-
stant column if the equations have already been solved, the elimina-
tion process is carried out for this column and its own new check
column, including the last step, for the term [ff/w]. The final sum
for the last term will be the reciprocal of the weight of the func-
tion,^ and from this and the mean square error of a single observa-
tion of unit weight, the mean square error of the function is ob-
tained by (237),
ep-
V^
(270)
IVf^
It will be seen that even though the weights of the original observa-
tions were equal, those of th(^ adjusted values may be unequal.
But if the original weights of certain of the observations are equal,
and also a, h, c, . . . are the same for all of these observations,
then the weights of the corresponding adjusted values will be
equal, since the /'s in these cases are unity. This will appear in
the following examples.
150. Examples: Differences of Elevation, (a) As an illus-
ti'ation of the method, lot us apply it to the problem of Art. 77,
Adjustment of Levels, and dotermino the mean square error of the
difference of elevation of the bcriehmarks A and D. The function
is, therefore,
F=Fo+F8 (271)
' Expressed in the Claussiuii form, this is,
_1_
[nP
2
'hf
2
r^'' ;1
Ijf
ir
— •1
(/■
IV
(1-
-•1
(■''
9
?/'
ir
■?/■
PRECISION OF OBSERVATIONS AND RESULTS
175
and the reciprocals of the weights of the observations are,
1/^6 = 2, and l/ws = l. Then /o=+l, and /8=+l, the other
/'s being zero. Referring to the condition equations (73), we find
^6 =
-1
08 =
0
be =
-1
68 =
0
cq =
0
C8=-
-1
Thus we obtain the terms.
of
= -2,
= — 9
= -h
= +3 (272)
UtiUzing the solution of the normal equations already made in
Art. 77, we write the /-terms in a new column and form a new check
column. To illustrate the second method of obtaining [wv^], stated
near the middle of page 173, the constant column will be included,
with the addition of the zero for the last term. The solution,
then, is as follows :
I
2
3
4
2
5
.1 i B
i
C
(/)
Check
Const.
+ 12
+6
+ 7
-4
0
+5
_2
_2
-1
+3
+ 12
+ 11
0
_ 2
+0.09
+ .08
- .08
0
a)x(-o,i2)
^2) +(5)
(I)X(+4/12)
(II)X(-2, 4)
(3) + (6) + (7)
fI)X(+2 12)
ai)X( + l/4)
(III)X
(+1.17/2.67/
+7
-3
0
+2
_2
+ 1
+ 11
- 6
+ .08
- .045
II
+4
+2
-1
+ o
+ .035
3
6
" 1
+5
-1.33
-1
-1
-0,67
+ 0.50
0
+ 4
-2.50
- .OS
+ .0300
- .0175
III 1
+2.67
-1.17
+ 1.50
- .0()75
0
- .0007
- .0003
- .0017
-0.0027
4
s
9
10
+3
-0.33
-0.25
-0.51
+2
+ 1.25
+0.66
IV
4-1 01
+ 1.91
176
Therefore,
and
PRACTICAL LEAST SQUARES
— =1.91
(273)
-[wv-]= -0.0027
From (76), wc evaluate (267) and obtain,
[M;y2]=_o.ooi2+0.0018+0.0021 = +0.0027
which agrees with the above and with the value determined
directly from the table of corrections, page 69.
Then,
l\om,2] /n 0097
(274)
7n' M 3
and from (272) and (273) we have,
= 0.03
e. = -;l=^£L = 0.02
Vw VI. 91
(275)
so that the best value of the difference of elevation from A to D,
from the given data, is, with its mean square error,
(F) = -6.36±0.02 (276)
(6) As a variation of the above, let us determine the precision
of the adjusted difference of elevation, A—F. The function is,
F'=Fo (277)
and from the data above,
1/wg = 2, /(3=+1, a6=— 1, 6g=— 1. and cg = 0.
so that
r^,/i r/i/i Vnfi r/ri
= +2.
The; only changes in the solution, therefore, are in the last two of the
/-terms. The result is
—-=1.41 (278)
whence,
e.-' = -4._ = ^-^ = ().02 (279)
In view of the statements at the close of Ai't. 149, it is evident
'oj
= -2,
V"
= -2,
'cl
= 0,
;r
L w_
livl
Iw]
Iwj
PRECISION OF OBSERVATIONS AND RESULTS 177
from an inspection of the tabulated condition equations (73)
that the weight and mean square error of V2 will be the same as
those of Vq, since these two columns in the table are alike.
151. Precision of Computed Quantities. As a result of the
adjustment of observations, the adopted values of the unknowns
are likely to be used in the computation of other quantities which
may be expressed as functions of the unknowns. Having inves-
tigated the precision of the unknowns, it may be desired to ascertain
the effect which the uncertainties in these values would have
upon the quantities computed from them.
For example, suppose the diameter of a cylindrical bar of
steel is measured with micrometer calipers at various points, from
which the mean diameter and its probable error are obtained;
the cross-sectional area computed from this mean diameter would
have a resulting uncertainty. Also, if the bar were tested in a
tension machine, the breaking stress per square inch would be
uncertain to a corresponding degree as a result of the uncertainty
in the measured diameter and computed area.
Again, suppose one side and the adjacent angles of a triangle
have been measured independently, resulting in an adopted mean
and a mean square error for each. If another side be computed
from these data, it will have an uncertainty due to the discrep-
ancies among the original measures of the given side and angles,
that is, to the uncertainties of the given means.
It must be emphasized that the determination of the best values
of the computed quantities is not involved in this question. Hav-
ing adjusted the observations, the resulting adopted values are
the best ones, as far as our knowledge goes, and quantities com-
puted fi-oin them are also the best we can determine froui the
given data. We are now concerned only with the precision of the
coniput(>d quantities, not witli the determination of the ([uantities
themselves.
Our j)i-o])leni is to determine the mean square (or probable)
error of afundion. of indepc^ndent, adjusted quantities of which the
mean scjuare (or probable) errors ai'e given. It will be ('onv(>ni(Mit
to assume that each of these given, adjusted values is the mean of
a lari'-c iiuiuIxm- of obs(M'\-ations. and that th(> cofresnoiuliim- indices
178 PRACTICAL LEAST SQUARES
of precision were determined by the formulas of Art. 141, although,
of course, they might result from indirect observations.
The errors in a linear function of independently observed quan-
tities occur in accordance with the same Law of Error as those
of the quantities themselves.^ Thus, the errors in the mean of a
set of observations occur in accordance with the usual Law of
Error. Such means, therefore, may be treated as original obser-
vations, as far as the occurrence of errors goes, as long as they do
not involve the same original observations, in which case they
would no longer be independent.^
This subject is usually called the Propagation of Error. We
shall consider it as divided into two parts, — the simple influence
of errors of one kind or character, and the compound effects of
errors of different kinds or resulting from different causes.
152. Simple Propagation of Error. Before attacking the gen-
eral case, a few special forms of functions will be considered in order
to illustrate the process of reasoning. Let F represent the func-
tion of the independent, adjusted quantities, x, y, . . . whose
mean square errors are ex, ey, . . . Let the original observations
of X be represented by Mi, M2, . . ., those of y by M'l, M'2, . . .,
etc., and let the true errors of these observations be represented
respectively by Ai, A2, . . ., A'l, A'2, . . ., etc. We may assume
an equal number of observations for each quantity, for simplicity.
(a) Consider first the sum or difference of two quantities.
Then,
F = x±y (280)
Taking the separate observations in pairs, the first of x with the
first of y, the second of x with the second of y, etc., each pair
gives a value of F, say Fi, F2, . . . Thus,
Fi=Mi±M'i
F2-M2:hM'2 (281)
^ For proof ( f this, see Wriglit and Ilayford, Adjustment of Olisorvations,
Art. 13.
- See Chauvenel par. 23, for treatment of tlie case of a function of functions.
PRECISION OF OBSERVATIONS AND RESULTS 179
Now, if we add to each M its true error, A, the resulting value of
F must be corrected by its corresponding error, and,
i^i+A^, = (ilfi+Ai)±(M'i+A'i)
/^2+A^,= (ilf2+A2)db(M'2+A'2) (282)
Subtracting (281) from (282), one by one, we have,
Af, = Ai±A'i
Af., = A2±A'2 (283)
Squaring each equation, adding, and dividing by their number, n,
M^[A2]^2[AAq^[A'2] ^2g^^
n n n n
But in a large number of observations, the positive and negative
errors occur with equal frequency, so that the sum of the products,
[AA'], would approximate to zero, — certainly so in comparison to
[A2] + [A'2], sothat,
l^.I^l+t^:!] (285)
n n n
or,
€/ = 6.2+e/ (286)
Obviously, the above process would apply likewise to a similar
function consisting of any number of quantities connected by
j)lus and minus signs, so that for
F = .T±i/±2± . . .
wc can write,
6/=e/+e/+6.2+ , , ^ (287)
and
6^ = Ve/+e/+e/+ . . . (288)
From the constant ratio of the })i-()babl(' error to the mean square
eri'or, it follows that
/V- = r,2 + /v- + /%2+ . . . (289)
This jii'incipk^ is vei'v iinpoilant and often used. Note that
tli(> si^i'iis in (287) and (2S9) an^ all ])ositive. The uncertainty in
tlu^ sum of two or more (juantities is thei'cfore the samc^ as in their
(liffei'('n('(>.
{U) In 1h(> next case, \v\ F^a.r^hijzt . . . (290)
180 PRACTICAL LEAST SQUARES
in which x and y are adjusted values from observations, and a
and b are known constants. As in (282), we may write,
Fi+A^. = a(Mi+Ai)±6(M'i+A'i)+ . . .
/^2+A^, = a(M2+A2)±6(M'2+A'2)+ . . . (291)
whence, as in (283),
Afi = aAi±&A'i+ . , .
A^, = aA2±6A'2+ . . . (292)
Squaring, adding, dividing by n, and omitting the products as
before, we have,
M^J^+^m+ . . . (293)
n n n
That is,
e/ = a2 6x2+&2e,2+ . . . (294)
or,
e^ = Va2 6x2+62 e,2+ . . . (295)
(c) Now we shall consider the general case in which F is amj
function of the quantities, x, y, z, etc.,
F=f{x,y,z, . . .) (296)
Since x, y, z, . . . are adjusted values, they may be assumed to
be nearly correct, so that their errors arc very small; let us rep-
resent them by differentials. Then, if A^- be the true error of F,
we have,
F+^J.=J{x+dx,y+dy,z+dz, . . .) (297)
Expanding this function by Taylor's Theorem, and omitting
terms which involve squares, products, and higher powers of the
differentials, we obtain,
F+^,=J{x,y,z, . . .)+%lx^-%hj-\-%h-\- . . . (298)
ox ay oz
whence, subtracting (290),
A,. = ^r/a: + ^r/7/+^f/2+ . . . (299)
9.T dy dz
This ccjuation has the same linear forin as (292), so that from (294)
we have directly,
PRECISION OF OBSERVATIONS AND RESULTS 181
in which the partial derivatives of the function correspond to the
constants, a, b, c, etc., of (294).
Thus we may state the general Rule: Express the given func-
tion in literal form. Differentiate it partially with respect to each
quantity for which the mean square error is given. Substitute in
these derivatives the given quantities (without reference to their
mean square errors, of course). Substitute in (300) and obtain ep-.
153. Example: Precision of the Mean. We shall now apply
the foregoing principles to determine the mean square error, to,
of the mean of n observations, when the mean square error of a
single observation is e.
The expression for the mean is,
F^^^^'l+Ml+ . . . +M. (301)
n n n
where M\, M2, etc., represent independent direct measures or
observations of the unknown quantity. This function has the
form of (290) and a = b = c^ . . . =l/n. From (294), there-
fore, we have
eo^ = \e^+—,e--h—,e^+ . . to n terms (302)
that is.
6(r-M-J=^ (303)
eo= --;- (304)
Vn
which states the very important principle that the precision of the
mean varies directly as the square root of the number of observa-
tions. To double the precision, — that is, to reduce the mean
square error of the mean to one-half its size, — it is necessary to
have four times as many ol)servations.
154. Compound Propagation of Error. The uncertainty in a
computed quantity may result from several sources which are not of
the same nature, and it may be impossible to state th(> quantity as
a single function of all these sources of error. For example, the
measiu'cment of a line with a steel tape involves the uncertainty
in the length of the tape itself and also the errors in the process of
measurement. We cannot express the length of the line as a func-
tion of the length of the tape and the " process of measurement!"
182
PRACTICAL LEAST SQUARES
From (283) to (287), we can state the principle that when the
error in the computed quantity is the algebraic sum of independent
errors from different sources, the total mean square error of the
computed quantity will be the square root of the sum of the squares
of the separate mean square errors of that quantity due to the
various causes.
In any given case, therefore, we determine the mean square
error of the computed quantity or function resulting from each
source, separately, by the methods of Art. 152, and then take the
square root of the sum of their squares as the total mean square error.
We shall now illustrate this subject by a series of typical
examples.
155. Examples: Propagation of Error. (1) The following
measures of the diameter of a cylindrical test-piece of metal were
made by means of micrometer calipers. The piece was then
broken in a testing-machine at a load of 20,000 lbs. Find the unit
breaking stress and its mean square error due to the uncertainty
in the measured diameter, D.
Inches
V
i
Inches
V
V-
Inches
V r-
0 . 6252
2
4
0.6251
1
1
0.6248
2
4
46
4
16
52
2
4
45
5
25
42
8
64
54
4
16
52
2
4
48
2
4
55
5
25
56
6
36
121
I2
= 10 ; Mean = 0 . 6240+0 . 0010 = 0 . 0250 = D
208
[.2] = 203, 60^ = ^ = 1.54, 60=1.24
Diameter (/)) =0.6250 ±0.0001 2. The function is,
20,000
Unit Ijreaking stress ■-
Area of section
20,000
80,000/ 1
3.14 \D-/
= 05,300 lbs. per sq. in.
(305)
PRECISION OF OBSERVATIONS AND RESULTS
183
Differentiating (305) with respect to D,
^_ 80,000
dD
3.14
- 160,000
-2\ -160,000
I>3/~3. 14X0.6253
= -208,900
0.766
From (300),
6^ = 208,900X0.00012 = 25.
Thus, the unit breaking stress = 65,300± 25 lbs. per square inch.
The uncertainty due to the variation among the measures of the
diameter is therefore neghgible when it is remembered that the
breaking load is seldom required within a range of a hundred
pounds.
(2) The length of a 50-meter tape is determined by comparison
with a 5-meter standard bar which is surrounded by chipped ice to
control its temperature. The length of the bar, as determined
from its standardization, is 5 = 5. 000060 ±0.000006 meters.
The following measures are made of the difference between the
length of the tape and ten lengths of the bar, the former being the
longer. It is required to find the length of the tape and its mean
square error due to the uncertainty in the length of the bar and
to the errors of measurement. The temperature is assumed con-
stant. The unit is in the sixth place of decimals, that is, a micron.
Interval (K)
.'■
r-
Obs.
0 . 002533
16
256
666
117
13689
529
20
400
444
105
11025
461
88
7744
5.53
4
16
638
89
7921
5137
18
324
Mean, 2549
41375
^ 41375
60"= ^^ =739.
oo
Interval (/v) = 2549±27
184 PRACTICAL LEAST SQUARES
The function is,
Length of tape (L) = 10 B-\-K (306)
= 50.000600+2549
= 50.003149 m.
This function corresponds to (290), so that, from (294),
6^2=100X62+739 = 3600+739 = 4339.
ez. = 66
Therefore, Length of tape (L) = 50. 003 149 ±0.000066 m.
An important principle is illustrated here. The larger source
of error in the length of the tape is that due to the error in the
length of the bar, amounting to ten times as much as the other.
It would be useless, therefore, to increase the above number of
observations with the idea of increasing the precision in the
length of the tape, since this part of the total error is almost
negligible. On the other hand, the above set of observations
might be diminished considerably without seriously affecting the
result. For example, suppose there were but one-half as many
observations, namely, 4. Dividing the number of observations
by 2 increases the square of the mean square error twofold. Thus,
we should have 60^=1478, and 6^2 = 3600+1478 = 5078. Then,
ei = 71, which is very little larger than 66. It must be remembered
however, that the number of observations should be sufficiently
large to justify the assumption that errors of observation follow
the Law of Error.
(3)^ A comparison of the two following cases will be instruct-
ive, (a) A line 400 feet long is measured with a 100-foot tape
of which the mean square error is 0.004 foot. The resulting
mean square error in the length of the line will be 0.016 foot, since
L = 4T.
(6) The same line is divided into four 100-foot sections and
each section is measured with a different 100-foot tape of which
the mean s(|uare error is 0.004 in each case. The resulting mean
1 Adapted from Craiidall's Cieodpsy and Least Squares.
PRECISION OF OBSERVATIONS AND RESULTS 185
square error in the length of the hne will be V4(0 . 004) = 0 . 008 foot,
since the function is, L = Ti + T2 + 7'3 + T4.
In the first case (a), whatever the true error in the tape may be,
it is constant and its effect is cumulative. In (b), on the other
hand, the actual errors in the different tapes are not the same even
though their mean square errors happen to be equal, and in con-
sequence they are likely to be both positive and negative so as
to neutralize to some extent. Therefore, the resulting error in
the length of the line would be smaller than in the former case. It
is important that this principle be well understood.
(4) Let it be required to compute the length and mean square
error of the side, h, of the triangle, A-B-C, from the side, a, and
the angles, A and B, given with their mean square errors as follows :
a = 4268 . 344 ±0 . 008 meter,
A = 56°37'42.4"±0.6"
5 = 70°26'54.3"±0.3"
The function is,
b = '^'^ (307)
sm A
from which we obtain, using the above data,
6 = 4816.349 m.
Differentiating (307) with respect to a, A, and B, in succession,
and reducing by means of (307),
9^ = :'^^ = ^ = 1.128
da sm A a
dh -a sin B cos A , , . ...^^
= . , , = — 0 cot .4 = — 3172
dA sm-' .1
dh a cos B
= -. = b cot y? = W 10
dB sin A
Subslituling in (300), and noting that it is necessary to nuilliply
e.i and e,i by sin 1" ( = 0.00000.")) in order to reduce them to
186 PRACTICAL LEAST SQUARES
abstract quantities so that each term may be expressed in the unit
of length, we have,
e,2=/^y,„2_|_(^ cot A)HeA sin l")2 + (6 cot Bfies sin 1")^
= (1.128X0. 008)2 + (3172X0. 6X0. 000005)2
+ (1710X0.3X0.000005)2
= (0 . 0090)2 + (0 . 0095)2 + (0 . 0026)2
= 0.00017801
and
€0 = 0.013
whence,
6 = 4816. 349 ±0.013 meters.
(5) Find the mean square error in a single measurement of an
angle, direct and reversed, with a direction theodolite having
three microscopes. Each reading consists of the mean of the
three microscope readings corresponding to a pointing upon one
object, and a measure of the angle is the difference between the
readings upon the two objects limiting the angle. This process is
repeated in the reversed position of the instrument and the mean
is taken. Suppose the mean square error of a pointing of the
telescope upon an object to be, ep = 0.04"; that of a reading of a
microscope to be, er = 0.00"; and that of a graduation-mark on
the circle to be, e^ = 0.03". The error in each microscope reading
will be the algebraic sum of the error of setting and reading the
microscope itself and that of the graduation, so that the moan
square error due to both causes will be v(er+e/). Then the
mean square error of the mean of the readings of the three micro-
scopes will be
V(er'+e-') \/(0.0045)
e«= --^--^ '= ■ -^ ^=V0.0015
v/3 \'^3
The error in a reading upon one object will be made up of tlu^ (^rrors
due to all three causes, that is, to the a])ov(> c()m})ined error and
the error of a single pointing, or.
e„ = V(e,,-+ e,r) = V (0 . 0015 + 0 . OOK)) = v'() ■ 00:5 1
Finally, tlu^ mc^an square error of the difference of the readings on
COMBINATION OF COMPUTED QUANTITIES 187
the two objects, which is that of the direct measurement of the
angle, will be,
V(eo2+6o-) = Vo.0062
and that of the mean of the direct and reversed results will be
,,=^^^»? = VOa31 =0.056"
V2
(6) A line 1000 feet long is measured eight times with a 100-
foot tape, and the mean square error of the mean of the eight
measures is found to be 0 . 004 foot. If the mean square error of the
length of the tape (resulting from its standardization) is 0.001,
what is the mean square error of the line, due both to errors of
manipulation and error in the tape length?
The mean square error of the line due to the tape error is
10X0.001=0.010. Since the total error is the algebraic sum
of both kinds, the mean square error due to both causes will be the
square root of the sum of the squares of the separate mean square
errors, that is,
e^ = V{0 . 0042+0 . 010-) =0.011
COMBINATION OF COMPUTED QUANTITIES
156. Weights from Mean Square or Probable Errors. In
Art. 143, it was demonstrated that weights are invcr.sely as the
squares of the corresponding mean square or probaljle errors.
Thus it is possible to combine the results computed from different
observations of a certain c^uantity, using thorn as weighted obser-
vations, when the mean square errors of these results are known
so that their relative weights may l)e determined. For example,
a certain angle in a triangulation may have been measured several
times, with a resulting nuvui and moan square error. Subse-
quently, in another season, perhaps, another s(M-ios of measures of
the same angle may be made, giving a cHffenMit result and mean
square (>rror. By giving to each result a weight equal to the
reciprocal of the sc}uai'e of its mean scjuare error, th(» weighted
mean of the two rosuhs may be taken as the best value of the angle
from all of the availa])le data.
188 PRACTICAL LEAST SQUARES
157. Limitations. It is obvious that this method assumes
that all of the original observations in the various groups are of the
same character, so that if they were known their mean could
reasonably be taken. The conditions under which they were
made should be similar, and especially is it assumed that constant
or systematic errors affect all of them in the same way.
On the other hand, it is seldom that these conditions are
fulfilled with any great degree of certainty. Frequently, nothing
at all is known about the observational methods or circum-
stances, except what is indicated by the mean square errors
as to the consistency of the original observations. Even in such
a case, however, it is probable that the weighted mean will be
as good as, or better than, any of the component results, so
that the method should not be discarded without careful con-
sideration.
Of especial importance in this connection, is the case in which
the observations resulting in one of the given values are known to
be of much greater precision than those which resulted in the other
value, without regard to their respective mean square errors.
For example, an angle might be measured with a direction theodo-
lite reading to a single second, and again by means of a transit
reading to half-minutes. Here, the judgment of the computer
may determine what weight, if anj^, shall be given to the transit
result in comparison with the other, in spite of the mean square
errors, provided, of course, the number of observations made,
with the theodolite is sufficient to reduce the effect of the
accidental errors. Should the two results be close together, how-
ever, the weights given by the mean square errors may still be
satisfactory.
When the results being compared are separated by a consider-
able interval in comparison with the given mean square errors,
the presence of systematic error may be indicated and should be
investigated. If the difference is not too great to be a reasonable
accidental error of observation, it may be considered safe to accept
the weights given by the mean square errors. But if the differ-
ence is too large to be thus considered, and the mean square errors
arc much smaller, there may be no reason for believing one of the
COMBINATION OF COMPUTED QUANTITIES
189
values to be* nearer the truth than the other, so that the arith-
metic mean of them may be adopted as the best value. Here,
again, the judgment of the computer must determine the method of
adjustment.^
158. Example: Weighted Mean of Computed Quantities.
Three independent series of observations give the following results
for the value of an angle; what is the best or most probable value
of the angle from these data?
Means, (xo)
e
6^
W = l/t''
w
WXti
72° 47' 43.18"
44.01
43.74
±0.06"
.10
.08
36
100
64
0.028
.010
.016
14
5
8
44.52
20.05
29.92
27
)94.49
3.50
Weighted mean, 72° 47' 43.50"
159. Precision of the Adjusted Value. Although the problem
has the nature of the determination of the weighted mean, the pre-
cision of the resulting value should not be computed as in the
case of the weighted mean owing to the small number of given
quantities, in general, and the fact that their individual mean
square errors or probable errors are given. On the contrary, the
method of propagation of error should be used. The two methods
will not usually give the same result. It is to l)e expected that the
final value will l)e better than the separate given values, and it
should generally have a smaller mean scjuare error although this
will not always be the case.
160. Example: Precision of the Adjusted Value. Applying
this process to the alcove example of Ai't. 158, we have the func-
tion,
X = -4 n4.ri+5.r2 + S.r:j)
^ Seo Johnsf)n's 'llu^ory of ]>rors, Chap. \'II, on this .^ubjert.
190 PRACTICAL LEAST SQUARES
whence, from (300),
1
27
1.3652_1.17^
272 272
l-l^- = 0.04
6.v2 = 74[(14X .06)2 + (5X .10)2 + (8X .08)2]
ex=^-
27
SO that the adjusted vahie is, 72° 47' 43.50"±0.04".
CHAPTER IX
CONCLUSION
161. Rejection of Observations. It is generally conceded
that an observer has the right to reject any observation, at the
time of making it, if he has reason to believe that he made a mis-
take in his setting or reading, or if the conditions were temporarily
so unfavorable as to indicate that the result was quite unreliable.
His attention may be drawn to the questionable observation
merely by its being discordant among the others of the series;
or he may question the observation as he makes it and mentally
decide to reject it if it proves to be very discordant. His power
is absolute but he is expected to exercise it with good judgment
and strict impartiality.
On the other hand, when the observations have been approved
by the observer and are turned over to the computer, or when
sufficient time has elapsed that the observer ceases to recall the
particular conditions under which each of the observations was
made, then the record must be regarded as inviolable, and must
not be changed without good reason,^ and this reason must be
evident from the records themselves.
If the observer has noted the unfavorable conditions and has
not indicated a resulting smallcM' weight for the corresponding
observation, the computer may feel justified in assigning such a
weight if tlu^ obsei'vation is ckvirly discordant. Ilow(>ver, if this
is necessary, it should have been dour by tlu> ()bs(M'V(n' in the field,
and the computer may wisely refi'ain from thus int(M'fering witli
th(> record unless with the consent of the obscM'ver himself, on
the gi'ound that this would haw been his action in the field.
' It is a rifiid rule t hat an oriuiiial I'ccord sliotild nov(M' b(> crasotl or ol)S('ure(l.
Chanel's sliould he so made as to show ch'arly tliat they an> clian^os, with
date and initials of the coniputcM-. and so as to leave the oi-iy;inal data legible.
(ienerall>', the oi'iiiinal will he in jxaicil, and not(\-: and computations will l)u
in ink. Jted ink may well be nsed for annotations.
191
192 PRACTICAL LEAST SQUARES
The assignment of weights to various observations is closely
associated with the question of the rejection of observations, since
a weight of zero is equivalent to rejection, and a diminished weight
means a partial rejection.
162. Criteria for Rejection of Observations. While the author
is of the opinion that weighting and rejection should be based
upon judgment rather than mere discrepancies among the ob-
servations, many writers and experienced computers have advo-
cated the rejection of all observations which deviate more than
a certain amount from the mean of the set. The mathemat-
ical basis for determining this maximum deviation is known as a
Criterion for the Rejection of Observations. Several of these methods
have been devised,^ but the following has the merit of simplicity.
It being assumed that the observations conform to the Law
of Error, the number of errors, or residuals, greater than a certain
size, to be expected in the given set, will be found by using Table
III, Appendix F, as stated in Art. 175. The table shows that the
probability of an error less than four times the probable error of a
single observation is 0.99; that is, 99 out of 100 residuals should be
less than that amount and only one out of 100 should be greater.
Therefore, if a greater residual occurs in a set of, say, 20 to 30
observations, it might be rejected as indicating a mistake. Having
computed the probable error, r, of a single observation, for the
given set, any individual observation whose residual from the
mean is greater than or equal to 4r, would be rejected, according
to this assumption.
Evidenth', the adoption of a certain criterion is a matter of
estimation and preference. The above value, 4r, would be con-
sidered conservative by many computers who believe in any kind
of a numerical criterion; 'Ar is sometimes used. Even the novice
will immediately suggest that the unusually large error might
h(i])pcn to occur in the small series of observations. If but one
veiy large residual occurs in the set, there may be more reason
for n^jecting it than if it be accompanied by a correspondingly
large one of the opposite sign, since the pair would neutralize
each otlun', to some extent, in the mean.
'Sec ('h;iu\'('iH't , Practicul and Spherical Astronomy.
CONCLUSION 193
163. Methods of Observing. One of the most important
uses of the Method of Least Squares Hes in the investigation of
methods of observing, with the idea of avoiding or ehminating
the effects of constant or systematic influences, of segregating the
sources of error which produce the greatest effect so that these
effects may be diminished, and of reducing the cost of securing
the desired degree of precision. ^
In Art. 154 it was shown how various sources of error combined
to affect the result; therefore, in arranging the observations, spe-
cial attention should be given to decreasing the errors which have
the greatest effect, since the final precision is dependent but little
upon the small errors. In reducing the errors from a certain
source, the design of the instrument and its support may require
study as well as the method of using it. Very important improve-
ments in instruments have resulted from the careful study of the
occurrence of the errors of observation.^
Constant and systematic errors may be due to the conditions
under which the observing is carried on. When such is the case,
it is desirable to so arrange the observing program that these con-
ditions will vary during the observations through a complete
cycle of changes, as far as practicable, in order that their effects
may neutralize one another, at least partially.
Finally, the matter of cost must be considered. This will
depend largely upon the number of the observations and their dis-
tribution during the day, after the instrumental equipment has
})een determined upon.
164. Precision Desired and Number of Observations. In
])lanning the obsc^rving ])rogi-ain, having a d(>finito end in view, it
is advisable to decide upon the degree of pi-ecision which is to be
sought in the result. This will ilepend to some extent upon the
insti'unuMits or apparatus available, but, with a given instrument
and an individual observer, the method of observing and the
number of ilie obstM-vations become of great imj)ortance in deter-
' For a more (wtcndod trcatinciit of this suliject. tli(^ r('a<lt>r is rc^fcrred to
Wright and Ilayford, Adjustment of Observations, ("ha!)ter IX.
-A notal)le iiistance of this was th(- (h-sigii of tlic Coast and (leodetic Survey
Precise Level in 1<)()(), l)y Mr. .1. F. Hayford, Chief <,f th(^ ( omputing Divi-
sion, and Mr. 11. Ci. I'iseher, Cliief of the Insiriiinent I)i\ is!o:i, U. S. C. i^ C!. 8.
194 PRACTICAL LEAST SQUARES
mining the precision. The observing program will frequently
take the form of a number of units, or parts, all of which are
alike with the exception of a change in the position of the instru-
ment, as in the case of horizontal angles measured with a direction
theodolite.
To attain the desired precision, then, the total number of obser-
vations must be considered. As a result of experience or experi-
ment,^ the precision (indicated by the mean square error, perhaps)
of each elementary observation is ascertained, and from these,
the precision of a unit observation. Then the number of observa-
tions necessary to obtain the desired precision in the result may be
computed from the relation that the precision of the mean varies
as the square root of the number of observations (Art. 141, page
163). That is, to double the precision (to divide the mean square
or probable error by two) four times as manj' observations must be
made. But how far can this process be continued? Is it possible
to reach any degree of precision by simply multiplying the observa-
tions?
165. Ultimate Limit of Precision and Accuracy. While in
theory the precision of the mean can be increased indefinitely by
increasing the number of observations, experience shows that a
limit is soon reached, beyond which it is not worth while to con-
tinue the observing; the theoretical increase in the precision as
indicated by the smaller probable error, for example, would
become quite misleading. Furthermore, after passing a certain
point, the number of observations would have to be enormously
increased in order to produce a very small decrease in the probable
error, so that this process would be very wasteful of time and
energy, and it is doubtful if the results would be much better.
After all, accuracy is desired rather than precision. The
observations are not made for the purpose of enjo\'ing the labor,
but in order to ascertain the truth as far as practicable. It is a
well-known fact that the mean of a small number of very consistcMit
observations, showmg a vcm'v small pro])able error, may be farther
' A theoretical discussion of the hinitations of the human eye in making
observations, antl lh(> increased j)ower resulting from jirojierly designed instru-
ments, will be found in Jordan, Ilandbuch der \'ermessungskunde, Band II,
§45.
CONCLUSION 195
from the truth than that of a larger number of observations which
vary over a considerable range. Cases can be cited in which a
value adopted as a result of many observations, by different
observers, extending over a long period of time, has been proved
to be incorrect by an amount greater than many times the prob-
able error. Of course, the conclusion is that we must not lose
sight of the fact that, however consistent the observations may be,
large systematic errors may be present and the observing methods
may not be such as to eliminate them, so that they directly affect
the results.
As to the limiting number of observations, then, we can safely
state that this should be large enough and so distributed as to
cover varying conditions as completely as practicable. Natu-
rally, it will be different in various kinds of work. However,
changes in the instrument and its supports are likely to take place
if the observations extend over too much time, so that it is gener-
ally advisable to observe as rapidly as is practicable without a
sacrifice of precision.
166. Indication of Systematic Errors. In order to discover
the presence of systematic errors, a careful study of the residuals
is essential. Unless the conditions causing these errors change
during the course of the observations, the errors fall into the class
of constant errors and will not be indicated at all by the discrep-
ancies or residuals. In this case, a different method of observing
might reveal them when the results of both methods were com-
pared.
By plotting the residuals in chronological order some regularity
or law may be rioted in their occurrence. Positive and negative
residuals may occur in separate groups or a curve drawn through
the i)lotted points may show a periodic character. Again, the
nunil)oi's of residuals of the various sizes may be plotted as in
Art. 17, to form a Curve of Error, and if the resulting curve
differs considerably at certain points from the theoretical form,
which may be ]:)lotted from Ta})l(> II or III in App(Mi(hx F, the
pres(Mice of syst(Mnati(' ei'rors may be indicated. Having thus
investigated the occun-(>nce of the residuals, it remains to seek
changes in the observing conditions which correspond to the
196 PRACTICAL LEAST SQUARES
variations in the residuals. The location of such changes should
serve to point out conditions responsible for part or all of the
systematic errors so detected.
167. Treatment of Discordant Observations. When the dis-
crepancies in a set of direct observations are unusually large,
the lack of precision will be indicated by a large probable error
or mean square error, and the mean remains as the best value
obtainable from the given measures. It sometimes happens,
however, that different sets of observations of the same quantity
will yield results which are so discordant as to indicate the pres-
ence of constant or systematic errors in one or both of the sets.
The problem may be further complicated by the fact that the
precisions of the results may be considerably different, so that if
their weighted mean were taken, as in Art. 156, it would give a
decided preference to one of them. The question arises as to
whether the results may not be so far apart as to make it advisable
to neglect their relative weights altogether and to take their
simple mean arbitrarily. This course is sometimes advocated.
Obviously, this is a matter of judgment rather than Least
Squares, and such action should be preceded by a careful investi-
gation of all the circumstances. However, it may be reasonably
contended that if such discordant results are to be used at all a
small difference in the adopted value would be of little moment
and the regular Least Squares process may well be followed
without considering the case as an exceptional one. Should
conditions or checks be found which would be satisfied much
more nearly by one of the results than by the other, the problem
is thereby altered and becomes one involving the assignment of
weights or perhaps the rejection of observations. The judgment
of the (!()mi)uter must be the determining factor.
168. Arbitrary Adjustments. The principles outlined in the
foregoing chaplcM's, especially in Chapters V and VI, will be found
of assistance in some problems where it may b(; deemed sufficient
to approximate to a rigid adjustment by assigning corrections to
the ()bs(M-ved (juantitios arbitrarily. While such a method can
hardly l)e defended in the hands of the comput(;r who is con-
versant with Least Squares, still it must be admitted that such a
CONCLUSION 197
computer is the only one who could be expected to carry out an
arbitrary adjustment consistently and reasonably. The usual
difficulty arises in satisfying all of the necessary conditions at the
same time without a distribution of the corrections which is clearly
unreasonable.
In certain problems, however, a distribution of arbitrary cor-
rections may be of use in preparation for a rigid adjustment. The
method consists in applying to the observations such preliminary
corrections, resulting from a detailed study of the condition
equations, as will reduce the amounts of the final corrections.
This advance study requires a clear understanding of the field
conditions as well as the methods of adjustment, but when care-
fully carried out is likely to diminish the labor of the computation
and to improve the adjustment by reducing the numerical quan-
tities involved. The method is analogous to the assumption of
approximate values for the unknowns in the adjustment of indirect
observations, Chapter III.
169. Use and Abuse of Least Squares. In view of the crit-
icisms which are sometimes directed at the use of Least Squares
for the adjustment of observations, a few words on the subject
may not be out of place here. While it is unquestionably true
that the method is sometimes used in an unwarranted manner, the
real difficulty probably arises from the placing of erroneous inter-
pretations upon, or the drawing of unreasonable conclusions
from, the results of the adjustments.
A great deal of misunderstanding in the minds of persons
unfamiliar with the fundamental principles of the method has
resulted from the use of the term " probable error," and such per-
sons are too apt to blame the method for the fruits of their misuse
of it. It is unfortunate that this term has come into use, since its
meaning in Least Squares is a technical one and not what would be
expected from the ordinary use of the word " probable." Some
of this trouble, to say the least, would have been avoided by using
the " mean square error."
A common criticism relates to the use of Least Squares in
connection with a very small number of observations, even as
small as two. The reply may well bo, " Whor(> is a bettor method?"
198 PRACTICAL LEAST SQUARES
The intelligent computer does not place the same reliance upon a
very small number of observations as upon a larger one, but having
only the small number he uses them as best he can. Hovv^ever, to
place great confidence in the precision of the mean of two observa-
tions is certainly questionable, although even that precision may be
very useful, in spite of its limitations, for purposes of comparison.
While investigations of the precision of observations and
results have been thus criticised, little or no objection has ever
been raised against the use of Least Squares for determining the
best values of the unknown quantities. Its advantages for this
purpose are evident even to those who are not familiar with its
details. It provides a method of adjustment which is consistent,
definite, and adaptable to the various kinds of problems and con-
ditions, and which conforms to the facts as to the occurrence of
errors of observation. Generally, also, it is simpler than an arbi-
trary adjustment; certainly it is more reliable.
170. Adjustments not Infallible. The beginner must not
make the error of assuming that the results of an adjustment
are correct. At the risk of repetition, this principle is emphasized,
—that the results are but approximations to the true or correct
values, the best obtainable from the given observations. Should
the observations be affected by constant errors, the results will be
likewise affected, without regard to their precision, which is deter-
mined from the discrepancies among them.
Also, as has been pointed out, different adjustments of the
,same observations by slightly different methods, perhaps, may
yield results which are not exactly the same, owing to the fact
that different sets of numerical quantities are used. If the
computations are carried out to one decimal place more or less,
slight variations in the final values may similarly occiu-. But it
should be kept in mind that any one of these various adjustments
will probably satisfy the requirements of the pro})lem within the
uncertainties among the observations, so that any one of them
can safely be adopted.
171. Other Laws of Error. When applying the method of
Least Squares to a new class of problems, it becomes necessary to
investigate the occurrence of the errors, particularly when these
CONCLUSION 199
are not actual errors of observation. It has been found by experi-
ment that the variations among many natural occurrences follow
the same law as the accidental errors of observation. Thus the
law is applied in studies of the growth of vegetables, and to the
occurrence of various characteristics among animals.
To illustrate errors which do not follow this law, we may
consider the errors in a table of logarithms. It is evident that
in a seven-place table, for example, the decimals following the
seventh place have been rejected when less than 5 in the eighth
place, while if the eighth place is greater than 5, the seventh place is
increased by unity. Therefore, instead of the three assumptions
upon which Least Squares is based (Art. 18), we have errors
occurring only between the limits 0.0 and 0.5, the unit being in the
last place of the logarithm, and in equal numbers without regard
to magnitude or sign. The probabilities of the occurrence of the
various errors between these limits would be equal, and the curve
of error would be a rectangle upon the axis of errors as a base and
limited by the ordinates at +0.5 and —0.5.
172. Review: Outline of Methods of Adjustment. In con-
clusion, a brief outline will be given covering the main classes
of problems which have been considered and the methods of solu-
tion.
Direct Observations of a Single Quantity.
AflJHstment. Take the mean or the weighted mean.
Indirect Observations.
Adjustment. "Write the observation equations and from them
the normal equations; the solution of the latter gives the unknown
quantities themselves or the corrections to their assumed approx-
imate values. The number of the observation equations will be
the same as that of the observations; the number of the normal
ecjuations will equal that of tlio unknown quantities, which must
always be less than that of the observations.
Conditioned Observations.
Adjusttneut. Write the condition equations in their general
form and tluMi in their simple form involving th(^ corrections.
From them form the normal ('(|ualions, the same in nunihcM- as the
200 PRACTICAL LEAST SQUARES
conditions. The solution of the normal equations gives a set of
factors, called correlates, one for each condition equation, from
which the desired corrections to the- observed quantities are deter-
mined.
Simple Propagation of Error.
Solution. Write the literal function whose mean square error
is desired. Differentiate it successively with respect to each of the
quantities for which mean square errors are given. Substitute
these partial derivatives and the given mean square errors in the
general equation of propagation of error to obtain the mean square
error of the function.
Compound Propagation of Error.
Solution. Find the mean square error of the function as above
for each of the different sources of error, and take the square root
of the sum of their squares.
Combination of Computed Quantities.
Adjustment. Give to each value a weight equal to the recip-
rocal of the square of its mean square error and take the weighted
mean as the best value of the quantity.
Empirical Formulas.
Solution. Plot the observations and sketch a smooth curve
through them. From this curve select the form of the desired
equation. Write an observation equation of the selected form for
each of the observations, reducing to the linear form if necessary.
Write normal equations and solve them as in Indirect Observations,
for the constants or coefficients of the formula.
APPENDIX A
HISTORY AND BIBLIOGRAPHY OF LEAST SQUARES
173. Historical Sketch.^ The principle of the arithmetic
mean is very old. But when the first indirect observations were
made, probably in astronomy, the necessity for adjustment became
apparent. Observation equations were written as early as 1748,
by Euler. In 1757 Simpson stated the axiom that positive and
negative errors occur with equal frequency, and in 1770 Lagrange
considered the occurrence of errors from the standpoint of the
theory of probability. Laplace, in 1774, in his " Mecanique
Celeste," further investigated the subject and laid the foundation
for the development of Least Squares.
It was not until the end of the 18th century, however, that
the Method of Least Squares was introduced. The first publi-
cation of the principle of least squares was by Legendre, in 1805,
in his " Nouvclles methodes pour la determination des orbites
des cometes," and by him the name was given, " Methode des
moindres quarres." Although there is no question as to the priority
of publication, it seems well established that Gauss had actually
developed and used the method itself since 1794, when he was a
student at the University of Gottingen. His first publication on
the subject, however, was not until 1809, in his classic work,
" Theoria motus corporum ccelestium." But Gauss deserves
more; credit than anyone else for the further development of the
Method of Least Squares, and as jMerriman states,^ " Few
liranches of science owe so large a pi'oportion of subject-matter to
the labors of one man."
The first publication of a theoretical derivation of the Law of
Error was made by Dr. R. Adrian, of Reading, Pa., in 1808, in
' For more dctiiiled information, the reader is referred to Jordan, Iland-
hueh der Vermessungskuiide, Hand I, KinleitiniK.
^ Merriman: Method of Least Squares.
201
202 PRACTICAL LEAST SQUARES
the " Analyst or Mathematical Museum," at Philadelphia.
Gauss published his in the next year, and various others have
followed.
174. Growth of the Literature. The development of the sub-
ject is indicated by the rate at which publications devoted to it
appeared. In 1877 Professor Merriman published an investiga-
tion 1 of the literature of Least Squares, as a result of which he
deduced some interesting statistics. The following data are
are based upon his work.
Prior to 1805, 22 titles were found. From that time on, aver-
aging by decades, the rate of publication increased steadily from
about two per year in 1810 to about ten per year in 1870. Alto-
gether, 408 titles were listed up to 1875. Of these, 153 were pub-
lished in German^', 78 in France, 56 in Great Britain, and 34 in
the United States, the remaining ones being scattered over eight
countries. The German language was used in 167 instances,
French in 110, and English in 90.
175. Bibliography. In addition to the paper by Merriman,
referred to above, Gore's Bibliography of Geodesy, in the Report
of the U. S. Coast and Geodetic Survey for 1887, will be found
ver}^ useful in an investigation of the literature of this subject,
although many important works have appeared since that time.
From the large number of books and parts of books devoted to
Least Squares and the Adjustment of Observations, the following
are selected for reference:
Wright: Adjustment of Observations. Van Xostrand, Xe-\v York, 1884.
This is the classic work in the English language on this subject. The
applications are principally geodetic. It has long been out of print, and
was succeeded by
Wright and Hayford: Adjustment of Observations. Van Xostrand, 1907.
Less comprehensive than the foregoing, but improved in many respects.
Mainly geodetic.
Jordan: Handl)uch der Vermessungskunde, I. Metzler, Stuttgart, 1910.
A very complete treatise, presented in a direct style which is easilj- read.
Most valuable for reference. Oeodetic.
Helmert: Ausgleichimgsrechnung. Teul)ner, Leipzig, 1907. Ooinprehen-
sive and scholarly, l)ut somewhat diflicult to read. The notation is un-
usual. Ocodetic and ))hysical.
1 Merriman: List of \\'ritings lielating to the Method of Least Squares,
published in the Transactions of the Connecticut Academy, Xew Haven, 1877.
HISTORY AND BIBLIOGRAPHY 203
Koll: Methode der kleinsten Quadrate. Berlin, 1893. Extensive and
practical with many applications.
Czuber: Theorie der Beobachtungsfehier. Leipzig, 1891. Largely theo-
retical, with applications to life insurance and statistics.
Merriman: Method of Least Squares. Wiley, New York, 1913. Geodetic
applications but general in scope.
Comstock: Method of Least Squares. Ginn, Boston, 1895. Astronomical
and general.
Bartlett: Method of Least Squares. Boston, 1915. Contains an exten-
sive list of examples for solution.
Weld: Theory of Errors and Least Squares. Macmillan, New York, 191G.
General and practical with many exercises for solution.
Johxson: Theory of Errors and Method of Least Squares. Wiley, 1892.
General; strong in illuminating explanations.
Bruxt: Combination of Observations. Cambridge University Press, 1917.
Theoretical.
Chauve.net: Practical and Spherical Astronomy. Lippincott, Philadaiphia,
1896.
Crandall: Geodesy and Least Squares. Wiley, New York, 1907.
Adams: Application of Least Squares to the Adjustment of Triangula'.ion.
Special Publication No. 28, U. S. C. & G. Survey, 1915. A verv im-
portant contribution to this subject.
APPENDIX B
PRINCIPLES OF PROBABILITY
176. Definition. If an event can occur in a ways, and can
fail to occur in b ways, the probability of its occurrence will be
, and that of its failure to occur will be , it being assumed
a-\-h a+b
that all the ways of occurrence or failure to occur are entirely
independent and equally likely. Thus, in one throw of a die, the
probability of a certain face lying upward is 1/6, and that of its
not being upward, that is, of any other face being upward, is 5/6.
The probability of throwing any face upward will be 6/6 = 1
in other words, certainty. Therefore, if the probability of the
occurrence of an event be p, then that of the failure of the event to
occur will be l—p, provided it is certain that the event must either
occur or fail.
First Principle. The probability of the occurrence of an
event is therefore a proper fraction between the limits zero (impos-
sibility) and unity (certainty), and may be defined ^as the ratio
between the number of ways in which the event may occur and the
number of ways in which it may either occur or fail.
177. Two Sources of Probability. The probability of an
event may be based upon theory or experience. The above case
of throwing a die is an example of the theoretical basis. We know
without question how many faces the die has and, therefore^, the
number of ways in which a certain face can lie upwards. The
numbers involved are known absolutely. In the second case, on
the other hand, the number of ways in which the event can occur
is assumed as a result of experiment or experience. For example,
if an event has occurred a times and failed b times out of a large
number a-f6, of trials, we may say that the probability of its
occurrence, under the same conditions, is a/(a-\-b), as ])efore.
Thus we may also define the probability of an event as the ratio
201
PRINCIPLES OF PROBABILITY 205
of the number of times it has occurred to the total number of
times it has occurred or failed; but the total number of cases, or
attempts, should be sufficiently large to justify their use as a basis
for generalization. To illustrate, suppose that statistics show
that in the long run the number of male children born is to that of
female children born as 21 to 20; then the probability that any
birth will be that of a male is 21/41.
178. Simple Probability. The above statements, relating to
the occurrence of a single event, illustrate simple probability.
The principle will be further amplified. Suppose a box to contain
w, white, h, black, and r red balls of the same weight and texture,
and that a single ball is drawn from the box at random. Then
the probability of drawing a ball of a certain color will be as follows :
White,
Black,
White or black,
Black or red,
w
w-\-h-{-r
b
w-{-h-\-r
w-\-h
w-\-h-\-r
6+r
w-\-b-]-r
White, black, or rod, = 1
iv-\-b-i-r
Yellow, ^ = 0
w-\-b-\-r
Thus we may state the Second Principle. If the ways in
which a single event can occur independently can be grouped in
differ(>nt sots or series, and the probability of its oecurrenco in
each scnios bo known, the total probability of its oceurroiico in any
coinbination of the series will b(> the .s^(o/i of the ('()ri-(\sp()n(hng
si^parate probabilities. In the above example, a single Ijall can
be drawn from the white on(\s with a probability of
w
206 PRACTICAL LEAST SQUARES
or from the black ones with a probabihty of
b
then the probabihty of drawing either a white or a black ball will
be the sum of the two probabilities, namely, .
w-\-b-\-r
179. Compound Probability. Independent Events. Suppose
we have, in addition to the above box, a second one containing iv'
white, h' black, and r' red balls, and that we draw a ball from each
box. Each of the w-\-b-\-r possible draws from the first box may
occur in combination with each of the w' + b'-\-7'' balls in the second
so that the total number of possible draws of two balls, one from
each box, wih be (w-\-b-\-r){w'+b'-{-r'). Also, each of the white
balls in the first box may be drawn with each of the white ones in
the second box, giving ww' possible pairs of white balls drawn
one from each box. Therefore the probability of drawing simul-
taneously, two balls of one color, one from each box, will be,
two white ball
rr'
Two red balls.
Two black balls,
(w + b^r){iv' + b' + r')
bb'
As a result of this reasoning, we can state the Third Principle:
If two or more independent events are to occur simultaneously,
and the ]:)r()bability of the separate occurrence of each is known,
that of the simultaneous occurrence of all of them wiU be the
product of the separate prol)abilities.
180. Compound Probability. Dependent Events. The prol)-
ability of drawing a black and wliitc^ I'^ii'', one ball from each of the
two boxes, is an example in which th(> events aix^ (k^pendent. For,
if a white ball werc^ drawn fi'om the first box, a black one would
necessarily hiwv to l)e drawn fi'om the second box in orck^- to make
the pair, and vice v(M'sa, so that the pi'obabilit}" of th(> scm'oiuI evcuit
would 1)(^ (liffenMit in the case of the failure of the first one than in
PRINCIPLES OF PROBABILITY 207
its occurrence. Then the number of possible black and white
pairs, one ball from each box, would be, wb'-\-w'b, and the prob-
ability of drawing such a pair would be,
wh'-\-w'b
(w-\-b-\-r){w'-\-b'-\-r')
Here we have the occurrence of a compound event in two sets or
series, so that the total probability is the sum of the separate
(compound) probabilities.
Events are dependent when the probability of the occurrence of
one of them depends upon the occurrence, or failure to occur, of
another. By a careful analj-sis of each problem, it will usually be
easy to so arrange or combine the events as to render them inde-
pendent. In the foregoing example, the case of drawing a black
and white pair, one ball from each box, is clearly one of dependent
events, but if we require the probability of drawing a white ball
from the first box simultaneously with a black one from the second
box, the events are independent and, from the preceding article,
the probability would be,
U'b'
(it'+6+rj(ir' + 6' + r')
Also, the probability of drawing a black ball from the first box
and a white one; from the second, simultaneously, would be
iv^
But each of these events, while compound, is independent of the
other. They arc of thc^ same charactcM- and may be considered
as a single compound event occurring in two s(>ts or seri(>^, so tli;il
the total prol)ability of its occurrence in either mannei- will be, as
in Art. 178, the sum of the two s('parat(^ probabilities, that is,
ivb'^ u/b^
This is evident, also, fi'om \hv first priiicii^le, when we note that
the total niunber of bhick and whit(^ pairs is irh'^w'b, while the
total number of possible ])airs, of all colors, is
(«- + /; + /■)(»•' + // + /■')
208 PRACTICAL LEAST SQUARES
181. Number of Occurrences. It follows from the definition
of probability (Art. 176), that the number of times an event occurs
may be determined by multiplying the total number of possible
occurrences and failures, in other words, trials or attempts, by
the probability of the occurrence of the event. In Least Squares,
for example, the number of errors less than a certain amount to be
expected in a given series of observations will be equal to the
probabiHty of an error less than that amount multiplied into the
total number of observations in the set.
APPENDIX C
DERIVATION OF THE LAW OF ERROR
182. The Law of Error, that is, the equation of the Error Curve,
(Art. 19), has been derived in several ways by different writers
since the original demonstration by Dr. Adrian in 1808, published
at Philadelphia in the '' Analyst." The most notable of these,
however, are the methods of Gauss (1809) and Hagen (1837).
The former of these two will now be explained.^
183. Assumptions. The Error Function. Gauss based his
derivation upon the assumption of the arithmetic mean as the most
probable value of a directly observed quantity when all of the
observations are made with the same care. Also, the occurrence
of the errors of observation is assumed to be in accordance with
the three axioms of Art. 18.
Since small errors are more numerous than large ones, and since
the probability of an error of a certain size is directly proportional
to the number of times that that error occurs in the given series of
observations, it is evident that the probability of an error is a
function of the error itself. Representing any error by A, the
probability of the occurrence of this error by Pa; and the prob-
abihty function by <?!)(A), we can write,
Pa = ^{A) (308)
Strictly speaking, consecutive errors will differ by small finite
amounts which are the least readings made with the given
instrument or by the method used. For example, the least reading
of a vernier on a circle may be 10", so that all the observations
might be made only to the nearest 10", and the errors themselves
' Soo Brunt's Combination of Observations, for tlio nietliods of Hagen,
Thomson and Tait, and luldington. Hagen's jjroof is given in many works
on Least Sciuares. Comstoek frankly assum(>s the Law of Error to be empir^
ical, wliich is a reasonable and practical method of attack.
209
210 PRACTICAL LEAST SQUARES
would differ by multiples of 10". So the ordinates to the error
curve, corresponding to the various errors, and the successive
points on the curve, would be separated by these intervals. How-
ever, as the precision of the observations increases, these ele-
mental differences decrease and so we may reasonably regard the
points as being so close together as to make the curve continuous.
Thus we may consider that the errors. A, var}^ continuously, and
that the function, 0(A), is a continuous one. The probability
of an error, A, is therefore equivalent to the probability of an error
between the limits, A and A + riA.
The probability of the occurrence of an error between two
limits is the sum of the separate probabilities of all the possible
errors h'ing between those limits.^ If we regard each probability
as the ordinate to the error curve, corresponding to its particular
error, the sum of these successive ordinates, when the curve is a
continuous one, will constitute the area between the limiting
ordinates, the curve, and the axis of A. Then, the probability of
an error between A and A-\-dA would be represented by the area
of the infinitesimal vertical strip of length 0(A) and of width r/A,
that is, by the area 0(A)r/A. Therefore, the probability of the
occurrence of an error between the limits a and b would be
/
b
0(A)rfA
If the limits be extended so as to include all possible errors,
namely, between — oo and + 20 , the probability of the occurrence
of any error between these limits would be unity, that is, certainty,
and this can be stated,
0(A)</A = 1 (809)
X
or, since th(^ area is symmetrical about the axis of probal)ility,
0(A)r/A=i (310)
1 S(H> Ai)i)('n(lix 15, Pi-inciples of Probability.
f:
LAW OF ERROR 211
184. Derivation of the Law of Error. We shall consider the
general case of indirect observations, since direct observations
form but a special case under it. The observed quantity is a
function of the unknown quantities. Let there be n observations
and m unknowns, n being greater than m (Art. 22). The observa-
tion equations may be written,
/i(X, Y,Z, . . .)=Mi
/2(X, Y,Z, . . .)=M2 (311)
UX, Y,Z, . . .)^M,
Let Ai, A2, A3, . . . A,„ be the respective errors of Mi, M2, M3,
, . . Mn, and let the probability of the occurrence of Ai be
^(Ai), that of A2 be <?i)(A2), etc. Then the probability of the sim-
ultaneous occurrence of this series of errors will be the product
of their separate probabilities, or,
P = (/)(Ai)(/)(A2)0(A;O . . . 0(A„) (312)
Taking the logarithm of each mem])er, this Ix'comes,
logP = log0(Ai)+l()g</>(A2)+ . . . +log0(A„) (313)
The most probable series of errors will be those for which the
above probaljility is a maximum, which also will l)e the case when
log P is a maximum. This is the condition for the best or most
probable values of the unknowns. Therefore, the first (l(M'ivativ(>
of (313) must equal zero, and since the unknowns X, Y, Z, . . ., in
the case of indirect observations are independent, it follows that
the separate partial doi'ivatives of log P witii respect to these
unknowns must equal zero. Thus we o])tain,
0 (A 1 ) (IX cpiSo) (IX 0 (^n)'IX
(314)
212 PRACTICAL LEAST SQUARES
Multiplying and dividing each fraction by the corresponding dA,
d(f)(Ai) dAi , d4>{A2) dA2 dcl>(A„) dAn_Q
0(Ai)dAi dX 4>{A2)dA2 dX </>(A„)dA„ dX
d(f)(Ai) dAi , f/0(A2) dA2, d4>(An) <iA„_ .
(t>{Ai)dAi dY </)(A2)dA2 dY <p{A„)dAn dY
Since the function, 0(A)dA, must be appHcable to any number
of unknowns, we shall make use of the case of one unknown,
directly observed, from which to determine the nature of the
function. Letting X represent the true value of the unknown,
and A the true error of the observation, M, we may write,
X-Mi=Ai
X-M2 = A2 (316)
X-Mn = An
Differentiating,
dM^dA2^dAz^ =^=1 C317)
dX dX dX ' ' ' dX ^ ^
Substituting in (315),
d<i>{Ai) , r/0(A2) , , d(/)(A„)
= 0 (318)
</.(Ai)dAi 0(A2)rfA2 </)(A„)f/A„
Multiplying and dividing each fraction by the corresponding A,
d«(A.) .^^+_MA^^^_^ ^ , ^ + <>'>'}^:\ A,. = 0 (319)
Ai(/)(Ai)dAi A2</)(A2)r/A2 " A„</)(A„)(/A„
But it is assumed in direct observations that the mean is the best
or most probable vahu; of the observed (quantity, and that, as the
number of observations increases indefinitely, the mean approaches
the true value as a limit. So we may write,
^JU+M..+ ^ ^ . +M„
n
or,
{X-My) + {X-M2)+ . . . +(X-3/„)=0 (321)
LAW OF ERROR 213
whence, from (316),
A1+A2+ . . . +An = 0 (322)
Both (319) and (322) hold good as the number of observations is
increased one by one. But in order that this condition may exist,
it is necessary that the coefficients of the A's in (319) be equal and
constant, so that they may be cancelled from that equation.
Therefore, we may write, in general,
d(P(A)
or,
Integrating,
whence,^
A0(A)dA
d<f)(A)
= a constant, say k (323)
M (324)
0(A)dA
log</>(A)=pA2+fc' (325)
0(A)=e^*^V (326)
But one of the original assumptions was that small errors are
more numerous and more probable than large ones. Thus, as A
decreases, (^(A) must increase, which requires that k must always
be negative. To effect this, we replace k/2 b}^ the new constant,
— h^. Then, replacing the constant factor, e^' , by the constant, C,
we obtain the expression for the probability of an error, A,
<PiA) = Ce-"'^' (327)
185. The Constant, C. It remains to determine the value of
the constant, C. Substituting (327) in (310),
f
Ce-"'^VA = § (328)
Let t = hA; then (U = hdA. Also, when A = 0, ^ = 0, and when
A = »^ , t= 00 . Therefore w(> may wiite (328) as follows.
j e-''^'hdA = ^ (329)
1 c is the base of Xapicrian logarithms.
214
But/
PRACTICAL LEAST SQUARES
/■
e-''dt
V^
(330)
(331)
so that, from (330) and (;^31),
h _Vx
2C 2~
whence,
h
C = -
Vtt
(332)
Therefore, we have from (327) the final expression for the Law of
Error,
0(A) =4--^-"^^^ (333)
Vtt
1 This definite integral may be evaluated in various ways. The following
method is given by Bartlett:
From the assumption, t = hA, we have,
itegrals, only, ar€
But when definite integrals, only, are used.
Multiplying (334) and (335),
(:-"!-!:.[
c-"^^^+^'^kdAdh
„ 2fl+A^)L Jo
1 r^ ds
-,-/,2(l+A2)(_2/),)(l+A2)(i/i
tan- 'A
Therefore,
i:
0 4
Vtt
e-"dt = -
(334)
(335)
(336)
(337)
(338)
(339)
(340)
(341)
LAW OF ERROR 215
186. Expansion of Law of Error in Series. The Law of Error
may be expressed in the form of series for convenience in evaluating
it for various values of A.^
Using the quantity, t = hA, as an auxiliary variable, we can
state (333) as follows :
«^(A)rfA = -^e-"'^'hdA = ^e-"dt (342)
Vtt Vtt
which is the probability of an error between A and A+c/A. The
probability of the occurrence of an error less than A will be that
of an error between the limits, —A and +A, that is, since t = hA,
+A r+A 1 r+t 9 r+t
p = <p(A)dA = -^\ e-'\lt = ^\ e-'\H (343)
-A J- A VirJ-t Vtt J
=;7?('-3Ti+5^-4 ■ ■ • ) (^^'
or,
for use with small values of t,
for use with large values of t.
187. Tables of the Law of Error. From the above formulas,
tables have been computed with the argument t, giving the prob-
al)ility of an ci-ror less than A, in a given set of observations.
Tabk^ I, in Api)endix F, has been formed in this manner. To use
such a table, the mean square error of a single observation, (e),
is computed fi'om the residuals of tlie mean. Then i is obtained
from the assumed error, A, by means of the relation,
t = }iA^ ^-^ (34())
eV2
siiic(>, from (209),
e\ 2
Finally, with / as an argmnenl, the tabular ])r()t)al)ilit >' is obtaiiUMJ.
' Sc(^ \\'njiht and HayfonK Art. 2iy Craiidall, Art. 114; dp ( 'hauvc-nct,
Vol. I, Art. ii:;.
216 PRACTICAL LEAST SQUARES
However, it is more convenient in many cases to express the
function in terms of A/ e or A/r directly, and this has been done in
Tables II and III, respectively, in Appendix F. The table gives
the probability of an error less than a certain fraction (A/e) of
the mean square error, or (A/r), of the probable error, of a single
observation. Thus, from Table II, the probability of an error less
than 0.4 of the mean square error is 0.3108, and the number of such
errors should be approximately 0.3108 times the number of obser-
vations, n, in the given series. Similarly, the number of errors
greater than 0.4 e would theoretically be, n(l— 0.3108). By com-
paring these theoretical numbers of errors with those actually
counted in the given set, it is possible to ascertain how closely
the observations conform to the theory (Art. 20).
APPENDIX D
OUTLINE OF A SHORT COURSE OF INSTRUCTION
188. General Plan. While it is desirable to devote a three-
hour course for one semester to the study and practice of Least
Squares and the Adjustment of Observations, with civil engineer-
ing students, the author presents the following outline of a one-
hour course which he conducted at Cornell University when, owing
to the demands of other courses, this was all the time which the
student could devote to the subject. He regards such a course
as very much worth while and believes that the students obtained
a good general knowledge of the methods of adjusting observa-
tions together with considerable practice in the solution of
problems.
The course was given in 16 lessons, and in addition to the
fifty-minute lecture, the student was expected to work two hours
at home upon the text and the assigned problem. The problems
were handed in at the next lecture, with a penalty for failing to do
so. It was considered essential that the problem be solved while
the topic was fresh in the student's mind. The problems were
carefully examined by comparison with standards and returned
for correction, if necessary, or retained until the end of the term.
The work was required to be neatly done with the idea that the
set of examples would be kept for reference.^
The lectures had to be limited to the essential parts of the
subject, owing to the limited lime, and especial attention was
given to the solution of the pr()l)l(MU at hand. Sometimes two
lectures intervened ])etween problems, and in tlu> ('as(> of the double^
problem of the adjustment of a (luadrilateral a lecluix^ was omittcnl
in order to give the studcMit more time for {hv solution. The
1 The jnipcr known as " Data SIkh'Is " was usod. It is SXlOj inclu\s and
ruled witli blue linos onc-fDUrtU inch apart ])arallcl to the slioi-t(>r (xlgo and
with periMMidicular r<'d lines I'orniin^ ten ecuial eohnnns. A blank margin is
left at the toj) and left-hand edges of the sheet.
217
218 PRACTICAL LEAST SQUARES
first lectures were devoted to a very careful consideration of the
occurrence of errors. Thence the order is indicated by the prob-
lems in the following list.
189. List of Problems. It was intended that each of the
ordinary problems would be of such length that the average
student could solve it in two hours, in connection with the accom-
panying text. The order here given may be varied, if desired, and
Nos. 9 and 10 may be combined. The inclusion of the topic of
index of precision and mean square error in the introductory lec-
tures will depend upon the preference of the instructor; it is not
necessary to introduce it until the propagation of error is to be
studied.
1. Simple and weighted means; precision and mean square
error.
2. Indirect observations; observation equations given; direct
solution for the unknowns.
3. Indirect observations; observation equations given; solu-
tion with approximate values of unknowns to find corrections to
those values.
4. Local adjustment of angles at a station.
5. Local adjustment; method of directions.
6. Adjustment of a level net.
7. Adjustment of a quadrilateral; method of angles.
8. Adjustment of a quadrilateral; method of directions.
(Problems 7 and 8 may be combined, using same data.)
9. Simple propagation of error.
10. Compound propagation of error.
11. Combination of results; weights from mean square errors.
APPENDIX E
TYPICAL CURVES FOR REFERENCE
TYPICAL CURVES
221
PLATE I
y=A + B X
A = Intercept en Y-axis
B= Tan gent of Slope, io X-axis,
— tan
STRAIGHT LIKES
C.S Log X
NOTE .' Sec next paces for these curves
plotted by X and y
222
PRACTICAL LEAST SQUARES
4
PLATE 11
:Ci.5T-
20 1,0 CO
H (-
H ^
100 X
PARABOLA
TYPICAL CURVES
223
PLATE III
224
PRACTICAL LEAST SQUARES
PLATE IV
PAIiAEOLAS
TYPICAL CURVES
225
PLATE V
i/ i
/
_->■
FARABOLAS
-iO-
226
PRACTICAL LEAST SQUARES
PLATE VI
TYPICAL CURVES
227
PLATE VII
228
PRACTICAL LEAST SQUARES
PLATE VIII
APPENDIX F
TABLES
TABLE I
Values of i» = — = | e~'^dt
2 n
of p = — = I e
(Arts. 136 and 187)
Argument is t = h^.
I
p
t
p
0.00
0.0000
1.00
0,8427
0.05
.0564
1.05
.8624
0.10
.1125
1,10
.8802
0.15
.1680
1.15
.8961
0.20
.2227
1.20
.9103
0.25
0.2763
1.25
0.9229
0.30
.3286
1.30
. 9340
0.35
.3794
1,35
.9438
0.40
.4284
1.40
.9523
0.45
. 4755
1.45
.9597
0.50
0 . 5205
1.50
0.9661
0.55
. 5633
1.60
.9763
0.60
.6039
1.70
. 9838
0.65
,6420
1.80
. 9891
0.70
.6778
1.90
. 9928
0.75
0,7112
2.00
0.9953
0.80
.7421
2,10
.9970
0.85
,7707
2.20
.9981
0.90
, 7969
2.30
.9989
0.95
. 8209 i
2.40
. 9993
1 no
0 . 8427
2 , ."in
n 9996
229
230
PRACTICAL LEAST SQUARES
TABLE II
Values of p ■
2/i
f
Jo
-"^^^iA in terms of —
€
Probability of the occurrence of an error less than A.
€ = A / , that is, the mean square error of a single observation.
\ 71 — 1
(Art. 187, J). 215,)
A
e
P
1
P
0.0
0.0000
2.0
0.9545
0.1
.0797
2.1
.9643
0.2
.1585
2.2
.9722
0.3
.2358
2.3
.9785
0.4
.3108
2.4
.9836
0.5
0 . 3829
1 2.5
0.9876
0.6
.4515
2.6
.9907
0.7
.5161
2.7
.9931
0.8
.5763
! 2.8
.9949
0.9
.6319
2.9
.9963
1.0
0.6827
3.0
0.9973
1.1
.7287
3 1
.9981
1.2
.7699
3 2
.9986
1.3
.8064
3.3
. 9990
1.4
.8385
3.4
. 9993
1.5
0.8664
I 3.5
0 . 9995
1.6
.8904
' 3.6
.9997
1.7
.9109
3.7
.9998
1.8
.9281
3.8
.9999
1.9
. 9426
3 9
. 9999
2.0
0 . 9545
4.0
0.9999
4 1
1 .oooo
TABLES
231
TABLE III
7- r^
ttJo
Values of p = —7= I e ''^^^dA in terms of
Probability of the occurrence of an error less than A.
r = 0.6745-» / = the probable error of a single observation.
H — 1
(Art. 187, p. 215)
A
r
P i
A
—
r
P
0.0
0.0000
2.5
0.9082
0.1
.0538
2,6
.9205
0.2
.1073
2.7
.9314
0.3
.1603
2,8
.9410
0.4
.2127
2,9
.9495
0.5
0.2641
3,0
0.9570
0.6
.3143
3.1
,9635
0.7
.3632
3.2
.9691
0.8
.4105
3.3
.9740
0.9
.4562
3.4
.9782
1.0
0.5000
3.5
0.9818
1.1
.5419
3,6
.9848
1.2
. 5817
3,7
.9874
1.3
.6194
3,8
.9896
1.4
,6550
3,9
.9915
1.5
0.6883
4.0
0.9930
1.6
.7195
4.1
.9943
1.7
. 7485
4,2
.9954
1.8
.7753
1 4,3
.9963
1.9
,8000
4,4
.9970
2.0
0.8227
4 . 5
0 . 9976
2.1
. 8433
4.6
.9981
2,2
.8622
4,7
.9985
2.3
. 8792
4,8
.9988
2.4
. 8945
4,9
.9991
2 . 5
(1.9()S2
5 0
0 , 9^)93
232
PRACTICAL LEAST SQUARES
TABLE IV
Factors for Computing Probable Errors from Bessel's Formulas.
(Arts. 140 and 141)
0 . 6745
0.6745
0.6745
0,6745
n
Vn-1
n
Vn-1
Vn(n-l)
Vn(n-l)
30
0.1252
0.0229
31
.1231
.0221
2
0.6745
0.4769
32
.1211
.0214
3
.4769
.2754 '
33
.1192
.0208
4
.3894
.1947
34
.1174
.0201
5
0.3372
0.1508
35
0.1157
0.0196
6
.3016
.1231 1
36
.1140
.0190
7
.2754
. 1041 t
37
.1124
.0185
8
.2549
.0901
38
.1109
,0180
9
.2385
.0795
39
.1094
.0175
10
0.2248
0.0711
40
0.1080
0.0171
11
.2133
.0643
41
.1066
.0167
12
.2029
.0587
42
, 1053
.0163
13
.1947
.0540
43
.1041
.0159
14
.1871
.0500
44
.1029
.0155
15
0.1803
0.0465
45
0.1017
0.0152
16
.1742
.0435
46
. 1005
.0148
17
.1686
. 0409
47
.0994
.0145
18
.1636
.0386
48
.0984
.0142
19
.1590
.0365
49
.0974
.0139
20
0.1547
0.0346
50
0,0904
0.0136
21
.1508
.0329
51
, 0954
.0134
22
.1472
.0314
52
,0944
.0131
23
.1438
.0300
53
,0935
.0128
24
.1406
,0287
54
,0926
.0126
25
0.1377
0.0275
55
0,0918
0.0124
26
. 1349
.0265
56
. 0909
.0122
27
. 1323
.0255
57
,0901
,0119
28
. 1298
.0245
58
.0893
,0117
29
. 1275
.0237
59
.0886
,0115
30
0.1252
0 . 0229
60
0,0S7S
0,0113
INDEX
(Numbers refer to pages)
Abridged method of solving normal
equations, 41, 47
Abuse of least squares, 197
Accidental errors, 7, 9, 10
Accuracy and precision, 151, 194
Adjustment, angles {see triangula-
tion)
arbitrary, 196
base lines, 130
by parts, 54, 97, 130
levels, 55, 64, 74, 174
necessity for, 3
trigonometric leveling, 81, 130
triangulation, 80, 127
between base lines or points of
control, 119, 127
figure, 81, 96, 107, 108
local, 56, 71, 76
quadrilateral, method of angles,
99
method of directions. 111
approximate method, 118
Adjustments not infallible, 78, 198
Aids in computation, 30, 51
Angle equations, 84, 86
Angle measurement with theodoHtc,
example, 186
Approximate method of adjusting
([uadrilatcral. lis
Approximate values of unknowns, use
of, 19, 34
Ar! itrary adjustments, 196
Arithmetic mean, 9 (s(r Mean)
Average error, l.")2, l.")7. loS, 163, 166
Axioms or assmnptions, 11, 209
Azimuth equation. 128
B
Base lines, adjustment of, 130
Bessel's formulas for mean square and
probable errors, 162, 164,
166, 169
Best values of the unknowns, 15
Bibliography of least squares, 202
Blunders, 7
Central-point figure, 83
side equation for, 92
central point unoccupied, 123
Coefficients, equalization of, 35
Comparison of indices of precision,
158
Comparison of observations and
theory, 11
Compound propagation of error, 181
Computation tables and machines, 30
Computed quantities, combination of,
187
precision of, 177
weighted mean of, 189
Conditioned observations, 53
adjustment by method of inde-
pendent unknowns. 75
precision of, 172
special case of one condition only,
73
Conditions, 53
angle, 81, S4, 97, 108
arrangement of. 70
ecjuations, 57. 63
ind(>pendent. 55, (')'), 85, 97
latitude, longitude, and azimuth,
128
233
234
INDEX
Conditions, length, 121, 128
local, 56, 81, 97
number of, 55, 57, 65, 82, 86
side, 81, 97, 108
Constant errors, 5, 193
Control or check, arithmetic mean, 20
weighted mean, 24
correlates, 62
formation of normal equations, 29,
62
solution of normal equations, 46,
62
final values of the unknowns, 50, 63
Corrections, 3
computation of, 64
used instead of errors, 3
Correctness unattainable, 2
Correlates, 61
method of, 59, 71, 106
Course of instruction, 217
Criteria for rejection of observations,
192
Curve of error, 10
Curves of empirical formulas, 219
D
Dependent quantities, 53
Derived equations, 43, 46
Direct observations of one unknown,
17
adjustment of, 18
precision of, 160
Directions, 105
list of, 107
method of, 97, 107
Discordant observations, 196
Discrepancies among observations, 1,
8
indicate errors, 2, 8
Doolittle, method of elimination. 40
Elimination, methods of , 40
Emi)irical formulas, 131
straiglit lines, 133, 141, 221
parabolas, 133. 143, 221-225
hyperbolas, 141, 221. 223. 226
jjcriodic functions. 134, blX, 22S
Empirical formulas, non-linear forms,
135
exponential functions, 135, 145, 227
logarithmic functions, 227
reduction to linear form, 137
test of, 139
use of, 140
Equations, angle, 84, 86
azimuth, latitude, and longitude, 128
base line, 128
condition, 57
derived, 43
length, 121, 128
normal, 28
observation, 27
residual, 27
side, 87
mechanical statement of, 93
simultaneous, general, 38
Error, average, 152, 157, 158, 162,
166
curve of, 10
law of, 12, 209
tables, 229-231
mean square, 152, 153, 158, 161,
164, 166
probable, 152, 155. 162, 164, 166
proi)agation of, 177
Errors, accidental, 7
constant, 5, 193
instrumental, 6
occurrence, 4, 10, 11
personal, 6
systematic, 5, 193, 195
theoretical, 5
Excess, .spherical, 53, 85
Exponential fimctions, 135, 145
F
Factors, Bessel's, f(jr ])robablc errors,
232
correlates, 61
Figure adjustment, 82. 96
l)('t\v('(>n fixed jjoints of control, 127
lucthod of angles, 82, 96
method of directions, 107, 111
to conform to fixed or adjusted
work, 119
INDEX
235
Figures, classification of, 83
Fixed, or controlling, data, 119, 127
Formulas, Bessel's, 162, 164, 166, 169
empirical, 131
Liiroth's, 169
Peters', 162, 164, 167, 169
Function, general, 26
linear, 26, 137
observations of a, 26
of the unknowns, 26
reduction to linear form, 26, 89, 137
G
Gauss, method of elimination, 40
Geometric mean, 9
H
History of least squares, 201
Hyperbolas, 141, 221, 223, 226
I
Independent conditions, 55, 65, 85, 97
Independent observations, 54
Independent unknowns, method of,
75, 106
Index of precision, 152
Indirect observations, 26
method of, 75, 106
Instruction, short course of, 217
Instrumental errors, 6
Interpolation formulas, 131
I.
Latitude, longitude, and azimuth
eciuations, 128
Law (if error, 12, 209
expansion in scries, 215
others than that of least sciuarcs,
19S
tables (,f, 13, 162, 215, 229-231
test of, 12
Law of pi'o])apitioi\ of error, 177
Least sciuares, 13
axioms or assuniiitions of, 11. 209
cliissificatioii of ])!'ol)leii!S, 15
priiH'iple of, 11
two uses of, 15
use and at)us(^ of, 197
Length equation, 121. 128
Levels, adjustment of, 55, 64, 74
precision of, 174
Limit of accuracy and precision, 163,
194
liinear function, reduction to, 26,
89, 137
Literature of least squares, 202
Local adjustment, 56
method of correlates, 71
method of independent unkaowns,
76
Logarithmic curves, 227
Logarithmic plotting, 146
Liiroth's formulas for mean square
and probable errors, 169
M
Machines for computation, 30, 51
Mean, 18
arithmetic, 9, 19
control, 20
assumed as best value, 9, 18
geometric, 9
weighted, 22
control, 24
Mean square error, 152, 153
compared with probable error, 158
of a single observation, 161
of arithmetic mean, 164, 181
of weighted mean, 166
of a function, 177
Mechanical aids in comi)utation, 30,
31
Mechanical statement of side equa-
tions, 93
M(>thods of observing, 5, 193
Micron, S
Mistakes, 7, l2
X
Xou-linear functions and curves, 135
Normal equations, 2S
formation of. 2S. 30, 32
control, 29
nunil)(>r of. 2S
redundauT tei'ins. 44
solution of. 40
236
INDEX
Normal equations, solution of,
abridged method, 41, 47
control, 46
Number of angle equations, 86
conditions, 55
local conditions, 82
observations, 14, 163, 193
occurrences, determined from prob-
ability, 208
side equations, 95
O
Observation equations, 27
Observations, conditioned, 53
direct, 17
discordant, 196
indirect, 26
number of, 14, 163, 193
precision of, 151
superfluous, 55
weighted, 20
weighting of, 21
Observing, methods of, 5, 193
Occurrence of errors, 4, 10
One condition only, 73
Outline of methods of adjustment, 199
Parabolas, 133, 143, 221-225
Partial adjustments, 54, 97, 130
Periodic curves, 134, 148, 228
Personal errors, 6
Peters' formulas for mean square
and ])robable errors, 162,
164, 167, 169
Pole of the side ecjuation, 93
Polygon fixed, with central point
unoccupied, 123
Precision, 151
and accuracy, 151, 194
increased by additional observa-
tions, 3, 193
index of, 152
of direct observations, 160
of a single observation, 161
of arithmetic mean, 163, 181
of weighted mean, 165
of indirect observations, 167
Precision of an observation of weight
unity, 169
of conditioned observations, 172
of a difference of elevation, 174
of a function, 177
of computed quantities, 177, 189
Principle of least squares, 14, 23
Probability, principles of, 205
simple 205
compound, 206
Probable error, 152, 155
approximate value of, 162
compared with mean square error,
158
of a single observation, 162
of arithmetic mean, 164
of weighted mean, 166
table of Bessel's factoid, 232
Problems, classification of, 15
list of, for a short course, 218
Propagation of error, 177
simple, 178
compound, 181
Quadrilateral, 83
adjustment, method of angles, 99
method of directions, 111
approximate method, 118
defined, 83
one triangle fixed, 122
side equation for, 88, 91
two sides and included angle fixed,
121
R
Readings, 1, 17
combination of, 17
Reduced condition equations, 58
Reduction to linear form, 26, 89, 137
Redundant terms in normal e(iua-
tions, 44
Refinement of computations, 50
Rejection of observations, 191
Relation between mean square, prob-
able, and average errors,
158
Residual equations, 27
INDEX
237
Residuals, 10, 195
from the mean, 20
not the same as errors, 10
sum of squares is a mii.imum, 14,
23
S
Side equations, 81, 87
formation of, 88
mechanical formation of, 93
number of, 95
reduction to linear form, 89
Simple propagation of error, 178
Simultaneous equations, solution by
means of normal equations,
38
Single observation, precision of, 161
Spherical excess, 53, 85
Straight lines, 133, 141, 221
Systematic errors, 5, 193, 195
T
Tables, Bessel's factors, 232
probability of error, 229-231
for computation, 30
Tape compari'^on, example, 183
Tape measurements, examples, 184
Test of empirical formulas, 139
Test of the law of error, 12
Test-piece, example, 182
Time by star transits, example, 36
Triangle errors, example, 185
Triangles, computation of, 118
Triangulation, adjustment of, 80 (see
Adjustment)
Trigonometric levehng, 81
True errors, 3
U
Unknowns, approximate values of, 19,
34
final check of, 50
Use and abuse of least squares, 197
Use of empirical formulas, 140
W
Weighted mean, 22
of computed quantities, 189
of two quantities, 24
Weighted observations, 20
Weights, 20
basis for, 21, 54, 187
determination of, 21
from mean square or probable
errors, 187
of the unknowns, 168
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