ramm
 
 ^ <X>^O<KXX><XXXXKXXXXX><XXX><><X><XXXHX>O<X:K><) -^
 
 UNIVERSITY OF CALIFORNIA 
 AT LOS ANGELES
 
 A TREATISE 
 
 ADJUSTMENT OF OBSERVATIONS, 
 
 APPLICATIONS TO GEODETIC WORK AND OTHER MEASURES 
 OF PRECISION. 
 
 T. W. WRIGHT, B.A., 
 
 Civil Engineer, 
 
 LATE ASSISTANT ENGINEER UNITED STATES LAKE SURVEY. 
 
 O Messkunst Zaum der Phantasie, 
 Wtr dir will folgeit irrct nie. 
 
 HALI.ER. 
 
 NEW YORK : 
 
 D. VAN NOSTRAND, 
 
 23 MURRAY STREET AND 27 WARREN STREET. 
 
 1884.
 
 Copyright, 
 
 1884. 
 T. W. WRIGHT. 
 
 PRINTED BY 
 
 H. J. HEWITT, 
 
 27 ROSE STREET, NEW YORK.
 
 Q. 
 
 
 PREFACE. 
 
 IN the following treatise I have endeavored to give a 
 systematic account of the method of adjusting observations 
 founded on the principle of the mean. The more important 
 applications, especially with reference to geodetic and 
 astronomical work, are fully discussed. 
 
 It has been my aim throughout to be practical. The 
 book originated and grew amid actual work, and hence 
 subjects that are interesting mainly because they are curious, 
 and methods of reduction that have become antiquated, are 
 not noticed. 
 
 Several of the views enunciated are not in the usual 
 strain, but they are, however, such as I think all experienced 
 observers, though perhaps not all mathematicians, will at 
 once assent to. 
 
 As regards notation I have been conservative, usually 
 following Encke's system as given by Chauvenet. Some 
 minor changes have been introduced which it is thought 
 will tend to greater clearness and uniformity of expression. 
 
 The examples and illustrations have been drawn chiefly 
 from American sources, for the reason that much valuable 
 material of this kind is to be had, and that thus far it has 
 not been used for this purpose. They have been taken 
 from records of work actually done, and principally irom 
 work with which I have been connected. 
 
 2 
 
 331 211005
 
 4 PREFACE. 
 
 In the applications to practical work 1 have aimed at 
 giving only so much of methods of observing as would 
 serve to make the methods of adjustment intelligible. It 
 has been difficult to do this succinctly and at the same time 
 satisfactorily, and accordingly references are given to books 
 where descriptions of instruments and modes of using them 
 can be found. 
 
 Special attention has been given to the explanation of 
 checks of computation, of approximate methods of adjust- 
 ment, and of approximate methods of finding the precision 
 of the adjusted values. But in order to see how far it is 
 allowable to use these short cuts the rigid methods must 
 first be derived. It is for this reason principally that the 
 subject of triangulation has been dwelt on at such length. 
 In general it is unnecessary to spend a great amount of 
 time in finding the probable error, when, after it has been 
 found, it in many cases tells so little. 
 
 I have been careful to give references to original author- 
 ities as far as I could ascertain them, and also to give lists 
 of memoirs on special subjects which will be of use to any 
 one desiring to follow those subjects farther. Of recent 
 writers I am indebted chiefly to Helmert and Zacharise. I 
 desire also to acknowledge my obligations to my old Lake 
 Survey friends, Messrs. C. C. Brown, J. H. Darling, E. S. 
 Wheeler, R. S. Woodward, and A. Ziwet, who have read 
 the manuscript in part and given me the benefit, of their 
 advice. Mr. Brown deserves special mention for assistance 
 rendered while the book was passing through the press.
 
 TABLE OF CONTENTS. 
 
 CHAPTER I. 
 
 Introduction. 
 
 General remarks on observing: 
 
 The instrument, .......... 
 
 External conditions, ......... 
 
 The observer, . . . 
 
 Synopsis of mathematical principles employed : 
 
 Theory of probability, ......... 18 
 
 Definite integrals. .......... 19 
 
 Taylor's theorem Examples 21 
 
 Interpolation, ........... 25 
 
 Periodic series, .......... 27 
 
 Notation, ............ 28 
 
 CHAPTER II. 
 
 T^i? Law of Error. 
 
 The arithmetic mean : 
 
 Quantity measured the quantity to be found, .... 29 
 
 The arithmetic mean the most plausible value. ... 31 
 
 When the arithmetic mean gives the true va'ue, ... 32 
 
 Inferences from the arithmetic mean, ..... 33 
 
 Quantity measured a function of the quantities to be found, . 34 
 
 The most plausible values of the unknowns, .... 37 
 
 Law of error of a single observed quantity, ..... 38 
 
 The principle of least squares, ....... 42 
 
 Reduction of observations to a common basis, .... 44 
 
 Combination of heterogeneous measures, ..... 44 
 
 Law of error of a linear function of independently observed quantities, 44
 
 6 THE ADJUSTMENT OF OBSERVATIONS. 
 
 PAGE 
 
 Comparison of the accuracy of different series of observations: 
 
 The mean-square error, ........ 46 
 
 The probable error, .......... 47 
 
 The average error, ......... 48 
 
 The probability curve, .......... 54 
 
 The law of erior applied to an actual series of observations: 
 
 Effect of extending the limiis of error to OD , .... 57 
 
 Various laws of error, ......... 59 
 
 Experimental proof of the exponential law, .... 65 
 
 General conclusions, ......... 66 
 
 Classification of observations, ........ 67 
 
 CHAPTER III. 
 
 Adjtistinent of Direct Observations of One Unknown. 
 
 Observed values of equal quality : 
 
 The most probable value The arithmetic mean, .... 69 
 
 Control of the arithmetic mean, ...... 71 
 
 Precision of the arithmetic mean : 
 
 Bessel's formula, ........ 73 
 
 Peters' formula, ........ 76 
 
 Approximate formulas, ....... 79 
 
 The law of error tested by experience, ..... 82 
 
 Caution as to the tests of precision, ...... 85 
 
 Constant error, .......... 89 
 
 Necessary closeness of computation, ...... gi 
 
 Observed values of different quality : 
 
 The most probable value The weighted mean, 9 2 
 
 Combining weights, ........ 94 
 
 Reduction of observations to a common standard, . . 95 
 
 Computation of weights, ......... 96 
 
 Control of the weighted mean, ...... 96 
 
 Precision of the weighted mean Control of [/z^'J, . . 99 
 Observed values multiples of the unknown : 
 
 The most probnble value, ........ 102 
 
 Precision of a linear function of independently observed values, .' 105 
 
 Miscellaneous examples, ......... no 
 
 NOTE I. On the weighting of observations, ..... 118 
 
 An approximate method, ........ 120 
 
 Weighting when constant error is present, 121 
 
 Assignment of weight arbitrarily, ...... 126 
 
 Combination of good and inferior work, ..... 127 
 
 The weight a function of our knowledge, ..... 128 
 
 General remarks, .......... 130 
 
 NOTE II. On the rejection of observations, ..... 131
 
 TABLE OF CONTENTS. 7 
 
 CHAPTER IV. 
 
 Adjustment of Indirect Oi>ser;>atioiis. 
 
 PAGE 
 
 Determination of the most probable values, ...... 139 
 
 Formation of the normal equations, ...... 145 
 
 Control of the formation, ........ 150 
 
 Forms of computing the normal equations: 
 
 With multiplication tables or a machine, . . . 151 
 
 With a table of logarithms, 152 
 
 With a table of squares, ...... 154 
 
 Solution of the normal equations: 
 
 The method of subsiitution, ....... 156 
 
 Controls of the solution, 158 
 
 Forms of solution : 
 
 Solution without logarithms, ..... 160 
 
 Solution with logarithms, ..... 163 
 
 The method of indirect elimination, ..... 165 
 
 Combination of the direct and indirect methods, . . 167 
 
 Time required to solve a ?et of equations, .... 172 
 
 Precision of the most probable (adjusted) values, .... 174 
 
 First method of finding the weights, ...... 177 
 
 Special case of 2 and 3 unknowns, ..... 178 
 
 Modifications of the general method, ..... 180 
 
 Second method of finding the weights, ..... 184 
 
 To find the m. s. e. of a single observation, ..... 186 
 
 Methods of computing [?'?']> ...... 187 
 
 Precision of any function of the adjusted values (three methods), . 193 
 Average value of the ratio of the weight of an observed value to its 
 
 adjusted value, .......... 198 
 
 Examples and artifices of elimination, ...... 201 
 
 CHAPTER V. 
 
 Adjustment of Condition Observations. 
 
 General statement, .......... 213 
 
 Direct solution Method of independent unknowns, . . . 214 
 
 Indirect solution Method of correlates, ...... 217 
 
 Precision of the adjusted values or of any function of them, . 224 
 
 Mean-square error of an observation of weight unity, . . 224 
 
 Weight of the function, ....... 227 
 
 Solution in two groups, ......... 238 
 
 Programme of solution, ........ 242 
 
 Precision of the adjusted values or of any function of them, . 243 
 
 Solution by successive approximaiion, ...... 247
 
 8 THE ADJUSTMENT OF OBSERVATIONS. 
 
 CHAPTER VI. 
 
 Application to the Adjustment of a Triangulation. 
 
 PAGE 
 
 General statement, . 250 
 
 The method of independent angles, ....... 252 
 
 The local adjustment, ......... 255 
 
 Number of local equations, . . . . . 258 
 
 The general adjustment, ........ 259 
 
 The angle equations, . . . . . . . . 259 
 
 Number of angle equations, ...... 261 
 
 The side equation's, ........ 263 
 
 Reduction to the linear form, ...... 266 
 
 Check computation, ....... 268 
 
 Position of pole, ........ 271 
 
 Number of side equations, ...... 272 
 
 Check of the total number of conditions, .... 273 
 
 Manner of selecting the angle and side equations, . . 273 
 Adjustment of a quadrilateral : 
 
 Solution by independent- unknowns, . . . . . 280 
 
 Precision of the adjusted values, ..... 282 
 
 Solution by correlates, ........ 284 
 
 Precision of the adjusted values, ..... 286 
 
 Solution in two groups, ........ 288 
 
 Precision of the adjusted values, ..... 293 
 
 Solution by successive approximation, ..... 298 
 
 Adjustment of a triangulation net, ...... 300 
 
 Artifices of solution, ........ 302 
 
 The local adjustment, ....... 302 
 
 The general adjustment, ....... 303 
 
 Approximate method of finding the precision, .... 3*3 
 
 The method of directions, ......... 315 
 
 The local adjustment, . . . . . . . . 316 
 
 Checks of the normal equations, 3 2 
 
 Precision of the adjusted values, ...... 3 22 
 
 The general adjustment, ........ 3 2 3 
 
 Approximate method of reduction, ...... 3 2 ^ 
 
 Modified rigorous solution : 
 
 General statement, .......... 33 r 
 
 Local adjustment, . . . 333.336 
 
 General adjustment, 33 6 > 339 
 
 On breaking a net into sections, ....... 34 1 
 
 Adjustment for closure of circuit, . . . . . 34 2 
 
 Discrepancy in azimuth, . 343 
 
 Disci epancy in bases, 347 
 
 Discrepancy in latitude and longitude, ..... 347
 
 TABLE OF CONTENTS. 9 
 
 CHAPTER VII. 
 
 Application to Base-Line Measurements. 
 
 PAGE 
 
 General statement 349 
 
 Precision of a base-line measurement, ...... 350 
 
 The Bonn Base, 353 
 
 The Chicago Base, 356 
 
 Length and number of bases necessary in a triangulation, . . . . 356 
 
 Conneciion of a base with the main trinngulation, .... 360 
 
 Adjustment of a triangulation when more than one base is considered, 362 
 
 Rigorous solution, .......... 363 
 
 Approximate solutions : 
 
 When the angles alone are changed, ..... 365 
 
 When the bases alone are changed, ..... 368 
 
 CHAPTER VIII. 
 
 Appliiation to Levelling. 
 
 Spirit levelling, ........... 371 
 
 Precision of a line of levels, ........ 374 
 
 Adjustment of a net of levels, ....... 377 
 
 Approximate methods of adjustment, ..... 378 
 
 Trigonometrical levelling, ........ 382 
 
 To find the refraction factors, ....... 384 
 
 To find the mean coefficient of refraction, ..... 386 
 
 To find the differences of height 386 
 
 Precision of the differences of height, .... 388 
 
 Adjustment of a net of trigonometrical levels, .... 388 
 
 Approximate methods of adjustment, ..... 392 
 
 CHAPTER IX. 
 
 Application to Errors of Graduation of Line Measures and to Calibration of 
 
 Thermometers. 
 
 Line measures, ........... 395 
 
 Calibration of thermometers, ........ 401 
 
 CHAPTER X. 
 
 Application to Empirical Formulas and Interpolation. 
 
 General statement, . . . . . . . . . . 408 
 
 Applications, .......... 413 
 
 Periodic phenomena, .......... 417 
 
 Applications, 421 
 
 APPENDIX I. Historical note, .... 427 
 
 APPENDIX II. The law of error, .... 429 
 
 TABLES, 435
 
 THE ADJUSTMENT OF OBSERVATIONS, 
 
 CHAPTER I. 
 
 INTRODUCTION. 
 
 THE factors that enter into the measurement of a quan- 
 tity are, the observer, the instrument employed, and the con- 
 ditions under which the measurement is made. 
 
 i. The Instrument. If the measure of a quantity is 
 determined by untrained estimation only, the result is of 
 little value. The many external influences at work hinder 
 the judgment from deciding correctly. For example, if we 
 compare the descriptions of the path of a meteor as given 
 by a number of people who saw the meteor and who try to 
 tell what they saw, it would be found impossible to locate 
 the path satisfactorily. The work of the earlier astrono- 
 mers was of this vague kind. There was no way of testing 
 assertions, and theories were consequently plentiful. 
 
 The first great advance in the science of observation was 
 in the introduction of instruments to aid the senses. The 
 instrument confined the attention of the observer to the 
 point at issue and helped the judgment in arriving at con- 
 clusions. As with a rude instrument different observers 
 would get the same result, it is not to be wondered at that 
 for a long time a single instrumental determination was 
 considered sufficient to give the value of the quantity 
 measured. 
 
 The next advance was in the questioning of the instru- 
 ment and in showing that a result better on the whole than 
 a single direct measurement could be found. This opened 
 
 3
 
 12 THE ADJUSTMENT OF OBSERVATIONS. 
 
 the way for better instruments and better methods of ob- 
 servation. For example, Gascoigne's introduction of cross- 
 hairs into the focus of the telescope led to better graduated 
 circles and to better methods of reading them, resulting 
 finally in the reading microscopes now almost universally 
 used. The culminating point was reached by Bessel, who, 
 by his systematic and thorough investigation of instrument- 
 al corrections and methods of observation, may be said to 
 have almost exhausted the subject. He confined himself, 
 it is true, to astronomical and geodetic instruments, but his 
 methods are of universal application. 
 
 The questioning of an instrument naturally arises from 
 noticing that there are discrepancies in repeated measure- 
 ments of a magnitude with the same instrument or in meas- 
 ures made with different instruments. Thus, if a distance 
 was measured with an ordinary chain, and then measured 
 with a standard whose length had been very carefully de- 
 termined, and the two measurements differed widely, we 
 should suspect the chain to be in error and proceed to ex- 
 amine it before further measuring. So discrepancies found 
 in measurements made with the same measure at different 
 temperatures have shown the necessity of finding the length 
 of the measure at some fixed temperature, and then apply- 
 ing a correction for the length at the temperature at which 
 the measurement is made. 
 
 Corrections to directly measured values are thus seen to 
 be necessary, and to be due to both internal and external 
 causes. The internal causes arising from the construction 
 of the instrument are seen to be in great measure capable 
 of elimination. From geometrical considerations the ob- 
 server can tell the arrangement of parts demanded by a 
 perfect instrument. He can compute the errors that would 
 be introduced by certain supposed irregularities in form 
 and changes of condition. The instrument-maker cannot, 
 it is true, fulfil the conditions necessary for a perfect instru- 
 ment, but he has been gradually approaching them more 
 and more closely. It is to be remembered that, even if an
 
 INTRODUCTION. 13 
 
 instrument could be made perfect at any instant, it would 
 not remain so for any great length of time. 
 
 It hence followed as the next great advance that the 
 instrument was made adjustable in most of its parts, so that 
 the relative positions of the parts are under the control of 
 the observer. This is getting to be more and more the 
 case with the better class of instruments. 
 
 Not only is error diminished by the improved construc- 
 tion of the instrument, but also by more refined methods of 
 handling it. It may be, indeed, that some contrivances 
 beyond those required to make necessary readings for the 
 measure of the quantity in question may be needed. Thus, 
 with a graduated circle regular or periodic errors of gradu- 
 ation may be expected. If the angle between two signals 
 were read with a theodolite, the reading on each signal, and 
 consequent value of the angle, would be influenced by the 
 periodic errors of the circle of the instrument. Though a 
 single vernier or microscope would suffice to read the cir- 
 cle when the telescope is directed to the signals, yet, as the 
 circle is incapable of adjustment, we can only get rid of the 
 influence of the periodicity by employing a number of ver- 
 niers or microscopes placed at equal intervals around the 
 circle. It happens that this same addition of microscopes' 
 eliminates eccentricity of the graduated circle as well. 
 
 This same principle of making the method of observation 
 eliminate the instrumental errors is carried through even 
 after the nicest adjustments have been made. Thus, in or- 
 dinary levelling, if the backsights and foresights are taken 
 exactly equal the instrumental adjustment may be poor and 
 still good work may be done. But good work is more 
 likely if the adjustments have been carefully made, as if for 
 unequal sights, and still the sights are taken equal. 
 
 Simplicity of construction in an instrument is also to be 
 aimed at. An instrument that theoretically ouglit to work 
 perfectly is often a great disappointment in practice. Two 
 striking examples are the compensating base-apparatus and 
 the repeating theodolite, both of which have been aban-
 
 14 THE ADJUSTMENT OF OBSERVATIONS. 
 
 doned on all the leading surveys. In both cases the me- 
 chanical difficulties in the way have proved insurmount- 
 able, and the instruments have been replaced by others of 
 simpler construction, to whose readings corrections can 
 either be computed and applied or the errors of the read- 
 ings can be eliminated by the me'thod of observation. In 
 this way no hopes of an accuracy which cannot be realized 
 are held out. 
 
 Such is the perfection now attained in the construction 
 of mathematical instruments, and the skill with which they 
 can be .manipulated, that comparatively little trouble in 
 making observations arises from the instrument itself. 
 
 2. External Conditions. The great obstacles to accu- 
 rate work arise from the influence of external conditions 
 conditions wholly beyond the observer's or instrument- 
 maker's control, and whose effect can, in general, neither 
 be satisfactorily computed nor certainly eliminated by the 
 method of observation. We have no means of finding the 
 complex laws of their action. Many of them can be avoided 
 by not observing while they operate in any marked degree. 
 Thus, if while an observer was reading horizontal angles 
 on a lofty station a strong wind should spring up, it would 
 be useless for him to continue the work. If the air com- 
 menced to "boil" he should stop. If the sun shone on one 
 side of his instrument its adjustments would be so disturbed 
 that good work could not be expected. So in comparisons 
 of standards. Comparisons made in a room subject to the 
 temperature variations of the outside air would be of little 
 value. The standards should not only not be exposed to 
 sudden temperature changes during comparisons, but at no 
 other time ; for it has been shown by recent experiments 
 that the same standard may have different lengths at the 
 same temperature after exposure to wide ranges of tem- 
 perature.* 
 
 The effects of external disturbances may sometimes be 
 eliminated, in part at least, by the method of observation. 
 
 * American Journal of Science, July, 1881.
 
 INTRODUCTION. 15 
 
 In the measurement of horizontal angles where the instru- 
 ment is placed on a lofty station, the influence of the sun 
 causes the centre post or tripod of the station to twist in 
 one direction during the day. When this influence is re- 
 moved at night the twist is in the opposite direction. As- 
 suming the twist to act uniformly, its effect on the results 
 is eliminated by taking the mean of the readings on the 
 signals observed in order of azimuth and then immediately 
 in the reverse order. 
 
 Atmospheric refraction is another case in point. In ob- 
 serving for time with an astronomical transit the effect of 
 refraction on the mean of the recorded readings is elimi- 
 nated by taking the star on the same number of threads on 
 each side of the middle thread. On the other hand, in the 
 measurement of horizontal angles, if long lines are sighted 
 over, or lines passing from land over large bodies of water 
 or over a country much broken, the effects of refraction are 
 apt to be very marked. As we have no means of eliminat- 
 ing the discordances arising in this way by the method of 
 observation, all we can do is, while planning a triangulation, 
 to avoid as far as possible the introduction of such lines. 
 
 It may happen that the effect of the external disturbances 
 on the observations can be computed approximately from 
 theoretical considerations assuming a certain law of opera- 
 tion. If the correction itself is small this is allowable. As 
 an example take the zenith telescope, with which the method 
 of observing for latitude is such that the correction for re- 
 fraction is so small that the error of the computed value is 
 not likely to exceed other errors which enter into the work. 
 
 3. The Observer. Lastly we come to the observer him- 
 self as the third element in making an observation. Like 
 the external conditions, he is a variable factor ; all new 
 observers certainly are. 
 
 The observer, having put his instrument in adjustment 
 and satisfied himself that the external conditions are favor- 
 able, should not begin work unless he considers that he him- 
 self is in his normal condition. If he is not in that condition
 
 16 THE ADJUSTMENT OF OBSERVATIONS. 
 
 he introduces an unknown disturbing element unnecessa- 
 rily. He is also more liable to make mistakes in his read- 
 ings and in his record. For the same reason he should not 
 continue a series of observations too long at one time, as 
 from fatigue the latter part of his work will not compare 
 favorably with the first. In time-determinations, for in- 
 stance, nothing is gained by observing from dark until 
 daylight. 
 
 The observer is supposed to have no bias. A good ob- 
 server, having taken all possible precautions with the ad- 
 justments of his instrument and knowing no reason for not 
 doing good work, will feel a certain amount of indifference 
 towards the results obtained. The man with a theory to 
 substantiate is rarely a good observer, unless, indeed, he 
 regards his theory as an enemy and not as a thing to be 
 fondled and petted. 
 
 The greater an observer's experience the more do his 
 habits of observation become fixed, and the more mechanic- 
 al does he become in certain parts of his work. But his 
 judgment may be constantly at fault. Thus with the as- 
 tronomical transit he may estimate the time of a star cross- 
 ing a wire in the focus of the telescope invariably too soon 
 or invariably too late, according to the nature of his tem- 
 perament. If he is doing comparison work involving mi- 
 crometer bisections, he may consider the graduation mark 
 sighted at to be exactly between the centre wires of the 
 microscope when it is constantly on the same side of the 
 centre. This fixed peculiarity, which none but experienced 
 observers have, is known as their personal error. 
 
 In combining one observer's results with those of another 
 observer we must either find by special experiment the dif- 
 ference of their personal errors and apply it as a correction 
 to the final result, or else eliminate it by the method of ob- 
 servation. Thus in longitude work the present practice 
 is to eliminate the effect of personal error from the final 
 result by having the observers change places at the middle 
 of the work.
 
 INTRODUCTION. I/ 
 
 It is always safer to eliminate the correction by the 
 method of observing rather than by computing for it. For 
 though it may happen that so long as instruments and 
 conditions are the same the relative personal error of two 
 observers may be constant, yet some apparently trifling 
 change of conditions, such, for example, as illuminating 
 the wires of the instrument differently, may cause it to be 
 altogether changed in character. 
 
 On account of personal error, if for no other reason, it 
 is evident that no number of sets of measures obtained in 
 the same way by a single observer ought to be taken as 
 furnishing a final determination of the value of a quantity. 
 We must either vary the form of making the observations 
 or else increase the number of observers, in the hope that 
 personal error will eliminate itself in the final combination 
 of the measures. 
 
 4. When all known corrections for instrument, for exter- 
 nal conditions, and for peculiarities of the observer have 
 been applied to a direct measure, have we obtained a cor- 
 rect value of the quantity measured? That we cannot say. 
 If the observation is repeated a number of times with equal 
 care different results will in general be obtained. 
 
 The reason why the different measures may be expected 
 to disagree with one another has been indicated in the pre- 
 ceding pages. There may have been no change in the con- 
 ditions of sufficient importance to have attracted the ob- 
 server's attention when making the observations, but he 
 may have handled his instrument differently, turned certain 
 screws with a more or less delicate touch, and the exter- 
 nal conditions may have been different. What the real dis- 
 turbing causes were he has no means of knowing fully. If 
 he had he could correct for them, and so bring the meas- 
 ures into accordance. Infinite knowledge alone could do 
 this. With our limited powers we must expect a residuum 
 of error in our best executed measures, and, instead of cer- 
 tainty in our results, look only for probability. 
 
 The discrepancies from the true value due to these un-
 
 1 8 THE ADJUSTMENT OF OBSERVATIONS. 
 
 explained disturbing causes we call errors. These errors 
 are accidental, being wholly beyond all our efforts to con- 
 trol. As soon as they are known to be constant, or we learn 
 the law of their operation, they cease to be classed as 
 errors. 
 
 A very troublesome source of discrepancies in measured 
 values arises from mistakes made by the observer in reading 
 his instrument or in recording his readings. Mistakes from 
 imperfect hearing, from transposition of figures and from 
 writing one figure when another is intended, from mistak- 
 ing one figure on a graduated scale for another, as 7 for 9, 
 3 for 8, etc., are not uncommon. These also must be classed 
 as accidental errors, theoretically at least. 
 
 Having, therefore, taken all possible precautions in mak- 
 ing the observations and applied all known corrections to 
 the observed values, the resulting values, which we shall in 
 future refer to as the observed values, may be assumed to 
 contain only accidental errors. We are, then, brought face 
 to face with the question, How shall the value of the quan- 
 tity sought be found from these different observed values? 
 
 Synopsis of Mathematical Principles Employed. 
 
 For convenience, and in order to avoid multiplicity of 
 references, the leading principles of pure mathematics made 
 use of in the further development of our subject are here 
 placed together. 
 
 5. Probability. (i) The probability of the occurrence of 
 an event is represented by the fraction whose denominator is tJie 
 number of possible occurrences, all of which are supposed to be 
 independent of one another and equally likely to happen, and 
 whose numerator is the number of these occurrences favorable to 
 the event in question. Thus if an event may happen in a ways 
 and fail in b ways, all equally likely to occur, the probability 
 
 of its happening is - -r,, and of its failing . , , certainty 
 
 tt L j U dr *"j U 
 
 being represented by unity.
 
 INTRODUCTION. 19 
 
 (2) If there are n events independent of each other, and the 
 probability of the first happening is if,\, of the second ip t , and so 
 on, the probability that all will happen is 
 
 Thus if an urn contains two white and seven black balls 
 the probability of drawing- white at each of the first two 
 trials, the ball not being replaced before the second trial, is 
 
 2 I _ I 
 
 9 X 8-^6 
 6. Definite Integrals. (a) To find the value of 
 
 Let / =: A'-sr / then regarding x as a constant in integrat- 
 ing, we have 
 
 ~f*dt = r~f**xdz 
 
 \J 
 
 Multiply each member by / e~*dx, which is equal to 
 
 J o 
 
 /e^dt, since the limits of integration are the same ; then 
 
 
 I y^.* \ 1 f**> /-. 
 
 / g+dt\ - I ds I <r**+* 
 
 ( ^/ ) v/ J 
 
 Hence f e^dt = | yC 
 
 J 
 
 (b) To find the value ol f
 
 20 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Integrating by parts, we have 
 
 / / fs>-aW\ r* tr a W 
 
 aff*dt = (-*!-} +/ L _ d(af) 
 \ 2a /- ^ J - 2 a* ^ ' 
 
 - 20* 
 
 Also, similarly, 
 
 (c) To find the value of 
 
 The value of this integral cannot be expressed exactly 
 in a finite form, but may be found approximately as fol- 
 lows : 
 
 Expanding e'* in a series and integrating each term 
 separately, we have 
 
 / 
 
 dt 
 
 a 3 . l a* 
 = a --- ---- 
 
 3 1.2 5 
 
 This series is convergent for all values of a; but the con- 
 vergence is only rapid enough for small values of a. 
 
 For large values of a it is better to proceed as follows: 
 Integrating by parts, 
 
 fe-*dt= f-^-de^ 
 
 J J 2t 
 
 i _ <2 i /V ,. 
 
 = -- a * -- / T dt 
 2t 2.J? 
 
 Hence 
 
 J a 
 
 ^ ~~ = 
 

 
 INTRODUCTION. 21 
 
 But 
 
 /a S** f^* 
 
 e-*dt -- I e~*dt I e^dt 
 J o *J a 
 
 VTT r" 
 '~~J * dt 
 . . finally, 
 
 J e " dt = ~? ' ^T I l ~~ i + ^y ~ Sf + | 
 
 It is easily shown that, by stopping the summation at 
 any term, the result will differ from the true value by less 
 than the term stopped at. 
 
 /(L 
 e'^dt may be 
 
 computed from the above formulas for any numerical 
 value of a. 
 
 7. Taylor's Theorem. (a) If / (x) is any function of 
 x,a.ndf(x-{-/i) is to be developed in ascending powers of//, 
 then 
 
 (O 
 
 A more rapid approximation is obtained by putting 
 the development in the following form : 
 
 By subtraction and transposition, 
 /( + A) = / W + /<
 
 22 THE ADJUSTMENT OF OBSERVATIONS. 
 
 (b) Let F denote a function of a series of quantities X, 
 F, . . . expressed by the relation 
 
 F=f(X, F, . . . ) 
 
 and let X', Y', . . . denote approximate values of X, Y, 
 . . . and x, y, . . . the corrections to these approximate 
 values, so that 
 
 XX'+x 
 
 Y=Y'+y 
 
 then 
 
 6'F ,, d'F , d'F , 
 
 ' ' 
 
 where . . . are the values found by diflferenti- 
 6X' 6 Y' 
 
 ating f (X, Y, ...),... with respect to X, F, . . . and 
 then substituting X', Y', . . . for X, F. . . . 
 
 If the corrections x, y, . . . are so small that their 
 squares and higher powers may be neglected and they are 
 written dX 1 , dY' t . . . and F f (X', Y', . . .) is written 
 dF, then 
 
 which is exactly the result found by differentiating F, which 
 is a function of X, Y, . . . with respect to these quanti- 
 ties. 
 
 We shall use one form or the other as may be most con- 
 venient. 
 
 Ex. i. If -v is a very small correction to the number N, required to express 
 log (N+v) in the linear form.
 
 INTRODUCTION. 23 
 
 We have 
 
 \og(W+v) =log^+ ^ 
 dN 
 
 where mod. is the modulus of the common system of logarithms. 
 With a seven-place table we may use the formula 
 
 log (N+ v) = log N + 5 N v 
 
 where <5 N is the tabular difference corresponding to one unit for the number. 
 
 For small numbers, however, it is better to take <5 N from the table for the 
 form 
 
 Thus from the table 
 
 log (6543.2 + ^) = 3.8157902 + 0.0000664 v 
 log (654.32 + v) 2.8157902 + 0.0006637 v 
 log (65.432 + ^) = 1.8157902 + 0.0066378 v 
 
 Ex, 2. In a ten-place log. table where angles are given at regular intervals, 
 required to find log sin (A+a) when A is given in the table and a is a num- 
 ber of seconds less than the tabular interval. 
 
 We have 
 
 log sin (A +a)=\og sin A +-T-J 1 log sin (A + - ) [ a 
 
 =log sin A + mod. sin i" a cot (A H ] 
 Now, 
 
 log mod. =: 9.6377843 10 
 log sin i" = 4.6855749 10 
 log io 7 = 7. 
 
 1.3233592 
 
 which expresses log (mod. sin i") in terms of the seventh place of decimals as 
 the unit. Hence 
 
 log sin (A + a) 
 
 = log sin ,4+log-' j i.3233592 + logrt+ (log cot A + - X diff. for i"j j- 
 
 With Vega's Thesaurus,* which gives the log. functions to single seconds 
 to ttie end of the first degree and afterwards for every io", log sin (A +a) can be 
 found from the above expression for values of A > 3, and also when// lies 
 between 20' and 2 to within less than unity in the tenth decimal place. Be- 
 tween 2 and 3 the difference may be as large as 3 units in the tenth phice 
 from the value found by carrying out the formula more exactly. But in the 
 
 * Thesaurus Logaritkmorum Contpletus. Lipsiae, 1794.
 
 24 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Thesaurus, in the trigonometrical part, " the uncertainty of the last figure 
 amounts to 4 units/'* Hence the above process is in general sufficient when 
 this table is used. 
 
 With a seven-place table of log. sines it would be, in general, sufficient to 
 take the tabular difference <5 A for i" for the angle A as the value of 
 
 TT 1 10 
 
 dA. ( 
 so that 
 
 log sin (A +a)=log sin A + 6 A a. 
 
 Ex. 3. If A is the approximate value of an angle, and v a correction to it 
 so small that its square and higher powers may be neglected, required to ex- 
 press log sin (A + v) in the linear form, using a ten-place table. 
 
 Let AI be the angle nearest to A in the table, and set 
 
 then 
 
 log sin (A +v) 
 
 =log sin (A i + a + v) 
 
 ( a \ 
 
 =log sin AI + mod. sin i" a cot ( AI + - \ + mod. sin i" cot (A\ + a) v 
 
 log sin AI + log" 1 1 1.3233592 + log a + (log cot AI +- X diff. for i") 
 
 + log- 1 1 1.3233592 + (log cot A i + a X diff. for i")| v 
 Expand log sin (68 16' 3: 
 
 1.3233592 
 
 log COt A! 9.6003780 
 
 A! 68 16' 30" 
 
 a = 2". 076 
 
 0.9237372 
 
 Xdiff. i" 127 
 
 0.9237245 8,3893 
 
 -?Xdiff. i" +64 
 
 2 
 
 0.9237309 
 
 logtf 0.3172273 
 
 1.2409582 I7,4l6 
 
 log sin 68 16' 30" 9.9680022,271 
 
 Hence log sin (68 16' 32" .076 + v)= 9.9680039,687 + 8,3893 v 
 
 when the difference is expressed in terms of the seventh decimal place as the 
 unit. 
 
 * Bremiker's edition of Vega, translated by Fischer. Preface, p. 10.
 
 INTRODUCTION. 2' 
 
 With a seven-place table, except for small angles or angles near 180 
 it will be sufficient to take 
 
 log sin (A i 
 
 =^o sn 
 
 when 6\ is the tabular difference corresponding to i" for the angle A\. It 
 can be taken by inspection from the table. Thus, 
 
 log sin (68 16' 32" + v)=(). 9680039 + 8,4 v 
 
 8. Interpolation. So far as interpolation is concerned, 
 we have mainly to deal with the logarithms of trigonometric 
 functions. The differences between the successive values 
 given in a table are first differences, and the differences be- 
 tween the successive first differences are second differences. 
 Beyond second differences we do not need to go. 
 
 This may be expressed in tabular form : 
 
 Function. 
 
 First diff. 
 
 Second diff. 
 
 f(A) 
 
 
 
 f(A+a) 
 
 
 4 
 
 f(A+2a) 
 
 
 
 Hence 
 
 f(A+a) := 
 
 Generally 
 
 f(A + na) =f(A) 
 which is Newton's formula. 
 
 Ex. In a ten-place table of log. sines in which values arc
 
 26 THE ADJUSTMENT OF OBSERVATIONS. 
 
 given to every 10 seconds, required log sin A when A is 
 any angle. 
 
 Let A l = the part of the given angled to the nearest 10 
 
 seconds that occurs in the table. 
 =ithe units of seconds and parts of seconds in the 
 
 given angle. 
 d^ d^=i the first and second tabular differences. 
 
 then 
 
 log sin A = log sin f^,-j --- . ioj 
 
 = logsin AA-a + tfaho a)-*-. (i) 
 
 10 100 
 
 Writing this in the form, 
 
 log sin ,4 = log sin ,4 ,+ j A -f (5 -^\ A i ( 2 ) 
 
 we have the convenient rule : Assume the second difference 
 constant throughout the interval a. Then from the first 
 and second differences find by simple interpolation the 
 value of the difference at the middle of the interval. This 
 difference multiplied by the interval gives the correction to 
 the tabular log. sine. 
 
 To find log sin 68 16' 32".O76 : 
 From the table, 
 
 log sin 68 i6'3o"= 9.9680022,271 
 
 di= 83,889 for 10" in units of the seventh decimal place 
 d%= 0,012 
 
 Hence, from equation (i), 
 
 Corr. to tab. value =2.076X8, 3889 + ^^ (10 2.076) 
 
 2 100 
 
 17,416 
 
 .'. log sin 68 16' 32". 076 = 9.9680039,687
 
 INTRODUCTION. 2J 
 
 The difference at 35"= 8,3889. 
 
 We want it at 31". 038, the middle of the interval. 
 
 Now, change of first difference = 0,00012 for i". 
 
 Hence corr. to first difference = (35 31.038) X 0,00012 
 
 = 0.0005 
 First diff. = 8,3889 
 
 Diff. required = 8,3894 
 And 2.076 X 8,3894 17,416 as before. 
 
 9. Periodic Series. To sum the series 
 
 Cos o -\- cos 8 -\- cos 2 -f- . . . -f- cos (;/ \)0 
 Sin (?-(-sin #-|-sin 20-\- . . . -|-sin (;/ 1)6 
 
 where 8=~ , n being an integer, we may proceed as fol- 
 
 lows : 
 
 If 6 is the angle which a line B 'OB makes with OA then 
 the projection of OB on OA is OB cos 0, and the projec- 
 tion of OB' is OB cos 6 if OB'= OB. The projections on 
 a line at right angles to OA are OB sin and OB sin 6 
 respectively. 
 
 If we divide the circumference of a circle into n equal 
 parts at the points A, B, . . . then each angle at the centre 
 
 ^60 
 
 O is ^ , or 6, and by projecting the lines OA, OB, ... on 
 n 
 
 the diameter through A we find the sum of the first series 
 to be zero, and by projecting the same lines on the diameter 
 perpendicular to OA we find the sum of the second series 
 to be also zero. 
 
 These results may be written : 
 
 - cos ;// = o - sn m = 
 
 where m assumes all values from o to n I. 
 Hence it follows that 
 
 - sin in0 cos wti= l/ 2 - sin in 2^ 
 
 v n i , ' v u n 
 
 _ cos i0 h/2- cos ;// 20 = 
 
 ~> 2 
 
 - sin 3 m0 - l /2- cos /// 20 -
 
 28 THE ADJUSTMENT OF OBSERVATIONS. 
 
 10. Notation. The following convenient notation, intro- 
 duced by Gauss, is now very generally used in the method 
 of least squares. 
 
 If a, 2 , . . . are quantities of the same kind, their alge- 
 braic sum is denoted by [a], and the sum of their squares by 
 [aa\ or [V], so that 
 
 [aa\ or [a*] = a? + a* + ... + a\ 
 Also, 
 
 ab~ = ab--a-- . , . a n b n 
 
 j_ _i_ ** i i n 
 
 L c J c, c^ c n 
 
 We shall use the symbol [a to denote the sum of a series 
 of quantities a all taken with the same sign.
 
 CHAPTER II. 
 
 THE LAW OF ERROR. 
 The Arithmetic Mean. 
 
 ii. (a) When the quantity measured, is the quan- 
 
 tity to be found. If M lt M t . . . M n are n direct and in- 
 dependent measures of a quantity, T, we may write 
 
 T-M=A 
 
 where J,, A v . . . A n indicate the differences between T and 
 the observed values, and are therefore the errors of obser- 
 vation. 
 
 We have here n equations and n -f- I unknowns. What 
 principle shall we call to our aid to solve these equations 
 and so find T, A^ A^ . . . J B ? In answering this question 
 I shall follow the order of natural development of the sub. 
 ject, which, in the main, is also the order of its historical 
 development. 
 
 The value sought must be some function of the ob- 
 served values and fall between the largest and smallest of 
 them. If the observed values are arranged according to 
 their magnitudes the}' will be found to cluster around a 
 central value. On first thoughts the value that would be 
 chosen as the value of T would be the central value in this 
 arrangement if the number of observations were odd, and 
 either of the two central values if the number were even. 
 In other words, a plausible value of the unknown would be 
 that observed value which had as many observed values 
 greater than it as it had less than it. Now, since a small 
 change in any of the observed values, other than the central
 
 3O THE ADJUSTMENT OF OBSERVATIONS. 
 
 value, would in general produce no change in the result, 
 the number of observations remaining the same, this method 
 of proceeding might be regarded as giving a plausible re- 
 sult, more especially if the observed values were widely 
 discrepant. 
 
 On the other hand, the taking of the central value is ob- 
 jectionable, because it gives the preference to a single one 
 of the observed values, while if these values are supposed 
 to be equally worth)' of confidence, as it is reasonable to 
 take them in the absence of all knowledge to the contrary, 
 each ought to exert an equal influence on the result. We 
 may, therefore, with more reason assume the value of T to 
 be a symmetrical function of the observed values.* Also, 
 since a change in the number of observations should pro- 
 duce no change in the form of this function, it follows that 
 the function must be of such a form as to satisfy the condi- 
 tion that when the observed values are all equal to one 
 another it will reduce to this common value ; that is, if 
 T=f(M 1 M, . . . M n ], and M,= M a = . . . M n = M, then 
 f(M,M, . . . M) = M. For if we had a single observa- 
 tion, then necessarily f(M) = M. 
 
 Let, then, V be a symmetrical function of M l M^ . . . M n> 
 and put 
 
 Expanding by Taylor's theorem : 
 
 = f(V,V,. . .)-\v\ 
 
 where //, K, . . . . are terms involving the second, third, 
 . . . powers of the small quantities, v lt v, . . . v n 
 
 * See Reuschle, Crelle Jour. Math., vol. xxvi. ; Schiaparelli, Rendiconti del R. Insti- 
 tute Lombards, 1868; Astron. Nachr., 2068,2097; Stone, Month. Not. Roy. Astron. Soc., 
 vol. xxxiii. ; Astron. Nachr., 2092; Ferrero, Expos, del Met. dei Min. Quacfr., Florence, 
 1876. Also Fechner, Ueber den Ausgangswerth der kl. Abweichungssumme, Leipzig, 1874.
 
 THE LAW Of ERROR. 31 
 
 The simplest symmetrical function of the observed val- 
 ues that can be chosen as the form for Fis their arithmetic 
 
 mean that is, '- ^. If we take V equal to this value, then 
 n 
 
 from equation (i) by addition [?>\o, and, neglecting powers 
 of v higher than the first, equation (3) is satisfied identically. 
 Thus far, therefore, it would appear that the arithmetic 
 mean may be taken as one solution. It may happen that the 
 values M 1} M v . . , J/ M are of such a nature that some other 
 symmetrical function than the arithmetic mean will satisfy 
 (2) better than will the arithmetic mean. That the arith- 
 metic mean is on the whole the best form for the function 
 f(M lt M v . . . M n ], when M lt M v . . . M n are direct meas- 
 ures of some phenomenon in the sciences of observation, 
 which sciences only we intend to consider, may be confirm- 
 ed by a comparison of results flowing from this hypothesis 
 with the records of experience. This we shall do later (see 
 Art. 24, 37, 51). 
 
 The older mathematicians, as Cotes and Simpson, laid 
 the foundations of our subject by announcing the principle 
 of the arithmetic mean. Gauss, to whom we owe the first 
 complete exposition, assumed the arithmetic mean as a 
 plausible hypothesis;* and Hansen, who made the next 
 great advances, started from it as an axiom. The princi- 
 ple itself may be stated as follows : If we have n observed 
 values of an unknown, all equally good so far as we know, 
 the most plausible value of the unknown (best value on the 
 whole) is the arithmetic mean of the observed values. 
 
 12. By adding equations (i), Art. n,and taking the mean, 
 we have 
 
 V .. 
 
 * \^ ' 
 
 n 
 
 The last term of this equation will become very small if, 
 
 * Gauss' words are, " Axiomatis loco haberi solet hypothesis." (Theoria Motus, lib. z, 
 sec. 3.)
 
 32 THE ADJUSTMENT OF OBSERVATIONS. 
 
 n being very large, the sum [J] of the errors remains 
 small. Now, if, after making one observation and before 
 making another, we readjust our instrument, determine 
 anew its corrections, choose the most favorable conditions 
 for observing, and vary the form of procedure as much as 
 possible, it is reasonable to suppose that the disturbing in- 
 fluences will balance one another in the result, following 
 from the proper combination of the observed values. It 
 may take an infinite number of trials to bring this about. 
 In the absence of all knowledge we cannot say that it will 
 take less. And, reckoning mistakes in reading the instru- 
 ment or in recording the readings as accidental errors, an 
 infinity of a higher order than the first may be required to 
 eliminate them. 
 
 In other words, there being no reason to suppose that 
 an error in excess (or positive error) is more likely to occur 
 on the whole than an error in defect (or negative error), we 
 
 may, when n is a very large number, consider LJ to be an 
 
 n 
 
 infinitesimal with respect to T. We may, therefore, in this 
 case put 
 
 V= T 
 
 that is, when the number of observed values is very great the 
 arithmetic mean is the true value. 
 
 13. From the principle of the arithmetic mean two im- 
 portant inferences may be derived. For, taking the arith- 
 metic mean, V, of n observed values of an unknown as the 
 most plausible value of that unknown, we may write our 
 observation-equations in the form, 
 
 V-M n =v n 
 
 where v^ t/ a , . . . v n are called the residual errors of observa- 
 tion, or simply the residuals.
 
 THE LAW OF ERROR. 33 
 
 (a) By addition 
 
 nV -[M~\ = [v] 
 
 and . . 
 
 [>]=0. (2) 
 
 that is, the sum of the residuals is zero ; in other words, the 
 sum of the positive residuals is equal to the sum of the negative 
 residuals. 
 
 There is a very marked correspondence between the 
 series in which n is infinitely great and T is the true value, 
 and a series in which n is finite and the arithmetic mean V 
 is taken as the best value attainable. For in the first case 
 the sum of the errors, J, divided by ;/, is zero, and in the sec- 
 ond the sum of the residuals, v, is zero. 
 
 (b) Let V be any assumed value of the unknown other 
 than the arithmetic mean, and put 
 
 V' M T'' 
 
 (3) 
 
 From equations (i) and (3), by squaring and adding, 
 
 [vv] = nVV-2V [M] + [MM] 
 [v'v f ] = n V V- 2 V [M] + [MM] 
 
 Hence by a simple reduction, 
 
 [t^=[ W ] + *(r' ") 
 
 Now, |F' -t J J , being a complete square, is always 
 
 positive. 
 
 .'. [W]> [>'?'] 
 
 that is, the sum of the squares of tlie residuals v, found by 
 taking the arithmetic mean, is a minimum. 
 
 Hence the name Method of Least Squares, which was first 
 given by Legendre. 
 
 Let us recapitulate the three forms of solution proposed
 
 34 THE ADJUSTMENT OF OBSERVATIONS. 
 
 for finding the most plausible value Fof the unknown from 
 the n equations : 
 
 (1) Find all possible values of Fand take the mean. The 
 values of Fare M^ M v . . . M n , since for each observation 
 considered singly the best value must be the directly ob- 
 
 served one, and the mean of these values is != J 
 
 n 
 
 (2) Solve simultaneously, making [v~\ O. This gives 
 
 or 
 
 n 
 
 the arithmetic mean. 
 
 (3) Solve simultaneously, making [w] = a minimum. 
 This gives 
 
 (V-M$+(V-M$+ . . . (V- M n )* = a min. 
 Differentiating with respect to F, 
 
 or 
 
 n 
 
 the arithmetic mean. 
 
 14. When the quantity measured is a function 
 of the quantities to be found. We pass now to the 
 more general but equally common case in which the ob- 
 servations, instead of being made directly on the quantity 
 to be determined, are made indirectly that is, made on a 
 quantity which is a function of the quantities whose values 
 are to be found.
 
 THE LAW OF ERROR. 35 
 
 Thus, let the function connecting the observed quantity 
 Tand the unknowns X, Y, . . . be 
 
 T=f(X, F, . . . ) (i) 
 
 in which the constants involved are given by theory for 
 each observation. 
 
 If M lt M t , . . . are the observed values of 7 1 , the equa- 
 tions for finding the unknowns, when reduced to the linear 
 form, may be written 
 
 (2) 
 
 in which a lt & lt ... Z,, . . . are known constants, and v lt 
 v v . . . are the residual errors of observation. 
 
 Now, if the number of equations, , is equal to the num- 
 ber of unknowns, ,-, the values of X, F, . . . may be found 
 by the ordinary algebraic methods, and if substituted in the 
 equations will satisfy them exactly. But if the number of 
 equations exceeds the number of unknowns, the values 
 found from a sufficient number of the equations will not 
 in general satisfy the remaining equations exactly. Many 
 such sets of values may be found, which are therefore all 
 possible solutions. But of all these possible sets some one 
 will satisfy the equations better than any of the others. We 
 have so far no means of knowing when we have found this 
 most plausible (best on the whole) set of values. With a 
 single unknown the arithmetic mean gives the most plausi- 
 ble result. Let us see if a method of finding means corre- 
 sponding to those of Art. 13 will apply to these equations. 
 
 For simplicity in writing take the three equations: 
 
 where a lt b^ . . . are known, and X, Fare to be found. 
 6
 
 36 THE ADJUSTMENT OF OBSERVATIONS. 
 
 (i) Find all possible values of X and Y, and combine 
 them. 
 
 To do this we form all possible sets of two equations 
 and solve each set. Thus, 
 
 whence at once 
 
 (a A - a A} X= b, M, - b,M, 
 (afa ,,) Y 0,M a a,M, 
 
 (a,d 3 a 3 d^ X= b t M^ b^M 3 
 (ajb t a&) Y= a,M 3 a 3 M, 
 
 (aj), a,b,) X= b 3 M, b,M, 
 (a& a&) Y a,M 3 a 3 M t . 
 
 In combining these values of Jf and of Y we are met by 
 a difficulty. It would not do to take the arithmetic means 
 as the most plausible values, for X and Fmay be better de- 
 termined from one set of equations than from another, and 
 the arithmetic mean gives the most plausible value only on 
 the assumption that all of the values combined in it are of 
 equal quality. It is necessary, therefore, to have a method 
 of combining observations of different quality before we can 
 find X and Fin this way. 
 
 (2) The simultaneous solution of the equations by mak- 
 ing the sum of the residuals equal to zero. 
 
 Hence X, Y should be found from 
 
 which is impossible, as this equation may be satisfied by an 
 infinite number of values of X and Y. The principle is, 
 therefore, insufficient. 
 
 (3) The simultaneous solution by making the sum of the 
 squares of the residuals a minimum. We have 
 
 (a,X '-\-b.Y- My + ... + ... a rain.
 
 THE LAW OF ERROR. 37 
 
 Differentiating with respect to X, Y as independent vari- 
 ables, we find 
 
 \_aa\X-\-\ab\ Y=[aM~\ 
 [ad] X+ [bb\ Y= [bM] 
 where 
 
 [aa\ = a, a, -f- aji, -f a 3 a 3 
 [ab] = ajbi -j- a& + aj, 
 
 This method gives as many equations as unknowns, and so 
 but one set of values of X and Fcan be found. 
 
 We cannot, however, say that we have found the most 
 plausible values of X and Y. All we can say is that the last 
 method employed reduces the number of equations to the 
 number of unknowns and gives us one set of values of X 
 and Y, and that the same principle applied to the special 
 case of one unknown gives the most plausible value of that 
 unknown, in that it gives the arithmetic mean. Analogy, 
 however, would lead us to suspect that we have found the 
 most plausible values of X and Y, With one unknown, if 
 the separate observed values, represented by lines of the 
 proper length, are plotted along a straight line from a cer- 
 tain assumed origin, and equal weights are placed at the end 
 points, the position of the centre of gravity of the weights 
 will coincide with the end of the line representing the 
 arithmetic mean of the distances, and the sum of the squares 
 of the distances of the weights from the centre of gravity is 
 a minimum. 
 
 Again, the centre of gravity of n equally well-observed 
 positions of a point in space would be the most plausible 
 mean position to take for the point. But it is a well-known 
 principle that, if equal weights be placed at points in 
 space, the centre of gravity of these weights is at a point 
 such that the sum of the squares of its distances from the 
 weights is a minimum.* On this principle Legendre found- 
 ed the rule of minimum squares, and he employed the rule 
 
 * See, for example, Todhunter's Statics^ Art. 138. 
 
 21101)5
 
 38 THE ADJUSTMENT OF OBSERVATIONS. 
 
 as giving a convenient method of solution in the class of prob- 
 lems under consideration. 
 
 The Law of Error of a Single Observed Quantity. 
 
 15. With a single unknown we have seen that the most 
 plausible value is the arithmetic mean of the independently 
 observed values, and that it can be found by making the 
 sum of the squares of the residuals a minimum. The 
 methods are equivalent. 
 
 With more than one unknown we have failed to find this 
 correspondence of methods. The reasoning from analogy 
 in the preceding article is well enough as far as it goes, but 
 it is not conclusive. The difficulty lies in combining values 
 not equally good. We must, therefore, devise some method 
 of combining such values before a rule for finding means 
 can be applied to several unknowns. 
 
 Now, when several independent measures of the same 
 quantity, all equally good, have been made, it must be grant- 
 ed that errors in excess and errors in defect are equally 
 likely to occur to the same amount that is, are equally 
 probable. Experience shows that in any well-made series 
 of observations small errors are likely to occur more fre- 
 quently than large ones, and that there is a limit to the 
 magnitude of the error to be expected. If, therefore, a de- 
 notes this limit or maximum error, we must consider all the 
 errors of the series to be ranged between -\-a and a, but 
 to be most numerous in the neighborhood of zero. Hence 
 the probability of the occurrence of an error may be as- 
 sumed to be a certain function of the error. 
 
 If, then, the probability that an error does not exceed 
 A be denoted by ?(J), the probability of an error between 
 J and A-\-dA is 
 
 tp(A)dd suppose. (i) 
 
 The function <p(A) is called the law of distribution of error, 
 or simply the law of error.
 
 THE LAW OF ERROR. 39 
 
 The probability that an error falls between any assigned 
 limits b and a is the sum of the probabilities <f(J) dA ex- 
 tending from b to a, and is expressed in the ordinary nota- 
 tion of the integral calculus by 
 
 A (2) 
 
 Hence it follows that the probability that an error does not 
 exceed the value a is 
 
 (3) 
 
 The properties of errors stated above are not sufficient 
 to determine the form of the function ^(J) definitely. 
 Among other forms that might be chosen to satisfy them 
 the simplest would be 
 
 (4) 
 
 where h is a constant. 
 
 This was, indeed, that selected by Laplace in his inves- 
 tigation Determiner le milieu que I' on doit pr entire entre f rot's 
 observations donnees d'un meme phtnomene (Mem. Acad. Paris, 
 1774). From this form may be readily derived the results 
 stated in Art. n. 
 
 The form of <p(J) may, however, be more satisfactorily 
 determined by calling in the aid of the calculus of probabil- 
 ities. For if ^>(4) dJv 9>(4i) ^4i> denote the prob- 
 abilities of the occurrence of errors in n observations be- 
 tween J, and Jj-j-^/Jj, 4, and A^-\-dA v . . . respectively, 
 the probability of the simultaneous occurrence of this sys- 
 tem of errors is proportional to the product (see Art. 5). 
 
 $0(4) ^(4,) . . . 9 r (4) 
 Denote this expression by </?, so that 
 
 Now, the true value T, and therefore the values of J,, J,, 
 . . . J w , are unknown. If we make the expression for </> a
 
 40 THE ADJUSTMENT OF OBSERVATIONS. 
 
 maximum, we should find the most probable value of the 
 unknown. But we have seen that the most plausible value 
 of the unknown is the arithmetic mean of the observed val- 
 ues, and that when the number of observations is very large 
 the arithmetic mean is the true value T. Calling, then, the 
 most plausible value the most probable value, we have, 
 when n is large, the true value by making </> a maximum. 
 The forfh of the function <p will, therefore, result from this 
 hypothesis. 
 
 Now, since log tp varies as </>, we must have log </> a maxi- 
 mum, and therefore by differentiation 
 
 <t>_b log y(4) v/J. b log 
 dT <>4 " dT ' K " dT ~ 
 
 or 
 
 6 log y>(4 
 
 i ffrL _ ^ IU ^ n^; , j j lu s w . , /g\ 
 
 / i *y^ ^i .^ A /f I 3 A f\ A 
 
 since from equation (i), Art. 11, 
 
 ^4 ^/4 ^4 
 
 dT~dT ~dT 
 
 But, from the principle of the arithmetic mean, when the 
 number of observations is very great, 
 
 4 + 4+. . . +4-0. (7) 
 
 Also, since equations (6) and (7) must be simultaneously 
 satisfied by the same value of the unknown, being the most 
 probable value in either case, and since the errors 4> 4 
 4 are connected only by the relation [J] = o, we necessari- 
 ly have, when n > 2* 
 
 6 log p(4) _ 6 log 
 
 4 d4 4 d J t 
 
 Clearing of fractions and integrating, 
 
 . . . k suppose. 
 
 * When = 2 it reduces to an identity.
 
 THE LAW OF ERROR. 4! 
 
 where e is the base of the Napierian system of logarithms 
 and c is a constant. 
 
 72 i 
 
 Now, since </> is to be a maximum, ^ must be negative. 
 But when <p is a maximum subject to the condition [J] = o, 
 
 72 / 
 
 then ; = nk$. Hence, since </> is positive, k must be 
 negative, and, putting it equal to - , we have 
 
 _ A 
 
 y( J) = ce ^ 
 
 the law of error sought. 
 
 16. In this expression there are two symbols undeter- 
 mined, c and IJL. To find c. Since it is certain that all of 
 the errors lie between the maximum errors -{-a and a, 
 we have 
 
 /+ 
 a 
 
 dA = 
 
 But as the values of a are different for different kinds of 
 observations, and as we cannot in general assign these val- 
 ues definitely, we must take -f- an d =o as the extreme 
 limits of error, so that c is found from 
 
 /+ 
 c I 
 
 J -00 
 
 and hence (see Art. 6) 
 
 H V2K 
 
 and the law of error may be written 
 
 i 
 
 ft \2 
 
 or by putting //" 
 *
 
 42 THE ADJUSTMENT OF OBSERVATIONS. 
 
 When this latter form is used it is only for greater conveni- 
 ence in writing, and h is to be looked on as a mere sym- 
 bol standing for 
 
 W 
 
 A" 
 
 As regards //", it is evident that for e~ 2 ^ to be a possible 
 
 J" 
 quantity must be an abstract number. Hence a is a 
 
 I? 
 quantity expressed in the same unit of measure as A. 
 
 Also, from the form of the function <f>(^}, it is evident 
 that the probability of an error J will be the larger the 
 larger JJL is, and vice versa. Hence // is a test of the 
 quality of observations of different series, the unit being 
 the same. 
 
 Again, the total number of errors in a series being n, the 
 number between J and A-\-dA will, from the definition of 
 probability, be n tp(d) dA. Hence the sum of the squares of 
 the errors A in the same interval will be equal to nJ 1 y(A) dd, 
 and the sum of the squares of the errors between the limits 
 of error -j- a and a will be 
 
 Extending the limits of error a to o t this expression 
 becomes, after substituting for <p(A) its value, 
 
 /"+0 
 
 I 
 
 J ~ 
 
 which (see Art. 6) reduces to //. 
 
 Hence // is the mean of the sum of the squares of the 
 errors A that occur in the series. It is called the mean- 
 square error, and will be referred to as the m. s. e. 
 
 17. The Principle of Least-Squares. Having de- 
 fined the symbols c and // in the expression for <p(A], let us 
 return to Art. 15, Eq. 5. 
 
 If the observed values are of the same quality through- 
 
 _ L A .!3 
 out, p. is constant and the product <p becomes c*e ^ . This
 
 THE LAW OF ERROR. 43 
 
 product is evidently a maximum when [J 11 ] is a minimum ; 
 that is, if we assume that each of a very large number of ob- 
 served values of a quantity is of the same quality, the most prob- 
 able value of the quantity is found by making the sum of the 
 squares of the errors a minimum. 
 
 If the observed values are not of the same quality, (i is 
 different for the different observations, and the most prob- 
 able value of the unknown would be found from the maxi- 
 
 - f 1 
 mum value of e L 2 ^J . that is, from the minimum value of 
 
 pf-| 
 
 I -~ a J. Thus if each of a large number of observed values of a 
 
 quantity is of different quality, the most probable value of the 
 quantity is found by dividing each error of observation by its 
 m. s. e. and making the sum of the squares of the quotients a 
 minimum. 
 
 This latter includes the case in which the observed 
 quantity is a function of several independent unknowns 
 whose values are to be found. For if 
 
 where the functions //,, . . . are of the same form and 
 differ only in the constants that enter; then if J,, 4,, . . . 
 denote the errors of observation, we have 
 
 4 =/'(*.* - O-^i 
 
 4 =/*(*,* ...)-^ 
 
 Hence since J,, J a , . . . are functions of the independent 
 variables x, y, . . . we must have 
 
 with respect to the variables .r, y, . . . But the differenti- 
 ation of this equation with respect to x t y, . . . and the 
 equating of the differential coefficients to zero, gives as 
 7
 
 44 THE ADJUSTMENT OF OBSERVATIONS. 
 
 many equations as unknowns, from which equations the 
 most probable values of ,r, y, . . . may be found. 
 
 1 8. Two other inferences from the preceding general 
 principles are important : 
 
 (a) Since in a series of observed values of different 
 quality the sum of the squares of the errors, divided by 
 their respective m. s. e. made a minimum, leads to the most 
 probable values, it follows that observed values of different 
 quality are put on a common basis for combination by 
 dividing by their respective m. s. e. This conclusion will 
 be found developed in Chapter III. 
 
 f . 
 
 (b) Since r is an abstract number, no matter what the 
 
 unit of measure in which the observed values are expressed, 
 it follows that heterogeneous measures may be combined in 
 the same minimum equation. For an example in which this 
 is fully brought out see Art. 169. i 
 
 The Law of Error of a Linear Function of Independently 
 Observed Quantities. 
 
 19. We have found the law of error in the case of a quan- 
 tity directly observed, and which may be a function of one 
 or more unknowns. There remains the question as to the 
 form the law of error assumes in the case of a quantity, F, 
 which is a linear function of several independently observed 
 quantities, M M . . . M n ; that is, when 
 
 where a lt a ..... are constants. 
 
 For simplicity in writing consider two observed quanti- 
 ties, M^ M^ only, and let // /* 2 be their m. s. e. The proba- 
 bility of the simultaneous occurrence of the errors J, in M^ 
 and 4, in M, is
 
 THE LAW OF ERROR. 45 
 
 Now, an error 4 in M l and an error J a in M^ produce an 
 error J in F, according- to the relation 
 
 J = *,4+,4 (2) 
 
 and this relation can always be satisfied by combining any 
 value of 4, with all values of J, ranging from oo to -f-co. 
 The probability, therefore, of an error J in F may be written 
 
 <p(J) dJ = H A <U t f 'Y*'*' 1 - W <W, 
 
 7T *s -w 
 
 But from (2), and since J 2 is independent of J,, 
 
 af J = ^J s 
 Hence 
 
 = h^d_J / ^-^v-vC-^OV 
 
 7T <7 2 ^/ -w 
 
 /, /, // w >- + =o v^ 
 _'tA J e --*w+w a s* I i 
 
 A ./ - 
 
 which is of the form 
 
 h 
 
 That is, the law of error of the function F is the same as that 
 of the independently measured quantities J/,, J/ 2 . 
 The m. s. e. of the function /MS found from 
 
 that is, from 
 
 Tliis theorem is one of the most important in the method of 
 least squares, and will be often referred to.
 
 46 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. To find the m. s. e. of the arithmetic mean of n equally well observed 
 values of a quantity : 
 We have 
 
 . . . +Af n ) 
 
 Let // = m. s. e. of the arithmetic mean 
 
 >u =: m. s. e. of each observed value 
 
 Then r 
 
 /* 2 = -5 (ju- + /< 5 + ... to n terms) 
 
 That is, the m. s. e. of the arithmetic mean of n observations is = part of that 
 
 Vn 
 of a single observation. 
 
 On the Comparison of the Accuracy of Different Series of 
 
 Observations. 
 
 20. The Mean-Square Error. We have seen in Art. 
 1 6 that the m. s. e. // affords a test of the relative accuracy 
 of different series of observations. This test was suggested 
 by the fundamental formula of the law of error, and is 
 naturally the first that would be taken for that purpose. 
 
 The value of// is the mean of the sum of the squares of 
 the errors in a series between the extreme limits of error, 
 and since the probability of an error is the number of cases 
 favorable to its occurrence divided by the total number of 
 cases, // is given by the expression 
 
 where -f- <* and a are the limits of error. 
 
 Hence if the number of errors n is a very large number 
 a close approximation to the value of // will be given by 
 
 A " A z A a 
 
 ,t* _L_ _L 2 _|_ -I- 
 
 7 - n n n 
 
 The difference in precision of these two values of // will be 
 pointed out later. (See Art. 23.)
 
 THE LAW OF ERROR. 47 
 
 21. The Probable Error. A second method of deter- 
 mining the relative precision of different series of observa- 
 tions is by comparing errors which occupy the same rela- 
 tive position in the different series when the errors are 
 arranged in order of magnitude. The errors which occupy 
 the middle places in each series are, for greater conveni- 
 ence, the ones chosen. 
 
 Let the errors in a series, arranged in order of magni- 
 tude, be 
 
 2a, . . . r, . . . o t 
 
 each error being written as many times as it occurs ; then 
 we give to that error r which occupies the middle place, 
 and which has as many errors numerically greater than it 
 as it has errors less than it, the name of probable error. If, 
 therefore, n is the total number of errors, the number lying 
 
 n 
 between -\- r and r is , and the number outside these 
 
 limits is also -. In other words, the probability that the 
 
 Z 
 
 error of a single observation in any system will fall between 
 the limits -\-r and r is -, and the probability that it will 
 
 fall outside these limits is also . We have, therefore, 
 
 V7r*S -r 
 
 from which to find r. 
 
 If we put // J = /, and the value t = p corresponds to 
 J = r, then 
 
 2 
 
 1 - 
 
 Expanding the integral in a series as in (c) Art. 6, we shall 
 find that approximately the resulting equation is satisfied by 
 
 p = 0.47694
 
 48 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Now, since 
 
 hr = p = 0.47694 and /ifj. \/ 2 z 
 
 it follows that 
 
 r = 0.67457* 
 
 = - n roughly. 
 
 
 
 Hence to find the probable error we compute first the 
 mean-square error and multiply it by 0.6745. 
 
 As a check, the error which occupies the middle place in 
 the series of errors arranged in order of magnitude may be 
 found. It will be nearly equal to the computed value, if 
 the series is of moderate length. 
 
 It is to be clearly understood that the term probable error 
 does not mean that that error is more probable than any 
 other, but only that in a future observation the probability 
 of committing an error greater than the probable error is 
 equal to the probability of committing an error less than 
 the probable error. Indeed, of any single error the most 
 probable is zero. Thus the probability of the error zero is 
 to that of the probable error r as 
 
 g~ l 
 
 Vx Vn 
 or 
 
 T ,,-( '47694) 2 
 
 i . e 
 or 
 
 i : 0.8 
 
 The idea of probable error is due to Bessel (Berlin. Astron. 
 Jahrb., 1818). The name is not a good one, on account of 
 the word probable being used in a sense altogether different 
 from its ordinary signification. It would be better to use 
 the term critical error, for example, as suggested by De 
 Morgan, or median error, as proposed by Cournot. 
 
 22. The Average Error. It naturally occurs, as a 
 third test of the accuracy of different series of observations, 
 to take the mean of all the positive errors and .the mean of 
 all the negative errors, and then, since in a large number of
 
 THE LAW OF ERROR. 49 
 
 observations there will be nearly the same number of each 
 kind, to take the mean of the two results without regard to 
 sign. This gives what may be termed the average error, 
 It is usually denoted by the Greek letter r t . 
 
 Reasoning as in Art. 20, we have approximately 
 
 \A 
 
 y = 
 
 n 
 
 where [J is the arithmetic sum of the errors. 
 
 An expression for ^ in terms of the mean-square error // 
 may be found as follows. The number of errors between J 
 and J -f- dJ is 
 
 and the sum of the positive errors in the series is 
 
 n C A <f(J) dJ 
 
 *s 
 
 The sum of the negative errors being the same, the sum of 
 all the errors is 
 
 /+00 
 A (p(A) dA 
 ) 
 
 Hence 
 
 2/1 f+o 
 
 ~7= / df 
 VnJ o 
 
 fe-'Vl 
 
 " h 
 
 the relation required. 
 
 The average error may, as stated above, be directly used 
 as a test of the relative accuracy of different series of ob- 
 servations. Indeed, I think it should be more used tor this 
 purpose than it is. The general custom is, however, to 
 employ it as a stepping-stone to find the mean-square and 
 probable errors. This can be done, for the reason that it is 
 more easy to compute [J than [J-]. 
 
 T 1 . C
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The formulas for // and r computed in this way are as 
 follows. From the last equation preceding 
 
 and from Art. 21 
 
 =1.2533- 
 
 n 
 
 r = 0.6745 n 
 = 0.8453^ 
 
 The relations connecting /JL, r, and rj are easily remembered 
 in the following form : 
 
 These relations may also be conveniently arranged in tabu- 
 lar form : 
 
 
 
 t* 
 
 r 
 
 ? 
 
 f* = 
 
 I.OOOO 
 
 1.4826 
 
 1-2533 
 
 r = 
 
 0.6745 
 
 I.OOOO 
 
 0.8453 
 
 7j = 
 
 0.7979 
 
 1.1829 
 
 I.OOOO 
 
 23. We have seen that the m. s. e. p. may be computed 
 from the sum of the squares of the errors and also from the 
 sum of the errors without regard to sign. In the deriva- 
 tion of each formula certain approximations have been 
 made. The question then arises which of the two methods 
 will give the more reliable result. This will be shown by 
 the ranges in the two determinations of the value of p. 
 
 (a) We proceed to find first the m. s. e. in the determi- 
 nation of //* by taking the approximate formula 
 
 -- (= /V suppose) 
 
 ft
 
 THE LAW OF ERROR. 51 
 
 instead of the rigid formula 
 
 If we put 
 
 l = ri-f 
 
 then, since // is the exact value, I will be the error of // 
 computed from the errors 4> J 3 . . . according to the 
 
 formula i -. Squaring, we have 
 n 
 
 Now, letting the errors J assume all possible values, the 
 average value of the fourth powers is (see Art. 6) 
 
 
 The number of the products 4*4*1 4*4' being the 
 number of combinations of n things, two at a time, is -' 
 and the average of the values is 
 
 n(n-i) \ 2h 
 2 
 
 that is (Art. 20), 
 
 "' P 
 
 j 21 r 
 
 \ Vx J o 
 
 - 
 
 The average value of ft LI is //*. Hence finally 
 
 n n* 2
 
 52 THE ADJUSTMENT OF OBSERVATIONS. 
 
 and 
 
 or 
 
 *=*(' 
 
 = /jLii ) when n is very large. 
 
 V V2H' 
 
 (b) In the second place, for the average error 37 we pro- 
 ceed in precisely the same way. We have 
 
 Let 
 Then 
 
 = /. 4. ^^ - 
 
 '" 
 
 n ' w a 2 TT 
 
 which gives the error in 37. 
 Also, since 
 
 the error in // is 
 
 a /7T . / 2 iU , . .A 2 
 ~= A/ - \f l -I that is, // A/ 
 
 I/TJJ ^ 2 K 7T f 2 
 
 Hence by this method of computing JJL the value of // is 
 contained between the mean limits
 
 THE LAW OF ERROR. 53 
 
 No\v, since r 2 > i, the limits in the latter case are the 
 larger, and we therefore conclude that the former method 
 of computing /* is the better of the two. 
 24. From the equation 
 
 we may derive a test of the validity of the law of error, and 
 a rather curious one. For /* and y may be determined 
 from measurements, and if the experimental values found 
 satisfy the equation 
 
 -* 
 
 r/ 2 
 
 we must conclude that the theory is correct. This may be 
 classed as an additional a posteriori proof to that given in 
 Art. 51. 
 
 25. Whether we should use the m. s. e. or the p. e. in 
 stating the precision is largely a matter of taste. Gauss 
 says : " The so-called probable error, since it depends on 
 hypothesis, I, for my part, would like to see altogether 
 banished ; it may, however, be computed from the mean by 
 multiplying by 0.6744897." On the other hand, the Inter- 
 national Committee of Weights and Measures decided in 
 favor of the probable error: " It has been thought best in 
 this work that the measure of precision of the values ob- 
 tained should always be referred to the probable error com- 
 puted from Gauss' formula, and not to the mean error." 
 (Proch Verbaux, 1879, P- 77-) 
 
 In the United States, in the Naval Observatory, the 
 Coast Survey, the Engineer Corps, and the principal ob- 
 servatories, the p. e. is used altogether. So, too, in Great 
 Britain, in the Greenwich Observatory, the Ordnance Sur- 
 vey, etc. In the G. T. Survey of India the m. s. e. is used, 
 for the reason given by Gauss above. Among German 
 geodelicians and astronomers the m. s. e. is very generally 
 employed. 
 
 In this book the m. s. e. will be used in the text, and the 
 m. s. e. and p. e. in the examples indifferently.
 
 54 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The Probability Curve. 
 
 26. The principles laid down in the preceding articles 
 may be illustrated geometrically as follows: 
 
 We have seen that in a series of observations the proba- 
 bility that an error will lie between the values x and x-\-dx 
 is given by the expression 
 
 h 
 
 dx 
 
 Fig.l 
 
 Now, if O is the origin of co-ordinates, and a series of errors, 
 
 x, are represented by the distances 
 from O along the axis of abscissas 
 OX, positive errors being taken to 
 the right of O and negative errors 
 to the left, then the probability, in 
 a future observation, of an error 
 falling between x and x -f- dx will 
 
 be represented by the rectangle whose height is ^ e~ h<t ^ 
 
 Vn 
 
 and width dx, or, more strictly, by the ratio of this rectangle 
 to the sum of all such rectangles between the extreme 
 limits of error. This sum we have for convenience already 
 denoted by unity. 
 
 Hence for a series of observations whose quality is 
 known, by giving to x all values from -j-oo to oo and draw- 
 ing the corresponding ordinates, we shall have a continuous 
 curve whose equation may be written 
 
 fi . _2~2 
 
 This curve is called the probability curve. 
 
 27. To Trace the Form of the Curve. Since x 
 enters to the second power, and y to the first power, the 
 curve is symmetrical with respect to the axis of y, and the 
 form of the equation shows that it lies altogether on one
 
 THE LAW OK ERROR. 55 
 
 side of the axis of x. Also, when x = o, - -=o; that is, the 
 
 ax 
 
 tangent at the vertex is parallel to the axis of x. 
 
 As x increases from o the values of y continually de- 
 crease. When x = co , then 
 
 j Jy 
 
 y = o and -f- o 
 ax 
 
 showing that the axis of x is an asymptote. 
 Again, since 
 
 -* - -- 
 
 dx t/jj. 
 
 there is a point of inflection when 
 
 i 
 
 X -:-=. = fJ. 
 
 h^ 2 
 and the m. s. e. is therefore the abscissa of the point of in- 
 
 flection. Also, when x = o, ^ is negative, showing that the 
 
 dx* 
 
 ordinate at the vertex is the maximum ordinate. Hence 
 the curve is of the form indicated in Fig. i, OA represent- 
 ing the maximum ordinate and CU/the m. s. e. 
 
 The values of //, that is, of - p., being different for diflfer- 
 
 l*V 2 
 
 ent series of observations, the form of the curve will change 
 for each series, and the curve may be plotted to scale from 
 values of y corresponding to assumed values of .r. 
 
 In plotting the curve, since the maximum ordinate at 
 
 the vertex - - enters as a factor into the values of each of 
 
 \' - 
 
 the other ordmates, its value may be arbitrarily assumed. 
 We may therefore adopt a scale for plotting the ordinates 
 different from the scale by which the abscissas are plotted, 
 in order to show the curve more clearly. 
 
 The form of the curve is in accordance with the prin- 
 ciples already laid down in deducing the law of error, and
 
 56 THE ADJUSTMENT OF OBSERVATIONS. 
 
 could have been derived from them directly. Thus that 
 small errors are more probable than large is indicated by 
 the element rectangle areas being greater for values of x 
 near zero than for values more distant ; that very large 
 errors have a very small probability is indicated by the 
 asymptotic form of the curve, and that positive and negative 
 errors are equally probable is indicated by its symmetrical 
 form with respect to the axis of y. 
 
 28. The area of the curve of probability being the sum 
 
 of the rectangles j=.e" kj ^dx, for values of x extending. 
 
 V 7T 
 
 from -j- oo to oo we have denoted by unity. If, then, we 
 represent by the area of this curve the total number of 
 errors that occur in a series of observations, it follows from 
 the definition of probability that the area included between 
 certain assigned limits would represent the number of 
 errors to be expected in the series between the values of 
 those limits. 
 
 Thus if O is the origin, the area 
 to the right of OA would represent 
 the number of positive errors and 
 the area to the left of OA the num- 
 ber of negative errors. The area 
 
 OPP'A would represent the number 
 
 P R X 
 
 of positive errors less than OP, the 
 
 area PRR'P the number that lie between OP and OR. 
 
 If the area AOPP' is equal to one-half the total area 
 AOX. then the number of positive errors less than OP 
 would be equal to the number greater than OP. Hence' 
 OP would represent the probable error. If OQ be taken 
 equal to OP, the area PQQ ' P' would represent the number 
 of errors numerically less than the probable error. 
 
 29. The average error 57 may be illustrated as follows: 
 If x^ is the abscissa of the centre of gravity of any curve 
 referred to the axes of Jf and Y, then 
 
 _ fyx dx
 
 THE LAW OF ERROR. 57 
 
 Applying this to the curve of probability and calling the 
 whole area unity, we have 
 
 h \/~ 
 = y, the average error. 
 
 The Law of Error applied to an Actual Series of Observa- 
 tions. 
 
 We here bridge over the gulf between the ideal series 
 from which we have derived the law of error and the actual 
 series with which we have to deal in practical work, and 
 which can only be expected to come partially within the 
 range of the law constructed for the ideal. 
 
 30. Effect of Extending the Limits of Error to 
 
 00. The expression -=e~ h *^dA gives the value of the 
 
 \ Tt 
 
 probability of an error between J and d-\-dJ in an ideal 
 series of observations where the values are continuous be- 
 tween limits infinitely great. In all actual series the possible 
 error is included within certain finite limits, and the proba- 
 bility of the occurrence of an error beyond those limits is 
 zero. Practically, however, the extension of the limits of 
 error to can make no appreciable difference in either 
 case, as the function <p(J) decreases so rapidly that we can 
 regard it as infinitesimal for large values of J ; in other 
 words, the greater number of errors is in the neighborhood 
 of zero, and therefore the most important part is the part 
 covered by both. This has been illustrated geometrically 
 in the discussion of the probability curve, and will now be 
 developed from another point of view.
 
 58 THE ADJUSTMENT OF OBSERVATIONS. 
 
 We have for the probability of the occurrence of an 
 error not greater than a, in a series of observations, 
 
 2h 
 
 This may be put in the form (/ = 
 
 2 
 
 and is usually denoted by the symbol 6(t). 
 
 The method of evaluating- S(f) has been explained in 
 (c) Art. 6. In Table I. will be found the values of the func- 
 tion S(t) corresponding to the argument , that is, -. The 
 
 P r 
 
 reason for arranging the table in this way is that it is more 
 
 convenient to compute than p. 
 
 r r 
 
 The probability that an error exceeds a certain error a 
 is i #(/), and may be found from Table I. by deducting 
 the tabular value from unity. Thus we have the probability 
 that a is greater than r is 0.5, than 2r is 0.1773, than y is 
 0.0430, than 4r is 0.0070, than ^r is 0.0007, than 6r is o.oooi. 
 
 Hence in 10,000 observations we should expect only one 
 error greater than 6r, in 1,000 only one greater than $r, in 
 100 only one greater than 4?% and in 25 only one greater 
 than y. If in any set of observations we found results 
 much at variance with these we could assume that they 
 arose from some unusual cause, and should, therefore, be 
 specially examined. As in practice the number of observa- 
 tions in any case is usually under 100, we are eminently safe 
 in taking the maximum error at about $r or 3//. 
 
 We are now in a position to estimate the change intro- 
 duced by replacing c as found from
 
 THE LAW OF ERROR. 59 
 
 h 
 by := as found from 
 
 in the investigation of the law of error. 
 These equations may be written 
 
 Hence 
 
 Jl 1 2/1 /*<*> \ 
 
 c= - -.(1+ =. I e - A ** 3 tU I approximately. 
 
 \ - \ \ - J a I 
 
 and taking a = $r, we have, from Table I., 
 
 // , 
 c "~^ ( 2 --9993) 
 
 h 
 = 1.001 - 
 
 T 7T 
 
 Hence the difference being less than ~T : of the quantity 
 
 I OOO 
 
 sought, the approximate value of c found by extending the 
 limits a to oo may be considered satisfactory. 
 
 31. Various Laws of Error. We have taken the 
 arithmetic mean of a series of observed values of a quantity 
 made under like conditions as the most plausible value of 
 the quantity. The supposition of each observed quantity 
 being subject to the same law of error leads to the mean as 
 the most probable value. " The method of least squares is, 
 in fact, a method of means, but with some peculiar char- 
 acters. The method proceeds upon this supposition, that 
 all errors are not equally probable, but that small errors are 
 more probable than large ones." * 
 
 Now, in an ordinary series we assume a good deal when 
 we take each observation of the series as subject to the same 
 
 * Whewell, History v/the Induclirc Sciences, vol. ii.
 
 60 THE ADJUSTMENT OF OBSERVATIONS. 
 
 special law of error the exponential law. We can certainly 
 conceive of laws different from this one. It is more probable 
 that each set of observations has its own law depending on 
 instrument, observer, and conditions. If we could go back 
 to the sources of error we could find this law in each case. 
 Let us follow out this idea in a few simple cases and see to 
 what it leads : 
 
 32. First take the case where all errors are equally prob- 
 able that is, where J can with the same probability as- 
 sume all values between -f- a and #, the extreme limits of 
 error. 
 
 Since a is the maximum error, 
 
 s* + a 
 
 (p(A) I dd=i, <p(A) being constant, 
 
 / -a 
 
 and therefore 
 
 K^)= 
 
 20. 
 
 the law of error. 
 
 For the m. s. e. // we have 
 
 r* 
 = / 
 
 J -a 
 
 3 
 Also 
 
 /* 
 a. 
 
 2a 
 
 2 
 
 The p. e. is found from the relation 
 
 or 
 
 r+ r dA i 
 
 / 
 
 J -r 2a 2
 
 THE LAW OF ERROR. 
 
 6l 
 
 and 
 
 Fig.3 
 
 that is, the p. e. is half the max. error. 
 
 This is also evident geometrically, for the curve of error 
 
 y = constant 
 
 is a straight line parallel to the 
 axis of abscissas. 
 
 Hence by definition we find 
 the p. e., by bisecting the area, to 
 
 be -. The p. e. would be repre- 
 
 o p x 
 
 sented in the figure by OP. 
 
 33. Next consider an error to arise from two independent 
 sources, x, y, each of which can with the same probability 
 assume all values between -f- a and a, so that for the total 
 error J we have 
 
 To find <f(J), the law of error: If J and J -f- dJ denote two 
 consecutive errors, then since x and y have values between 
 -\-a and a with interval dJ, each of the quantities x, y has 
 
 ' possible values, and the whole number of possible causes 
 
 The causes which are favor- 
 
 r . 2a 2a 4<i 
 
 oferroris Z7 x Z( or (ZO" 
 
 able for an error of the magnitude -f- -/ answer to all possible 
 values of x, y which together give -\- J ; namely, 
 
 for x, J a, J a -f- dJ, A a -(- 2dJ, ... a dJ, a 
 for y, a, a dJ, a 2dJ, . . . J a -\- dJ, J a 
 
 and whose number is therefore . 
 
 In the same way we find the number of causes favorable 
 for the error of the magnitude J to be ~-^-
 
 62 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hence the probability <p(A) dd of an error between 
 and J -f- */J is given by 
 
 that is, 
 
 = it when J lies between 2 and o 
 4a 
 
 O// _ A 
 
 = - - when J lies between o and -{- 2a. 
 
 For the m. s. e. we have 
 
 r 3. ^ r> f ~ + J 
 
 J o 4 2 ' J _ 2a 4 
 
 For the p. e., 
 
 f r 2 
 / 
 / o 
 
 s*o 2a -\- A 
 / -~ 
 / - r 4 
 
 Geometrically, the equations 
 
 J 2d! X 
 
 y = 2a -f- x 
 
 Fig.4 
 
 represent two straight lines 
 which cut the axes of x and y 
 at an angle of 45, and there- 
 fore the curve of probability 
 is as in the figure. If in the 
 figure OA 2a, then OP rep- 
 resents the p. e. 
 34. From the preceding we may derive an important 
 practical point. In an ordinary seven-place log. table the 
 seventh place is never in error by more than o 5. Hence, 
 
 P
 
 THE LAW OF ERROR. 63 
 
 this being the maximum error, the p. e. of a log. as given in 
 the tables is 0.25 in the seventh place. The interpolated 
 value at the greatest distance from a tabular value is the 
 mean between two tabular values. Its p. e., from Art. 33, is 
 
 (2 1/2) X 0.25 =0.15 
 
 Hence the p. e. of the log. of a number may be taken 0.2 in 
 the seventh decimal place. The p. e. of the number corre- 
 sponding to this log. is (Ex. i, Art. 7) 
 
 = approx. 
 
 io 8 X mod. 22 X 10 
 
 Suppose now that we are computing a chain of triangles 
 starting from a measured base. The p. e. of the base may 
 
 be taken as of its length. Hence the error arising 
 
 1,000,000 
 
 from this source is 22 times that to be expected from the 
 log. tables. Again, the triangulation will be most exact, and 
 therefore the test most severe, when the angles of each tri- 
 angle are equal to 60. Now, the change in log sin 60 
 corresponding to a change of i /r is 12.2 in units of the seventh 
 decimal place. And in a primary triangulation an angle 
 may, with the instruments now in use, be measured with a 
 p. e. of o".25- Hence 
 
 p. e. of log sin 60= 12.2 X 0.25 
 
 = 3.0 in units of the seventh decimal place, 
 
 which p. e. is 15 times greater than that, arising from the 
 log. tables. 
 
 For the solution of triangles, therefore, we conclude that, 
 with our present means of measurement, seven-place tables 
 are sufficient. The common practice is to carry out to 
 eight places to give greater accuracv in the seventh place, 
 and then drop the eighth place in stating the final result. 
 (See Struve, Arc du Mcridicn, vol. i. p. 94.) 
 
 35. If an error J arises from three independent sources 
 of the same kind, each of which can with the same proba- 
 bility assume all values between + a and -- a, then, the
 
 64 THE ADJUSTMENT OF OBSERVATIONS. 
 
 maximum error being 3*?, we have, from similar reasoning 
 to that employed in Art. 33, 
 
 <p(A) = -^ ' ' when J is between -f- 3 and -J- a 
 lua 
 
 2 (2 
 
 ^( J) = -3_ I when J is between -f- a and # 
 8rt 
 
 = ^= 7r*~ wnen ^ ^ between a and 3# 
 Also 
 
 = 2 A 
 
 Jo 
 
 and 
 
 /; 
 
 Sl>* 
 
 The curve of probability consists of three parts, as in 
 the figure : 
 
 OA = OB 30 OC=ODa 
 There are common tangents to 
 the two branches at E and F, and 
 the curve touches the axis of X 
 
 , at A and B. The p. e. is repre- 
 
 PC \ A 
 
 sented by OP. 
 
 36. A consideration of the results obtained in Arts. 31-35 
 will show that the more numerous the sources of error as- 
 sumed the nearer we approach the results obtained from 
 the Gaussian law of error. Thus for 
 
 r T 
 
 one source 1= 0.87 - = i.oo 
 
 y T 
 
 two sources - = 0.72 - = 0.88 
 
 T y 
 
 three sources - = 0.71 = 0.87 
 
 Gaussian law = 0.67 =10.85
 
 THE LAW OF ERROR. 
 
 The forms of the curves of probability show the same 
 approach to coincidence. Starting with a straight line as 
 the curve for a single source of error, we approach quite 
 closely to the Gaussian probability curve, even with so 
 small a number of sources of error as three. Hence we 
 should expect that, starting from the postulate of an error 
 being derived from the combined influence of a very large 
 number of independent sources of error, we should arrive at 
 the Gaussian law of error. A complete demonstration of 
 this by Bessel, to whom the idea itself is due, will be found 
 in Astron. Nachr., Nos. 358, 359. The elementary proof 
 given in Art. 33 for the simple case of two sources of error 
 is due to Zachariae. 
 
 37. Experimental Proof of the Law of Error. 
 The same point was brought out experimentally in a series 
 of researches by Prof. C. S. Peirce, of the U. S. Coast Sur- 
 vey.* He employed a young man, who had had no previous 
 experience whatever in observing, to answer a signal consist- 
 ing of a sharp sound like a rap, the answer being made upon 
 a telegraph operator's key nicely adjusted. Five hundred 
 observations were made on each of twenty-four days. The 
 results for the first and last days are plotted below. In the 
 
 Fig.6 
 
 0?35 
 
 0?10 
 
 0*30 
 
 Fig.7 
 
 0?20 0?25 0*30 
 
 * Coast Surrey Report, 1870, appendix 21.
 
 66 THE ADJUSTMENT OF OBSERVATIONS. 
 
 figures the abscissas represent the interval of time between 
 the signal and the answer, the ordinates the number of ob- 
 servations. The curve is a mean curve for every day, drawn 
 by eye so as to eliminate irregularities entirely. After the 
 first two or three days the curve differed very little from 
 that derived from the theory of least squares. On the first 
 day, when the observer was entirely inexperienced, the ob- 
 servations scattered to such an extent that the curve had to 
 be drawn on a different scale from that of the other days. 
 38. General Conclusions. On the whole, though we 
 
 cannot say that the formula -= e~^^ will truly represent 
 
 V 7T 
 
 the law of error in any given series of observations, we can 
 say that it is a close approximation. 
 
 When in a series of observations we have exhausted 
 all of our resources in finding |;he corrections, and have 
 applied them to the measured values, the residuum of error 
 may fairly be supposed to have arisen from many sources ; 
 and we conclude from the foregoing investigations that, of 
 any one single law, the best to which we can consider the 
 residual errors subject, and the best to be applied to a set of 
 observations not yet made, is the exponential law of error. 
 
 The general theorem of Art. 17 may therefore be applied 
 to a limited series and be written : If the observed values of a 
 quantity are of different quality, the most probable value is found 
 by dividing each residual error by the m. s. e. and making 
 the sum of the squares of the quotients a minimum ; if of the 
 same quality, the most probable value is the arithmetic mean of 
 the observed values. 
 
 If a set of observations shows a marked divergence from 
 this law a rigid examination will reveal the necessity, in 
 general, of applying some hitherto unknown correction. 
 Thus in the earlier differential comparisons of the compen- 
 sating base-apparatus of the United States Lake Survey with 
 the standard bar packed in ice, the observed differences did 
 not follow the law of error, as it was fair to suppose that 
 they should, the bars being compensating. There was in-
 
 THE LAW OF ERROR. 67 
 
 stead a regular daily cycle : some one source of error so far 
 exceeded the others that it overshadowed them. A study 
 of the results was made, and the law of daily change dis- 
 covered, which gave a means of applying a further correc- 
 tion. The work done later, after taking account of this 
 new correction, showed nothing unusual. 
 
 
 Classification of Observations. 
 
 39. For purposes of reduction observations may be di- 
 vided into two classes those which are independent, being 
 subject to no conditions except those fixed by the observa- 
 tions themselves, and those which are subject to certain 
 conditions outside of the observations, as well as to the con- 
 ditions fixed by the observations. In the former class, be- 
 fore the observations are made, any one assumed set of 
 values is as likely as any other; in the latter no set of values 
 can be assumed to satisfy approximately the observation 
 equations which does not exactly satisfy the a priori condi- 
 tions. 
 
 For example, suppose that at a station O the angles 
 A OB, AOC are measured. If the measures of each angle 
 are independent of those of the other, the angles are found 
 directly. 
 
 The angle BOC could be determined from the relation 
 
 AOC=AO + BOC 
 
 The unknown in this case may be said to be observed in- 
 directly, and therefore independent observations may be 
 classed as direct and indirect. The former class is a special 
 class of the latter. 
 
 But if the angle BOC is observed directly as well as 
 A OB, A OC, then these angles are no longer independent, 
 but are subject to the condition that when adjusted 
 
 AOC= AOB + BOC 
 
 andno set of values can be assumed as possible which does 
 not exactly satisfy this condition. 
 
 10
 
 68 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The observations in this case are said to be conditioned. 
 Though we have, therefore, strictly speaking, only two 
 classes of observations, we shall, for simplicity, divide the 
 first into two and consider in order the adjustment of 
 
 (1) Direct observations of one unknown. 
 
 (2) Indirect observations of several independent un- 
 knowns. 
 
 (3) Condition observations.
 
 CHAPTER III. 
 
 ON THE ADJUSTMENT OF DIRECT OBSERVATIONS OF ONE 
 UNKNOWN QUANTITY. 
 
 IN the application of the ideal formulas of Chapter II. to 
 an actual series of observations we shall begin with a single 
 quantity which has been directly observed. We shall con- 
 sider two cases first, when all of the observed values are 
 of equal quality, and, next, when they are not all of equal 
 quality. 
 
 A. Observed Values of Equal Quality. 
 
 40. The Most Probable Value ; the Arithmetic 
 Mean. We have seen that in a series of directly observed 
 values M lt M^ . . . M n of equal quality the most probable 
 value Fof the observed quantity is found by taking the 
 arithmetic mean of these values; that is, 
 
 It has also been shown that the same result will follow 
 by making the sum of the squares of the residual errors a 
 minimum. Thus the observations give the equations, 
 
 (2) 
 
 and V is to be found from 
 
 ,' + .+ . . . + *>'=: a min. (3) 
 
 that is, from 
 
 . . . +(F-J/ M )' = amin. (4)
 
 70 THE ADJUSTMENT OF OBSERVATIONS. 
 
 By differentiation of (4) 
 
 o (5) 
 
 and V (6) 
 
 n 
 
 In practice it would evidently be simpler to find the value 
 of the unknown by taking the arithmetic mean of the 
 observed values directly rather than to form the obser- 
 vation equations and find it by making the sum of the 
 squares of the residuals a minimum. 
 
 It is useful to notice, for purposes of checking, that Eq. 
 (5) may be written 
 
 M=o (7) 
 
 41. As the observed values M are often numerically 
 large and not widely different, the arithmetical work of 
 finding the mean may be shortened as follows: 
 
 A cursory examination of the observations will show 
 about what the mean value V must be. Let X' denote this 
 approximate value of F, which may conveniently be taken 
 some round number. Subtract X' from each of the ob- 
 served values J/j, M t , . . . M n in succession, and call the 
 differences / t , /,... / respectively. Then 
 
 M l X t =l 1 
 
 M,-X' = l s (i) 
 
 jf.--r=4 
 
 By addition, 
 
 n 
 X'-{-.v' suppose. (2) 
 
 Hence all that we have to do is to take the mean x' of the 
 small quantities / / 2 , . . . /, and add the assumed value X' 
 to the result.
 
 DIRECT OBSERVATIONS. 71 
 
 Ex. The measured values of an angle are 
 
 177" 21' 5". 80 
 
 177 21' 7".35 
 
 177 2l' 4". 28 
 
 find the mean. 
 
 It is sufficient to find the mean of the seconds and carry in the degrees 
 and minutes unchanged. 
 
 42. Control of the Arithmetic Mean. In least 
 squares, as in all computations, it is important to have a 
 check or control of the numerical work. This is specially 
 desirable when a computation takes several weeks, or it 
 may be months, to complete it. In long computations it is 
 better for two computers to work together, using different 
 methods whenever possible, and to compare results at in- 
 tervals. But even this is not an absolute safeguard against 
 mistakes, as it sometimes happens that both make the same 
 slip, as, for example, writing -f- for , or vice versa. Hence, 
 even if the computation is made in duplicate, it is advisable 
 to carry through an independent check which may be re- 
 ferred to -occasionally. In computations not duplicated a 
 control is essential. 
 
 A control of the accuracy of the arithmetic mean of a set 
 of observed values of the same quantity is afforded by the 
 relation 
 
 M=o 
 
 that is, that the sum of the positive residuals should be 
 equal to the sum of the negative residuals. 
 
 If, however, in finding the arithmetic mean, the sum [J/] 
 of the observed quantities was not exactly divisible by their 
 number n, the sums of the positive and negative residuals 
 would not be equal, but the amount of the discrepancy 
 could easily be estimated and allowed for. For if the value 
 of the mean taken were too large by a certain amount, the 
 positive residuals would each be too large, and the negative 
 residuals also too large, by that amount. Hence the dis- 
 crepancy to be expected would be n times the amount that 
 the approximate quotient taken as the mean differed from 
 the exact quotient.
 
 72 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. In the telegraphic determination of the difference of longitude 
 between St. Paul and Duluth, Minn., June 15, 1871, the following were the 
 corrections found for chronometer Bond No. 176 at ish. 5im. sidereal time 
 from the observations of 21 time stars. (Report Chief of Engineers U. S. A.^ 
 1871.) 
 
 M 
 
 V 
 
 vv 
 
 s. 
 
 s. 
 
 
 8.78 
 
 + 0.04 
 
 0.0016 
 
 .76 
 
 , + .02 
 
 4 
 
 .85 
 
 + .11 
 
 121 
 
 .78 
 
 + .04 s. 
 
 16 
 
 51 
 
 O.23 
 
 529 
 
 .64 
 
 .IO 
 
 IOO 
 
 .68 
 
 .06 
 
 36 
 
 .63 
 
 - .11 
 
 121 
 
 58 
 
 - .16 
 
 256 
 
 .80 
 
 + .06 
 
 36 
 
 75 
 
 + .01 
 
 I 
 
 .78 
 
 + .04 
 
 16 
 
 .96 
 
 + .22 
 
 484 
 
 .64 
 
 - .IO 
 
 IOO 
 
 65 
 
 - .09 
 
 81 
 
 83 
 
 + .09 
 
 81 
 
 .70 
 
 - -04 
 
 16 
 
 .64 
 
 O. IO 
 
 IOO 
 
 79 
 
 + .05 
 
 25 
 
 .90 
 
 + .16 
 
 256 
 
 -8.93 
 
 + 0.19 
 
 0.0361 
 
 Mean 8.74 
 
 + 1.03 0.99 
 
 [z/z/] 0.2756 
 
 
 \y = 2.02 
 
 
 Taking the observations as of equal precision, we find the arithmetic 
 mean to be 8.74. This is the most probable value of the correction. 
 
 The residuals v are found by subtracting each observed value from the 
 most probable value according to the relation 
 
 V-M-v 
 They are written in two columns for convenience in applying the check 
 
 The exact mean being 8.74/ T> the quantity 2 4 T X 21 = 4 should be, as it is, the 
 numerical difference between the + and residuals. Hence we may con- 
 sider the mean to be correctly found.
 
 DIRECT OBSERVATIONS. 73 
 
 Precision of the Arithmetic Mean. The degree 
 of confidence to be placed in the most probable value of the 
 unknown is shown by its mean- square or probable error. 
 
 43. (a) BesseTs Formula. 
 
 If we knew the true value T of the unknown, and conse- 
 quently the true errors 44 we should have, as in Art. 
 20, for the m. s. e. of an observation, 
 
 n 
 
 But we have only the most probable value V and the 
 residual errors i\, ?' . . . v n instead of the true values 
 
 T, 4, 4 . . . 4,. Now, 
 
 y-v 1 =M 1 = r- 4 
 
 V-v t = M^= 7-- 4 (i) 
 
 By addition, remembering that [?']=: o, 
 
 nV=nT-[J] (2) 
 
 Substitute for Fin equations (i) and 
 
 nv l = (n i)4 4 
 
 WT'.,=: -4 -)-(_ 1)4- . . . 
 
 Squaring, 
 
 w X'=(-i)'4' + 4' + - .-2(-i)44-. - - 
 v,- = 4' + ( - 0*4' + - - 2( - 04 J a - 
 
 By addition, assuming that the double products destroy 
 each other, positive and negative errors being equally 
 probable, 
 
 
 = (- 
 
 = : _t^ 
 w 
 
 which gives the m. s. e. of an observation. 
 
 and //' = : _^ (3) 
 
 w i
 
 74 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Now, from Art. 19, 
 
 /' = 
 
 which gives the m. s. e. of the arithmetic mean of n observa- 
 tions of equal precision. 
 
 The result, // -^ - , might have been inferred a priori, 
 n i 
 
 For the series of residuals i\,v^ . . . found from the arith- 
 metic mean Fof the observed values approximates closely 
 to the true series of errors J from which the law of error 
 was derived. Hence we conclude that the formula 
 
 // = M (4) 
 
 would be a close approximation to the m. s. e. of an observa- 
 tion. It is, however, not satisfactory, from the fact that it 
 ought to become indeterminate when n = i, which it does 
 not. For when = i, z> o, and unless the denominator of 
 (4) is equal to o, // would be equal to o; that is, the first 
 observation would give the true value of the unknown, 
 which is absurd. Hence we should expect the formula to 
 be of the form 
 
 ,,_. 
 
 1 
 
 n-l 
 
 which becomes of the indeterminate form -g- when n= i. 
 44. As in Art. 23, we may show from the expansion of 
 
 //) that the square of the m. s. e. ofu 9 = - is 
 
 n i ' / n i 
 
 equal to // 4/^_ We have, therefore, 
 
 r n i
 
 DIRECT OBSERVATIONS. 75 
 
 and when n is very large 
 
 / [w] / I \ 
 
 '-V^'V z: -fi&=$ 
 
 that is, the mean uncertainty of /* is 
 
 \'2(n T) 
 
 45. From the constant relation existing between the 
 m. s. e. and p. e. given in Art. 22 we have for the p. e. of 
 an observation and of the arithmetic mean of n observations 
 respectively, 
 
 r p 4/2 n 
 
 where /> 1/2 = 0.6745 nearly. 
 
 46. If we consider the m. s. e. <j. of the arithmetic mean 
 of ;/ observations as the true error of the mean V, the results 
 
 fa-- and u*=- !- may be derived very neatly as fol- 
 
 \'n n ~ l 
 
 lows : 
 
 We have 
 
 By addition, taking the mean
 
 ?6 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Error of F=I (4 + 4 + + 4) 
 and ^='(4 + 4+... +4)' 
 
 But from Art. 20 
 
 ,_ 
 7 
 
 
 
 Vn 
 Again, 
 
 A=TM 
 
 V+JL-.M 
 
 ' Vn 
 
 and 
 
 Hence, remembering that [v~\ = o, 
 
 47. (b) Peters Formula. 
 
 The m. s. e. and p. e. of a series of' observed values may 
 be more rapidly computed from the sum of the errors rather 
 than from the sum of their squares by means of the con- 
 venient formula first given by Dr. Peters.* 
 
 From the equation 
 
 
 * Astronomische Nachrichtin, No. 1034.
 
 DIRECT OBSERVATIONS. 77 
 
 we have approximately, without regard to sign, 
 
 Adding and dividing by ;/, 
 
 But from Art. 22 
 
 1.2533 
 = . = \y nearly. 
 
 For a demonstration of this formula more rigorous in form 
 see Astron. Nachr., No. 2039. 
 
 As in Art. 23, it follows that the precision of this formula 
 is expressed by the complete form 
 
 1-2533 
 
 the last term giving the mean uncertainty of /*. 
 
 For the p. e. of an observation and of the arithmetic mean 
 of observations we have respectively 
 
 0-8453 
 
 - 
 
 Vn(n 
 
 0.8453 
 
 r = - -- \y 
 
 
 The mode of deriving Peters' formula given in the pre- 
 ceding is approximate, and the formula itself is not very
 
 /8 THE ADJUSTMENT OF OBSERVATIONS. 
 
 close for small values of n. Thus when n = 2, if ^/denotes the 
 difference of the observed values, then by Bessel's formula 
 
 d d 
 
 I* = ^, and by Peters' formula it = 1.25 -=., which is one- 
 V2 V2 
 
 fourth greater. The corresponding probable errors are in 
 the same ratio. 
 
 A formula for the probable error of the mean which 
 answers better than Peters' for small values of n has been 
 derived by Fechner (Poggendorff, Annalen, Jubelband, 1874) 
 as follows : 
 
 r _ I - I 9SS , \v_ 
 
 T ~ A/2 n 0.8548 n 
 
 As this formula is troublesome to compute, and as it gives 
 results agreeing closely with those found from Peters' 
 formula when n is a moderately large number, there is no 
 advantage to be derived from the use of it in ordinary 
 work. 
 
 48. Collecting the formulas for finding the p. e. of a 
 sinle observation and of the arithmetic mean of n observa- 
 
 
 tions, we have 
 
 
 r = 0.6745 yl. r = 0.8453 -7= 
 
 n i vn(n i) 
 
 /-> Q A f 1 
 
 r = 0.6745 y ! u _ l} r " =a84 53,, v;/ _ L 
 
 To save labor in the numerical work I have computed 
 tables containing the values of the coefficients of V[^'] and 
 [v in these equations for values of n from 2 to 100. (See 
 Appendix, Tables II., III.) 
 
 If Bessel's formula is used compute first [vv~], then V[^ ; ] 
 can be taken from a table of squares closely enough. This 
 square-root number multiplied by the number in Table II.. 
 corresponding to the given value of n gives the p. e. sought. 
 If Peters' formula is used multiply the sum of the residuals, 
 without regard to sign, by the numbers in Table III. cor- 
 responding to the argument n.
 
 DIRECT OBSERVATIONS. 79 
 
 49. Control of [?'?'] A control is afforded by the 
 derivation of [i'?>\ from the observed values and the arith- 
 metic mean directly. 
 
 We have 
 
 v n = V-M H 
 Square and add, 
 
 [w] = n F 2 - 2 F[J/ ] + [>/'] 
 
 = [j/']_[j/]r. (i) 
 
 since nV=Jf 
 
 The values of J/ 2 may be found from a table of squares 
 or from Crelle's tables, or, if the numbers M are large, an 
 arithmometer, or machine for multiplying- and dividing, may 
 be employed with advantage. 
 
 The computation may often be much abbreviated by the 
 artifice of Art. 41. Substituting the values of J/ p M^ . . . M n 
 from that article in (i), we find, after a simple reduction, 
 
 = [//]-[/>' 
 
 50. Approximate Method of Finding the Pre- 
 cision. A connection between the p. e. of a single observa- 
 tion and the greatest error committed in the series may be 
 established approximately by the aid of the principle proved 
 in Art. 30. There we saw that in a large series the actual 
 errors may be expected to range between zero and 4 or 
 5 times the p. e. of an observation. If, then, we find from 
 the observations a p. e. of an amount, say, r, we may assert 
 that the greatest actual error is not likely to be more than 
 5r. The probability of its being as large as this is only 
 about T oVo. 
 
 The same principle will enable us to estimate roughly 
 the p. e. in a series of observations. A glance at the
 
 80 THE ADJUSTMENT OF OBSERVATIONS. 
 
 measured results will show the largest and smallest, and 
 their difference may be taken as the range in the results, 
 and half the difference as the maximum error. Hence, 
 since in an ordinary series of from 25 to 100 observations 
 the maximum error may be expected to be from 3 to 4 times 
 the p. e., we may take the p. e. to be from \ to \ of the range 
 of the errors of observation. 
 
 This result may be confirmed as follows : Expanding 
 the exponential function <p(d) in a series, we may write 
 
 P QJ? -f RJ - . . . 
 
 where P, Q, R, . . . are constants. 
 
 If a denotes the maximum error, then, since the proba- 
 bility of the occurrence of an error between the limits 
 -\-a and a is certainty, we have, taking two terms of the 
 series, 
 
 f 
 
 j -< 
 
 Also, since a is the maximum error, its probability is zero. 
 
 From these two equations P and Q may be found in terms 
 of a. 
 
 To find the p. e. r we have by definition (see Art. 21). 
 
 or Pr - Qr> - 
 
 3 2 
 
 Substituting for P and Q their values, and solving for r, we 
 find 
 
 a , 
 
 r - nearly 
 
 3 
 
 that is, the p. e. is approximately \ of the maximum error, or 
 of the range of the errors of observation.
 
 DIRECT OBSERVATIONS. 8 1 
 
 A closer approximation would be found by taking three 
 terms of the series for ^(J). We should then find 
 
 r = - nearly. 
 
 4 
 
 See note by Capt. Basevi, R.E., in G. T. Survey of India, 
 vol. iv. ; also Helmert in Zeitschr.fiir Verviess., vol. vi. 
 
 Ex, We shall now apply the preceding formulas to the example in Art. 
 42 to find the m. s. e. and p. e. of the arithmetic mean and of a single 
 observation. 
 
 (i) The m. s. e. and p. e, of the arithmetic mean. 
 
 These we may find in two ways : 
 
 (a) From the sum of the squares of the residuals (Art. 43) : 
 
 ft = |/_If^L 
 
 ' n(n i) 
 
 21 X 2O 
 
 = 0.026 
 r a = 0.6745 X O.O26 
 
 = 0.017 
 or from Table II. at once : 
 
 ;- =0.525 X 0.033 
 
 = 0.017 
 
 (b) From the sum \v of the residuals (Art. 47): 
 
 The multiplier in Table III. corresponding to the number 21 is 0.009. 
 . . r a = 2.02 X 0.009 
 
 = o.oiS 
 
 (2) The p. e. of a single observation. 
 From Tables II. and III. directly: 
 
 r = o. 525 X 0.151= 0.079 
 r=2.O2 X 0.041 =0.082 
 
 Check (). Let the residuals be arranged in order of magnitude. They are: 
 0.23 0.22 0.19 0.16 0.16 o.n o.n o.io o.io o.io 0.09 
 0.09 0.06 0.06 0.05 0.04 0.04 0.04 0.04 0.02 o.oi 
 
 The residual 0.09 occupies the middle place, and is therefore the p. e. r of a 
 single result (Art. 21). The computation above gives 0.08. 
 Check (ft). See Art. 50. 
 
 Range = 0.22 + 0.23 = 0.45 
 
 .-.r = -i- 5 =o.oS. 
 6 
 
 The values found by the different methods agree reasonably well.
 
 82 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 51. The Law of Error Tested by Experience. We 
 
 shall now test our example and see how closely it conforms 
 to the law of error, and hence be in a better position to 
 judge of how far the law of error itself is applicable in 
 practice. This is the h. posteriori proof intimated in Art. 11 
 as necessary for the demonstration of the law. 
 
 (1) The number of -(- residuals is 12, and the number of 
 
 - residuals is 9. 
 
 (2) The sum of the -f- residuals is 1.03, and the sum of the 
 
 - residuals is 0.99. 
 
 (3) The sum of the squares of the -f- residuals is 1417, 
 and of the residuals is 1339. 
 
 (4) The p. e. of a single observation is 0.08. To find the 
 number of observations we should expect whose residual 
 errors are not greater than o. 10, we enter Table I. with the 
 
 argument ' 1.25 and find 0.60. This multiplied by 21 
 0.08 
 
 gives 13 as the number of errors to be expected not greater 
 than o. 10. By actual count we find the number observed 
 to be 14. 
 
 To find the number to be expected between o. 10 and 0.20 
 
 2O 
 
 we enter the table with the argument -- 2. 50 and find 0.91. 
 
 .08 
 
 From this deduct 0.60 and multiply the remainder by 21. 
 This gives 6. The number observed is 5. 
 
 The number to be expected over 0.20 is, by theory, 2. 
 The number observed is 2. 
 
 The preceding results are collected in the following 
 table : 
 
 Limits of Error. 
 
 Number of Errors. 
 
 Theory. 
 
 Observation. 
 
 s. s. 
 
 o.oo to o. 10 
 
 13 
 
 14 
 
 0.10 tO 0.20 
 
 6 
 
 5 
 
 over o. 20 
 
 2 
 
 2
 
 DIRECT OBSERVATIONS. 
 
 Table I., it will be remembered, is founded on the sup- 
 position that the number of observations in a given set is 
 very large. In our example the number is only 21. Per- 
 fect accordance between the number of errors given by 
 theory and the number given by observation is, therefore, 
 not to be expected. For longer examples of this kind see 
 Chauvenet's Least Squares, p. 489 ; Airy's Theory of Errors 
 of Observations, third ed., appendix. 
 
 Comparisons between the number of errors within given 
 limits that actually occur in a series of observations and the 
 number to be expected from theory in the same series show 
 the degree of confidence we may place in the law of error. 
 It is the final criterion, and forms the second part of the 
 proof of the law as stated in Art. 11. 
 
 The law of error has been so thoroughly tested in this 
 way, so far as the sciences of observation (for which, in- 
 deed, it was framed) are concerned, that if in a series of 
 observations we find that the errors do not conform to it 
 we may suspect the presence of other than accidental 
 sources of error. For example, Bessel * found, from his re- 
 duction of a series of 300 observations made by Bradley on 
 the declinations of a Tauri, etc., that the numbers ot errors 
 that actually occurred and the numbers given by theory 
 within specified limits were as follows: 
 
 Limits. 
 
 Number of Errors. 
 
 
 Experience. 
 
 Theory. 
 
 o".o to o" 
 
 4 
 
 66 
 
 65 
 
 o".4 to o" 
 
 S 5 S 
 
 to 
 
 O\8 [0 l" 
 
 2 
 
 55 
 
 53 
 
 I .2 tO l" 
 
 6 
 
 28 
 
 41 
 
 i".6 to 2" 
 
 o 27 
 
 30 
 
 2 .0 to 2 
 
 4 23 
 
 21 
 
 2". 4 to 2" 
 
 S 10 
 
 13 
 
 2".S to 3" 
 
 2 15 
 
 S 
 
 3". 2 to 3' 
 
 6 
 
 5 
 
 3"-6 to 4" 
 
 o 4 
 
 2 
 
 over 4 
 
 6 
 
 - 
 
 * f'urti/antentii As(r<>n<>>Nitr, pp. 19, 20.
 
 84 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The agreement, especially for the larger errors, is not 
 very close. Prof. Safford, to whom I referred this ex- 
 ample, states that in a new reduction of Bradley's observa- 
 tions made by Auwers a much better agreement between 
 experience and theory is found. Three sources of sys- 
 tematic error enter into the observations which were not 
 taken into account in Bessel's reduction : 
 
 (a) Personal equation. Bradley was not the only ob- 
 server, Mason and Green being the others. 
 
 (b) Local deviation of the plumb-line. 
 
 (c) The assumption that the collimation of Bradley's 
 Quadrant did not vary irregularly. 
 
 Auwers has in a great degree overcome these difficulties 
 in his reduction. He finds, for example, the number of 
 errors over 4" to be 2 instead of 6, thus agreeing with the 
 theoretical number. 
 
 For another illustration see Helmert's discussion of the 
 errors in Koppe's triangulation for determining the axis of 
 the St. Gothard Tunnel. (ZeitscJir. fur Vermess,, vol. v. 
 pp. 146 seq.) 
 
 52. As the Gaussian law of error is found to apply 
 reasonably well to other phenomena, such, for example, as 
 statistical questions, guesses, etc., it has been often rashly 
 assumed to be of universal application ; and when prediction 
 and experience are found not to agree, the validity of the 
 law in any case has been as rashly impugned. 
 
 In the fundamental investigation in Art. 15 the hypoth- 
 eses there made are satisfied by other functions of the 
 measured values besides the arithmetic mean. Thus, tor 
 example, taking the geometric mean g, the resulting law of 
 error would be of the form 
 
 where c and h are constants. 
 
 This form appears to apply to many statistical questions 
 better than the Gaussian law. That it does so is, how 
 
 * Gallon, Proc. Roy, Sac. Land., 1879, pp. 365 seq.
 
 DIRECT OBSERVATIONS. 85 
 
 ever, no argument against the Gaussian law in its own 
 territory. 
 
 A curious illustration of the preceding remarks, in the 
 misapplication of the Gaussian law of error to a case 
 where it would appear at first sight that it ought to apply 
 directly, is to be found in the essay on " Target-Shoot- 
 ing " in Sir J. F. \V. Herschel's Familiar Lectures on Scientific 
 Subjects. 
 
 53. Caution as to the Application of the Tests 
 of Precision. In the preceding article we have given 
 several cautions with regard to the strict application of the 
 law of error in practice. We shall now perform a similar 
 service for the tests of precision the m. s. e. and the p. e. 
 
 The m.s. e. and p. e. of series of observations have been 
 defined as measures of their relative accuracy. With ideal 
 series that is, series which do not contain systematic 
 errors, and in which the accidental errors are continuous in 
 magnitude between the extreme limits of error this is true ; 
 and in actual series, in good work, it is on the whole true. 
 But in any actual series selected at random we must apply 
 these tests with caution. It is a common mistake to over- 
 look the distinction between observations which conform to 
 the law of error and those which only apparently do so, and 
 hence to condemn the m. s. e. and p. e. as not only worth- 
 less but misleading. 
 
 For example, in levelling, if the same line is run over in 
 duplicate in the same direction, a good agreement may be 
 expected at the several bench-marks where comparisons 
 are made. The m. s. e. of observation will consequently be 
 small. If the line is levelled in opposite directions experi- 
 ence shows that the agreement would not be so good. The 
 m. s. e. would be larger than before. We might, therefore, 
 hastily conclude that the first work would give the better 
 result. But when we reflect that the main differences arise 
 from such causes as the refraction of light, the personal bias 
 of the observer, etc., which causes are less likely to be 
 mutually destructive and more likely to be cumulative if
 
 86 THE ADJUSTMENT OF OBSERVATIONS. 
 
 the lines are run in the same direction, it is to be expected 
 that the final result obtained from measurements in opposite 
 directions will be nearer the truth. The conclusion arrived 
 at by trusting to the m. s. e. alone would be illusory. 
 
 Again, in measuring a horizontal angle, if the same part 
 of the limb of the instrument is used in making the readings 
 the results may be very accordant and the m. s. e. conse- 
 quently small, but the angle itself may not be anywhere 
 near the truth. This would be shown, for example, by 
 the large discrepancies in the sums of angles of triangles 
 measured in this way from the theoretical sums. For a 
 long time this contradiction was a source of much per- 
 plexity, and many good instruments were most unjustly 
 condemned.* At last the discovery was made that it was 
 mainly owing to periodic errors of graduation of the limb 
 which, when corrected for, made the remaining errors fairly 
 subject to the law of error. This most important discovery 
 may be said to have revolutionized the art of measuring 
 horizontal angles. 
 
 The difficulty may be explained in this way. In the 
 derivation of the m. s. e. from a series of n observed quanti- 
 ties M lf M^ . . . we had the observation equations 
 
 \vv\ 
 Also // = 
 
 n i 
 
 Now, if we suppose each of the observed quantities to be 
 changed by the same amount c, which may be of the nature 
 of a constant error or correction, so that they become 
 M^-{-c t M^-\-c, . . . the most probable value, instead of 
 being V, will be 'V-\-c. Also since 
 
 = V-M 
 the residual errors will be the same as before. 
 
 * See, for example, G. T. Survey of India^ vol. ii. pp. 51, 96.
 
 DIRECT OBSERVATIONS. 87 
 
 Hence if is unchanged, and we see, therefore, that the 
 m. s. e. makes no allowance for constant errors or correc- 
 tions to the observed quantity. These are supposed to be 
 eliminated or corrected for before the most probable value 
 and its precision are sought. 
 
 Another common misapprehension is the following: 
 From Arts. 19 or 46 the relation between the m. s. e. ol a 
 single observation n and the m. s. e. of the mean of n obser- 
 vations is 
 
 This formula shows that by repeating the measurement a 
 sufficient number of times we can make the m. s. e. of the 
 final result as small as we please. Nothing would, there- 
 fore, seem to be in the way of our getting an exact result, 
 and that we could do as good work with a rude or imper- 
 fect instrument as with a good one by sufficiently increasing 
 the number of observations. 
 
 Experience, however, shows that in a long series of 
 measurements we are never certain that our result is nearer 
 the truth than the smallest quantity the instrument will 
 measure. If an instrument measures seconds we cannot be 
 sure that by repeating the observations we can get the 
 nearest hundredth or tenth. In a word, we cannot measure 
 what we cannot see. 
 
 Take an example: With the meridian circle Prof. Rogers 
 found the p. e. of a single complete observation in declina- 
 tion to be o".36, and the p. e. of a single complete observa- 
 tion in right ascension for an equatorial star to be + o s .O26. 
 He says: " If, therefore, the p. e. can be taken as a measure 
 of the accuracy of the observations, there ought to be no 
 difficulty in obtaining from a moderate number of observa- 
 tions the right ascension within o s .O2 and the declination 
 within o".2. Yet it is doubtful, after continuous observations 
 in all parts of the world for more than a century, if there is 
 a single star in the heavens whose absolute co-ordinates are
 
 88 THE ADJUSTMENT OF OBSERVATIONS. 
 
 known within these limits." The explanation is, as inti- 
 mated, that constant errors are not eliminated by increasing 
 the number of observations. Accidental errors are elimi- 
 nated by so doing ; and if a number of observations ex- 
 pressed by an infinity of a sufficiently high order could be 
 taken, so that the constant errors entering in the different 
 series could be classed as accidental, these errors would 
 mutually balance in the reduction and we should arrive at 
 the true result. 
 
 Closely allied to the preceding is the common idea that 
 if we have a poor set of observations good results can be 
 derived from them by adjusting them according to the 
 method of least squares, or that if work has been coarsely 
 done such an adjustment will bring out results of a higher 
 grade. A seeming accuracy is obtained in this way, but it 
 is a very misleading one. The method of least squares is 
 no philosopher's stone: it has no power to evolve reliable 
 results from inferior work. 
 
 A third source of uncertainty from the same cause may 
 be mentioned. It may happen that the value obtained of 
 the p. e. is numerically greater than that of the observed 
 quantity itself. It is then a question whether in subsequent 
 investigations we should use the value of the observed 
 quantity as found or neglect it. This depends on circum- 
 stances. It is ever a principle in least squares to make use 
 of all the knowledge on hand of the point at issue. If we 
 have strong a priori reasons for expecting the value zero it 
 would be better to take this value. Thus if we ran a line 
 of levels between two points on the surface of a lake we 
 should expect the difference of height to be zero. If the 
 p. e. of the result found were greater than the result itself 
 it would be allowable in this case to reject the determina- 
 tion. On the other hand, when we have no a priori know- 
 ledge, as in determinations of stellar parallax, for example, 
 if the p. e. of the value found were in excess of the value 
 itself, as is sometimes the case,f we could do nothing but 
 
 * Proc. Amcr. Acad, Sci., 1878, p. 174. tSee, for example, Newcomb, Astronomy, app. vii.
 
 DIRECT OBSERVATIONS. 89 
 
 take the value resulting from the observations, unless, in- 
 deed, it came out with a negative sign, and then its unre- 
 liable character would be evident. 
 
 54. Constant Error. The remarks on constant error 
 in the preceding articles lead us to notice an example or 
 two of the detection and treatment of this great bugbear of 
 observation. 
 
 We suspect the presence of constant errors in a series of 
 observations from the large range in the results a range 
 greater than would naturally be expected after all known 
 corrections have been applied. Great caution is necessary 
 in dealing with such cases, and one should be in no hurry to 
 jump at conclusions. 
 
 Sometimes the sources of error are detected without 
 much trouble. Thus in measuring an angle with a theod- 
 olite, if the instrument is placed on a stone pillar firmly 
 embedded in the ground, the range in results, if targets are 
 the signals pointed at, would not usually be over 10" in 
 primary work ; and on reading to a number of signals in 
 order round the horizon the final reading on closing the 
 horizon would be nearly the same as the initial reading on 
 the same signal. If next the instrument were placed on a 
 wooden post or tripod, and readings made to signals in 
 order round the horizon in the same way as before, the 
 final reading might differ from the initial by a large amount. 
 The observations might also show that the longer the time 
 taken in going around the greater the resulting discrepancy. 
 The natural interence would be that in some way the 
 wooden post had to do with the discrepancy in the results. 
 In an actual case* of this kind examination showed the 
 change to be most uniform on a clav when the sun shone 
 
 t> 
 
 brightlv. Measurements were then made at night, using 
 
 * At U. S. Lake Survey station Emle 1 , Lake Superior, many observations were taken during 
 both day and night in July, 1871, to determine the rate of twi-t of centre-post on which the theod- 
 olite used in measuring angles was placed. The conclusion arrived at was that " during a day of 
 uniform sunshine and clear atmosphere this twist seemed to be quite regular, and at the rate of 
 about ant second of arc per minute of tite, reaching a maximum about 7 P.M. and a minimum 
 about 7 A.M., during the month of July. On partially cloudy days there was no regularity in the 
 twist, being sometimes in one direction and again in the opposite.''
 
 9O THE ADJUSTMENT OF OBSERVATIONS. 
 
 lamps as signals on the distant stations, and the same 
 change was observed, only it was in the opposite direction. 
 
 The effect on the value of an 'angle of this twist of sta- 
 tion, assuming it to act uniformly in the same direction 
 during the time of observation, can be eliminated by the 
 method of observation : first reading to the signals in one 
 direction and then immediately in the opposite direction, 
 and calling the mean of the difference of the two sets of 
 readings a single value of the angle. 'So also in azimuth 
 work the mean of the difference of the readings, star to 
 mark, and mark to star, gives a single value free from 
 station-twist. 
 
 This mode of procedure is in accordance with the general 
 principle to eliminate a correction, when possible, by the 
 method of observation, rather than to compute and apply it. 
 See Art. 2. 
 
 The effort to avoid systematic error causes in general a 
 considerable increase of labor, and sometimes this is very 
 marked. For example, in the micrometric comparison ol 
 two line measures belonging to the U. S. Engineers the 
 results found by different observers showed large dis- 
 crepancies. The micrometer microscopes used were of low 
 power, with a range of about one mm. between the upper 
 and lower limits of distinct vision. Examination showed 
 that the discrepancies arose mainly from focusing, each ob- 
 server's results being tolerably constant for his own focus. 
 As the value of a revolution of the micrometer screw 
 entered into the reduction of the comparison work, and as 
 this value was obtained from readings on a space of known 
 value, error of focusing entered from this source. Hence a 
 value of the screw had to be determined from a special set 
 of readings taken at each adjustment, and this value used 
 in reducing the regular observations made with the same 
 focus. Had the microscopes been of high power it would 
 have been sufficient to have determined the value of the 
 screw once for all, since the error arising from change of 
 focus could have been classed as accidental.
 
 DIRECT OBSERVATIONS. 9 1 
 
 In trying to avoid systematic error the observer will, as 
 he gains in experience, take precautions which would at 
 first seem to be almost childish. Good work can only be 
 had at the cost of eternal vigilance. 
 
 55. Necessary Closeness of Computation. The 
 number of observations necessary to the proper determina- 
 tion of a quantity may be approximated to by referring to 
 the m. s. e. of these observations. If, after planning the 
 method of observing with the view of eliminating "con- 
 stant error," we find that increasing the number of obser- 
 vations beyond a certain limit does not sensibly affect the 
 m.s. e. , we ma}' conclude that we have a sufficient number. 
 It is evident that increasing or decreasing this number will 
 affect the result more or less. But we cannot say that we 
 are nearer the truth in either case. 
 
 Hence the folly of a too rigorous computation. An 
 approximate value being the best that we can have at any 
 rate, no weight is added to it by carrying out that value to 
 a great many decimal places. Thus if the most probable 
 value of a quantity computed in an approximate way is V, 
 whereas the value found from the same observations by a 
 rigorous computation is V-\-c, we may estimate the allow- 
 able value of c as follows. Let J p J 3 , . . . J n be the errors 
 of F, then J l -\- c, J^-\-c, . . . J n -\-c are the errors of V-\-c. 
 Hence 
 
 and 
 
 /V+c=#p|i-| --| approxi match'. 
 
 V 2 /v/ 
 
 Now, we may safely allow the difference between // r . , and 
 13
 
 92 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hv to be --^. Hence there will be no appreciable error in- 
 troduced by computing in such a way that 
 
 -- < 
 
 2 fiy " 100 
 
 that is, when the error c committed is, roughly, ^ of the 
 m. s. e. 
 
 B. Observed Values of Different Quality. 
 
 56. The Most Probable Value: the Weighted 
 Mean. It has been shown in Art. 17 that if the directly 
 observed values M^ M v . . . M n of a quantity are of dif- 
 ferent quality, the most probable value is found by multiply- 
 ing each residual error of observation by the reciprocal of 
 its m. s. e., and making the sum of the squares of the pro- 
 ducts a minimum ; that is, with the usual notation, 
 
 * + .. !+ - +^i = amin. (i) 
 
 /V /V !*n 
 
 or 
 
 =a min - 
 
 By differentiation and reduction, 
 
 (3) 
 
 We have, therefore, the equivalent rule: 
 
 If the observed values of a quantity are of different quality, 
 the mast probable value is found by multiplying each observed 
 value by the reciprocal of the square of its m. s. e. , and dividing 
 the sum of the products by the sum of the reciprocals. 
 
 The form of the expression for V suggests another 
 stand-point from which to consider it. Let p^p . . . /
 
 DIRECT OBSERVATIONS. 93 
 
 be the numerical parts of- , such that each is 
 
 /v /v /v 
 
 of the type 
 
 (unit of measure) 2 
 
 P 
 
 I? 
 
 then equation (3) may be written 
 
 Also, since , , . . . - - are similarly involved in the 
 
 f*i t** P* 
 
 numerator and denominator of the value of V, this value 
 will remain the same if/,,/.,, . . . / are taken any num- 
 
 bers whatever in the same proportion to , , . . . 
 
 /V /V f'-n 
 
 that is, if/>,'A Pn satisfy the relations 
 
 (5) 
 
 where // is an arbitrary constant. 
 
 57. Let us look at this question from another point of 
 view. It is in accordance with our fundamental assump- 
 tions that observations of a quantity made under the same 
 conditions, so that there is no a priori reason for choosing 
 one before another, are of the same quality. They require 
 the same expenditure of time, labor, money, etc. If, there- 
 fore, we represent the quality of a single observation of a 
 certain series by unity, the quality of the arithmetic mean 
 of/ such observations, as it would require / times the ex- 
 penditure to attain it, would be represented by />. Let. us 
 suppose, then, that the arithmetic mean of /, observed 
 values of a certain quality is J/, ; of /., other observed 
 values of the same quality it is J/,, and so on. The total 
 number of observed values is [/]. All of the observed 
 values being of the same quality, the most probable value
 
 94 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Fof the unknown is given by the arithmetic mean; that 
 is, by 
 
 y _ sum of the values of all the sets 
 ~sum of number of obs. in each set 
 
 A+A+ +A 
 
 which is of the same form as Eq. 4, above. 
 
 The numbers A A ..... p n are called the weights, or, 
 better, the combining weights, of the observed values, and 
 the mean value Fis called the weighted mean. 
 
 In view of this definition, the general principle stated in 
 Art. 56 may be replaced by the following necessarily equiva- 
 lent one : 
 
 If the 'observed values of a quantity are of different weights, 
 the most probable value is found by multiplying the square of 
 each residual error of observation by its weight, and making the 
 sum of the products a minimum. 
 
 Thus the most probable value V is found from 
 
 \_pvv\ = a min. 
 that is, from 
 
 By differentiation, 
 
 p l (V 
 whence 
 
 This form of the value of V leads to the rule : 
 
 If the observed values of a quantity are of different weights, 
 the most probable value is found by multiplying each observed 
 value by its weight, and dividing the sum of the products by 
 the sum of the weights.
 
 DIRECT OBSERVATIONS. 95 
 
 As in the case of the arithmetic mean (Art. 40), it is evi- 
 dently simpler in practice to find the weighted mean directly 
 by this rule rather than from the minimum equation. 
 
 If the observed values J/are numerically large we may 
 lighten the arithmetical work by finding V by the method 
 of Art. 41. Proceeding as there indicated, we have 
 
 = X' -\- x" suppose 
 
 58. Reduction of Observed Values to a Common 
 Standard. The principle of the weighted mean is evi- 
 dently an extension of that of the arithmetic mean, as was 
 pointed out long ago by Cotes, Simpson, and others. It 
 merely amounts to finding a mean of several series of 
 means, the unit of measure being the same in each. As 
 soon, therefore, as results of different weights are changed 
 into others having a common standard of weight, the rules 
 for combining and finding the precision of observed quantities 
 of the same weight can be applied to weighted quantities. 
 
 This change we are enabled to make by means of the 
 relation (5), Art. 56, which may be written 
 
 
 
 VA A/A VA 
 
 Now, since /Jt t , /* . . . tt n are the m. s. e. of M lt M v . . . M n , 
 the m. s. e. of M^p^M^/p^ . . . J/ w V/ would each be the 
 same quantity n. 
 
 Hence if a serits of observed values M t , M^ . . . M n have tJie 
 weights A A> A> they are reduced to the same standard by 
 multiplying by A/A, A/A> VA respectively. 
 
 For example, given the observation equations, 
 
 V J/, =v t weight A 
 V-M, = i>, " A 
 
 V-M n =v n " A 
 to find the most probable value of V.
 
 g6 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Reducing to the same standard of weight, we have the 
 equations 
 
 A/A v- VAM, = A/A^ 
 
 Vp n v A/A, ^4= 
 
 and the most probable value of Fis found by making 
 
 (A/A <0* + (A/A fO* + - + ( VA *)' = a rain. 
 that is, by making 
 
 V- A/A ^) 
 
 Reducing this equation, \ve find, as before, 
 
 59. Computation of the Weights. If A> A A 
 
 represent a series of weights corresponding to the m. s. e. 
 /^j, // 2 , . . . /i n , then we have the relations 
 
 where the value of // is entirely arbitrary. The combining 
 weights are, therefore, known when the m.s.e. f^,/^,/^, .../* 
 are known. 
 
 These relations suggest that it would be convenient to 
 define // as the m. s. e. of a single observation assumed to be 
 of the weight unity. 
 
 We shall define 11 in this way, so that in future it is 
 understood that the standard to which observations of dif- 
 ferent weights are reduced for comparison and combination 
 is the fictitious observation whose weight is unity and 
 whose m. s. e. is /x 
 
 60. Control of the Weighted Mean. Eq. 2, Art. 57, 
 may be written 
 
 [pv] = o
 
 DIRECT OBSERVATIONS. 97 
 
 Hence if the weighted mean V has been computed correctly, 
 the sum of the products of each residual error by its 
 weight is equal to zero. 
 
 Usually, however, Fis not an exact quotient that is, 
 [/J/] is not exactly divisible by [/>] and then the dis- 
 crepancy of [/7'J from zero is evidently equal to the prod- 
 uct of the sum of the weights and the difference between 
 the exact value of Fand the approximate value used. See 
 Art. 42. 
 
 Ex. i. Deduce the relation [/^] o from the observation equations 
 directly. 
 
 [Multiply each observation equation by its weight, and add the products.] 
 
 Ex. 2. Find the most probable value of the velocity of light from the 
 following determinations by Fizeau and others: 
 
 298000 kil. 1000 kil. 
 298500 " looo " 
 299990 " 200 " 
 300100 " 1000 " 
 299930 " 100 " 
 
 {Amer. Jour. Sci., vol. xix.) 
 
 The weights, being inversely as the squares of the probable errors, are as 
 the numbers i, i, 25, i, 100. (Art. 59.) 
 
 (a) Direct solution. 
 
 , M p 
 
 pM 
 
 298000 
 
 i 
 
 298000 
 
 298500 
 
 i 
 
 298500 
 
 299990 
 
 25 
 
 7499750 
 
 300100 
 
 i 
 
 300100 
 
 299930 
 
 IOO 
 
 29993000 
 
 
 128 
 
 38389350 
 
 V = -j. :- = 299917 kil. approx.. 
 
 the exact value being 2999 16|| kil.
 
 THE ADJUSTMENT OF OBSERVATIONS, 
 (b) Solution according to Art. 57. 
 
 Assume X' = 298000 
 
 / 
 
 / 
 
 pi 
 
 o 
 
 I 
 
 o 
 
 500 
 
 I 
 
 500 
 
 1990' 
 
 25 
 
 49750 
 
 2IOO 
 
 I 
 
 2100 
 
 1930 
 
 IOO 
 
 I93OOO 
 
 
 128 
 
 245350 
 
 and V = 298000 + 1916^ = 299916!^, as before. 
 
 Control. Take V= 299917, and proceed to find [/']. (See Art. 60.) 
 
 V 
 
 P 
 
 pv 
 
 1917 
 
 i 
 
 1917 
 
 1417 
 
 i 
 
 1417 
 
 - 73 
 
 25 
 
 1825 
 
 -183 
 
 i 
 
 - 183 
 
 - 13 
 
 IOO 
 
 1300 
 
 
 
 26 
 
 The discrepancy should be 
 
 128 (299917 299916^) = 26 
 which it is.
 
 DIRECT OBSERVATIONS. 99 
 
 61. The Precision of the Weighted Mean. Since 
 the weighted mean Fis the arithmetic mean of [/>] observa- 
 tions of the unit of weight, its weight is [/]. Hence the 
 m. s. e. Y of Fis found from 
 
 where n is the m. s. e. of an observation of the unit of weight 
 (standard observation). 
 
 According to Art. 58, the value of }t may be found by 
 writing \X/>, 7-,, VA ?' 2 > . . . for v lt i\, ... in the formulas 
 derived for observations of the same weight. Hence, sub- 
 stituting in Bessel's and in Peters' formulas Arts. 43, 47, we 
 have 
 
 := 1.2533 
 
 n i \/n(n i) 
 
 and therefore 
 
 ..3533- 
 
 These expressions reduce to those for the arithmetic mean 
 where the observed values are of the same weight by 
 putting [/] = up. 
 
 62. Control of \pi'v\. A control of the accuracy of [/>?'?'] 
 is afforded by the derivation of this quantity from the 
 observed values directly. 
 
 The observation equations are 
 
 Vl VM, weight/, 
 v t =V-M t " A 
 
 Hence
 
 100 THE ADJUSTMENT OF OBSERVATIONS. 
 
 By addition, 
 
 \_pirj] = [>] V* - 
 
 since 
 
 In using this formula the troublesome term is \_pMM\ 
 With large values of M it is better to deduce it from the 
 column pM) already computed in finding the weighted mean. 
 This is specially advisable if one has a machine for perform- 
 ing multiplications. With small values of M a table of 
 squares is best. 
 
 With large values of M we may perhaps proceed still 
 more conveniently by the method explained in Arts. 41, 57. 
 With the same notation and reasoning as there employed it 
 is found that 
 
 where the quantities /are numerically small. 
 
 Ex. The linear values found for the space o".oo to O*.O5 of inch \ab\ 
 on the standard steel foot I F. of the G. T. Survey ol India were as follows: 
 o". 050027, o.O4997i, o'. 050019, o'. 050079, o 1 '". 050021, o. 050011. The num- 
 bers of measures in these determinations were 6, 6, 15, 15, 8, 8 respec- 
 tively. 
 
 Taking the numbers of measures as the weights of the respective determi- 
 nation-, required the most probable value of the space and its p. e. 
 
 The direct solution presents no difficulty. The value of F may be found 
 as in Ex. Art. 60, and thence the residuals v. The m. s. e. or p. e. follows 
 from the formulas of Art. 61. 
 
 We shall give the solution according to the methods of Arts. 57 and 62, 
 which are advantageous in this case on account of the large numbers that 
 enter.
 
 DIRECT OBSERVATIONS. 
 
 Assume X' = 0.049971 
 
 101 
 
 / 
 
 P 
 
 // 
 
 /// 
 
 56 
 
 6 
 
 336 
 
 18816 
 
 o 
 
 6 
 
 o 
 
 o 
 
 48 
 
 15 
 
 720 
 
 34560 
 
 108 
 
 15 
 
 1620 
 
 1 74960 
 
 50 
 
 8 
 
 400 
 
 2OOOO 
 
 40 
 
 8 
 
 320 
 
 I2SOO 
 
 58 
 
 3396 
 
 26II36 = [///] 
 198842 = [//].r" 
 
 62294 = [pvv] 
 
 = 0.6745 A/ 622 94 
 
 r 58(6-1) 
 
 V 0.049971 + 0.000059 
 = 0.050030 
 
 V = O"*. 
 
 = o.ooooio 
 o".c>oooio 
 
 By choosing the approximate value X' equal to the smallest of the measures 
 or equal to the greatest of them, all of the remainders / have the same sign, 
 which is a great convenience in computation. 
 
 In this example an important practical point occurs, 
 and one often overlooked. The p. e. is not computed from 
 the original observations, but from these observations 
 grouped in six sets of means. These means we have treated 
 as if original observations of certain weights. Had the 
 original observations been accessible we should have used 
 them, and would most probably have found a different 
 value of the p. e. from that which we have obtained. This 
 arises from the small number of observations in the several 
 sets. In good \vork the difference to be expected between 
 the value of the p. e. found from the means and that found 
 from the original observations would be small. Still, when- 
 ever there is a choice, the p. e. should always be deduced
 
 102 THE ADJUSTMENT OF OBSERVATIONS. 
 
 from the original observations rather than from any com- 
 binations of them. 
 
 The weighted mean value V would evidently be the 
 same whether computed from the partial means or from the 
 original observations. 
 
 Observed Values Multiples of the Unknown. 
 
 63. Let the observed values M lt M^ . . . M K be multi- 
 ples of the same unknown X ' ; that is, be of the form 
 a t X, a^X, . . . a H X, where a^ a v . . . a n are constants given 
 
 by theory for each observation. The values \ ?, . . . ? 
 
 , #* a n 
 
 of X may be regarded as directly observed values of unequal 
 weight. If fi. is the m. s. e. of an observation, that is, 
 
 of MM V . . ., then, since the m. s. e. of _' is ^,of ? 
 
 a l a 1 a^ 
 
 is^, . . . the weights of these assumed observations are 
 a, 
 
 proportional to a*, a*, . . . Hence taking the weighted 
 mean 
 
 Also, since [V] is the weight of X, 
 
 *_ P* 
 = ['J 
 
 Cotes,* in solving this problem, reasons that, since for 
 the same error of M the greater a is, the less is the error of 
 X, we may take the coefficients a to express the relative 
 weights of the values of X. Now, placing the coefficients 
 a t , a^ . . . as weights along a straight line at distances 
 
 _', ^?, . . . from one end of the line, the most probable 
 a, a, 
 
 * Harmonia Mensurarum. Cambridge, 1722. (Quoted by Hultman, Minsta Qvadr. 
 Stockholm, 1860.)
 
 DIRECT OBSERVATION'S. 
 
 103 
 
 value will be the distance X of the centre of gravity of 
 these weights from the end point, and will be found by 
 taking moments about this point ; that is, 
 
 and therefore 
 
 Ex. To test the power of the telescope of the great theodolite (3 ft.) of the 
 English Ordnance Survey, and find the p. e. of an observation, a wooden 
 framework was set up 12,462 ft. distant from the theodolite when at station 
 Ben More, Scotland. It was so arranged that when projected against the 
 sky a fine vertical line of light, the breadth of which was regulated by the 
 sliding of a board, was shown to the observer. The breadth of this opening 
 was varied by half-inches from i^ in. to 6 in. during the observations; which 
 were as follows : * 
 
 No. of obs. 
 
 Width. 
 
 Side of opening. 
 
 Mean of 
 micr. readings. 
 
 I 
 
 6.0 
 
 ( left. 
 < right. 
 
 ( 28.00 
 1 37oO 
 
 2 
 
 5-5 
 
 11: 
 
 (28.50 
 
 1 37-t*. 
 
 3 
 
 5.0 
 
 11: 
 
 ( 29.16 
 ( 37-16 
 
 4 
 
 4-5 
 
 U: 
 
 j 30.16 
 f 36.66 
 
 5 
 
 4.0 
 
 11: 
 
 j 30.50 
 < 37-16 
 
 6 
 
 3-5 
 
 f i. 
 
 }r. 
 
 j 31.16 
 
 1 37-00 
 
 7 
 
 3-o 
 
 11: 
 
 ( 32.66 
 I 36.83 
 
 8 
 
 2-5 
 
 il: 
 
 < 33-50 
 < 36.83 
 
 9 
 
 2.0 
 
 jl: 
 
 \ 33-83 
 / 37-00 
 
 
 
 ( l. 
 
 < 35-50 
 
 10 
 
 i-5 
 
 < r. 
 
 1 37-16 
 
 * Account of the Principal Triangulation, pp. 54, 55.
 
 104 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Let X = the most probable value of the angle subtending an opening of 
 i inch. Then we have the observation equations 
 
 6 X 9.50 = vi 3.5^5.84 = 2/8 
 
 5.5^8.50 = 2/3 3 A" 4.17 = 2/7 
 
 5 X 8.00 = 2/3 2.5 X 3.33 = z/8 
 
 4.5 Af 6.50 = 2/4 2. X 3.17 = 2/8 
 
 4 X ' 6.66 = 2/6 1.5 X 1.66 = z/ 10 
 
 From the preceding we have for the individual values of A' and their 
 weights 
 
 X = 1.58 weight 6 2 
 JT= 1.55 weight 5-5 2 
 
 1.58X6' + 1-55 X5-5 2 + 
 .'. weighted mean = - - - - 
 
 = 1-55 
 or making the sum of the squares of the residuals v a minimum, that is, 
 
 (6A- - 9.50)* + (5. 5 A" - 8. so) 2 + . . . = a min. 
 we find by differentiation that 
 
 AT=i.55 
 as before. 
 
 The practical rule following from either method is the same, and may be 
 stated thus : Multiply each observation equation by the coefficient of A" in 
 that equation, and add the products. The resulting equation gives the value 
 of X. 
 
 We again see that the principle of minimum squares is more general than 
 that of the arithmetic mean, and why it was that we failed in solving the equa- 
 tions of Art. 14. In other words, we cannot write in the preceding example 
 
 X= 1.58 
 
 and take the mean of these values as the value of x, because these equations 
 should be written strictly 
 
 where the errors are not lessened in the same proportion throughout the 
 equations.
 
 DIRECT OBSERVATIONS. IO5 
 
 Precision of a Linear Function of Independently Observed 
 
 Quantities. 
 
 64. Suppose that there are n independently observed 
 quantities J/,, J/ a , . . . whose m. s. e. are //,, /^, ... re- 
 spectively, to find the m. s. e. /j. of F where 
 
 F=a l M l + a t M t + . . . +a n Af n (i) 
 
 # a v . . . a n being constants. 
 
 This has been already solved in Art. 19, where it was 
 shown that 
 
 2 r 2 an 
 
 H = [a [t. ] 
 
 On account of the great importance of this result we add 
 another method of deriving it. 
 
 If J,, J a , . . . denote the errors of M lt M^ . . . we shall 
 have the true value T of F by writing ^/, + -/,, J/.,-)- J 2 , . . . 
 for J/,, M 9 , ... in the above expression for F ; that is, 
 
 , + 4)+- . . +.(-*/. +40 
 
 Call J the error of F; then, since T= F-\- J, we have 
 
 J = ^ 1 J 1 + ^. 2 J 2 +. . . +a n J H 
 and .*. 
 
 ff = aW + af4+. . .+2^,44+- 
 
 Let the number of sets of M lt M^, . . . required to find T 
 be n, and suppose J" summed for all the sets ol values of 
 J,, 4,, . . . and the mean taken, then attending to Art. 20, 
 
 In forming all possible values of J,-/,, J 2 J S , . . ., the 
 number of values being very large, there will probably be 
 as many -j- as products of each form, and we therefore 
 assume 
 
 [J 1 J S ] = [J 1 J I ]=. . . =o 
 Hence 
 
 ^=[>y] (3)
 
 Io6 THE ADJUSTMENT OP OBSERVATIONS. 
 
 Ex. i. The Kevveenaw Base was measured with two measuring tubes 
 placed end to end in succession. Tube i was placed in position 967 times, 
 and tube 2 966 times. Given the p. e. of the length of tube i = o*.ooo34, 
 and of tube 2 = o'".ooo37, find the p. e. in the length of the line arising 
 from the uncertainties in the length of the tubes. 
 
 [ p. e. from tube i = 967 X 0.00034 = o*.329 
 p. e. from tube 2 = 966 X 0.00037 = o"*.357 
 .'. p. e. of line = Vo.^2(f + 0.3572 
 
 Ex, 2. In the Keweenaw Base the p. e. of one measurement of 94 tubes, 
 deduced from the discrepancies of six measurements of these 94 tubes, was 
 found to be o'.o3 . Show that the p. e. in the length of the line of 1933 tubes 
 arising from the same causes ma}' be estimated at o*".i36. 
 
 [ p. e. of i measurement of i tube = '-^ 
 
 1/94 
 
 0.03 
 p. e. of base of 1933 tubes = - 
 
 = 0.136 J 
 
 Attention is called to these two problems, from the im- 
 portance of the principles illustrated. In Ex. i the p. e. of 
 a tube was multiplied by the whole number of tubes to find 
 the p. e. of the base from that cause, for the reason that 
 with whatever error the tube is affected it is cumulative 
 throughout the measurement. 
 
 In Ex. 2 the p. e. of one tube is multiplied by the square 
 root of the number of tubes, because each measurement is 
 independent of every other, and the errors are as likely to 
 be in excess as in defect, and, therefore, may be expected to 
 destroy one another in the final result. 
 
 Ex. 3. The m. s. e. of aM\, /< being the m. s. e. of M\ and a a constant, is 
 equal to aju, but the m. s. e. of the sum of the a independently observed 
 quantities Mi, Mi . . . M a ; that is, of Mi + M* + . . . + M a , the m.s. e. of 
 each being ju, is Vaju. Explain. 
 
 65. If the function F whose m. s. e. is required is not 
 in the linear form, we first reduce it to that form, as in Art. 
 7, and apply Eq. 3, Art. 64. Thus, if
 
 DIRECT OBSERVATIONS. 
 
 the true value T of F will result if we write 
 M^-^-dM^, . . . for J/ p J/,, . . . the differentials representing 
 the errors of these quantities. Then 
 
 Expanding by Taylor's theorem and retaining only the 
 first powers of the small quantities ^/J/,, dM.^ . . . we have 
 
 T = f + 
 
 or 
 
 Error of F 
 
 where 
 
 This expression is of the same form as (i), Art. 64. Hence 
 
 HP = a?ti? -f tf,V/,* + . . . -f a^i'-n fay ] 
 
 The still more general case of the m. s. e. of a function of 
 quantities which are themselves functions of the same ob- 
 served quantities may be readily reduced to the form of 
 Eq. i. The whole point is to express the error of F 
 as a linear function of the errors of the independently ob- 
 served quantities J/,, J/, . . . J\I n . 
 
 It is useful to note that Eq. i results from differ- 
 entiating the function equation directly, as has been al- 
 ready pointed out in Art. /. 
 
 Ex. i. If /i,, it., are the m. s. e. of the measured side? AB, BC of a rect- 
 angle ABCD, find the m. s. e. of the area of the rectangle. 
 [Here 
 
 F-A^M-, 
 .'. by differentiation 
 
 ,/F=Jlf l <Uf. 1 + J/,,/J/, 
 and 
 
 HP- /I/, -//..- t Af-i'n t '-\ 
 
 15
 
 108 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. 2. The expansions of the steel and zinc bars of tube i of the Repsold 
 base apparatus of the U. S. Lake Survey for i Fahr. are approximately 
 
 5 = 0.0248 o.oooi 
 Z ' 0.0617 0.0003 
 
 Show that 
 
 S 2 i 
 
 -- nearly. 
 
 Z 5 400 
 
 [For F= 
 
 and 
 
 (p. e.)- = , (0.0001)- + 4 (o.ooos) 2 ] 
 
 Ex. 3/iThe base b and the adjacent angles A, C of a triangle ABC are 
 measured. If their m. s. e. are respectively //*, J.IA, Uc, find the m. s. e. of the 
 angle B and of the side a. 
 
 To find /.IB- 
 
 We have 
 
 y? = xSo + A C 
 
 where denotes the spherical excess of the triangle. 
 Hence, A and C being independent of one another, 
 
 MB"- HA* + l-ic'- 
 To find jit a . 
 
 , sin A 
 
 a = b- - - 
 
 sin B 
 
 By differentiation, 
 
 sin A , sin (C e) . 
 
 da =. --db+l> - sin I dA + a cot B sin i dC 
 
 sin B sin* B 
 
 and therefore 
 
 sin 2 A !> sin 2 (C s) sin 2 i" 
 
 "a" = -^"T-f>t"> + ~ -T^- - I- 1 * + a cot B sm : /'C 2 
 
 sin^ B sin 4 B 
 
 Ex. 4. Given the base b and the angles A,B of a triangle with m. s. e. 
 Mb, HA, MB respectively, to find the m. s. e. f.i a of the side a. 
 
 We have 
 
 , sin A 
 a = b - (i) 
 
 sin B 
 
 This might be expanded as in the preceding example, but more conveniently 
 as follows : 
 
 Take logarithms of both members. Then 
 
 log a = log b + log sin A log sin B (2)
 
 DIRECT OBSERVATIONS. 109 
 
 (a) By differentiation. 
 
 da = - db + a cot A sin I "dA u cot B sin I "dB 
 
 b 
 Hence 
 
 a 1 
 Ha" 2 Hb* + o' 2 cof 2 ,-/ sin^i"//,! 7 + <x 8 cot j .fl sin 2 i>s 2 (3) 
 
 0" 
 
 If, as is usually assumed in practice, 
 
 MA = MB = H, and /m, = o 
 then 
 
 f.t a = a sin i" f.i V'cot'M + cotfB (4) 
 
 (b) Using log differences as explained in Chapter I., we have by differen- 
 tiating (2) 
 
 S a da = db db + f) A dA - t) B dB (5) 
 
 where ft a , di> are the differences corresponding to one unit for the numbers a 
 and b in a table of logarithms, and dA, SB are the differences for i" for the 
 angles A and B in a table of log sines. Hence 
 
 ' 
 
 The two equations (3) and (6) may be used to check one another. 
 
 Ex. 5. The following example is given for the sake of showing the form 
 of solution by the method of logarithmic differences. 
 
 In the triangulation of Lake Superior there were measured in the triangle 
 Middle, Crebassa, Traverse Id. (ABC) 
 
 Z A = 57 04' 51". 4 HA = o".30 
 Z B 67 15' 39". 2 /IB o".29 
 
 The side Middle-Traverse Id. as computed from the Keweenaw Base is 
 16894.9 yards. Taking fti, = 0.05 yd., find //< and //c 
 
 We have 
 
 , sin A 
 
 a = b 
 
 sin B 
 
 . . log (a + da) = log (b + dt>) + log sin (A + dA) log sin (B + dB) 
 
 Then (see Ex. i, Art. 7), the differences being expressed in units of the 
 seventh decimal place, 
 log (b + db) = 4.2277556 + 257 db 
 
 log sin (A + dA) = 9.9239892 + 14 dA 
 colog sin (B + dB) . 0.0351398 9 dB 
 . . log a + $ a da = 4. 1868846 + 257 db + 14 dA 9 dB 
 and 283 da 257 db + 14 dA ^dB. 
 
 Since log a = 4.1868846, and 283 is the difference S a as given in the 
 
 table.
 
 IIO THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hence ,. , 
 
 = 0.0024 
 
 and Ha = 0.05 
 
 Also 
 
 = (?g)' (.05)' + (--)' (.30) 
 
 = V'(o.29) 2 + (0.30)- 
 = o".42 
 
 Miscellaneous Examples. 
 
 Y >" 
 66. Examples of Mean-Square and Probable Error. 
 
 .>. i. If in a theodolite read by 2 verniers the p. e. of a reading (mean of 
 vernier readings) is 2", show that if it is read by 3 verniers the p. e. of a 
 reading will be a little over i".5, and if read by 4 verniers a little less than i".5. 
 
 Ex. 2. The p. e. of an angle of a triangle is r ; show that the p. e. of the 
 triangle-error is ;- \ 3, all of the angles being equally well measured. 
 
 [Error = 180 - (A + B + Q.] 
 
 Ex. 3. The expansion of a bar for i C. is ga 9/7 show that for i Fahr. 
 it is $a $r. 
 
 Ex. 4. The length of a measuring bar at the beginning of a measurement 
 was a >'i. After x measures had been made it was b ;- 2 . Show that the 
 length of the th measure, the length being supposed to change uniformly 
 witli the distance measured, is 
 
 .~^-^ 
 
 a+ 6- 
 
 [For if da is the error of a, and db of />, then the error of a -\ (b a) is 
 
 (i -- ]da + - db. and the above p. e. follows. 
 x) x 
 
 It is a common mistake to write the error in the form da + - (db da), 
 
 / 
 
 ,'^ and hence to infer that the p. e. is {/ r^ -\ -(r^ + r 2 ' 2 ). ] 
 
 Ex. 5. Prove that the p. e. of the mean of two observations whose dif- 
 ference is d is 0.337 d, and the p. e. of each observation is 0.477 d.
 
 DIRECT OBSERVATIONS. Ill 
 
 Ex. (i. The line Monadnock-Gunstock (94469 m.) was computed from the 
 Massachusetts Base (17326 w.) through the intervening triangulation. The 
 p. e. of the line arising from the triangulation is o m .3i7, and the p. e. of 
 the base is 0^.0358 ; find the total p. e. of the line. 
 
 [p.e. = f/(^ X 0.0358)* +(0.317)- = o.372] 
 
 Ex. 7. The Minnesota Point Base reduced to sea-level is 
 
 1325 X 15 ft. bar at 32 4- n".3i4 o'.42i 
 and 15 ft. bar at 32 = 179'". 95438 o'.oooi2 
 
 show that the p. e. of the base is o.45o. 
 
 [p. e. =: 1(1325 X 0.00012)- + (0.421)- = o'.45o. We multiply 
 o'. 00012 by 1325 : uncertain which sign it is ; but whichever it is, it is 
 constant all the way through.] 
 
 E.\\ S. If the zenith distance of a star is observed ;/i times at upper 
 culmination, and trfe zenith distance T of the same star is observed 2 times 
 at lower culmination, show that the m. s. e. of the latitude of the place of 
 observation is 
 
 V-+- 
 
 2 ' HI n-z 
 
 H being the m. s. e. of a single observation. 
 
 [Latitude = 90 f (? + ?')! 
 
 Ex. 9. Given the telegraphic longitude results, 
 
 h. m. s. s. 
 
 Cambridge west of Greenwich = 4 44 30.99 0.23 
 Omaha west of Cambridge = i 39 15.04 0.06 
 Springfield east of Omaha 25 08.69 o. n 
 
 show that 
 
 Springfield west of Greenwich = 5 58 37.34 0.26 
 
 [p. e. =: t .23- + .06- + .11- = 0.26] 
 E.\. 10. Given mass of earth + mass of moon = 
 
 305879 2271 
 
 and mass of moon = mass of earth 
 
 prove mass of earth 
 
 309635 2299 
 
 [For (305879 22-1) X ^ - = 309635 2299] 
 c i . 44
 
 112 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. ii. In measuring an angle suppose 
 
 r^ = p. e. of a pointing at a signal, 
 
 r-i = p. e. of a reading of the limb of the instrument, 
 
 e =. error of graduation of the arc read on, 
 
 then, assuming that these result from the only sources of error not eliminated, 
 show if the limb has been changed m times and n readings taken in each 
 position, that 
 
 p. e. of angle = 
 [For one position of the limb 
 
 /2(?V + rj) 
 -- + c 2 
 
 as the error of graduation remains constant throughout each set of // readings]. 
 
 Ex. 12. The distance o i mm on a graduated line-measure is read with 
 a micrometer; show that the p. e. of the mean of two results is equal to the 
 p. e. of a single reading. 
 
 [For distance o i mm , 
 
 = ^ | (first + second rdg.) at o (first + second rdg.) at i mm \ 
 ..(p.e.) s =i{4(p.e.) s ofardg.}] 
 
 Ex. 13. In the comparison of a mm. space on two standards placed 
 side by side and read with a micrometer, the p. e. of a single micrometer read- 
 ing being a, show that the p. e. of the difference of the results of M combined 
 
 /2 
 measurements (each being the mean of two measurements) is 4/ -a. 
 
 [For p. e. of a reading = a 
 
 p. e. of a combined measurement = a 
 
 and p. e. of mean of n combined measurements = , etc.] 
 
 Vn 
 
 Ex. 14. A theodolite is furnished with n reading microscopes, all of the 
 same precision. A graduation-mark on the limb is read on m times with a 
 single microscope, giving the p. e. of a single reading to be r. The telescope 
 is then pointed at an object m times, and the p. e. of the mean of the micro- 
 scope readings is found to be r 2 . Show that the p. e. of a pointing is 
 
 [p. e. of reading (mean of verniers) with n microscopes = 
 
 Vn 
 Total error = error of reading + error of pointing 
 
 .'. rj = + (p. e. of ptg.) 5 , etc.] 
 n
 
 DIRECT OBSERVATIONS. 113 
 
 Ex, 15. If //ii, n-ibi are the m. s. e. of the base measurements, and fi t \ 
 the m. s. e. of the ratio A, given by the triangulation, of a base b- t to a base b\, 
 show that the m. s. e. of the discrepancy between the computed and measured 
 values of b* is b t V[> s ]. 
 
 [Discrepancy = 2 />]A = / suppose 
 
 db*-l> l d\Xdl> l =dl 
 and fist** + /'i s (;< 3 A) 4 + A-(/<,*,) 2 = n", 
 or /'./W+/<3 2 + ,,-) = /<-] 
 
 Ex. 16. At the time t\ the correction to a chronometer was af rj, and 
 at the time / 2 it was a-f r a y show that the p. e. of the rate of the chronometer 
 
 is - and find the p. e. of the correction to the chronometer at an 
 
 /a /i 
 
 interpolated time /'. 
 
 I",-. a i a > , 
 Correction = a -\ (/ A) at time / , 
 
 L I* ^i 
 
 .*. p. e. = 
 
 Ex. 17. Given x cos a = A n 
 
 x sin a = / 2 r a 
 find p. e. of .v and of or. 
 
 , .//I'-vr- + / 2 ->./- 
 
 .'. p.e. of JT I/ -- r^ . Similarly p. e. of n: = 
 
 M" + l-i~ 
 
 "JT. iS. Given on a line measure the p. e. of a distance OA measured 
 from to be r\, and of OB, also measured from O, to be r t ; find the p.e. of 
 OD when D is the middle point of AB. 
 
 + OB) 
 
 .-. r = i i TV' + TV 
 If /-, = r 2 = i**, for example, then r o*.S, when // = one micron. 
 
 It may at first sight appear paradoxical that the p. e. of the computed 
 quantity may be smaller than the p. e. of the measured. It is evident, how- 
 ever, that the error of OD is one-half the sum of the errors of OA and OB. 
 If the signs of the errors are alike the error of OD is never greater than the 
 larger of the errors ; if the signs are different it is always less.J
 
 114 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. 19. Given the p. e. of .r to be r; find the p. e. of log x. 
 
 r j i mod. , 
 
 a log x = _ dx 
 
 L ' x 
 
 mod. ~l 
 
 . . p. e. log x = r 
 
 a- J 
 
 Ex. 20. 'In the measurement of the Massachusetts base line, consisting of 
 2165 boxes, the p. e. of a box, as derived from comparisons with the standard 
 meter, was o'.ooooo55, the p. e. from instability of microscopes in meas- 
 uring a box was o m .oooi27, and the p. e. of the base from temperature cor- 
 rections was o"*.o332. Show that the p. e. of the base arising from these 
 independent causes combined is o'*.o358. 
 
 [p. e. y (2165 x o^.oooooss) 9 -\- (o".oooi27 V 2165)" -\-(o m . 0332)- 
 = o>.035S] 
 
 Ex. 21. Given the length of the Massachusetts base to be 17326. 3763 
 o m .O3$&; show that the corresponding value of the p. e. of its logarithm is 
 8.973 in units of the seventh place of decimals. 
 
 [ log (b 0.0358) = log b ~ (0.0358) See Art. 7. 
 
 log mod. 9.6377843 
 log 0.0358 8.5538830 
 
 8.1916673 
 log b 4.2387077 
 
 0.0000008973 3.9529596 ] 
 
 Ex. 22. The m. s. e. of the log. of a number A^ in units of the seventh 
 decimal place is 10.6 ; find the ratio of the m. s. e. to the number. 
 
 [ log (N + v) = log N + ~ rr 1 v 
 
 /mod.Y . 
 ) 
 
 /'-log (N + )= I 
 
 mod. 
 . . TV- /' = 10.6 -5- io 7 
 
 // T n 
 
 and 
 
 N 4iooooJ 
 
 67. Examplc of Weighting. 
 
 Ex. i. The weights of the independently measured angles BAC, CAD, 
 DAE are 3, 3, i respectively ; find weight of the sum-angle BAE. 
 
 ti ill 
 - =-+- + -, .'. wt. -0.6 
 wt. 3 3 i -J
 
 DIRECT OBSERVATIONS. 115 
 
 Ex. 2. If X = a t xi + <ix.2 + . . . + a n x n , and p\, p-i, . . . p n are the weights 
 of xi, xi, . . . x n , and P the weight of X, show that 
 
 -u r -!* ^"'^.^ I 
 
 , ^ O. . ' \ ^ ol -p 
 
 r~ > f/- 
 
 .J!' V * ' -"-. 3. Prof. Hall found, from observations of the satellites of Mars, that 
 
 from Deimos, Mass of Mars = , and from Phobos, Mass of 
 
 3095313 3435 
 
 Mars = the mass being expressed in the common unit. 
 
 3078456 10104 
 
 Show that, taking the weighted mean, we have approximately 
 
 Mass of Mars = 
 
 3093500 3295 
 
 Ex. 4. On a graduated bar the space o i m -is measured anci found to be 
 im. with a weight i, and the space o 2 m is measured and found to be -2m. 
 with a weight 2; required the value of the space i m 2 m and its weight P. 
 
 [Space i m 2 m = im. It makes no difference what the weights are so 
 far as the value of the space is concerned. 
 
 To find P. (i m 2'" ) = (o 2"' ) (o i'" ) 
 
 1113 
 
 - = - + - = - and P= 
 
 P i 2 2 
 
 2 -i 
 = 
 
 3 J 
 
 E.\. 5. Given the weight of .\ ~ = f, show that 
 weight of log * = 
 
 y 
 
 Ex. 6. If .v = and the weight of v is/, then 
 weight of .V = c- p 
 
 Ex. 7. Given the results for difference of longitude, Washington and 
 Key West, 
 
 m. s. s. 
 
 1873, Dec. 24, 19 01.42 0.044 
 Dec. 26, 1.37 .037 
 Dec. 30, 1.38 .036 
 Dec. 31, 1.45 .036 
 
 1874, Jan. 9, i. 60 .046 
 Jan. 10, 1.55 .045 
 Jan. ii, 19 01.57 0.047 
 
 show that 
 
 m. s. t. 
 
 weighted mean = 19 01.460 0.016 
 
 weighted mean of first four nights =n 19 01.404 0.019 
 weighted mean of last three nights = 19 01.573 0.027 
 
 and from the last two results check the first. 
 
 10
 
 Il6 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. 8. In the trianguiation connecting the Kent Id. Base, Md., and the 
 Craney Id. Base, Va., the length of the line of junction computed from 
 
 w. m. / 
 
 Kent Id. Base = 26758.432 0.38 ^ 
 
 Craney Id. Base = 26758.176 0.43 
 Show that 
 
 m. m. .V A, 
 
 (1) Discrepancy of computed values = 0.256 0.57 ' 
 
 (2) Most prob. length of junction line = 26758.32 0.028 
 
 Ex. g. In latitude work with the zenith telescope, if n north stars are 
 combined with s south stars, giving ns pairs, to find the weight of the combina- 
 tion, that of an ordinary pair, one north and one south, being unity. 
 
 [Let // = the m. s. e. of an observation of one north star or of one south 
 star. 
 
 Then, as though combining the mean of n north stars with the mean of 
 s south stars, the wt./ of the combination is 
 
 
 2ns ~[ 
 .'. i> 
 
 f n + s . 
 
 "The combination of more than two stars gave some trouble. In one 
 case there were 3 north and 4 south, which would give 12 pairs, but with a 
 
 3X4 
 weight of 27- only. In this and al! similar cases I treated the whole com- 
 
 3~r4 
 
 bination as one pair ; that is, I inserted in the blank provided the half-sum of 
 the mean of the declinations of north stars ;md of the mean of the declinations 
 of south stars, and gave the result a higher weight. This is the only logical 
 method." (Safford, Report, Chief of Engineers U. S. A., 1879, p. 1987.) 
 
 For a series of examples by Airy on the weights to be given to the 
 separate results for terrestrial longitude determined by the observations of 
 transits of the rnoon and fixed stars, see Atem. Roy. Astron. Soc., vol. xix. 
 
 Ex. 10. If a close zenith star is observed with a zenith telescope first as a 
 north star, and immediately after as a south star, show that the weight of the 
 resulting latitude is less than that found from observing an ordinary pair. 
 
 Ex. ii. In the trianguiation of Lake Ontario the angle Walworth-Palmyra- 
 Sodus was measured as follows : 
 
 In 1875, with theodolite P. and M. No. i, 
 
 74 25' 05". 429 o".2g, mean of 16 results ; 
 In 1877, with theodolite T. and S. No. 3, 
 
 74 25' O4".6ii o".22, mean of 24 results ; 
 required the most probable value of the angle and its probable error.
 
 DIRECT OBSERVATIONS. 117 
 
 [With first theodolite p. e. of a single obs. = o".2() I 16 = i".i6 
 Wiih second theodolite p. e. of a single obs. = o".22 ^'24 = i".oS 
 
 Let a single result with the first theodolite be taken as unit of weight, 
 then mean of 16 results has weight 16. 
 
 Let a single result with the second theodolite have a weight/, re f erred to 
 the same unit as the first, then mean of 24 results has weight 24 p. The value 
 of/ is found from the relation 
 
 i Vi.oS/ 
 Also 
 
 r".429 X 16 4- o".6n X 
 
 most prob. value of angle = 74' 25' 04" + 
 
 16 + 24/ 
 and 
 
 weight of this value = 16 + 24 / ] 
 
 Note. If, instead of being two measurements of the same angle, the above 
 were the measurements of two angles side by side, then 
 
 total angle = 148 50' io".O4o 
 
 because, no matier how much better one is measured than the other, we can 
 do nothing but take the sum of the two values. The weight P of the result 
 would be found from 
 
 .v. 12. An angle is measured n times with a repeating theodolite, and 
 also n times with a non-repeating theodolite, the precision of a single reading 
 and of a single pointing being the same in both cases; compare the weights of 
 the results. 
 
 [ //i, /'a the m. s. e. of a single pointing and of a single reading. 
 
 With a non-repeating theodolite each measurement of the angle contains 
 
 (pointing + reading) (pointing 4- reading) 
 .'. (m. s. e.)- of one measurement = 2//, 5 + 2// 2 - 
 
 and (m. s. e.)- of mean of ;; measurements = (2//r + 2// 2 2 ). 
 
 
 
 With a repeating theodolite the successive measurements of the angle are 
 
 (pointing + reading) pointing 
 pointing pointing 
 
 pointing (pointing 4- reading) 
 .'. (m. s. e.)- of times the angle = 2/<, 2 + 2fti- 
 
 and (m. s. e.)- of the angle = (aw//, 1 4- 2// 7 ! )
 
 Il8 THE ADJUSTMENT OF OBSERVATIONS. 
 
 If, then, p\,pi denote the weights of an angle resulting from n reiterations or 
 from n repetitions, ^ ,. ., . 
 
 and hence it would seem that the method of repetition is to be preferred to 
 the method of reiteration. This advantage is so much less, the smaller -^ is ; 
 
 that is, the more the precision of the circle reading increases in proportion to 
 the precision of the pointing. 
 
 This result is contradicted by experience so much so that in all of the 
 leading surveys repeating theodolites are no longer used in primary work. 
 Where, then, is the fault? Is the theory of least squares false? By no means. 
 We have only another example of a point which occurs over and over again, 
 and which is so apt to be overlooked. (See Arts. 53, 54.) 
 
 The result obtained is true on the hypothesis that only accidental errors 
 enter. We have assumed a perfect instrument . But the instrument-maker 
 cannot give what the geometer demands. From various mechanical reasons 
 the systematic error in a repeating theodolite increases with the number of 
 observations, whereas in the reiterating theodolite it disappears. This sys- 
 tematic error, in whatever way it arises, causes the trouble. It is useless to 
 discuss accidental errors until it is out of the way; and as no means have yet 
 been devised of getting rid of it, the instrument itself has been abandoned. 
 
 Cf. Struve, Arc dtt Meridien, vol. i. ; Louis Cruls, Discussion stir les 
 Me'thodes de repetition et reiteration, Gand, 1875 ; Herschel, Outlines of A str. n- 
 omy, Art. 188 ; Coast Survey Report, 1876, App. 20.] 
 
 Ex. 13. An angle is measured with two repeating theodolites. With the 
 first are made n, repetitions, //i, /</ being the m. s. e. of a pointing and of a 
 reading ; with the second are made n^ repetitions, // 2 , Hi' being the m. s. e. of 
 a reading and of a pointing. Show that if i, 2 are large the weights of the 
 results are 'as 
 
 NOTE I. 
 
 ON THE WEIGHTING OF OBSERVATIONS. 
 
 68. When the sources of error are of such kinds that, so 
 far as we know, they cannot be separated, the m. s. e. and 
 consequent weight are found as described in the preceding 
 sections. The weight has been defined as a number repre- 
 senting the relative goodness of an observation, and as a
 
 WEIGHTING OF OBSERVATIONS. I 19 
 
 number inversely proportional to the square of the m. s. e. 
 of an observation. If in a series of observations the con- 
 ditions required for the determination of the law of error 
 could be strictly fulfilled, these two statements would lead 
 to the same result. In actual cases, however, this is only 
 approximately true. Thus two separate determinations of 
 a millimeter space, made in the same way, gave 
 
 n M 
 
 looo. i 0.40, mean of 20 readings. 
 
 1000.3 -33> mean of 30 readings. 
 
 To find the weighted mean of these two sets of measure- 
 ments we may proceed in two ways. The number' of re- 
 sults in the first measurement is 20, and the number in the 
 second is 30. Hence, taking the weights proportional to 
 the number of results, the mean . 
 
 l = IOQ0.220 
 
 20 + 30 
 
 Again, since the p. e. of the measurements are 0.40 and 0.33, 
 
 their weights are as to , that is, as 1080 to 1600. and 
 
 40' 33' 
 
 the resulting weighted mean is 1000.219, agreeing, within 
 the limits of the p. e., with the other value. 
 
 In this example the two methods agree as nearly as 
 could be expected from the small number of observations. 
 But it is not always so. Some " run of luck," or balancing 
 of error, or constant conditions might have made the ob- 
 servations of one set fall very closely together, in which 
 case the weight as found from the p. e. would have been 
 very large, while varying conditions might have caused 
 wide ranges, giving a small weight. A great deal, there- 
 fore, depends on the judgment of the computer in deciding 
 what weight is to be given, it being constantly kept in mind 
 that the strict formulas which are correct in an ideal case 
 must not be pressed too far in practice. Thus in the second 
 set of observations above the first three results were 999.8,
 
 120 THE ADJUSTMENT OF OBSERVATIONS. 
 
 999.8, 999.8. The p. e. computed from these would be zero, 
 and the consequent weight infinite. But no one will doubt 
 but that the mean of the 30 results is more reliable than the 
 mean of these three results. 
 
 69. An Approximate Method of Weighting. A 
 long-continued series of observations will show the kind of 
 work an instrument is capable of doing under favorable con- 
 ditions ; and if work is done only when the conditions are 
 favorable, the p. e. derived from a certain number of results 
 will generally fall within limits that can be assigned d priori. 
 For example, with the Lake Survey primary theodolites, 
 which read to single seconds, the tenths being estimated, 
 the work of several seasons showed that the p. e. of the 
 mean of from 16 to 20 results of the value of a horizontal 
 angle, each result being the mean of a reading with telescope 
 direct and of a reading witJi telescope reverse, need not be 
 expected to be greater than d '.3. If, therefore, after having 
 measured a series of angles in a triangulation net with these 
 instruments, the p. e. all fell within o".3, it was considered 
 sufficiently accurate to assign to each angle the same 
 weight. 
 
 The objection to this is that "an instrument which has a 
 large periodic error may, if properly used, give as good 
 results as 1 if it had none ; but the discrepancies between its 
 combined results for an angle and their mean may be large, 
 thus giving an apparently large probable error to the mean. 
 Moreover, a given number of results over short lines, or 
 lines over which the distant signals are habitually steady 
 when seen in the telescope, will give a resulting value for 
 the angle of much greater weight than the same number of 
 combined results between two stations which are habitually 
 unsteady." * 
 
 The same method of weighting was employed by the 
 Northern Boundary Commission in their latitude work. 
 " The standard number of observations [for a latitude de- 
 termination] was finally fixed at about 60, it being found 
 
 * Professional Papers of the Corps of Engineers U. S. A., No. 24, p. 354.
 
 WEIGHTING OF OBSERVATIONS. 121 
 
 that with the 32-in. instrument 60 observations would give 
 a mean result of which the p. e. would be about 4 feet." 
 70. Weighting when Constant Error is Present. 
 
 The preceding leads us to the case where the error of ob- 
 servation can be separated into two parts, one of which is 
 due to accidental causes and the other to causes which are 
 constant throughout the observations. The total error e 
 would, therefore, be of the form 
 
 This case has been discussed already in general terms in 
 Art. 53 in explaining the well-known fact that an increase in 
 the number of observations with a given instrument does 
 not lead to a corresponding increase of accuracy in the 
 result obtained. 
 Let 
 
 //, = the m. s. e. of the observation arising from the 
 
 accidental causes, 
 ^ = the error peculiar to the observation arising from 
 
 the constant causes. 
 
 Then /./,, ^, being independent, the total m. s. e. [j. of obser- 
 vation may be assumed 
 
 !' = /V + /V 
 
 If // observations have been made we shall have for the 
 m. s. e. fjL g of their mean, since //,, is constant, 
 
 It is evident that when ;/ is large ./ becomes the important 
 term, and that in any case the value of // and consequent 
 weight can be but little improved by increasing the num- 
 ber of observations. 
 
 For the purpose of finding the value of the m. s. e. 
 arising from the constant sources of error a special series of 
 observations is in general necessary. After this series has 
 
 * Kffort, Su >";) <>/ ///< y,-//i,-rn l^uihiary, p. 86.
 
 122 THE ADJUSTMENT OF OBSERVATIONS. 
 
 been made the value of ^ found from it can be applied in 
 the determination of the value of /^ in any other series made 
 under like conditions. 
 
 For illustration let us consider a latitude determination 
 with the zenith telescope. It is well known that with this 
 instrument a latitude result found from two observations of 
 a single star either north or south of the zenith is inferior to 
 one found from a combination of a north and a south star. 
 This arises, not from any difference in the mode of observa- 
 tion, but from the errors in declination as given in the star 
 catalogue being cumulative in the one case and balancing 
 in the long run in the other. 
 
 The zenith-distance, , of each star being observed, the 
 half-difference of zenith-distances for each pair may be com- 
 puted, and each of these computed values may be con- 
 sidered an observed value. The values of the declinations 
 3 are taken from a catalogue of stars. The errors of 3 are, 
 therefore, independent of those of , and are constant for the 
 same pair of stars. The latitude <p from one pair is given by 
 
 Let 
 
 fa = the m. s. e. of (' ) for one observation of one pair, 
 
 fji s = the m. s. e. of % (3' -f- d) for this pair, 
 
 /4j, the m. s. e. of the resulting latitude <p from one pair, 
 
 then for a single observation of this pair 
 
 and for n observations of this pair 
 
 The quantity // will be found from repeated observa- 
 tions of the same pair of stars, as the error in declination 
 will not influence the result. A better value will, of course, 
 be obtained from several pairs than from a single pair. Let, 
 then, many pairs of stars be observed night after night for
 
 WEIGHTING OF OBSERVATIONS. 
 
 123 
 
 a considerable period. Collect into groups the latitudes 
 resulting from the observed values of each separate pair. 
 Let ;/., n. 2 , . . . v m be the number of results in the several 
 groups, the number in any group being at least two. Form 
 the residuals for each group and compute the m. s. e. in the 
 usual way. We have: 
 
 
 First Pair. 
 
 Second Pair. . . . 
 
 No. of 
 
 
 
 Night. 
 
 
 
 
 
 
 Results. 
 
 z> 
 
 Results. 
 
 1 . . . 
 
 I 
 
 f,' 
 
 v, 7 
 
 f.' 
 
 './ 
 
 . . . 
 
 2 
 
 n 
 
 z , i " 
 
 ?; 
 
 ,. /' 
 
 
 
 
 
 
 
 
 3 
 
 f'" 
 
 .. 
 
 <p* 
 
 7' 12 
 
 
 Means 
 
 f, 
 
 f, 
 
 ... 
 
 Now, assuming that, the m. s. e. of observation of each pair 
 is the same, 
 
 n, i 
 
 If, then, n is the total number of results, and ;// the number 
 of groups, by adding the above equations there results 
 
 In finding //^ we assume that though errors of declina- 
 tion are constant for each star, still for a latitude found 
 from many pairs in the same catalogue these errors mav be
 
 124 THE ADJUSTMENT OF OBSERVATIONS. 
 
 regarded as accidental. Let, then, many different pairs of 
 stars be observed on each of n nights at m' places, no star 
 being observed at more than one place. Collect the means 
 of the single results of each separate pair and form' the 
 residuals v' for each place, tajdng the differences between 
 these means considered as single results and their mean for 
 that place. Then, reasoning as above, the m. s. e. of a 
 latitude resulting from n observations on a single pair of 
 stars is 
 
 QV] 
 
 where n' is the number of different pairs of stars observed 
 and m' is the number of places occupied. 
 Now, /2s is found from 
 
 and is, therefore, known for the star catalogue used. This 
 value may be taken in future work in finding /^ from 
 
 and the consequent combining weight of y will be as 
 
 i 
 
 /V 
 
 An example of a similar kind is afforded in finding the 
 weights of the angles measured with a theodolite in a tri- 
 angulation where more rigid values are required than would 
 be found by Art. 69. The actual error of a measured value 
 of an angle arises from two main sources, errors of gradua- 
 tion and errors of observation The former are constant 
 for different parts of the limb read on, and correspond to 
 the declination errors above, while the latter are incapable 
 of classification, and are, therefore, assumed to be accidental. 
 The periodic errors of graduation are supposed to have
 
 WEIGHTING OF OBSERVATIONS. 125 
 
 been eliminated by proper shiftings of the circle. The re- 
 sultant m.s. e. IJL of a single measurement is found from 
 
 and the m. s. e. /* of the mean of n measurements made on 
 the same part of the limb from 
 
 , !+? 
 
 -=*+= 
 
 where //// are the m.s. e. of graduation and observation 
 respectively. The method of treating this problem is quite 
 similar to that of the preceding; // 2 is found by reading the 
 same graduation-mark on the limb many times, and n, by 
 reading the angle between two fixed signals many times, 
 the limb being changed after each reading. Thence ( 1 is 
 known for the instrument in question, and the combining 
 weights of angles measured with this instrument are at once 
 found. 
 
 The foregoing leads to another important practical point 
 in the measurement of angles. If the weight of a single 
 observation is unity, then the weight of the mean of ;/ ob- 
 servations made with the limb in one position is 
 
 /=<"'+"' 
 
 /v + -%- 
 
 Experience has shown that we may safely assume 
 and therefore it follows that 
 
 Hence, no matter how many observations we make in one 
 position of the limb, we never reach the precision of the 
 mean of two observations made with the limb in different 
 positions. 
 
 It might fairly be inferred that the limb should be 
 shifted after each single reading of an angle, and the rea-
 
 126 THE ADJUSTMENT OF OBSERVATIONS. 
 
 sons for not doing so are to guard against mistakes in 
 reading and to eliminate twist of station, as explained else- 
 where. (Art. 54.) 
 
 71. Assignment of Weight Arbitrarily. So far we 
 
 have deduced the combining weights from the observed 
 values themselves, or from them in connection with a 
 special series of observations. But this may not always be 
 the best way of finding the weights. The observations may 
 not be our only source of information, and, indeed, not the 
 most reliable source. If, for example, some phenomenon 
 has been observed by many persons in different parts of the 
 country, and the observations are sent to one place for com- 
 parison and reduction, it would not be proper for the com- 
 puter to deduce a weight for each series from the observa- 
 tions themselves independent of other sources of information 
 he might have. Some of the most inexperienced observers 
 with the poorest instruments might have apparently better 
 results than the most experienced with good instruments. 
 In such a case the computer must exercise his own judg- 
 ment in classing the observations. He should consider the 
 experience of the observer, his previous record for accurate 
 work, the kind of instrument used, the conditions, and the 
 observer's record of what he saw whether it is clear and 
 precise or hazy in its statements. An arbitrary scale of 
 weights may then be constructed, and to each set of obser- 
 vations be assigned a weight from this scale according to 
 the computer's estimate of its value. No two computers 
 would be likely to assign precisely the same weights, but if 
 done by one of experience and good judgment the result 
 obtained from weighting in this way will undoubtedly be of 
 more value than that found by the strict application of the 
 formulas of least squares. 
 
 The point is simply this. The class of observations con- 
 sidered may be expected to contain systematic errors which 
 cannot be determined, and is therefore not capable of being 
 treated by the method of least squares. As we have no 
 direct means of eliminating this kind of error, we must do so
 
 WEIGHTING OF OBSERVATIONS. 1 27 
 
 indirectly as best we can, and that is what the system of 
 weighting- mentioned seeks to accomplish. 
 
 An example will be found in the discussion of the Tele- 
 scopic Observations of the Transit of Mercury, May 5-6, 1878, 
 Washington, 1879, where, of 109 observations sent in, to only 
 18 was the hig-hest weight assigned. Prof. Eastman, under 
 
 o o o 
 
 whose direction they were reduced, says: "... Several 
 instances may be found where small weight is given to ob- 
 servations that apparently agree well with those to which 
 the highest weight is assigned, but in most cases the ob- 
 server's remarks indicate the uncertain character ol the 
 observation." 
 
 72. Combination of Good and Inferior Work. 
 It is strictly in accordance with the idea of weight that if 
 we have two results of very different degrees of accuracy, 
 a result better on the whole than either ma)- be found 
 by combining both with their proper weights. But the 
 proper weights may be difficult to find. On this account it 
 depends on circumstances whether it is advisable to reduce 
 a set of observations poorly made, in order to combine them 
 with a well-made set. If the quantity isavailable for observ- 
 ing again it might not cost any more to do this than to reduce 
 the poor observations. Even if it did the result would be 
 more satisfactory. The committee of the Royal Society of 
 England which was appointed to examine Col. Lambton's 
 geodetic work in India reported that " Col. Lambton's 
 surveys, though executed with the greatest care and 
 ability, were carried on under serious difficulties, and at a 
 time when instrumental appliances were far less complete 
 than at present. There is no doubt that at the present time 
 the surveys admit of being improved in every part. The 
 standards of length are better ascertained than formerly, 
 and all uncertainty on the unit of measure may be re- 
 moved. The base-measuring apparatus can be improved. 
 The instruments for horizontal angles used by Col. Lamb- 
 ton were inferior to those now in use. . . . The com- 
 mittee express the strong hope that the whole of Col.
 
 128 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Lambton's survey may be repeated with the best modern 
 appliances." * 
 
 73. The Weight a Function of our Knowledge. 
 
 If a quantity is not available for observing again, as, for 
 example, some transient phenomenon, all of the material on 
 hand must be used, and the best weights possible assigned 
 to the separate values in order to combine them. The 
 point is that where systematic or constant error has not been 
 eliminated the weight to be assigned is a function of the 
 state of our knowledge is, in fact, a matter of individual 
 judgment. 
 
 This is brought out very fully in the methods used in 
 combining the older star catalogues with the more modern 
 ones. Thus Safford (Catalogue of Mean Declinations of 2018 
 Stars, Washington, 1879) says: "In computing positions I 
 have generally employed Argelander's rule giving to a 
 modern determination from 
 
 1 observation a weight |, 
 
 2 observations a weight , 
 
 3 to 8 observations a weight i, 
 
 ** o 
 
 9 or more observations a weight li or 2. 
 
 Argelander generally gives Piazzi a weight equal to unity ; 
 the value \ is much nearer the truth ; in general he assigns 
 rather a larger relative weight to the older and poorer ob- 
 servations than they deserve. But this is mostly compen- 
 sated for by the number of determinations." 
 
 The weight of a quantity being a function of our know- 
 ledge may have assigned to it a certain value at one time 
 and another value at another time when our knowledge of 
 it has increased. Thus in the Fond du Lac (Wis.) base, 
 measured in 1872 with the Bache-Wiirdemann compensating 
 apparatus, a portion was measured seven times. The re- 
 sults differed widely, far beyond what was expected with 
 the apparatus. No reason could be assigned at the time for 
 the discordances. At this stage, then, one would have been 
 
 * G. T. Survey of India, vol. ii. p. 70.
 
 WEIGHTING OF OBSERVATIONS. 129 
 
 justified in assigning a small weight to the value of the 
 base. 
 
 The Keweenaw base was next measured with the same 
 apparatus, and the same trouble came in. Next the Sandy 
 Creek base and then the Buffalo base were measured. Dur- 
 ing all this time (four years) material had been accumulating 
 for the explanation of the behavior of the apparatus. When 
 the law of its behavior was discovered it was found that 
 good work not only could be done but had been done 
 with it. 
 
 Hence the systematic error being got rid of, one would 
 be justified in increasing the weights of the bases measured 
 with this apparatus in comparison with bases measured with 
 an apparatus of a* different kind. Had the later work not 
 been done the Fond du Lac base would still have had 
 assigned to it the low weight. 
 
 Take another instance. Sir G. B. Airy, in 1847, sa j s f 
 the Mason and Dixon arc (Encyc. Metrop., p. 209) : " The 
 results of this measure must, we think, be received as equal 
 in authority to those of any other measure." This may have 
 been true when written ; but Mr. Schott, in 1877, in his 
 note on the determination of the figure of the earth from 
 American sources, says of this same arc (U. S. C. S. Report, 
 1877, p. 95): " It is, therefore, only owing to the increased 
 perfection of instrumental means and methods that we now 
 dismiss from further consideration the first measured North 
 American arc, which, moreover, is now superseded by the 
 present measures." 
 
 As a third illustration we may consider the weights to 
 be assigned to a system of differences of longitudes in which 
 the connections of the stations occupied are interlaced as in 
 a triangulation net, and the whole svstem is to be adjusted 
 so as to remove existing contradictions. 
 
 If the longitude work has been carried out on one plan, 
 with instruments and observers of about the same quality, 
 tnen the m. s. e. of each determination mav be computed 
 from the measures of the separate nights, and in the adjust-
 
 130 THE ADJUSTMENT OF OBSERVATIONS. 
 
 ment the weights may be taken inversely as the squares of 
 these m. s. e. 
 
 But if this has not been done, if in the older work in- 
 struments, observers, and methods were poorer than later 
 and the two have to be combined in the adjustment, the 
 computer must estimate as best he can their relative 
 weights. Thus in a system in Germany, France, and Aus- 
 tria reduced by Dr. Albrecht* the observations were made 
 between the years 1863 and 1876. The methods of obser- 
 vation had been much improved in this interval. In as- 
 signing the relative weights a scale of weights was first 
 formed from a consideration of all the knowledge on hand, 
 taking the march of improvement from year to year into 
 account, and the separate determinations, placed in one or 
 other of these classes. Thus, for example, 
 
 Weight i No change of observers ; few observations; non- 
 adjustment of electric current ; 
 
 Weight 2 No change of observers ; usual variety of obser- 
 vations ; non-adjustment of electric current ; 
 
 Weight 3 Change ot observers; usual variety of observa- 
 tions ; non-adjustment of electric current, 
 
 and so on. 
 
 Similarly Dr. Bruhns in Verhandlungen der europdisclien 
 Gradmessung, 1880. See also Coast Survey Report, 1880, 
 Appendix 6. 
 
 74. General Remarks. -The subject of the weighting 
 of observations is confessedly a difficult one. In general it 
 may be affirmed that the less experienced a computer is 
 the more closely he will adhere to the rigorous formulas 
 without considering whether systematic errors enter or not. 
 As he adds to his experience he will consider outside evi- 
 dence as well as the evidence afforded by the observations 
 themselves. This will be specially true if he has any prac- 
 tical knowledge of how observations are made. Indeed, 
 it is doubtful if a computer can apply the principles of 
 
 * Astronomische Nachricliten, 2132.
 
 REJECTION OF OBSERVATIONS. 131 
 
 least squares properly unless he is at least an average ob- 
 server. 
 
 Great caution, however, is necessary in assigning weights, 
 because it is sometimes possible so to choose them as to 
 make observations tell anything desired. They should 
 always be chosen from a consideration of all the evidence 
 on hand, and may be changed as additional evidence is pre- 
 sented, so that a result is never final, but is ever open for 
 improvement. The original records of the observations 
 and the methods of reduction are quite as desirable, if in- 
 deed not more so, than the results deduced. The best plan, 
 therefore, is to publish all of the data along with the reduc- 
 tion, when the reader, if he wishes, can make a reduction 
 for himself. He can then form a more intelligent opinion 
 of the computer's skill and judgment, and also of the value 
 of the work. 
 
 NOTE II. 
 
 ON THE REJECTION OF OBSERVATIONS. 
 
 75. There is nothing in the \vhole theory of errors more 
 perplexing than the question of what shall be done with an 
 observation of a series which differs widely from the others. 
 In making a series of observations the observer is given full 
 power. He can vary the arrangements, choose his own 
 time for working, reject any result or set of results; he can 
 do anything, in fact, that in his best judgment will tend to 
 give the best value of the observed quantity. But when he 
 has finished observing and goes to computing, has he the 
 same power? Can he alter, reject, manipulate in such a 
 way as in his best judgment will give a result of maximum 
 probability ? As observer he was supreme ; as computer is 
 he supreme, or only in leading-strings? Various answers 
 
 iS
 
 132 THE ADJUSTMENT OF OBSERVATIONS. 
 
 may be given to this question, as we look at it Irom one 
 point of view or another. 
 
 In the hypothetical case on which the exponential law 
 of error was founded there were no discontinuous observa- 
 tions taken into account. There we contemplated not only 
 observations made with the best instruments and by the 
 most experienced observers, but observations of all grades, 
 from this highest grade down to those made with the poorest 
 instruments and by the most ignorant and careless observers 
 conceivable. It is only in this way that errors continuous 
 all the way from -f- co to QQ could arise. In the cases 
 occurring in ordinary work we confine our attention to one 
 section of the observations only that made with the good 
 instrument and by the skilful observer. This, to "be sure, is 
 the most important, and, as shown in Art. 30, the result 
 following from it differs ordinarily but little from that found 
 in the ideal case. But we are naturally confronted with 
 difficulty when we try to deal with a very incomplete series. 
 Extra assumptions must be made, and it is not to be won- 
 dered at that no solution yet offered is regarded as entirely 
 satisfactory. 
 
 76. A common summary method of disposing of the sub- 
 ject is contained in the following statement: "The weights 
 [of the angles] would have been materially increased in 
 many instances by rejecting what would appear bad obser- 
 vations ; but the rule has been never to reject any unless the 
 observer has made a remark to the effect that it ought to 
 be rejected." This statement, however, does not cover the 
 whole ground. Those who reason in this way make a dis- 
 tinction between mistakes and errors of observation. Mis- 
 takes are rejected. But the great difficulty is to tell just 
 where mistakes end and errors begin. 
 
 Given a set of measures involving discordances unlocked 
 for, and which the observer's remarks do not cover, how 
 shall we proceed? Two views may be taken. In the first 
 place, the computer, from a consideration of the measures 
 themselves and from all other evidence bearing on them
 
 REJECTION OF OBSERVATIONS. 133 
 
 that he can discover, may make a distinction between meas- 
 ures and mistakes which will do for the set before him. 
 With observations of another kind he might have a different 
 mode of procedure. Another computer might have differ- 
 ent rules altogether, precisely as in the case of weighting 
 as explained in Arts. 71-74. 
 
 To put the discrepant values with the other values, and 
 take the arithmetic mean of all, would give a result con- 
 siderably different from what would be found by omitting 
 them. It would take a great many good observations to 
 balance the effect of a single widely discrepant one. It is, 
 then, for the computer to judge whether this discrepant 
 observation shall have the same weight as the others or a 
 different weight. 
 
 It may happen, indeed, that the discrepancy is so evi- 
 dently a "natural mistake" that it may be corrected with- 
 out a doubt from the evidence furnished by the other 
 observations, and the discrepant observation changed so 
 that it may be treated as a good one. Thus an angle may 
 be read 5' or 10' wrong, or a micrometer screw may be read 
 5 or 10 revolutions out of the way, as shown by the rest of 
 the observations, and the like. 
 
 Or the observations may be arranged in well-defined 
 groups, and if the computer finds that he cannot account 
 for the presence of unusual discrepancies in a certain group, 
 he may decide to reject the whole group. For example, in 
 longitude work it may happen that one of the time stars may 
 give a clock correction differing, say, one second of time 
 from that given by fifteen or twentv others observed on the 
 same night. Instead of rejecting the single star it would 
 perhaps be better to reject the whole night's work. If 
 necessary, an extra set of observations may be made to fill 
 the blank. By rejecting a group no hidden law can be 
 slighted, for if any exists it will continue to reappear in 
 further observations, and finally to reveal itself. 
 
 77. In the second place, the computer, instead of trusting 
 to his judgment, may call in the aid of the calculus of proba-
 
 134 THE ADJUSTMENT OF OBSERVATIONS. 
 
 bilities and seek to establish a test or criterion for the rejec- 
 tion of observations which will serve for all kinds of obser- 
 vations. Of the criterions which have been proposed the 
 earliest is due to Prof. Peirce. It is as follows: "Observa- 
 tions should be rejected when the probability of the system 
 of errors obtained by retaining them is less than that of the 
 system of errors obtained by their rejection multiplied by 
 the probability of making so many and no more abnormal 
 observations." A proof by Dr. Gould will be found in the 
 U. S. Coast Survey Report, 1854, pp. 131, 132. It is founded 
 on the assumption of the Gaussian law of error. 
 
 Another criterion " for the rejection of one doubtful ob- 
 servation " is given by Chauvenet in his Astronomy, vol. ii. 
 p. 565. " We have seen that the function [Art. 30] 
 
 / f '~ r /,'A* 
 '" 
 
 represents in general the number of errors less than a which 
 may be expected to occur in any extended series of obser- 
 vations when the whole number of observations is taken as 
 unity, r being the p. e. of an observation. If this be multi- 
 plied by the number of observations n, we shall have the 
 actual number of errors less than a; and hence the quantity 
 
 n-nO(f) = n\i- 8(t}\ 
 
 expresses the number of errors to be expected greater than 
 the limit a. But if this quantity is less than \ it will follow 
 that an error of the magnitude a will have a greater proba- 
 bility against it than for it, and may, therefore, be rejected. 
 The limit of rejection of a single doubtful observation is, there- 
 fore, obtained from the equation 
 
 t = 
 
 2n
 
 REJECTION OF OBSERVATIONS. 13$ 
 
 A third criterion was proposed by Mr. Stone, Radcliffe 
 observerat Oxford, Eng., in Month. Not. Roy.Astron.Soc., 1868, 
 1873, in these terms : " I assume that a particular person, with 
 definite instrumental means and under given circumstances, 
 is likely to make, on an average, one mistake in the making 
 and registering n observations of a given class. The proba- 
 bility, therefore, is that any record of his of this class of ob- 
 servations as a mistake is - n . From the average discord- 
 ances among the registered observations of this class we 
 can find the p. e. of an observation in the usual way, and 
 also the probability of an error greater than a given quan- 
 tity, as C. Then if the probability in favor of a discordance 
 as large as C is less than that of a mistake, or^, the discord- 
 ant observation is rejected." 
 
 The least objectionable criterion based on mathematical 
 principles may, I think, be developed from the principle 
 laid down in Art. 50, where the maximum error was esti- 
 mated at about five times the p. e. or three times the m. s. e. 
 If, therefore, an observation differs from the general run of 
 the series by more than this amount it should at least be 
 bracketed and attention be called to it. 
 
 78. It may be stated that, as a general rule, criterions are 
 apt to be most highly esteemed by those who look at the 
 observations from a purely mathematical rather than from 
 the practical observer's point of view. The latter is, with- 
 out doubt, the true standpoint. Every observer will, con- 
 sciously or unconsciously, construct a criterion suited to 
 the sort of work he is engaged in. This criterion will not 
 necessarily be founded altogether on mathematical for- 
 mulas. Indeed, most likely it will not be. Nor does it- 
 follow that the criterion adopted in any special series is of 
 universal application or will receive universal assent. In 
 the process of weighting the observer will not assign 
 weights always as the inverse square of the m. s. e. It is 
 often better to assign them arbitrarily from a feeling 
 founded on a general grasp of all the circumstances con- 
 nected with the making of the observations. In like man-
 
 136 THE ADJUSTMENT OF OBSERVATIONS. 
 
 ner, and on this same feeling 1 , he will found his criterion for 
 rejection. There is no uniform rule for weighting-, neither 
 is there one for rejection. 
 
 A criterion such as Stone's, for example, would be very 
 useful in the work for which it was proposed, and in its 
 proposer's hands would be of much value. But that any 
 one without the insight given by long familiarity into the 
 special kind of work from which this criterion arose could 
 apply it properly is not to be expected. As already pointed 
 out, in the case of weighting the only thing for the com- 
 puter to do is to publish all of the observations, including 
 those rejected, along with his reduction, when, if more 
 light can at any time be thrown on them, a new reduction 
 can be made. Take, for example, Bradley 's observations as 
 reduced by Bessel and Auwers (Art. 51). It cannot be too 
 strongly insisted on that a result deduced from a series of 
 observations is never to be looked on as final, but as ever 
 open for improvement. 
 
 79. The difficulty in combining single observations lies 
 in assigning to them their proper weights. We have as- 
 sumed the arithmetic mean to give the most probable value. 
 If the sources of error could be separated, so that to each 
 single observation could be ascribed its proper weight, the 
 resulting weighted mean would be nearer the truth than 
 the direct arithmetic mean. We can, therefore, conceive of 
 a better value than the arithmetic mean in certain cases.* 
 This has been already pointed out in Art. n. 
 
 We may, therefore, consider whether, when discrepant 
 observations occur we may not get a more satisfactory 
 result by ignoring the arithmetic mean altogether. Sup- 
 pose, for example, that we had three observed values, 100, 
 60, 61, and that we had no means of getting any further 
 observations. These values would seem to show that the 
 true value was likely to be nearer 60 than 100. Just 
 
 * De Morgan (Encyc. Metrop., " Theory of Probability," p. 456) suggested that the combin- 
 ing weights might be found from the observed values themselves, but he did not develop his plan, 
 and it is apparently fruitless.
 
 REJECTION OF OBSERVATIONS. 137 
 
 how much nearer is the question. To reject the observa- 
 tion 100 would be without reason, and to take the arith- 
 metic mean of all three would ignore the evidence afforded 
 by the observations themselves. 
 
 The observations being discontinuous, the question is 
 removed from the theory of least squares, which presup- 
 poses continuity (see Art. 30), and must be treated by other 
 methods. Laplace* discussed this problem long before 
 Legendre and Gauss developed the method of least squares. 
 The subsequent confusion in the introduction of criterions 
 has arisen from trying to fasten to the method of least 
 squares what is in reality a very different question. 
 
 When the observed values are discontinuous we cannot 
 reasonably assume the value of the unknown sought to be a 
 symmetrical function of them (see Art. 11). A plausible 
 result has been given (Arts 11, 15) as that observed value 
 which has as many observed values greater than it as it has 
 less than it. Thus in the preceding example a good value 
 to choose would be 61. 
 
 In discussing such problems, so long as not much greater 
 plausibility can be assigned to one method of combination 
 than to another, the question of convenience of computation 
 comes in. In respect to this the propriety of selecting the 
 middle term stands pre-eminent. 
 
 80. It is ever to be kept in mind that unexpected dis- 
 crepancies in his results do not always prove to the observer 
 that his work is bad, any more than a close agreement 
 among them shows it to be good. When either occurs 
 great caution is necessary. It is unsafe to have a rigid rule 
 of any kind for sifting the observed values, not allowing 
 the computer to make use of evidence outside of the ob- 
 servations. By following such rules we are apt to bar 
 the way to discovery of new truths, or at least to hinder 
 progress in that direction. See, for example, the history 
 of the discovery of personal equation, American Cyclopicdia, 
 vol. xiii. 
 
 * M4in. Acati. Paris, vol. vi. p. 634.
 
 138 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Throughout this discussion it has been assumed that the 
 observations have been reduced by the observer himself or 
 by a computer who is at the same time a competent ob- 
 server. A computer who is not an observer must of ne- 
 cessity employ the same criterion always; and in this case 
 the criterion derived from Art. 50, as stated on page 135, is 
 to be preferred. 
 
 The following memoirs may be consulted in addition to 
 those already mentioned : Gergonne, Amiales de Math., 
 vol. xii. pp. 181 seq. ; Peirce in Gould's A sir on. Jour., 
 vol. ii. pp. 161 seq.; Airy in do., vol. iv. pp 137, 138; Win- 
 lock in do., vol. iv. pp. 145-147 ; Stone, Month. Not. Roy. 
 Astron. Soc., vol. xxviii. pp. 165 seq., vol. xxxiv. pp. 9 seq. ; 
 Glaisher in do., vol. xxxiii. pp 391 seq., also in Mem. Roy. 
 Astron. Soc., vol. xxxix. pp. 75 seq.
 
 CHAPTER IV. 
 
 ADJUSTMENT OF INDIRECT OBSERVATIONS. 
 Determination of tJie Most Probable Values. 
 
 81. If direct measurements of a quantity have been made 
 under the same circumstances, we have seen that the arith- 
 metic mean of these measures gives the most probable value 
 of the quantity. We now come to the case where the 
 quantity measured is not the unknown required, but is a 
 linear function of one or more unknowns whose values are 
 to be found. This is the more general form, and its solution 
 has been carried a certain distance in Art. 14. The point 
 stopped at was the combining of observations of different 
 weights. As by the aid of the law of error this can now be 
 done, we proceed to finish the solution. 
 
 Let, as in Art. 14, the equations connecting a series of 
 observed quantities J/j, M. 2 . . . M n , n in number, and the 
 independent unknowns X, V, . . . , n { in number (n > ;/,-), be 
 
 ^r+/,,F+. . . -L^M> + v, 
 
 a,X+b, F+ . . . - L, = J\L + , (i) 
 
 where #,. /;,, . . . /,,, . . . L n are constants given by theory 
 for each observation, and i\, ?',, ...?' are the residual 
 errors of observation. 
 
 In practice the labor of handling these equations will be 
 much lessened by using an artifice we have several times 
 already employed (see Art. 41). Let A", Y' , ... be close 
 approximations to the value of A', F, . . . found by ordinary 
 elimination from a sufficient number of the equations, or by 
 some other method, as by trial, for example, and put 
 
 x-x^x, r- Y =j>,... 
 19
 
 140 THE ADJUSTMENT OF OBSERVATIONS. 
 
 where ,r, y, . . . are the corrections required to reduce the 
 approximate values to the most probable values. 
 Then the observation equations reduce to 
 
 a,x -f b,y -f . . . /, = v, 
 
 a,x -\- b,y + . . . / = v, (2) 
 
 where 
 
 + . . . -L n -M n 
 
 and are, therefore, known quantities. 
 
 It is more convenient in practice to omit the residuals 
 and write the observation equations in the form 
 
 a^x -\-b.y-\- . . . =/, 
 
 "S + b t y 4- . = 4 (3) 
 
 <*** + b n y -f . . . = 4 
 
 always keeping in mind that the strict form is as in (2). 
 The observation equations 3 may be written 
 
 a 1 x = / l ' 
 
 a^x // (4) 
 
 a n x l n ' 
 
 if the values of y, z, . . . are supposed to be known. 
 
 Now, it has been shown in Art. 63 that the most probable 
 value of x would be found from these equations by taking 
 
 I' I ' I ' 
 
 the weighted mean of the separate values -L, JL, . . . JL. 
 
 a 1 a^ a n 
 
 The weights of these values are in the same section shown 
 to be as a*, a*, . . . a n \ We have, therefore, 
 
 _[aT\ 
 -
 
 INDIRECT OBSERVATIONS. 14! 
 
 or, by putting for //, //, . . . their values, 
 
 _ {al} \ab~\y \ac~\z . . . 
 
 * p =j 
 
 [aa] 
 
 Similarly if the values of x, z, . . . are supposed to be 
 known, the most probable value of y is found from 
 
 _\bl-\-\ba\ X -\bc-\z-. . . 
 
 [W] 
 and so on for ^, . . . 
 
 Hence we should obtain the weighted mean values of 
 x, y, . . . , that is, their most probable values, from the 
 simultaneous solution of the equations 
 
 [aa] 
 _ [/] - \ba~\x - \bc~\z - ... 
 
 ~\bbT 
 
 that is, from the simultaneous solution of the equations 
 
 [>*>+[^]j,+ . . . = [*/] 
 
 (5) 
 
 which equations are equal in number to the number of un- 
 knowns. They are called normal equations, or, better, final 
 equations. We have thus found the most probable values of 
 the unknowns in a series of observation equations by taking 
 the mean.* 
 
 * Another method of solving a series of observation equations, due to Richelot, is worthy of 
 notice. 
 
 Take the simple case of n equations involving two unknowns. Let the equations ( > 2) be 
 written in the strict form 
 
 to find the values of .r, y which satisfy them best. 
 
 Multiply the equations in order by the undetermined factors /(-,, X- 2 , ..., and add ; then if 
 ^i, 2 , . . . k n satisfy the condition \,bk\ o, we have 
 
 [*/] [**] 
 [*] ^ [*] 
 
 The best value of a- must be that in which the second member is as small as possible, and this
 
 142 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The preceding result may be derived in a manner which 
 will perhaps show still more clearly that we have taken the 
 means of the separate values of the unknowns 
 
 For simplicity in writing consider the case of the three 
 observation equations, 
 
 a,x + bj /, 
 a,x + b,y = I, 
 a*x -f b, y = / 3 
 
 already given in Art. 14. 
 
 Solve in sets of two in all possible ways by the method 
 of determinants, and 
 
 OA - a.b^x = bj, - bj, 
 0/ 3 a z b^x = bj, bj, 
 (a& a.b,}x = bj, - bj, 
 
 Take the weighted mean of the values of x according to 
 Art. 63, and the result is 
 
 _ (bill bil^a^bi a^i) + (b 3 l\ bil 3 )(aib 3 a 3 i>i) + (b 3 Zi b^l 3 }(a^b 3 a 3 b^) 
 
 ' 
 
 [aajbb]- [abjab 
 Similarly 
 
 _[ fl /][^]-MM 
 
 \adbb~\-\ab\ab~\ 
 
 happens when \kd\ is as great as possible. But \_ka\ as a linear function has no maximum unless 
 a condition exists among the /t's of at least two dimensions. The simplest such condition is 
 
 \kK\ = i 
 If we now compute the maximum value of [/(] subject to the conditions 
 
 [bk] - o [M] = i 
 we shall find as the best value of x (Todhunter, Diff. Calc., chap, xvi.) 
 
 \aa\\bb~\ - \ab~\\aK\ 
 which agrees with the value resulting from the normal equations 5.
 
 INDIRECT OBSERVATIONS. 143 
 
 These are the same values as would be derived from the 
 solution of the equations 
 
 \aa\x 
 \bd\x 
 
 Hence if in a series of n observation equations containing 
 i^ independent unknowns (it > w f ) -we solve all possible sets 
 of , equations by the method of determinants, the weighted 
 means of the separate values found will be the most probable 
 values of the unknowns. 
 
 This method of solution would be very troublesome 
 when the number of equations is large, and accordingly we 
 must look for a more convenient but necessarily equivalent 
 method. 
 
 Since the principle that the sum of the squares of the 
 residual errors is a minimum holds whether the observed 
 quantity is a function of one or of several unknowns (Art. 17), 
 we can apply it to the simultaneous solution of the equations 
 2. The residual errors must satisfy the relation 
 
 <', 2 + ''/ + - + v* = a mill. 
 that is, we must make 
 
 + . . . + aKX + b n y + . . . - / = a mn. 
 
 Now, the variables x,y, . . . being independent, the dif- 
 ferential coefficients of the expression for the minimum with 
 respect to each of them in succession must be equal to zero. 
 Hence 
 
 ..- = o (6) 
 
 or, collecting the coefficients of x, y, ... in each equation, 
 
 + =[<! 
 
 c\s + . . .
 
 144 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The number of these equations is the same as the number of 
 unknowns ; that is, n f . Their solution will give the most 
 probable values of x, y, . . ., and, adding these values to the 
 approximate values X' , Y' . .'. already known, the most 
 probable values of X, Y, . . . will result. 
 
 The equations 7 are identical in form with equations 
 5, found by taking the mean of all possible values of x and 
 y. As it has now been abundantly shown that the principle 
 of the mean and that of minimum squares lead to the same 
 results whatever be the number of independent unknowns, 
 we shall use whichever promises to be most convenient in 
 any particular case. 
 
 It is useful to notice that equations 6 may be written 
 
 [av] = o 
 
 r>]= (8) 
 
 These relations correspond to \v\ = o in the case of the 
 arithmetic mean, and may be used as a check on the com- 
 putation of the values of the residuals. 
 
 Ex. Given the elevation of Ogden above the ocean by C. P. R. R. levels 
 to be 4301 feet, and the elevation of Cheyenne to be 6075 feet ; also the ele- 
 vation of Cheyenne above Ogden by U. P. R. R. levels to be 1749 feet; find 
 the adjusted elevations of Ogden and Cheyenne above the ocean, supposing 
 the given results to be of equal value. 
 
 First solution : 
 
 (a) Ogden 
 
 Direct determination 4301 weight i 
 Indirect determination 4326 " \ 
 
 .'. weighted mean = 4309 feet. 
 
 (b) Cheyenne 
 
 Direct determination 6075 weight i 
 Indirect determination 6050 " \ 
 
 .'. weighted mean = 6067 feet. 
 
 Second solution : 
 
 Let X, Y denote the elevations of Ogden and Cheyenne respectively. 
 Then 
 
 (^- Y+
 
 INDIRECT OBSERVATIONS. 145 
 
 Differentiate with respect to X, Fin succession, and 
 
 2X- 7=2552 
 
 -X + 2^=7824 
 
 .' . X 4309 feet. 
 
 Y= 6067 feet. 
 
 82. If the observation equations are of" different weights 
 Pv A A then, reducing each equation to the same unit 
 of weight by multiplying it by the square root of its weight, 
 we have 
 
 V A 
 
 with 
 
 [/t'f] ^=a min. 
 
 Substituting the values of \ / p l ?\, Vpv n _, ... in the mini 
 mum equation, and differentiating with respect to x, y, . . 
 as independent variables, we have the normal equations 
 
 ...--= [pbl] (2) 
 
 from which ,r, y, . . . may be found. 
 
 The relations for weighted equations corresponding to 
 those of Eq. 6, Art. Si, are evidently 
 
 \par~] =o, [X>''l=o, ... (3) 
 
 Formation of the Normal Equations. 
 
 83. Instead of forming the minimum equation and dif- 
 ferentiating with respect to the unknowns in succession, it 
 is more convenient to proceed according to the following 
 plans suggested by the form of the normal equations them- 
 selves.
 
 146 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The first, from equations 6, Art. 81, may be stated as fol- 
 lows : To form the normal equation in x multiply each obser- 
 vation equation by the coefficient of x in that equation, and 
 add the results. To form the normal equation in y multiply 
 each observation equation by the coefficient of y in that 
 equation, and add the results. Similarly for the remaining 
 unknowns. 
 
 The second is suggested by the complete form of the nor- 
 mal equations as given in equations 7, Art. 81. Accord- 
 ing to this plan we compute the quantities [aai], [ab], . . . [/], 
 etc., separately, and write in their proper places in the 
 equations. 
 
 The equality of the coefficients of the normal equations 
 in the horizontal and vertical rows leads to a considerable 
 shortening of the numerical work in computing these quan- 
 tities. Thus with three unknowns, x, y, z, all the unlike 
 coefficients are contained in 
 
 4- \_ad\x -f- \_ab~\y -f~ \_ac\z = [/] 
 
 >==[*/] 
 
 Instead, therefore, of computing 12 quantities, only 9 are 
 necessary, as the remaining 3 can be at once written down. 
 With n unknowns the saving of computation amounts to 
 
 1+2+3+ - - .+(*- !) = *(*- I) 
 
 quantities. 
 
 If the observation equations are of different weights the 
 formation of the normal equations may be carried out pre- 
 cisely in the same way as in the preceding as soon as the 
 observation equations have been reduced to the same unit 
 of weight. 
 
 The form of the weighted normal equations, however, 
 shows that it is not necessary, in order to obtain the coef- 
 ficients [paa], \_pab~], ... to multiply the observation equa- 
 tions by the square roots of their weights, and form the aux-
 
 INDIRECT OBSERVATIONS. 147 
 
 iliary equations I, Art. 82, since the products aa, ab, . . . 
 multiplied by the weights of the respective equations from 
 which they are formed and summed, give [paa\, [pab\, . . . 
 directly. This is important because labor-saving. 
 
 Ex. i. Given the-observation (or error) equations, all of equal weight, 
 
 x = i 
 
 x + y = 3 
 
 x y + z = 2 
 
 xy+z=l 
 
 show that the normal equations are 
 
 4-*' + y =5 
 
 Ex. 2. The expansions .Vi, x- 2 , ,v 3 , x 4 for i Fahr. of four standards of length 
 were found by special experiment to be connected by the following relations 
 at a temperature of 62 rj Fahr. (// one micron.) 
 
 iJ- 
 
 + -Vi = 39-945 weight i 
 
 -l-viv. = 5.932 " 1 6 
 
 + -v 3 5-371 " 4 
 
 + -r 2 1.0937^3 = 0.006 " 8 
 
 + 4-v a --v, = - 1.335 " 3 
 
 + a, .v 4 = + 14.833 " 6 
 
 find their most probable values. 
 [The normal equations are 
 
 + 7.r, - 6.r 4 = + 128.943 
 
 + 72.000 A-O 8.750^-3 I2.r 4 = + 78.940 
 
 - 8.750^2 + 13.569.1-3 = + 21.432 
 
 + (_).r 4 = 84.993 
 
 and x, = 39.913, .r 2 = 5-932, -v 3 = 5-45> -V4 = 25.075] 
 
 An example will now be given to illustrate the method 
 of forming a series of observation equations : 
 
 E.\\ 3. At Washington the meridian transits of the following stars were 
 observed to determine the correction and rate of sidereal clock Kessels No. 
 132^1, April 12, 1870. at 11 hours clock time. 
 20
 
 148 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Star. 
 
 Observed 
 clock time of transit, T. 
 
 Right ascension 
 of star, a. 
 
 r Leonis. 
 
 h. m. s. 
 II 21 17.98 
 
 s. 
 
 16.00 
 
 v Leonis. 
 
 II 30 20.41 
 
 18.51 
 
 ft Leonis. 
 
 II 42 28.57 
 
 26.57 
 
 o Virginis. 
 
 II 58 38.15 
 
 36.20 
 
 ?/ Virginis. 
 
 12 13 18.37 
 
 16-37 
 
 6 Virginis. 
 
 13 3 16.36 
 
 14-39 
 
 Let x = corr. of clock at n hours clock time, 
 y = rate per hour of clock. 
 
 Now, from theoretical considerations* it is known that the equation 
 x + y(T n) = a T 
 
 gives the relation between the clock correction and rate and the clock time of 
 transit of each star observed. 
 
 Hence the observation equations are 
 
 ^ + 0.35^= 1.98 
 x + 0.507= 1.90 
 
 X + O.7I_J/= 2.OO 
 * + 0.987= 1.95 
 X + 1.22}'= 2.OO 
 
 x + 2.057= !-97 
 For the remainder of the solution see Art. 104. 
 
 Ex. 4. In the triangulation of Lake Superior executed by the U. S. 
 Engineers there were measured at station Sawteeth East the angles 
 
 Farquhar-Porcupine 62 59' 40". 33 weight 5 
 
 Farquhar-Outer 64 n' 34". 92 " 7 
 
 Farquhar-Bayfield 100 20' 29". 12 " 4 
 
 Porcupine-Bayfield 37 20' 49". 55 " 7 
 
 Outer-Bayfield 36 08' 55". 86 " 4 
 
 required the adjusted values of the angles. 
 
 All of the angles may evidently be expressed in 
 terms of FSP, FSO, FSB. Let X, Y, Z denote the most 
 probable values of these angle?, and let X', Y', Z' be 
 assumed approximate values of these most probable 
 values, and x,y, z their most probable corrections. Denoting the measured 
 
 Fig.9 
 
 * See Chauvenet, Astronomy, vol. ii. chap. v.
 
 INDIRECT OBSERVATIONS. 149 
 
 angles in order by M\, M^ . . . M-,, and their most probable corrections by 
 Vi, v-i, ... ^ 5 , we have 
 
 X' +jc= X = Mi + vi 
 
 Y' +y = Y = J/ 2 + v* 
 
 Z' + Z= Z = M 3 + V 3 
 
 X' x+Z' + z = X + Z= M< + v t 
 - V -y + Z' + Z--V + Z=M 6 + v* 
 
 For simplicity the assumed approximate values may be taken equal to the 
 observed values of the angles, so that we have the reduced observation 
 equations 
 
 + x = z/i weight 5 
 
 + y =v* "7 
 
 + z =v* "4 
 
 x + 2 0.76 = 7/4 " 7 
 
 y + z 1.66 = 7' 5 " 4 
 
 Hence the normal equations 
 
 izx - 72 5.32 
 
 + iiy 42= 6.64 
 
 7-r 47 + 152= +11.96 
 
 Solving these equations, we find 
 
 x= o".os, y = o".36 2=+o".68 
 
 Hence z>i = o".O5, t/ 2 = o".36, ^ 3 =+o".6S, z' 4 = o".O3, v b = o".62, 
 and the adjusted values of the angles are 
 
 62 59' 40". 28 
 
 64 n' 34". 56 
 
 1 00 20' 29 ".So 
 
 37 20' 49". 52 
 
 36 08' 55". 24 
 
 We might have used z/i, c' 2 , . . . v-, for the corrections without introducing the 
 symbols x, y, z at all. 
 
 Ex. 5. If the unknown x occurs in each of the n observation equations 
 
 x + b\y + f i z + . . . = A weight i 
 
 x + l>iy + c? z + . . . = l-i " i 
 
 these equations are equivalent to the reduced observation equations 
 
 b l y + dz-\- ...=/, weight i 
 k->y 4- Ci z + . . . = / 2 i
 
 150 THE ADJUSTMENT OF OBSERVATIONS. 
 
 [For the normal equations found from the first set after eliminating x are the 
 same as the normal equations formed from the second set directly.] 
 
 Ex. 6. Instead of the observation equation 
 
 ax + by + cz + . . . = I weight / 
 we may write 
 
 qax + qby +...=/ weigh t ^ 
 
 84. Control of the Formation of the Normal 
 Equations. A very convenient check or control is the 
 following. Add as an extra term to each observation equa- 
 tion the sum of the coefficients of x, y, . . . and of the abso- 
 lute term / in that equation, and treat these added terms just 
 as we do the absolute terms. Thus let j,, s a _, . . . s n denote 
 the sums, so that 
 
 ! + b, + c, + . . . + /, = s, 
 
 Multiply each of these expressions by its a and add the 
 products, each by its b and add, and so on ; then 
 
 . . .+[/?]=[&] 
 
 If these equations are satisfied the normal equations are 
 correct. Thus each normal equation is tested as soon as it 
 is formed. 
 
 Since \aa\, \ab\, . . . \al\ have been computed in forming 
 the normal equations, the only new terms to be computed 
 in applying the check are [as], \_bs\, . . . \ls], [//]. 
 
 Various modifications may readily be applied to suit 
 individual tastes. Thus the absolute term may be placed
 
 INDIRECT OBSERVATIONS. 151 
 
 on the other side of the sign of equality ; or the sign of the 
 check may be changed so as to make the sum of each hori- 
 zontal row equal to zero. 
 
 85. Forms of Computing: the Normal Equations. 
 
 When the number of unknowns in the observation equa- 
 tions is large, or when their coefficients contain several 
 figures, it is convenient to have a fixed form for the compu- 
 tation of the terms of the normal equations. It lightens the 
 labor much either in forming, solving, or in finding the pre- 
 cision of the unknowns from these equations, if the computa- 
 tion is so arranged that a check can at all times be applied 
 and the whole process proceed in a uniform and mechanical 
 manner. 
 
 The aids in the arithmetical work are a table of squares, 
 a table of products, a table of reciprocals, a table of log- 
 arithms, and an arithmometer, or machine for performing 
 multiplications and divisions. The latter is of the greatest 
 use in computations of this kind. With it the drudgery of 
 computation is in great measure got rid of. On the Lake 
 Survey two forms of machine were used, the Grant and the 
 Thomas. A series of trials showed that with either ma- 
 chine multiplications could be performed in from one-half 
 to one-third of the time required with a log. table, and with 
 much less liability to error. About as good speed can be 
 made tor a short time with Crelle's multiplication-tables, but 
 it cannot be kept up. 
 
 Form (a). With Crelle's tables, or with a machine, the 
 products aa, ab, . . . are found directly, and all that is then 
 to be done is to \\rite them in columns and take their sums 
 |<7<?|, [tf/>l, . . . With a Thomas machine, however, each 
 product mav be added to all that precede, so that the final 
 values result at once. 
 
 Let us, for example, take the observation equations 
 
 - i . 2.1- -|- o. 2y -j- 0.9 =n 7-, 
 + 3.0* 2.n-f- 1.1=1',
 
 152 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Arrange as follows, the headings indicating the nature of 
 the numbers underneath : 
 
 a 
 
 b 
 
 / 
 
 J 
 
 1.2 
 
 + O.2 
 
 + O.Q 
 
 O.I 
 
 + 3-0 
 
 2.1 
 
 + I.I 
 
 + 2.0 
 
 + 0.7 
 
 + 1.6 
 
 -4.0 
 
 -i-7 
 
 act 
 
 all 
 
 al 
 
 as 
 
 1.44 
 9.00 
 0.49 
 
 0.24 
 6.30 
 
 + I. 12 
 
 1. 08 
 + 3-30 
 -2.80 
 
 + O. 12 
 
 + 6.00 
 -1.19 
 
 + 10 93 
 
 -5-42 
 
 0.58 
 
 + 4-93 
 
 
 bb 
 
 bl 
 
 bs 
 
 - 5-42 
 
 0.04 
 4.41 
 2.56 
 
 + 0.18 
 
 2.31 
 6.40 
 
 O.O2 
 4. 2O 
 -2.72 
 
 + 7.01 
 
 -8.53 
 
 -6.94 
 
 
 
 11 
 
 Is 
 
 - 0.58 
 
 -8-53 
 
 0.81 
 
 I. 21 
 
 16.00 
 
 0.09 
 
 + 2.2O 
 + 6.80 
 
 + 18.02 
 
 + 8. 9 I 
 
 Hence the normal equations, with the check all ready for 
 solution, are 
 
 <?=-+ 1 0.93.*- 5.427-0.58 +4-93 
 
 o- 5.42^+7.017-8.53 -6.94 
 
 Form (b). If logarithms alone are used, form a table of 
 the log. coefficients of the observation equations as follows : 
 
 log a lt log lt . . . log / t , log s, 
 log log &,,... log / log s, 
 
 log a nt log b nt . . . log /, log s n 
 
 Write log a l on a slip of paper and carry it along the top 
 row, forming the products 
 
 log a,a lt log afa . . ., log ,/ log a,s,
 
 INDIRECT OBSERVATIONS. 
 
 153 
 
 Similarly with log a 3 form the products from the second 
 row, 
 
 log a,a,, log aj) v . . ., log aj v log a^ 
 
 and so on till log a n is reached. 
 
 The numbers corresponding to these logarithms are next 
 found, so that we have 
 
 By addition we find 
 
 \aa], [ab~\, . . . [>/], DW] 
 
 the coefficients of the unknowns in the first normal equation. 
 Proceed in a precisely similar way with log b^ log /> 
 . . . log b H , omitting the term [ab~\ already found; with 
 log c lt log c v . . . log c n , omitting the terms \_ac\ [be] already 
 found ; and so on till the last quantity is reached, 
 
 logo 
 
 log b 
 
 log/ 
 
 log j 
 
 0.47712 
 9.84510 
 
 9.30103 
 
 O.32222;; 
 O.2O4I2 
 
 9-95424 
 0.04139 
 
 0. 6O2O6 M 
 
 9.00000;; 
 0.30103 
 
 log aa 
 0.15836 
 0.95424 
 9.69020 
 
 log ab 
 
 9.38021;; 
 
 o. 79934'* 
 0.04922 
 
 log al 
 0.03342;; 
 
 0.51851 
 
 log as 
 9.07918 
 0.77815 
 
 
 log bb 
 8.60206 
 0.64444 
 0.40824 
 
 log l>l 
 
 0.36361;; 
 0.806 1 8 
 
 log i>s 
 0.62325;; 
 
 
 log// 
 9.90848 
 0.08278 
 1.20412 
 
 log/j 
 8.95424;; 
 0.34242 
 0.83251 
 
 and the numbers corresponding to these logs, are exactly 
 the same as those in form (a). The remainder of the com- 
 putation is the same as there given.
 
 154 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Form (c). If we wish to use a table of squares altogether, 
 then since 
 
 ab --%\(a-\-l>y-a- lf\ 
 
 and therefore 
 
 [fl=itf(* + *)T- [a*] -[**]} (i) 
 
 we form the square sums 
 
 /)], 
 
 [//],[(/+ *)'] 
 
 and perform the necessary subtractions. 
 
 In doing this, first take from the table of squares the 
 squares aa, bb, ...//, ss, and sum them ; next write the co- 
 efficients a of A- on a slip of paper and carry them over the 
 coefficients of y, 3, . . ., forming the sums <?,-)- /;,,,-{- ,, ; 
 a, -\-b tfo-f-^.,, Take out the squares of these numbers 
 and sum them. Proceed similarly with the coefficients of 
 y, z, . . . Finish as indicated in (i). 
 
 Thus in the preceding example, 
 
 aa 
 
 M 
 
 // 
 
 JJ 
 
 1.44 
 
 0.04 
 
 o.Si 
 
 O.OI 
 
 9.00 
 0.49 
 
 10.93 
 
 4.41 
 2.56 
 
 I .21 
 
 16.00 
 
 4.00 
 
 2.89 
 
 7.01 
 
 18.02 
 
 6 . 90 
 
 a + b 
 
 (H-tf 
 
 a + l 
 
 (a + I? 
 
 a + s 
 
 (a + s)" 
 
 I.O 
 
 i .00 
 
 0.3 
 
 0.09 
 
 1-3 
 
 1 .69 
 
 0.9 
 
 0.81 
 
 4.1 
 
 16.81 
 
 5-0 
 
 25.00 
 
 2-3 
 
 5-29 
 
 3-3 
 
 10.89 
 
 I.O 
 
 I.OO 
 
 
 7.10 
 
 
 27.79 
 
 
 27.69 
 
 [aa] + 
 
 [W] = I 7 . 94 
 
 [aa\ + [ 
 
 
 [aa] + [ss] 
 
 = 17.83 
 
 
 10.84 
 
 
 1.16 
 
 
 9.86 
 
 
 - 5-42 
 
 
 0.58 
 
 
 4.93 
 
 
 
 
 = [*/] 
 
 
 = [<"J 
 
 giving the same results as before.
 
 INDIRECT OBSERVATIONS. I 55 
 
 This form, which is very neat analytically, was first given 
 by Bessel in the Astron. NacJir., No. 399. 
 
 A consideration of the simple case of three observation 
 equations, each involving two unknowns, will show that to 
 form the normal equations, using a log. table only, 24 entries 
 in the table are required, while by this method we only 
 need to enter a table of squares 18 times, thus effecting a 
 saving of 6 entries. The Bessel method has also the ad- 
 vantage that, as we deal with squares, all thought with 
 regard to sign is done away with. Besides, if the table of 
 squares is a very extended one, accuracy can be had to a 
 greater number of decimal places than with an ordinary 
 log. table. As compared with the logarithmic form, then, 
 this method is to be preferred, more especially when the 
 coefficients are not very different. 
 
 On the other hand, if Crelle's tables or a computing 
 machine is to be had, the direct process explained in (a) is 
 much to be preferred to either, as experience will show. 
 
 It is worth noticing that whichever method of formation 
 of the normal equations is adopted, labor will be saved by 
 changing the units in which the unknowns are expressed if 
 the coefficients of the different unknowns are very different. 
 Thus, suppose we had the observation equations, 
 
 Check sums. 
 
 looo.r -(-0.0001^ = 4. 1 1 1004.1101 
 999.v -(- O.OOO2J/ = 3.93 1002.9302 
 
 from which to find x and y. 
 By placing 
 
 x' -- i oo.i', y' = o. o \y 
 
 the equations reduce to 
 
 Check sums. 
 
 i o.i'' -f-o.oiy = 4. 1 1 14.12 
 
 9.99.1-' -f 0.02/ = 3.93 13.94 
 
 which are in more manageable shape for solution.
 
 156 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Solution of tJie Normal Equations. 
 
 Before beginning the solution of a series of normal equa- 
 tions we should consider whether the object is to find 
 
 (1) the unknowns only, or 
 
 (2) the unknowns and their weights ; 
 and, in the latter case, 
 
 (a) whether the number of unknowns is large, 
 
 (b) whether many of the coefficients of the unknowns in 
 
 the normal equations are wanting. 
 
 Normal equations may be solved by the ordinary algebraic 
 methods for the elimination of linear equations or by the 
 method of determinants. When, however, they are numer- 
 ous the methods of substitution and of indirect elimination, 
 both introduced by Gauss, are more suitable. Each has its 
 advantages, which will be pointed out as we proceed. The 
 method of substitution is quite mechanical, and is well 
 suited for use with an arithmometer, which is as great a 
 help in solving as it is in forming the normal equations. 
 
 86. The Method of Substitution. For convenience 
 in writing, take three unknowns, x, y, z, the process being 
 the same whatever the number. 
 
 The normal equations are 
 
 ]j +[>=[/] 
 
 From the first equation 
 
 [ah] [tfr] . [at] 
 
 _r i l<o t -s-\-- - (2) 
 
 [aa\ y \aa\ r \aa\ 
 
 Substitute this value in the remaining equations, and, in 
 the convenient notation of Gauss, there result 
 
 \bb.i\y + \bc.\\s = [/;/. i] 
 
 >=|W.i] (3)
 
 INDIRECT OBSERVATIONS. 157 
 
 where 
 
 \bb.i ] = m -[^[^.1 
 
 \aa\ 
 
 (4) 
 
 Again, from the first of equations 3, 
 
 _[^i] [*/.i] 
 [W.i]~ r [^.i] 
 
 which value substituted in the second equation gives 
 
 =t^l (6) 
 
 where 
 
 (7) 
 
 Having thus found ^, we have y at once by substituting in 
 (5), and thence .r by substituting for y and z their values 
 in (2). 
 
 The first equations of the successive groups in the elimi- 
 nation collected are 
 
 \CC.2\Z =\cl.2\ 
 
 These are called the derived normal equations.
 
 158 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Divide each of these equations by the coefficient of its 
 first unknown, and 
 
 i \ a " \ , \ac\ 
 
 x H- - - y -f - .? 
 
 [<7rt] [] [] 
 
 [^<:. ij _ [^/. i] / N 
 
 p.l]" = [^7i] 
 
 ; 87. Controls of the Solution. In solving a set of 
 normal equations a control is essential. It is sometimes 
 recommended to solve the equations arranged in the reverse 
 order, when, if the work is correct, the same results will be 
 found as before. But what is wanted in a control is a 
 means of checking the work at each step, and not at the 
 conclusion, it may be, of several weeks' work, when, if 
 the results do not agree, all that is known is that there 
 is a mistake somewhere without being able to locate 
 it. 
 
 (a) Continuation of the formation control. Experience 
 has shown that it is convenient to carry on through the 
 solution the check used in forming the equations. It merely 
 consists in placing as an extra term to each equation the 
 sums \as], [bs], . . . \_ls~], and operating on them in the same 
 way as on the absolute terms [#/], [#/], . . . The sum of the 
 terms in every line, after each elimination of an unknown, 
 must be each equal to the check sum numerically ; the 
 closeness of the agreement depending on the number of 
 decimal places employed. 
 
 This check may be applied at every step and mistakes 
 be weeded out. 
 
 (b) The diagonal coefficients \aa\, \_bb~], ... of the normal 
 equations, and [aa~\, \bb. i], \_cc.2~], . . . of the derived normal 
 equations, are always positive.
 
 INDIRECT OBSERVATIONS. 159 
 
 For [aa], \bb. i], . . . being the sums of squares, are posi- 
 tive. Also 
 
 Mc-'Hsj:E] = ':'+ 
 
 a positive quantity. 
 
 Similarly for \cc.2\ [dd.$\ t . . . 
 
 The principle may be of use as a check in the solution 
 of a series of normal equations which are apparently cor- 
 rect, but which 'have been improperly formed. The follow- 
 ing normal equations, which came up once in my own ex- 
 perience, will show this : 
 
 7,r+ 77-9.1,2=26 
 
 jx + 287 1 2.3- =64 
 
 - 9-i.r 127+ 12^ = -39 
 
 The derived normal equations are 
 
 jx + 77 9. i s = 26 
 + 217 2.9^=38 
 
 showing by the presence of the negative diagonal term that 
 there was a mistake somewhere. An examination of the 
 data from which the normal equations were derived showed 
 a condition wanting and a condition incorrectly expressed. 
 After the proper corrections had been made the solution 
 was carried through without trouble. 
 
 (c) By equations 8, Art. Si, the residuals found by sub- 
 stituting for x, y, s their values in the observation equations 
 must satisfy the relations 
 
 [ar] = [bi>~\ = . . . = o 
 
 (d) A very complete check is afforded by the different 
 methods of computing [IT], the sum of the squares of the 
 residuals. (See Art. 100.) 
 
 88. Forms of Solution. In applying the method of 
 substitution to any special example it is important that the
 
 :6o 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 arrangement of the computation be convenient and that 
 every step be written down. Experience teaches that sim- 
 plicity and uniformity of operation are great safeguards 
 against mistakes. 
 
 Form (a). Solution without logarithms. 
 
 The following form has been found by experience to be 
 convenient. It is well fitted for use with the arithmometer 
 or any other rapid method of multiplication. The form 
 can be readily modified to suit computer's tastes. 
 
 For illustration let us take, as before, three unknowns, 
 x, y, z. The computation is divided into sections, each sec- 
 tion being formed in a precisely similar way, and in each 
 section one unknown is eliminated. 
 
 Given the normal equations, 
 
 No. 
 
 .r 
 
 y 
 
 2 
 
 
 Check. 
 
 Remarks. 
 
 I. 
 II. 
 III. 
 
 \aa\ 
 
 H 
 [at] 
 
 Ml 
 
 \bb} 
 [be] 
 
 
 ac\ 
 bc\ 
 cc\ 
 
 [/] 
 ten 
 
 [el] 
 
 
 as] 
 faj 
 ] 
 
 
 Solution. 
 
 IV. 
 
 I 
 
 M 
 
 [*] 
 
 [] 
 [M] 
 
 fl 
 
 i.*M 
 
 V. 
 
 II. 
 
 
 [66] 
 
 [ac] 
 
 t*flfeH 
 
 T ' ] 
 
 1 fcl 
 
 IV. X [*] 
 
 ii. 
 
 VI. 
 
 
 [66.1] 
 
 [Ar.i] 
 
 [.i] 
 
 [*r.i] 
 
 II. -V. 
 
 VII. 
 
 
 M E^3 
 
 [ac] EffJ 
 
 MT 
 
 M~ 
 
 IV. X [rtf] 
 
 III. 
 
 VIII. 
 IX. 
 
 
 [6c] 
 
 [! 
 
 let 
 
 [] 
 
 III. 
 
 III. -VII. 
 
 [*.i] 
 
 [..] 
 
 [rf.I] 
 
 [.i] 
 
 
 i 
 
 ^ 
 
 [/'/ . 1 ] 
 L^V^. I J 
 
 L.i] 
 
 X. 
 
 
 ^^ffirl 
 
 [ fc - x lf&Tl 
 
 ^^[T] 
 
 IX. X [fc.i] 
 
 VIII. 
 
 
 U-.i] 
 
 U'li 
 
 I.'x] 
 
 
 XI. 
 
 
 [.a] 
 
 [C/.2] 
 
 [..] 
 
 VIII. -X. 
 
 XI.n-[V<r.2] 
 
 
 i 
 
 [.*] 
 
 iSf!
 
 INDIRECT OBSERVATIONS. l6l 
 
 From Eq. IX. 
 
 v _ -rfr.i] , [*/.i] 
 [^.i]^~[^.i] 
 
 From Eq. IV. 
 
 [**] JiMltW 
 y \aa\ [aa]~ r [aa'] 
 
 To eliminate the first unknown, x. In the first line write 
 
 the quotients LfLJ, LffJ, . . . that is, the coefficients of the 
 \aa\ \aa\ 
 
 first normal equation divided by [aa], the coefficient of x in 
 that equation. 
 
 The first line is now multiplied in order by [a&], \_ac\, 
 forming the second and fifth lines. 
 
 In the third and sixth lines write equations II. and III. 
 
 The fourth line is the sum of the second and third, and 
 the seventh the sum of the fifth and sixth. 
 
 This concludes the elimination of ,r, and the results in the 
 fourth and seventh lines involve y and z only. 
 
 Take now these results and proceed in a precisely simi- 
 lar way to eliminate y. 
 
 The value of the last unknown, ,7, next results. 
 
 Now proceed to find y and .r. Thus substitute for z its 
 value in the eighth line, and we have y ; and for y and z their 
 values in the first line, and we have x. 
 
 In carrying this solution into practice there are three 
 points that deserve special notice: 
 
 (i) In order to render the work mechanical, and so 
 lighten the labor, the number of different operations should 
 be made as small as possible. Instead, therefore, of divid- 
 ing by [aa], \bb.\~\, \cc.2], it is better to multiply by the 
 reciprocals of these quantities, and, in order to avoid sub- 
 tractions, to first change the signs of the reciprocals. We 
 shall then have to perform only two simple operations 
 multiplication and addition. By transferring the terms 
 [/], [/>/], [V/] to the left-hand side of the equations before 
 beginning the solution, the values of the unknowns will 
 come out with their proper signs.
 
 1 62 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 (2) Equations VI. and VIII. are the normal equations with 
 x eliminated. An inspection of them shows that the co- 
 efficients of the unknowns follow the same law as the co- 
 efficients of the unknowns in the original normal equa- 
 tions with respect to symmetry of vertical and horizontal 
 columns. Hence in the elimination it is unnecessary to 
 compute these common terms more than once. Thus [bc.i] 
 from Eq. VI. may be written down as the first term of Eq. 
 VIII. This principle is of great use in shortening the 
 work when the number of unknowns is large. 
 
 (3) In a numerical example it is evident that since [aa], 
 [bb.i~\, \cc.2~] do not in general divide exactly into the other 
 coefficients of their respective equations, and that only ap- 
 proximate values of the unknowns can at best be obtained, 
 it will give a closer result to divide by the larger coeffi- 
 cients and multiply by the smaller than vice versa. Atten- 
 tion to this by a proper arrangement of the coefficients before 
 beginning the solution results in a considerable saving of 
 labor, as the successive coefficients in the course of the 
 elimination need not be carried to as many places of deci- 
 mals to insure the same accuracy that a different arrange- 
 ment would require. 
 
 Ex. To make the preceding perfectly plain we shall solve in full the 
 normal equations formed in Art. 85. 
 
 (i) Write the absolute term on the right of the sign of equality, and make 
 the check sum equal to the sum of the other terms in each horizontal row. 
 
 
 X 
 
 y 
 
 / 
 
 Check. 
 
 Remarks. 
 
 I. 
 II. 
 
 + 10.93 
 - 5.42 
 
 - 5.42 
 + 7.01 
 
 + 0.58 
 + 853 
 
 + 6.09 
 
 + 10.12 
 
 
 III. 
 
 + I. 
 
 - 0.496 
 
 + 0.053 
 
 + 0.557 
 
 I.-H 10.93 
 
 IV. 
 
 II 
 
 V. 
 
 
 + 1.688 
 + 7.01 
 
 - 0.288 
 + 8.53 
 
 - 3-019 
 
 + IO.12O 
 
 III. x - ,4, 
 II. - IV. 
 
 + 4-322 
 
 + 8.818 
 
 + 13.139 
 
 VI. 
 
 
 + I. 
 
 + 2.040 = y 
 
 + 3-040 
 
 V. H- 4 322 
 
 VII. 
 
 III. 
 
 VIII. 
 
 + I. 
 
 - 0.496 
 - 0.496 
 
 - I. 012 
 + 0.053 
 
 - 1.508 
 + 0.557 
 
 VI. x - c.496 
 
 III. - VII. 
 
 + I. 
 
 o. 
 
 + 1.065 = x 
 
 + 2.O65 
 
 Hence 
 
 x = 1. 06 
 
 y = 2.04
 
 INDIRECT OBSERVATIONS. 
 
 (2) Write the constant term on the left of the sign of equality, and form the 
 check so as to make the sum of the terms in each horizontal line equal to zero. 
 
 
 Reciprocals. 
 
 X 
 
 , 
 
 I 
 
 Check. 
 
 Remarks. 
 
 I. 
 II. 
 
 0.0515 
 
 + 10 03 
 
 - 5-42 
 
 - 5.42 
 + 7.01 
 
 - o.s8 
 
 - 8.53 
 
 - 4-93 
 
 + 6.94 
 
 
 III. 
 
 
 - I. 
 
 + 0.496 
 
 * 0.053 
 
 + 0.451 
 
 I. ' - 0.091; 
 
 IV. 
 
 II. 
 
 V. 
 
 0.2314 
 
 
 - 2.688 
 + 7.010 
 
 - 0.288 
 - 8.^30 
 
 - 2.445 
 
 + 6.940 
 
 III.- -5 42 
 II. + IV. 
 
 + 4.322 
 
 - 8.818 
 
 + 4-495 
 
 VI. 
 
 
 
 - i. 
 
 + 2.040 =y 
 
 - 1.040 
 
 V. - 02314 
 
 VII. 
 
 III. 
 
 VIII. 
 
 
 - I. 
 
 - 0.456 
 + 0.496 
 
 + I. 012 
 + 0.053 
 
 - o.m6 
 
 -> o 451 
 
 VI. ' 0.456 
 
 III. 
 
 III. t VII. 
 
 - I. 
 
 
 + 1.065 = * 
 
 - 0.065 
 
 In order to find the values of the unkno%vns to two places of decimals ihe 
 computation should be carried through to three places, and ihe third pLce 
 dropped in the final result. 
 
 Form (b). The logarithmic solution. 
 
 As an example of the logarithmic method let us take the 
 general form of the preceding example, when R and 6" are 
 substituted for the absolute terms 0.58 and 8.53 respec- 
 tively. 
 
 In the numerical work it is better to convert all the 
 divisions into multiplications. Therefore write down the 
 complementary logs, of the divisors with the signs changed. 
 Each multiplier may now be written as needed on a slip of 
 paper and carried over each logarithm to be operated on. 
 Thus for the first operation the slip would have on it 
 8.961 38/7, where the ;/ indicates that the number is negative. 
 
 Paper ruled into small squares, so as to bring the figures 
 in the same vertical columns and facilitate additions and 
 subtractions, renders the work more mechanical and is con- 
 sequently an assistance to the computer. 
 
 In general solutions, when the number of unknowns is 
 large, it will be found much better to carry a double check, 
 one for the coefficients of .r, y, . . . and the other for the 
 coefficients of A', S, . . . Though unnecessary in our ex- 
 ample, it is inserted for illustration.
 
 164 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 It will be noticed that the coefficients of J?, S in the 
 values of x, y follow the same law of symmetry as the nor- 
 mal equations. A little consideration will show that this is 
 always the case. 
 
 Hence, attending to this, we may shorten the computa- 
 tion by leaving out the common terms. We have, there- 
 fore, one term less to compute for each unknown, pro- 
 ceeding from the last to the first. The case is precisely 
 analogous to that of Art. 83. 
 
 
 X 
 
 y 
 
 Check. 
 
 R 
 
 S 
 
 Check. 
 
 Remarks. 
 
 I. 
 II. 
 
 1093 
 
 - 5.42 
 
 - 5 42 
 + 7.01 
 
 - 5-5' 
 - 1-59 
 
 - l. 
 
 - i. 
 
 + i. 
 + i. 
 
 
 III. 
 
 IV. 
 
 i. 03862 
 (8.96i38) 
 
 0.73400 
 9.69538 
 
 o.74H5 
 9.70253 
 
 0. 
 
 8.96138 
 + 0.091 
 
 
 o. 
 8.96is8 
 - 0.091 
 
 log I. 
 III. - log 10. 93 
 
 Nos. 
 
 V. 
 VI. 
 
 II. 
 
 VII. 
 
 
 o.42938 
 - 2.6t3 
 + 7.010 
 
 o.436s3 
 - 2.732 
 - 1-590 
 
 9.69538 
 - 0.496 
 
 - i. 
 
 9.69538 
 + 0.496 
 + i. 
 
 IV. - log 5.42 
 Nos. 
 II. 
 
 VI. + II. 
 
 + 4-322 
 
 - 4-322 
 
 - 0.496 
 
 - i. 
 
 + 1496 
 
 vn;. 
 
 IX. 
 X. 
 
 
 063568 
 (9-36432) 
 - i. 
 
 o.6}568 
 o.ooooo 
 + i. 
 
 0.69 548 
 9.05980 
 + 0.115 
 
 0.0. 
 
 9-36432 
 + o 231 
 
 0.17493 
 9- 53925 
 - 0.346 
 
 log VII. 
 VIII. - log 4.322 
 Nos. 
 
 XI. 
 XII. 
 
 
 
 
 8.75518 
 + 0.057 
 + 0.091 
 
 9.05970 
 
 + 0.115 
 
 9.23463 
 - 0.172 
 - 0.091 
 
 IX. + log 5 ' 42 
 Nos. JO -93 
 
 XIII. 
 
 - 1. 
 
 
 + i. 
 
 + 0.148 
 
 + 0.115 
 
 - 0.263 
 
 
 .'. x = o.i4R + 0.1155 
 y = 0.115/1'-)- 0.2315 
 
 Substituting for R and S their values, we have, as before, 
 
 x = i. 06 
 y- 2.04 
 
 Ex. i. In the elimination of n normal equations by the method of substitu- 
 tion, show that the total number of independent coefficients in the original 
 
 5) 
 
 . 
 and derived normal equations is - 
 
 [The sum is \ \ i . 4 + 2 . 5 + . . . n(n + 3)|] 
 
 Ex. 2. If the elimination of the unknowns in the normal equations is car- 
 ried out by the method of substitution, the product 
 
 [aa] [M.i\ [fc.2\ . . . 
 hns the same value whatever order has been followed.
 
 INDIRECT OBSERVATIONS. 165 
 
 [For it is the determinant 
 
 [at], 
 
 89. The Method of Indirect Elimination. This 
 method is often of service when the number of unknowns is 
 large and many of the terms in the normal equations are 
 wanting. The principle involved in the solution is very 
 simple. If x',y', 3' are approximate values of x, y, z, and 
 x \* y\i s i t ne corrections to these values, so that 
 
 x = x' -f- x, 
 
 y=y' +7, 
 _ ,_/ i ^ 
 -" ' i *i 
 
 then, substituting these values in the normal equations, we 
 have 
 
 C*^, = [*/], 
 
 where the new absolute terms [#/] [/] [//], will be, on the 
 whole, smaller than the original terms [a/], [/], [r/]. A 
 second approximation will tend to decrease the absolute 
 terms still farther. The approximations are continued till 
 the absolute terms either vanish or are sufficiently small. 
 Then 
 
 x x'-^x"-^ . . .; yy'-\-y"-\- ...;... 
 
 In finding the approximate values it is best to consider 
 one unknown at a time, and preferably the equations should 
 be operated upon in the order of magnitude of the absolute 
 terms. Thus suppose [a/] the largest absolute term ; then, 
 since the diagonal term is in general the important term in 
 a normal equation, we may neglect all but it in the equation 
 
 [aa] 
 Substitute for x this value in the three equations, when a 
 
 and call the first approximation to the value of 
 
 [aa]
 
 l66 THE ADJUSTMENT OF OBSERVATIONS. 
 
 value of y or of z may be similarly found. Or the form 
 of the equations may be such that first approximations to 
 the values of both x and y, or of x, y, and s, may be ad- 
 vantageously found from the original equations before 
 making any substitutions. These and similar points must 
 be decided according to the circumstances of the case in 
 hand. 
 
 If the diagonal terms in the normal equations, instead 
 of being much larger than the other terms, are but little 
 larger than several of them, considerable difficulty will 
 be found in making the approximations. An approxima- 
 tion made for one unknown may counterbalance those 
 made for several others, and the whole process will be 
 found tedious and troublesome. Various expedients have 
 been suggested for getting over this difficulty ; but in all 
 cases where the normal equations are not very loosely 
 connected (that is, where many terms are wanting) and 
 the diagonal coefficients large, my experience has been 
 that it is much better to use the method of substitution, 
 or, in simple cases, the ordinary algebraic methods of 
 elimination. 
 
 While employed on the adjustment of the primary 
 triangulation of Lake Superior we came across Von Free- 
 den's glowing account* of the advantages of this method 
 of solution, and determined to give it a trial. The 
 number of equations to be solved was 32. Where the 
 connection of the unknowns was loose it worked well 
 enough, but where they fell in groups it was very 
 troublesome and slow. On the whole, the work was 
 more fatiguing and took longer time. We never tried 
 the method again, though, as it works so nicely in 
 simple cases, we were at one time very much in its 
 favor. 
 
 Gauss, the author of this form of solution, describes it as 
 in general tedious (langwierig), and very justly. For the 
 contrary view see Coast Survey Report, 1855, App.^f; Von 
 
 * Die Praxis dcr Mcthodc dcr k2cinsten Quadrate. Braunschweig, 1863.
 
 INDIRECT OBSERVATIONS. 
 
 I6 7 
 
 Freeden, loc. cit. ; Vogler, Grundziige der Ausgleicliungsrech- 
 nutig, p. 129. 
 
 Ex. Required to solve the equations formed in Art. 85. 
 
 Solution. 
 
 
 Check. 
 
 _r 
 
 y 
 
 10.93 
 
 - 5-42 
 
 - 5-51 
 
 
 
 - 5.42 
 
 + 7-oi 
 
 - 1-59 
 
 
 
 0.58 
 
 -3.53 
 
 + 9-II 
 
 
 
 - 5.420 
 
 + 5-465 
 
 + 7-oi 
 - 2.71 
 
 - 1-59 
 
 - 2.755 . 
 
 0-5 
 
 I .0 
 
 i .0 
 
 - 0.535 
 + 5.465 
 
 -4.230 
 
 - 2. "I 
 
 + 4-765 
 
 - 2.755 
 
 - 5.420 
 
 4- 7-01 
 
 - 1-59 
 
 0.5 
 0.05 
 
 0.03 
 
 -0.490 
 + 0.5465 
 
 0.1626 
 
 4- 0.070 
 0.271 
 + O.2IO3 
 
 4- 0.420 
 
 - 0-2755 
 - o 0477 
 
 o. 1061 
 
 + 0.0093 
 
 + 0.0968 
 
 ->r 0.1093 
 
 0.0542 
 
 -0.0551 
 
 O.OI 
 
 
 0.0542 
 
 + 0.0701 
 
 0.0159 
 
 0.005 
 
 O.OI 
 
 0.0510 
 + 0.0546 
 
 4- 0.0252 
 0.0271 
 
 + 0.0258 
 0.0275 
 
 + 0.0036 
 
 0.0019 
 
 0.0017 
 
 1 .065 
 
 2.04 
 
 The first two lines contain the coefficients of x, y, together with the check 
 sums ; the third line the absolute terms and their check sum. 
 
 The first approximations are taken y=\ and x o. 5. The second and 
 first lines are multiplied by these numbers and the products written in the 
 fourth and fifth lines. The third, fourth, and fifth lines are summed, and 
 y = i, j; = o.5 are taken as the next approximations. The sum of the x 
 column of approximations gives the value of .r, and of the y column the 
 value of^. 
 
 90. Combination of the Direct and Indirect 
 Methods of Solution. A numerical example illustrating 
 an ingenious combination of the direct and indirect methods 
 of elimination is given in the Coast Survey Report tor 1878,
 
 1 68 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Appendix 8. For performing the multiplications necessary 
 Crelle's tables are used altogether. 
 
 In order to make the process employed readily followed 
 I will give in general terms the solution of the three normal 
 equations 
 
 \_ad\x -f \ab~\y + \ac~\s [>/] 
 
 \_ab~\x + \bb~\y + \bc\z = 
 
 according to this form. 
 
 The coefficients and absolute term of the first equation 
 are written in line i, Table A. The reciprocal of the diago- 
 nal coefficient [aa] is taken from a table of reciprocals and 
 entered in the front column with the minus sign prefixed. 
 The remaining terms of line I are multiplied by this 
 reciprocal, and the products written in line 2. This gives 
 x as an explicit function off and z. 
 
 The coefficients and absolute term of Eq. 2 (omitting the 
 coefficient of x) are written in line i, Table B. The terms 
 in line 2, Table A, beginning with that under y, are multiplied 
 by \ab\ the coefficient of y, and the products set down in 
 line 2, Table B. The sum of lines i, 2, Table B, is now writ- 
 ten in line 3, Table A. 
 
 Line 4, Table A, is found from line 3 in the same way as 
 line 2 was found from line i. This gives y as an explicit 
 function of z. 
 
 The coefficients and absolute term of Eq. 3 (omitting the 
 coefficients of x and y] are written in line 3, Table B. The 
 terms in lines 2, 4, Table A, beginning with those under z, 
 are multiplied by [ac], [&M], the coefficients of z in lines i, 3 
 respectively, and the products set down in lines 4, 5, Table 
 B. The sum of lines 3, 4, 5, Table B, is written in line 5, 
 Table A. 
 
 Line 6, Table A, gives the value of z. 
 
 The next step is to find y and x. The coefficients of 
 the explicit functions are written in Table C. The abso- 
 lute terms of the explicit functions are written in the first
 
 INDIRECT OBSERVATIONS. 169 
 
 line of Table D. The value of z is multiplied by the 
 coefficients of z in Table C, and the products written in 
 the second line of Table D. The sum of the numbers in 
 column y gives the value of/ written underneath in line 
 3. The value of y is multiplied by the coefficients of y in 
 Table C, and x the products written in the third line of 
 Table D. The sum of the numbers in column x gives the 
 value of x. 
 
 The values of x, y, z are now found to three places 
 of decimals. Denote them by x', y', z'. If these values 
 are not sufficiently close a second approximation must be 
 made. This we proceed to describe. 
 
 First substitute the values obtained in the original nor- 
 mal equations, and carry out to a sufficient number of deci- 
 mal places. The residuals are written in the first line of 
 Table E. The coefficients in line i, Table C, are multiplied 
 by [#/],, and the products written in line 2, Table E. 
 The first reciprocal in Table A is multiplied by the same 
 residual, and the product written in column x, line I, 
 Table F. The sum of the numbers in column 2, Table E, is 
 written underneath, as \bl. ij,. 
 
 The coefficient in line 2, Table C, is multiplied by 
 - \bl. i], and the product written in line 3, Table E. The 
 second reciprocal in Table A is multiplied by the same 
 residual, and the product written in column y, line i, Table F. 
 The sum of the numbers in column 3, Table E, is written 
 underneath, as [V/.2],. 
 
 The third reciprocal of Table A is multiplied by this 
 residual, and the product written in column z, line i, 
 Table F. This gives the correction to the value of z. 
 The first line of Table F corresponds to the first line 
 of Table D, and exactly the same process employed in 
 Table D will complete Table F. The total values now are 
 
 y=y 
 
 . / i 
 
 "* ' i
 
 I/O THE ADJUSTMENT OF OBSERVATIONS. 
 
 B 
 
 Recip. 
 
 [W.i] 
 
 X 
 
 y 
 
 z 
 
 
 + [-] 
 
 + [*] 
 
 + [-] 
 
 -M 
 
 * = 
 
 [ab] 
 [aa] 
 
 [at] 
 
 ^&! 
 
 [W.i] 
 
 + [fc-.l] 
 
 -[W.i] 
 
 
 y- 
 
 " [*i] 
 
 + [] 
 
 z 
 
 [rf.3] 
 1 [<V.2] 
 
 [a * ] 
 
 [be] 
 
 D 
 
 .V 
 
 z 
 
 
 
 
 y 
 
 z 
 
 -0 
 
 [at] 
 [aa] 
 
 
 + [<l 
 
 + [*dil 
 
 + [^1 
 
 
 [fci] 
 
 
 
 
 z' 
 
 
 "[Ail] 
 
 
 " [aa] / 
 
 " [S] 2 ' 
 
 
 
 
 
 - r a ^]J / ' 
 
 y 
 
 
 
 *' 
 
 E 
 
 - 
 
 , 
 
 z 
 
 
 X 
 
 y 
 
 z 
 
 -M, 
 
 -[W], 
 
 -M, 
 
 
 + [/], 
 
 [W.i], 
 
 [rf2]] 
 
 
 [a*] 
 
 [ac] 
 
 
 [aa] 
 
 ' [W.i] 
 
 + [-2] 
 
 
 + L^] [a ' J] 
 
 + T^l [a/]l 
 
 
 ~ r V" 
 
 
 " 
 
 
 [W.i], 
 
 [if l] 
 
 
 [aa] 
 
 [W.i]- 
 
 
 
 
 + [All] [W ' l] ' 
 
 
 _ f?y 
 
 y" 
 
 
 
 
 - [ c ' 2] 
 
 
 [aa] 
 
 
 
 
 
 
 
 jr" 
 

 
 INDIRECT OBSERVATIONS. \J\ 
 
 Addition of New Equations. It often happens that after 
 the adjustment of a long series of observations additional 
 observations are made leading to additional condition equa- 
 tions. To make a solution de noi'o is necessary, but the 
 work may be very materially shortened by the process just 
 given. Suppose, for simplicity, that one new condition has 
 been established. This will give one additional normal 
 equation, which may be written 
 
 \_ad~\x + \bd~\y + \cd~\z + \dd]w = \dl] (i) 
 
 w being the new unknown. 
 
 The extra term to each of the other normal equations 
 may be written down at sight. The complete equations are 
 
 \aei\x + \ab\y + \ac\z + [ad]w = [a/] 
 
 -f- [_dd~\iv = [til] 
 
 Now, values of x, y, z have been already found from the 
 normal equations resulting from the original condition 
 equations, and these values may be taken as first approxi- 
 mations to the values of x^y,z resulting from the above 
 four normal equations. Substitute in (i), and 
 
 \_ad~\x + \bd~\y + \cd~\s' + \dd~\w = [<*/]' (2) 
 
 where .r', y' , z are corrections to the approximate values of 
 .v, }', z. The solution is now finished as follows: 
 
 Form Table C (a) by adding the extra column w to 
 
 Table C. The term - '^j is found by multiplying \_ctd \ by 
 
 the first reciprocal. The coefficients of the new equation, 
 (2), are written in the first line of Table G. Since correc- 
 tions to values already found are required, the method of 
 proceeding must be similar to that employed in Table E. 
 The notation in Tables C (a) and G explains this. 
 
 The reciprocal of the sum of column :;', that is, \dd.^\, in 
 23
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Table G is written last in the column of reciprocals of 
 Table C (a) with the minus sign. The product of this 
 reciprocal and the absolute term \dl'\ of the new equation, 
 
 that is, A -., is an approximate value of w. This value of w 
 
 is multiplied by the terms in the last column ot Table C (a), 
 and the products are written in the first line of Table II. 
 Column s gives the correction to z. Table II is now com- 
 pleted in the same way as Tables D and F. 
 
 C (a) G 
 
 
 y 
 
 z 
 
 . 
 
 I 
 [oa] 
 
 [a*] 
 [aa] 
 
 [ac\ 
 [aa] 
 
 M 
 [aa] 
 
 i 
 ~ [**T] 
 
 
 [fer.i] 
 [W.i] 
 
 [M.i] 
 [W.i] 
 
 i 
 
 " i^.2] 
 
 
 
 [frf.2] 
 
 'tec. 2] 
 
 1 
 
 " JVW.3] 
 
 
 
 
 X 
 
 y 
 
 
 
 7W 
 
 M 
 
 CM] 
 
 [g 
 
 [rfrf] 
 
 
 - KH 
 
 - S arf] 
 
 [arfj 
 
 
 fiZT] 
 
 [ * C ' 1] M i] 
 
 [M '' M 
 
 
 
 [W.i] 
 
 [W.ij 
 
 
 
 M.2] 
 
 _ [f^-.2], frf 2] 
 
 ftu.ti 
 
 [ad] 
 
 H 
 
 [W.i] 
 
 - 
 
 [Cd2] 
 ;. ,7 
 
 [cc.a] 
 
 91. Time required to Solve a Set of Equations. 
 
 The labor involved in solving a series of normal equations, 
 and the consequent time employed, increases enormously 
 with an increase in the number of normal equations. To 
 any one who has never been engaged in such work it will
 
 INDIRECT OBSERVATIONS. 173 
 
 seem out of all reason. Thus Dr. Hug-el,'- of Hessen, Ger- 
 many, states that he has solved 10 normal equations in from 
 10 to 12 hours, using a log. table, but that 29 equations took 
 him 7 weeks. 
 
 The following are examples of rapid work: Gen. 
 Baeyer, in the Kiistcnvcrmessung (Vorwort, p. vii.) mentions 
 that Herr Dase solved 86 normal equations between the 
 first of June and the middle of September; and Mr. Doo- 
 little,f of the U. S. Coast Survey, solved 41 normal equa- 
 tions in 5^ days, or 36 working hours. 
 
 A great deal depends, so far as speed is concerned, on 
 the form of solution and on the mechanical aids used. 
 With a machine or with Crelle's tables much better time 
 can be made than by the logarithmic method, which is by 
 far the most roundabout. Mr. Doolittle used Crelle's ta- 
 bles and the form explained in the preceding article. 
 
 In comparisons such as these it is important to know 
 how many of the terms of the normal equations are want- 
 ing. Herr Dase's and Mr. Doolittle's equations were both 
 derived from triangulation work. The latter had only 430 
 terms in his 41 equations, while the former had 3141 differ- 
 ent terms in his 86 equations. 
 
 A machine for the solution of simultaneous linear equa- 
 tions has been invented by Sir \V. Thomson, of Glasgow, 
 Scotland. The results are obtained, by a series of approxi- 
 mations, to any accuracy required. A description of the 
 machine and of the mathematical principles underlying its 
 construction will be found in Thomson and Tait's Natural 
 PJiilosopJiy, vol. i. Appendix 13. Sir W. Thomson says: 
 " The exceeding ease of each application of the machine 
 promises well for its real usefulness, whether for cases in 
 which a single application suffices or for others in which 
 the requisite accuracy is reached after two or three or more 
 of successive approximations." 
 
 With regard to a machine for performing multiplica- 
 
 * C,i-nfral-Bcricht iiber die curofaisrhen Cr,i,hes:siin^. iS'.j. p. 109. 
 tAY/,>r/ r. .V. Co.tsf Surrey, 1878, Appendix S.
 
 1/4 THE ADJUSTMENT OF OBSERVATIONS. 
 
 tions and divisions, I am certain that no one will ever prefer 
 to use a log. table in solving a set of equations by the 
 method of substitution who has ever used any of the forms 
 of arithmometer. 
 
 92. For Jacobi's method of elimination, and the applica- 
 tion of determinants to the solution of normal equations, see 
 Astron. Nachr., 404, 1960; Month. Not. Roy. Astron. Soc., 
 vols. xxxiv., xl.; Proc. Amer. Assoc. for Adv. of Science, 1881. 
 
 The Precision of the Most Probable (Adjusted} Values. 
 
 93. The problem now before us is to find the m. s. e. of 
 the unknowns x, y, ... as determined from a series of 
 normal equations. If the observation equations are re- 
 duced to the same unit of weight, which we shall take to 
 be unity for convenience, the general form of the normal 
 equations is 
 
 \ad\x -(- \ati\y -\- . . . =[_al'] 
 
 Wx + \bb-\y + . . . ==[*/] (i) 
 
 Let fjt= the m. s. e. of a single observation. 
 /jt x , (jL y , . . . = the m. s. e. of x, y, . . . 
 p x , /,,... = the weights of x, y, . . . 
 
 From Art. 56 we have 
 
 P*ti=P 3 ti = ...=// (2) 
 
 In order, therefore, to determine /Ji x , [Jt y , . . . we must make 
 two computations, one of the weights p x , p y , . . . and the 
 other of//, the m. s. e. of a single observation. 
 
 It is evident from an inspection of the normal equations 
 that ^r, y, . . . are linear functions of / / 2 , . . . Let, then, 
 x aj, -f / 2 -f . . . -f aj n = [/] 
 
 y = M + M + . + /U = [/3/] (3) 
 
 in which 2 , . . . ; &, /9 a , ...;... are functions of 
 a lt b a . . .;,&. . . ; . . . their values being as yet un- 
 determined.
 
 INDIRECT OBSERVATIONS. 175 
 
 Now, n being the m. s. e. of each of the observed quanti- 
 ties J/,,J/,, . . . J/ M , must be also the m.s. e. of //, .../, 
 which differ from M lt M v . . . M n by known amounts (see 
 Art. 81). Hence since / /.,,... / are independent of 
 each other (Art. 19), 
 
 H* /*"[], ti = l?[tf~\, (4) 
 
 and therefore 
 
 We shall first of all determine the weights/,,/,, . . . 
 
 Before proceeding with the general proof let us con- 
 sider the simple case of two unknowns, x,y, which are to 
 be found from the three observation equations 
 
 a,x + b l y = l l 
 a n x -f- b.j = /., 
 <** + b *y I* 
 all of the same weight. 
 
 The normal equations are 
 
 \ad\x -f- [afr] y = [/] 
 ^ab\-c -\-\bti\y = \bl\ 
 
 Their solution by the method of substitution gives 
 
 Hence since /,, / /, are independent of one another, 
 
 = rrl--, (6)
 
 1/6 THE ADJUSTMENT OF OBSERVATIONS. 
 
 that is, the weight of y is the denominator of the value of y 
 found from the solution of the normal equations by-tJie Gaussian 
 method of substitution. 
 
 The general demonstration may be carried out more 
 simply by the application of the principles of undetermined 
 coefficients. Thus substitute [/], [/?/], for x, y, . . . 
 in the normal equations (i), and 
 
 M [/] H- [*][#]+. =[>/] 
 
 9/]+. .=[*/] I 7.1 
 
 or, arranging according to /,, /,... 
 
 I [], + [V^R +. . -,}/, 
 
 + I [WK + [^R + 
 
 + { [*], + WA + - - W + 
 
 Tlie unknown quantities i; .,, . . . may be so determined 
 that the coefficients of /,, / g , . . . shall each equal zero. 
 Hence the several sets of equations 
 
 < 
 ( 
 
 i + [^] + . ^ o 
 
 , -f 0/41 + . . . - a, o 
 
 ... (8) 
 
 are simultaneously satisfied by the same values of
 
 IXfJlRKCT OBSERVATIONS. 1/7 
 
 Multiply the equations of each set by a^ a.,, ... in order, 
 and add ; then necessarily 
 
 \aa\ = i , [a ft \ = o,[ ay | = O, . . . (y) 
 
 In a similar way, multiplying by />,, A,, . . . ; r,, r.., . . ., etc., 
 and adding, there result 
 
 [l,a\ = , [A? |=i, |?/} = o, . . . 
 \ca \ o, [eft \ = o, [cy ] = i , . . . 
 
 Again, multiply the first set by '/.,, .,, . . ., the second by 
 flu fl > an d so on, and add, and we have the sets of 
 equations 
 
 i an aa -- ab 9 -- = ' = T 
 
 V. 
 
 (10) 
 
 from which equations [], [^|, . . . may be found. 
 
 It is plain that the coefficients of [], [a^J, . . . 
 1/^5], ... in these equations are the same as those ot 
 -i', y, ... in the normal equations, and that the absolute 
 terms are i, o, ...; o, i, ...;... instead of [a! ], 
 |/>/], . . . Hence, 
 
 94. To Find the Weights of the Unknowns./;/ 
 tlic first normal equation write \ for a I j, and in the otlier 
 normal equations put o for each of />/],[</, . . . ; the value 
 of x found from these equations will be the reciprocal of t lie 
 weight of x, and t lie values of y, z, . . . will be the values of 
 [^], ['/;'], . . . In the second normal equation ft/rite i for\bl , 
 and in the other equations put o for each of al , \cl \ tin- 
 value of y found from these equations will be the reciprocal of 
 the weight of y, and the values of z, . . . found will be the
 
 1 7 8 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 values of [$K], . . . Similarly for each of the unknowns in 
 succession. For example, the weight equations for three 
 unknowns are 
 
 [] 
 
 W] 
 
 [y] 
 
 
 <- [aa] 
 
 + [a*] 
 
 + [ac] 
 
 i 
 
 >- [a*] 
 
 + [A*] 
 
 f [*C] 
 
 o 
 
 i [ac] 
 
 + M 
 
 + M 
 
 o 
 
 [fl 
 
 [J3/3J 
 
 [W 
 
 
 + [aa] 
 
 + [a*] 
 
 + [ac] 
 
 o 
 
 + [a*] 
 
 + [W] 
 
 + [Ac] 
 
 I 
 
 + [af] 
 
 + [Ac] 
 
 + [cc] 
 
 
 
 [] 
 
 [6y] 
 
 [VYl 
 
 
 + [aa] 
 
 + [a*] 
 
 t [ac] 
 
 o 
 
 + [a*] 
 
 + [W] 
 
 + [Ac] 
 
 
 
 + M 
 
 + M 
 
 + [cc] 
 
 I 
 
 The quantities [;9], [^], . are necessary when the 
 weight of a linear function of the unknowns is required, as 
 will be seen presently. (See Art. 101.) 
 
 It is evident from the form of the weight equations that 
 if the elimination is carried through by the method of sub- 
 stitution, the successive steps to the left of the sign of 
 equality are the same as in Art. 88. Hence if the equations 
 are arranged so that the unknown whose weight is re- 
 quired xr, for example is found first, we should have the 
 forms 
 
 X 
 
 y 
 
 z 
 
 
 [aa] 
 
 + [ai] 
 
 \- [ac] 
 
 [a/] 
 
 
 + [*ft.i] 
 
 f [if. i] 
 
 [W.i] 
 
 
 
 + [CC.2] 
 
 [Cl.2] 
 
 .'. [. 2 ]s = [r/.2] 
 
 [rl 
 
 [Oy] 
 
 [yy] 
 
 
 [aa] 
 
 + [a*] 
 
 + [ac] 
 
 
 
 
 H- [AA.i] 
 
 + [Ac.:] 
 
 
 
 
 
 + Ice. 2^ 
 
 I 
 
 ' ["2][yyj - i 
 
 Hence the coefficient of the unknown first found in the 
 ordinary solution of the normal equations is the weight of 
 that unknown. By a separate elimination for each unknown 
 the weight of that unknown could be found as above, but 
 the process would be intolerably tedious. 
 
 95. Special Cases of Two and Three Unknowns. We may, 
 however, from the preceding derive formulas for the weights
 
 INDIRECT OBSERVATIONS. 179 
 
 in a series of normal equations containing not more than 
 three unknowns, which are easy oF application. 
 
 Thus with two unknowns, x and y, y being found first, 
 
 [aa] 
 In the reverse order, x being found first, 
 
 or 
 
 where 
 
 With three unknowns, x, y, rj, performing the elimina- 
 tion of the normal equations in the order ,7, y, x, we have 
 
 \cc.i\ 
 
 -\bU\cc\-\bc\bc\ 
 which expressions are easily transformed into 
 
 A = , \\hb\-\ t il>\' 
 
 iy i ., ii .... i HTTp* 
 
 where 
 
 / = [^]1 M>\\cc\ -f 2\<il>}\lx--}\n ( -\ - \aa\bc\- - \l>l>\\<ic\> - \cc\
 
 180 THE ADJUSTMENT OF OBSERVATIONS. 
 
 From these formulas the weights of the unknowns can be 
 found directly without solving the normal equations. If 
 the normal equations have simple coefficients it is much 
 more rapid to find the weights in this way and solve the 
 equations by ordinary algebra rather than by the Gaussian 
 method. But when the number of unknowns exceeds three 
 this becomes too cumbersome. 
 
 Ex. To find the weights of the adjusted angles in Ex. 4, Art. 83. 
 Here 
 
 A = 12 X ii X 15 12 X 16 ii X 49 
 
 = 1249 
 and 
 
 1249 
 
 Px - = 8.4 
 
 149 
 
 1240 
 
 /v = = Q ; 
 131 
 
 If u x , u y , ii z denote the reciprocals of p x , p y , p z respectively, then 
 
 Uy = O.IO49 
 
 Uz O.IO57 
 
 96. Modification of General Method. To carry out the 
 method of Art. 94 directly as stated would be excessively 
 troublesome, and various modifications have been proposed. 
 The following scheme, which consists in running the weight 
 equations together, will be found very convenient. 
 
 Take, for simplicity in writing, three unknowns, x, y, r, 
 and to the ordinary form of the normal equations as ar- 
 ranged for solution add the columns 
 
 i o o 
 o i o 
 o o i 
 
 the check being carried throughout.
 
 INDIRECT OBSERVATIONS. 
 
 181 
 
 Perform the elimination exactly as stated in Art 88, and 
 find the values of the unknowns in the usual way. We 
 have then 
 
 X 
 
 y 
 
 X 
 
 
 * 
 
 s 
 
 r 
 
 Check. 
 
 [aa] 
 
 M 
 
 [ac] 
 
 [ail 
 \bb\ 
 [be] 
 
 n 
 
 [an 
 
 bl 
 
 f * 
 | 
 } 
 
 o 
 
 i 
 o 
 
 
 
 
 I 
 
 [as] + i 
 
 fej : ; 
 
 i 
 
 [a*] 
 
 [ac] 
 
 [a/] 
 
 , 
 
 
 
 
 
 [aa] 
 
 [aa] 
 
 [aa] 
 
 [ ./i.' 1 
 
 
 
 
 
 [44. i] 
 
 
 [6/ . i ] 
 
 JP 
 
 i 
 
 
 
 
 k.i] 
 
 tccij 
 
 K'-i] 
 
 f [ffl 
 
 
 
 1 
 
 
 
 
 
 
 j "[<<] 
 
 1 
 
 
 
 
 i 
 
 [*] 
 
 [.l] 
 
 I *i 
 
 ! 
 
 
 
 
 
 [M.ll 
 
 [6*.i] 
 
 LM.i] 
 
 [W.i] 
 
 
 
 
 
 [<r<r.2] 
 
 [^.2] 
 
 r "2 
 
 f S* 
 
 f I 
 
 
 
 
 I 
 
 [C/.2] 
 
 ) 
 
 
 ; i 
 
 
 where 
 
 A', 
 
 O FT^ ^T "T~ *J<) 
 
 \bb.i\ 
 
 Now, taking the first column in the table under the 
 heading R, and attending to Art. 94, we have 
 
 A', \bc.\ 
 
 ^Ti-j-^. 
 
 A 5 
 
 + [J.2] 
 
 U - -- J U a 
 
 [rtrt] [W.l] \cc.2
 
 1 82 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Similarly for the column under S, 
 
 - I 3 
 
 ~\bb.i}^ \_cc.2] 
 
 and for the column under T, 
 
 "=irr] = ^ 
 
 Also it is evident that 
 
 
 , . , 
 
 " 1 " r 
 
 The forms of the expressions for [], [/5|5], [K7], show 
 that these quantities may be conveniently computed from 
 the preceding tabular elimination scheme. Thus the sum 
 of the products of each pair of numbers bracketed under 
 the heads R, S, T will give u x , u y , u z respectively. 
 
 The convenience of this form is seen in such a case as 
 the following, which is of common occurrence. In a set of, 
 say, 40 normal equations the weights of 10 of the unknowns 
 may be required. These 10 would be placed last in the 
 solution of the equations, and the extra columns R, S, . . . 
 added after 30 of the unknowns had been eliminated, 
 thus giving the weights required, with a trifling increase 
 of work.
 
 INDIRECT OBSERVATIONS. 
 
 Ex. i. Given the normal equations 
 
 1 2A- - 73 = K 
 
 + i y - 4z = S 
 
 - ix ~ 4y + 153 = 7' 
 
 to find the weights of y and s. 
 
 X 
 
 y 
 
 z 
 
 S 
 
 T 
 
 + 12 
 
 - 7 
 
 
 
 + II 
 - 4 
 
 - 1 
 
 - 4 
 + 15 
 
 
 
 i 
 
 o 
 
 - 0.5333 
 
 
 
 
 + ii 
 
 -4 
 + 10 9169 
 
 + ' 1 
 
 + i 
 
 
 i 
 
 - 0.3636 
 
 + 0.0909 j 
 
 
 
 
 + 9-4^25 
 
 + 0.3636 ) 
 
 *' i 
 
 
 
 i 
 
 + 0.0384 \ 
 
 + 0.1057 1 
 
 Hence 
 
 HZ i X 0.1057 
 
 Ky = I X O.O9O9 -f- 
 
 agrceing with the values in Art. 95. 
 
 = 0.1057 
 
 X O.03S4 = O.IO49 
 
 Ex. 2. Show that 
 
 .i] = [a/]A' 1 + [M] 
 /. 2 J = [,//] A'., + [_t,l\S. i+ \cl\ 
 
 Lx. 3. Show that the multipliers A'i, A'..., . . . satisfy the conditions 
 [rta]A"i 4- [fl^]= o 
 \na\ ft? + \til>} S-t + [<jc] - o 
 
 [,ia]A' 3 + [a6]S 3 + [<:{} 7' 3 + \a<i\ o 
 [a/.] A', + [M]S, + [/-el 7', f [ M\ = o 
 [ac] A' 3 + [be ] S 3 + [ f f] Ti + [at} = o
 
 184 THE ADJUSTMENT OF OBSERVATIONS. 
 
 97. Second Method of Finding the Weights of 
 the Unknowns. If we multiply the first of the normal 
 equations i, Art. 93, by [a], the second by [/3], the third 
 by [a/], and so on ; add the products, and attend to equa- 
 tions 10, Art. 93, we obtain 
 
 Similarly 
 
 y = \ap\\_al-] + WW\ + [MM + . 
 
 Hence since [], [/9/9], . . . are the reciprocals of the 
 weights of x, y, . . ., this method of finding the weights 
 may be stated as follows : 
 
 In any given series of observations, Jiaving formed the normal 
 equations, replace the numerical absolute terms by the general 
 symbols [#/],[#/], . . . and find by any method of elimination 
 the vahies of x, y, . . ., in terms of [a 1], [/], . . . / then the 
 ^veight of x is the reciprocal of tlie coefficient of\al~\ in the value 
 of x, the weight of y is the reciprocal of the coefficient of \_bl~\ in 
 the value of y, and so on. 
 
 The coefficients of the remaining symbols for the abso- 
 lute terms in the expressions for x, y, . . . give the values 
 of [a/9], [7], . . . ; [/fy], ...;... and the numerical values 
 of the unknowns x, y, . . . may be found by substituting 
 for [#/], \bl~\, . . . their numerical values. 
 
 In this method of computing the values of the unknowns 
 and their weights a machine can be used with great ad- 
 vantage. 
 
 The formulas of Art. 96 are easily derived from the pre- 
 ceding principles. For solving the normal equations 
 
 \_ad\x -f \_ab~\y + \_ac\s [>/]
 
 INDIRECT OBSERVATIONS. 185 
 
 by the method of substitution we have for the first un- 
 known 
 
 ~ 
 
 i J I-,/-. _ 
 
 [<r.2]/ H J [-.2 
 
 Comparing this with the general expression for x in Eq. i, 
 
 f"xy,"l I I A A T 1 I 
 
 [W.IJ ' \JC.2] 
 
 Similarly for j and . 
 
 jr. To find the weights of the unknowns in Ex. 4, Art. 83. 
 Solving the normal equations in general terms, 
 
 -r = 0.1193 [a/] +0.0224 [I'll +0.0616 [</] 
 y ~ 0.0224 [<*t] + o. 1049 [/;/] + 0.0384 [c /] 
 = 0.0616 [/]+ 0.0384 [/'/]+ 0.1057 [<-/] 
 Hence 
 
 Z'*= [] = O.II93 
 
 Uy - [ft ft} =0.1049 
 
 M * = [?'r] = 0.105 7 
 
 as found in Art. 95. 
 
 98. In deducing the formulas for the precision of the 
 adjusted values in a series of normal equations we have, for 
 convenience in writing, taken the observation equations to 
 be reduced to weight unity, and the normal equations, con- 
 sequently, to be of the form 
 
 \aa\v +\ab}y+. . .=[/]
 
 1 86 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The formulas with the weight symbols introduced, cor- 
 responding to those found in the preceding articles, are 
 easily derived from them by writing a \/~p b \/ T . . . / |/J~ 
 for a, /?,... /, and a \fy /? \/^ . . . for , fl f . . . respec- 
 
 tively. (See Art. 58.) 
 
 Thus, for example, from the normal equations 
 
 [paa\x + \_paV\y + . . . = [//] 
 
 |>0> + \_pbb -\y-\- . . . = [>*/] (2) 
 
 we should have 
 
 x [/] = [?/][/>/] -|- [>/?][>/] + . . 
 
 (3) 
 
 and by equating coefficients of /,, /,,... in the first 
 expression, 
 
 [uaa]a l -(- [/9]^, -)- . = ,, 
 
 (4) 
 
 99. To Find the in. s. e. // of a Single Observation. 
 
 If the errors A were known that is, if the n observation 
 equations were 
 
 where X H , y tt , . . . are the true values of the unknowns we 
 should have at once 
 
 But we have only the residuals i> with the observation 
 equations 
 
 tf. r r + b,y -f . . . - / v, (2)
 
 INDIRECT OBSERVATIONS. 187 
 
 where x, y, . . . are the most probable values of the un- 
 knowns. We must, therefore, express [JJJ in terms of the 
 residuals i> in order to find it. 
 
 From the two sets of equations, by subtracting in pairs, 
 
 Now, taking the m. s. e. /i x , /Jt y , ... to be the errors of 
 A-, y, . . . , that is, to be equal to x a x, y y, . . . , we 
 have from Eq. 4, Art. 93, 
 
 x, - x 
 and therefore 
 
 J, = V , + //(, 4/[J + /;. Vtffi + . . . ) 
 
 Squaring, adding, and attending to equations 10, Art. 93, we 
 have approximately, ;/,- being the number of unknowns, 
 
 [^] + ; W r (4) 
 
 Putting [JJJ = ;//r, there results 
 
 //-J^i- (5) 
 
 n ;/,- 
 the expression required. 
 
 Reasoning as in Peters' formula, Art. 47, we easily de- 
 duce from (4) 
 
 /'= i. 2533- ( 6 ) 
 
 Vn(nnf) 
 
 which is known as Liiroth's formula (Astron. Nadir., 1740). 
 
 When ,-= i, equations 5 and 6 reduce to Bessel's and 
 Peters' formulas respectively (Arts. 43, 47). 
 
 100. J\Iethods of Computing [n>]. (a) The ordinary method 
 is to substitute the values of the unknowns found from the 
 solution of the normal equations in the observation equa-
 
 188 THE ADJUSTMENT OF OBSERVATIONS. 
 
 tions, and thence find v lt v . . . The sum of the squares of 
 these residuals will give [vv~\. 
 
 The residuals having- to be found, for the purpose of 
 testing the quality of the work this method of computing 
 [?'?>] is on the whole as short as any. 
 
 As checks on the values of [_vv\ found in this way the 
 following are of value : 
 
 (b) If we multiply each observation equation by its v 
 and take the sum of the products, then, remembering that 
 [<77'] = o, [bv~\ = o, . . ., we find 
 
 M= -M 
 
 (c) If we multiply each of the observation equations by 
 its / and take the sum of the products, 
 
 - - -[//] = [*>/] 
 
 (d) We have for two unknowns, x and y, 
 
 = \ad\x" + 2\at].iy + [/;%* - 2[al}x - 
 
 , firi] [rt/]Y . /..,, [* 
 + LJ/ - L-j) + ([M] - 
 
 and generally for m unknowns
 
 INDIRECT OBSERVATIONS. 
 
 189 
 
 Now, from (9), Art. 86, the coefficients of [aa~], 
 are each equal to zero. Hence 
 
 ' [.l] \_CC.2~] 
 
 This expression was first given by Gauss (De Element is 
 Ellipticis Palladis, Art. 13). Its form suggests that if we add 
 an extra column to the normal equations, as shown in the 
 following scheme, we shall find [z'z'J at the same time as the 
 first unknown. This is analytically very elegant, and, as 
 the check (see Art. 87) can be carried with this column 
 through the solution of the normal equations, it may be 
 used for finding [?'?/], if one is computing alone. Only one 
 extra term [//] has to be computed while forming the 
 normal equations. 
 
 The scheme is as follows: 
 
 X 
 
 y 
 
 z 
 
 
 [aa] 
 
 + Fa6] 
 * [6*J 
 
 : m 
 
 [an 
 
 [6/1 
 
 
 
 , [a] 
 
 \ff] 
 [11} 
 
 i 
 
 [a6] 
 [aa] 
 
 far] 
 [aa] 
 
 ] 
 
 
 . [66.1] 
 
 * f6<Ml 
 
 [W.I] 
 
 
 
 + [.] 
 
 [//:!] = [//] - ("'][./] 
 
 
 
 
 [aa] 
 
 
 
 [*.i] 
 
 [6/.I] 
 
 
 i 
 
 [W.i] 
 
 [66.,] 
 
 
 
 [.,] 
 
 I r/2 l rw ii 
 
 [^ I , [ //,]-^;-; [H ,,
 
 1 90 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. i. To find the m. s. e. of the adjusted values of the unknowns found 
 in Ex. 4, Art. 83. 
 
 The first step is to find [pvv]. This we shall do in the four ways in- 
 dicated. 
 
 (a) 
 
 V 
 
 P 
 
 pvv 
 
 0.05 
 
 5 
 
 O.OI 
 
 0.36 
 
 7 
 
 0.91 
 
 + 0.68 
 
 4 
 
 1.85 
 
 0.03 
 
 7 
 
 O.OI 
 
 0.62 
 
 4 
 
 r. 54 
 
 
 
 4.32 = [pvv] 
 
 (b) 
 
 p\l\v\ o 
 
 pllrfJl = O 
 
 pzl s v 3 = o 
 
 pJtVi = 7 X 0.76 X 0.03 
 6 / 6 z/ 5 = 4Xi.66X 0.62 
 
 o. 16 
 
 4.12 
 
 4.28 = + [pvl] = [pvv] 
 
 (c) 
 
 [pal~\x = 5.32 X 0.05 = 0.27 
 [pl>l~\y - - 6.64 X 0.36 = 2.39 
 \_pcl~\z = + 11.96 X + 0.68 = 8.13 
 
 7 X (o.76) 2 
 4 X (i.66) 2 
 
 4-04 
 = 11.02 
 
 10.79 
 
 15.06 
 4.27 = [pvv\ 
 
 (d) We find [///] = 15.06.
 
 INDIRECT OBSERVATIONS. 
 
 The solution of the normal equations, with the extra column for [///] 
 added, would be, according to the foregoing scheme, 
 
 X 
 
 y 
 
 z 
 
 
 12 
 
 o 
 + II 
 
 - 1 
 
 - 4 
 + 15 
 
 - 5-32 1 
 - 6.64 
 +11.96 
 + 15.06 > 2.36 
 
 I 
 
 o 
 
 - 0.583 
 
 - 0.443 J 
 
 
 + II 
 
 - 4 
 + 10.917 
 
 6.64 
 + 8.857 
 + 12.70 (=15.06 2.36) 
 
 
 I 
 
 - 0.364 
 
 - o . 604 
 
 
 
 + 9.462 
 
 + 6.422 
 + 8.69 (=12.704.01) 
 
 
 i 
 
 + 0.68 
 
 
 \pw]= + 4.30 (= 8.69-4.39) 
 
 Mean value of [/ZT] = 4.29. 
 Hence (Art. 99) 
 
 and (see Ex., Art. 97) 
 
 =, -.4, 
 
 //*= T.47 I'o.ii93, // y = i".47 t '0.1049, /' = i"-47 * o. 1057 
 = o".5i = o".4S =0^.48 
 
 Ex. 2. Show that 
 
 \_Av\ = [vv] 
 
 [Multiply equations i, Art. 99, by v t , v?, . . . and add. Then since 
 
 [av\ = o, [bv] = o, . . . 
 ..[JiQ = -[/*] ]
 
 192 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. 3. Show that 
 
 [Form the normal equations from equations 3, Art. 99, and 
 
 [aa](x - x) + \ab}(y -y) + . . . - [a/J] 
 
 since [_av~\ = [&v] = . . . = o 
 Hence from Art. 100 
 
 Ex. 4. From the equation 
 
 \al}x + \bl]y +...-[//]=- \vv] 
 
 and 
 
 
 deduce 
 
 ^AT. 5. Prove that [<xz>] = [ySz/] = . . . = o 
 
 Ex. 6. From 
 
 deduce the formula
 
 INDIRECT OBSERVATIONS. 193 
 
 1 01. To Find the Precision of any Function of 
 the Adjusted Values A' K, . . .This is the more gen- 
 eral case of the problem just discussed. The method of 
 solution is : 
 
 First, to find //, the m. s. e. of an observation of weight 
 unity, and next p F the weight of the function, whence the 
 m. s. e. ot the function is given by 
 
 where n F is the reciprocal of/ F 
 
 The value of p is computed from (5) or (6), Art. 99. 
 Next, to find n F . Let the function be 
 
 F=f(X, F, . . . ) (i) 
 
 in which A', Y, . . . are functions of the independently 
 observed quantities M^ Jlf. 2 , . . . KI n . 
 By differentiation 
 
 and therefore, since J/,, M v . . . are independent, 
 
 where ;/,, . . . Jt n are the reciprocals of the weights of the 
 observed quantities. 
 
 (a) Instead, however, of using this general formula 
 directly, it is, in general, a great saving of labor to compute 
 from a modified form as follows, by means of which much 
 of the work already done in solving the normal equations 
 may be utilized. 
 
 Reducing the function to the linear form, we have, 
 adopting the notation of Art. 81,
 
 IQ4 THE ADJUSTMENT OF OBSERVATIONS. 
 
 F=f (*' + *, Y'+y, . . . ) 
 
 or, as it may be written, 
 
 dF=Gs+G.y+... (3) 
 
 Now, since ,r, y, . . . are not independent, but are con- 
 nected by the equations 
 
 [*]*+ [*b+. = [/] 
 
 we must get rid of this entanglement by expressing these 
 quantities x, y, ... in terms of /,, / 2 , . . . , which are inde- 
 pendent of each other. From Art. 93 we may write 
 
 where ,, s , . . . ; /3 1 ,/3 2 , ...;... are functions of <7,, b^ . . . ; 
 a^ b v ...;... Hence, substituting in (3) 
 
 and, therefore (Arts/ 19, 56), since the observation equations 
 have been reduced to weight unity, 
 
 [/9,9] + . . . (4) 
 
 where [], [;9] . . . may be found in the manner indicated 
 in Arts. 94, 96, or 97. Hence /is known. 
 (b) Eq. 4 may be written 
 
 F=G& + G&+. . . +G.Q n (5)
 
 INDIRECT OBSERVATIONS. 
 
 w here 
 
 a = [*], + 
 
 that is, where (see Eq. i, Art. 97) 
 
 + 
 
 . (7) 
 
 Hence the weight of a function 
 
 G>+6^+. . . 
 of several independent unknowns x, y, ... is found from 
 
 jr=[<2] 
 
 where (2,, (2 2 , . satisfy the equations 
 
 Therefore we conclude that, z/~ / # scries of observation 
 equations the values of the unknowns x, }>, . . . are required, as 
 well as tJieir weights or the weight of any function of them, 
 these results can be found at one time by making a solution of 
 the normal equations for finding .v, }',... in general terms, and 
 then substituting for [W], [/>/], their numerical values on 
 the one hand and the values of G^ G z , . . . on the other. 
 
 (c) This result may be stated in other forms. Thus from 
 Eq. 4, by substituting for [], [^], . . . their values from 
 Art. 96, or by substituting for (9,, Q.,, . . . their values in (6) 
 as expressed in Art. 96, we have, alter a simple reduction, 
 
 G?_ (G 1 R A G^ (C.fl.+ C.S. + g.)' 
 Uf ~[aa^ [M.i] \cc.2\ 
 
 Comparing this expression with (d), Art. 100, it is evident 
 that the several terms are such as would result from the 
 26
 
 196 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 following elimination (Ex. three unknowns) by finding the 
 products of the quantities bracketed : 
 
 M 
 
 M 
 
 M 
 
 [*] 
 
 
 [fc.1] 
 
 [aa\ 
 G-jA', + 
 
 (d) The expression (8) for ii p may be easily transformed 
 
 into 
 
 where 
 
 Circumstances must decide which of the four forms given 
 should be chosen in any special case. A machine can be 
 used to the best advantage with the second and third forms. 
 The third form is also convenient when the weights alone 
 are required, without the values of the unknowns, and the 
 second when the values of the unknowns can be found by 
 an easier method of solution than the Gaussian method of 
 substitution.
 
 INDIRECT OBSERVATIONS. 
 
 I 9 7 
 
 Ex. i. In Ex. 4, Art. 83, it is required to find the m. s. e. of the angle 
 PSB. 
 
 The function is 
 
 dF= -x + z 
 
 First Solution. From Eq. 4, 
 
 up - [aa] - 2[ay] + [yy] 
 
 = 0.1193 2 X 0.0616 4- 0.1057 (from Ex., Art. 97). 
 = o. 1018 
 
 Second Solution. From equations 7, 
 
 Hence 
 and 
 
 - 4(?3= o 
 
 (?,= 0.0577, (?3 = + 0.0440 
 
 p= - i X 0.0577 + i X 0.0440 
 = 0.1017 
 
 Third Solution. Add the extra column G to the solution of the normal 
 equations, which would give the scheme 
 
 X 
 
 y 
 
 z 
 
 G 
 
 + 12 
 
 
 - 7 
 
 i 
 
 
 
 + ii 
 
 4 
 
 o 
 
 
 
 
 + 15 
 
 + i 
 -0.0833 
 
 
 + I 
 
 
 -0.5833 
 
 
 + ii 
 
 4- 
 + 10.9169 
 
 
 
 + 0.4169 
 
 
 + 0.4169 - 
 
 + 0.0441 
 
 1 
 1 
 
 \ 
 
 
 + i 
 
 0.3636 
 + 9-4625 
 
 
 + i 
 
 Hence 
 
 UP - ix 0.0833 + 0.4169 X 0.0441 
 0.1017
 
 198 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Fourth Solution. 
 
 I2/ =: + I 
 
 II/&! = O 
 
 .'. k = 0.0833, ^1=0, k-i 0.0440 
 
 up (0.0833)- X 12 + (0.0440)^ X 9-4625 
 
 o. 1016 
 Also 
 
 j.ip= i"-47 Vo.102 
 
 Ex. 2. Given the observation equations 
 
 diX + b^y = A 
 a?x + b?y = I?. 
 
 a n x + b n y / 
 
 to find the weight of/Jc + gy. 
 [The normal equations are 
 
 \aa~\x + [ali\y = [>/] 
 
 1 02. To find the average value of the ratio of the weight of 
 the observed value of a quantity to that of its adjusted value in 
 a system of independently observed quantities. 
 
 The adjusted value of the first observed quantity M l is 
 M l -f v,. From Art. Si it follows that the weight of M 1 -f v, 
 is the same as the weight of /, -\- v t . Now, 
 
 Ii+v l = a 1 * + t> t y+ (0 
 
 Hence if P l is the weight of the adjusted value Jlf t -\-v^ that 
 is, is the weight of the function a^x -f- l^y -f- . . ., and 
 pvpv . . . are the weights of / 1? l v . . ., we have 
 
 where (see Eq. 4, Art. 98, (b) Art. 101) 
 
 Q l 
 
 Q, = [>k + [^]^ + . . . - A (3)
 
 INDIRECT OBSERVATIONS. 199 
 
 Therefore by substitution of Q lt (? 2 , . . . in (ty, 
 
 J5 = ,, + &j t + - - - 
 Similarly 
 
 Hence by addition 
 
 ] = M + [4?] + ... to ;/,. terms 
 
 = ;/,-, the number of independent unknowns. 
 
 This result may be very readily derived directly as 
 follows: In (i) put [/] for x, [,?/] for/, . . . , and we have 
 
 Hence, since /,, /,... are independent, 
 
 as before. 
 
 JT. To check the weights of the adjusted values uf the angles found in 
 Ex. 4, Art. 83. 
 
 The weights of the measured values of the angles are 
 
 5, 7- 4, 7, 4 
 The weights of the adjusted values are (Ex., Art. <)5 ; Ex. i, Art. 101) 
 
 8.4, 9.5, 9.5, 9.8, 7.5 
 Also 
 
 ^+- + - + ^ + - 
 b.4 9.5 9.5 9.h 7.5 
 
 = 3 
 
 = the number of independent unknowns, 
 as it should.
 
 200 THE ADJUSTMENT OF OBSERVATIONS. 
 
 103. From the principle just proved we may derive a 
 proof of the formula found in Art. 99 for the m. s. e. of an 
 observation of weight unity in a series of n observations in- 
 volving fif independent unknowns. It is analogous to the 
 proof given for the m. s. e. of a single observation in a series 
 of n observations of one unknown. 
 
 The errors of the observed values M 1} M v . . . M H are 
 J u J 2 , . . . 4,, and the errors of the most probable values 
 V v V v . . . V n may be assumed to be //,-,, /v s , . . . p Vn re- 
 spectively. Hence since 
 
 M= V-v 
 we have 
 
 4 = t^i v i 
 = - v 
 
 and, therefore, 
 
 [/ J J] = [frwr] + [/ mi\ ( I ) 
 
 Again, since P 1 is the weight of 
 
 where jj> is the m. s. e. of an observation of weight unity. 
 Hence 
 
 A/V-4" 2 
 
 * 
 
 t n p v *= / 
 
 * i *n -- r> ' 
 
 *H 
 
 By addition, 
 
 (2)
 
 INDIRECT OBSERVATIONS. 2OI 
 
 But by definition 
 
 11 
 Substituting in (i), we have finally 
 
 I? \-P vv \ 
 
 71 tti 
 
 the result required. 
 
 Miscellaneous Examples and Artifices of Elimination. 
 
 In this section will be discussed several problems of im- 
 portance, and also certain artifices of elimination which may 
 often be employed with advantage in the solution of obser- 
 vation equations. 
 
 104. The labor of solving and finding the values of the 
 unknowns may often be shortened by taking advantage of 
 some principle inherent in the form of the observation 
 equations themselves. For example, if we have a series of 
 observation equations containing two unknowns, and of 
 which the coefficient of the first unknown is unity, instead 
 of solving in the usual way we may reduce the observation 
 equations to equations containing the second unknown only, 
 and thus solve more readily. 
 
 Given 
 
 *+^J' = A weight/, 
 
 * + t>*y = t, " A 
 
 Forming the normal equations in the usual way, we have 
 
 whence eliminating x, 
 
 Now, if the first normal equation is divided by [/>], so that 
 
 , M r _[ 
 
 ~'
 
 2O2 THE ADJUSTMENT OF OBSERVATIONS. 
 
 and from this equation each of the observation equations in 
 succession is subtracted, there result the equations 
 
 The normal equation for finding y from these equations is 
 
 the same as results from the elimination of x in the normal 
 equations. 
 
 This process is specially convenient if the original ob- 
 servation equations are numerous and the coefficients l\, 
 b v . . . and the terms t lt /,... large and not widely dif- 
 ferent. 
 
 Ex. To solve the equations in Ex. 3, Art. 83. 
 The mean of the equations is 
 
 *- 0.97.7 = - 1.97 
 Subtract each equation from this mean equation, and 
 
 + O.62.7 = + O.OI 
 
 + 0.477 = -0.07 
 + 0.267 = + 0.03 
 
 0.017 = 0.02 
 
 0.257= + o -3 
 
 i.o8ji'= o.oo 
 
 The normal equation formed from these equations is 
 
 + i.9U'= 0.27 
 and . ' . y= o x .oi4 
 
 Substitute for^j' its value in the mean equation, and 
 
 105. Again, we may take advantage of the presence in 
 the problem of some arbitrary quantity to which a con- 
 venient value may be assigned. Thus, to find the difference 
 of the coefficients of expansion of two standards A and B
 
 INDIRECT OBSERVATIONS. 203 
 
 from observed differences of length at certain fixed tem- 
 peratures. 
 
 Let x - the excess in length of A over B at an arbitrary 
 
 temperature f a . 
 y = the excess of the coefficient of expansion of A 
 
 over that of B, 
 /,, /,... = the observed differences in length at tempera- 
 
 tures / 1( / 2 , . . . and whose weights are /,, />.,, . . . 
 
 We have then the observation equations 
 
 -r + (', - t )j' - /, = < -, weight A 
 
 x + (/ a - /) y - L = 7- a weight A ( i ) 
 
 and the normal equations 
 
 <Y) y = \.C - 0//1 (2) 
 
 As the value of / is arbitrary, the normal equations will 
 be simplified by taking it equal to the weighted mean of 
 the temperatures ; that is, 
 
 "AJ 
 and they will then become 
 
 O'lJ' - [(' - OXJ (4) 
 
 from which .i' and y are found at once, with their weights at 
 the mean temperature /. 
 
 If the values of /are numerically large it will lessen the 
 labor of finding the value of y if the mean value of .r found 
 from 
 
 is substituted in the observation equations before the nor- 
 mal equation in y is formed. We should then have 
 
 from which to find j.
 
 2O4 THE ADJUSTMENT OF OBSERVATIONS. 
 
 It is evident that the value of y found in this way is the 
 same as before. For 
 
 OC - ')('- *)] = EX* - '.)'] - 
 
 since the coefficient of x is equal to zero from Eq. 3. 
 
 The quantity I x comes in very conveniently in com- 
 puting the residuals v in finding the precision. 
 
 The precision. If n is the number of observations, the 
 number of unknowns being 2, we have for the m. s. e. // of a 
 single observation 
 
 and 
 
 The length at any temperature /' is 
 
 * + (t-*t.) 
 
 and its m. s. e. //.,., is found from 
 
 The weight is greatest when /i x . is least, that is, when 
 
 Ex. The following were among the observations made for the determina- 
 tion of the difference of length between the Lake Survey Standard Bar and 
 Yard ; and also for the difference between their coefficients of expansion. 
 The unit is Tn7 ? fl?f7r inch.
 
 INDIRECT OBSERVATIONS. 
 
 205 
 
 Required the difference of length at 62 Fahr. and at any other tempera- 
 ture /. 
 
 Date. 
 
 Observed temp. (/). 
 
 Bar- Yard (/). 
 
 Weight </>). 
 
 1872, March 5 
 
 24. 7 
 
 791 
 
 5 
 
 " M 
 
 37-i 
 
 SlI 
 
 I 
 
 " 26 
 
 6i. 7 
 
 833 
 
 6 
 
 April 4 
 
 49-3 
 
 820 
 
 6 
 
 " 12 
 
 66. 8 
 
 847 
 
 8 
 
 " 20 
 
 7i. 5 
 
 849 
 
 8 
 
 Let x = the most probable difference between Bar and Yard at 62^ Fahr., 
 y = the most probable difference between coefficients of expansion of Bar 
 
 and Yard. 
 The observation equations will be of the form 
 
 x + (t 62)y I = v 
 The computation is arranged in tabular form as follows : 
 
 p 
 
 ft 
 
 pi 
 
 t-t 
 
 l-x 
 
 5 
 
 123.5 
 
 3955 
 
 - 32.2 
 
 40 
 
 i 
 
 37-1 
 
 Sn 
 
 - 19.8 
 
 20 
 
 6 
 
 370.2 
 
 499 s 
 
 4- 4-8 
 
 + 2 
 
 6 
 
 295.8 
 
 4920 
 
 - 7-6 
 
 II 
 
 8 
 
 534-4 
 
 6776 
 
 + 9-9 
 
 4- 16 
 
 8 
 
 572.0 
 
 6792 
 
 + 14.6 
 
 + 18 
 
 34 
 
 1933.0 
 
 28252 
 
 
 
 
 > = 56 .9 
 
 x 831 
 
 
 
 P(t-ttf ' 
 
 /(/-/)(/-*) 
 
 y(t - /) 
 
 V 
 
 pw 
 
 5184.20 
 
 6440.0 
 
 40.6 
 
 0.6 
 
 i.S 
 
 392.04 
 
 396.0 
 
 - 24.9 
 
 -4-9 
 
 24.0 
 
 138-24 
 
 57.6 
 
 + 6.0 
 
 + 4.0 
 
 96 . o 
 
 34<>-5 ( '> 
 
 501.6 
 
 - 9-6 
 
 + 1.4 
 
 II .8 
 
 784.08 
 
 1267.2 
 
 + 12.5 
 
 -3-? 
 
 98 . o 
 
 1705.28 
 
 2102.4 + 1^-4 
 
 + 0.4 
 
 i .3 
 
 8550.40 
 
 10764.8 
 
 
 
 232.9 
 
 y = 1.26 
 
 

 
 2C>6 THE ADJUSTMENT OF OBSERVATIONS. 
 
 7232.9 
 
 =V 6^2 = 
 7-6 
 
 7.6 
 
 \ 8550.40 
 
 Value of .r = 831 at 56. 9 
 6.4 5- 1 
 
 = 837.4 62. o 
 // ea = (i.3)+ (5. i) 2 X (o.i) 2 = 1.9 
 
 These values may be checked by computing by the ordinary process. 
 The application of the preceding method to the solution of this problem 
 is due to Mr. E. S. Wheeler. 
 
 106. It often happens that of two series of observed 
 quantities one is a function of the other, and that, therefore, 
 the observed quantities may be regarded as co-ordinates of 
 points in the curve which represents the relation connect- 
 ing them. From the plot of the points the general form of 
 the curve can be judged, and then the special form which 
 satisfies the observations may be found as follows. 
 
 Let us investigate the important practical case where the 
 co-ordinates have been equally well measured and the curve 
 is a straight line, both co-ordinates being regarded fallible. 
 
 Let x^y^ : x y^ : . . . x n , y n be the observed values of 
 the co-ordinates, and X^ F, : X v F 2 : . . . X n , Y H their most 
 probable values. 
 
 The conditions to be satisfied are: 
 
 (i) 
 
 where a and b are constants, to be found from the observa- 
 tions. Also 
 
 (X t - *,)' + ( F, -;0* + ... - a min. (2) 
 
 Eliminate the F's by substituting from equations i in (2), 
 and then 
 
 (X, x^f -J- (aX l -f- b }\} y -(- . . . = a minimum.
 
 INDIRECT OBSERVATIONS. 
 
 207 
 
 Differentiate this equation with respect to the independent 
 variables X lt X.,, . . . X H , and 
 
 X, ,r a -f a(aX, -f b 
 
 = o 
 
 from which equations X lt X. 2t . . . X H may be expressed in 
 terms of a, b and the known quantities ,\\, }\ ; . . . ; x n , y n . 
 Again, differentiate with respect to a, b, and we find 
 
 Hence a and /; are known, and therefore the equation of the 
 straight line is known. 
 
 The same mode of reasoning may be applied to the pass- 
 ing of a curve of a specified form through n points, whether 
 their co-ordinates have been equally well measured or not. 
 
 The problem of passing a line which should deviate as 
 little as possible from the positions of several given points 
 was discussed by Lambert* as long ago as 1765, thirty 
 years before Gauss first made use of the method of least 
 squares. Lambert reasoned that if the line is supposed 
 drawn the points should deviate as much on the one side as 
 on the other, and hence that the line would pass through 
 the centre of gravity of the points. 
 
 107. Potheiiot's Problem. In topographical work the 
 three-point problem, usually called 
 Fothenot's problem, is of great im- 
 portance. It may be stated as fol- 
 lows : 
 
 Let P lt P.., P 3 be three points whose 
 positions are known- as, for example, 
 by their co-ordinates with reference to 
 known axes and P a point at which 
 the angles P^PP., = c, : PJ'P., = cr, are o 
 
 Fig. 10 
 
 * Jicylragc ~ui Gclo'auchc dcr Matlieiimtik. Berlin, 1765.
 
 2O8 THE ADJUSTMENT OF OBSERVATIONS. 
 
 observed. It is required from these data to determine the 
 position of P. 
 
 The geometrical solution is simple. Plot the points 
 P lt PV P y Describe on the line 1\P^ a circle containing an 
 angle equal to <p lt and on P^P 3 a circle containing an angle 
 equal to <p z . The intersection of these circles will be the 
 point required. 
 
 A solution much used in practice as, for example, in 
 plotting soundings is to lay off on a piece of tracing-linen 
 two adjacent angles equal to <f> 1} y v and then move this 
 figure over the map on which the points 1\, P, P a are 
 plotted until the directions PP lt PP V PP 3 lie over the points 
 P n P v P 3 respectively. The position of P is then pricked 
 through. 
 
 So far as the plotting of soundings is concerned, the above 
 is in general sufficient; but in the location of important 
 secondary points with reference to known primary points 
 greater precision is necessary. 
 
 The position of P may be computed trigonometrically 
 as follows: Since the positions of the points F lt P v P 3 are 
 known, the lengths of the lines 1\P P^P^ PJ\ ^re known, 
 and, therefore, the angle P^P* can be found. Call its 
 value ft. 
 
 Denote the angles PP^P V PP S P~ 2 by j, 2 respectively. 
 
 The sum of the angles of each of the triangles PP t P 3 , 
 bein 180, we have 
 
 and, therefore, 
 
 , + .< = 3 6o - ?, - f - /? 
 a known quantity. 
 Again, 
 
 PP, />./, PP n _ P^_ 
 
 sin j sin ^, sin ., sin ^ 
 
 sin 2 _ P^ sin ^ 2 
 
 . 
 sin a 
 
 .., ,, 
 
 ' " . - - tan y suppose, 
 PJ sin > 
 
 3
 
 INDIRECT OBSERVATIONS. 209 
 
 and 
 
 sin a 1 -\- sin a _ I -f- tan y 
 sin a, sin ,^ i tan y 
 or 
 
 cot I (a, - ,) = tan (y + 45) cot I (a, -f 2 ) 
 
 and, therefore, , , is known. Combining- with the value 
 of , -(- . 2 , we find , and .,. 
 
 Mence the position of Pis completely determined. 
 
 It often happens, as in the determination of the position 
 of a light-house, for example, that more than three points 
 are sighted at from the point occupied, and that there are 
 more observations than are necessary to locate the point. 
 Its most probable position can consequently be found. We 
 proceed to show the method of finding it. 
 
 Let P lt P. 2 , . . . be the known points in order of azi- 
 muth, as seen from the unknown point P. X^ Y l : X^ Y. 2 ; . . . 
 the co-ordinates of/ 3 ,, /*... referred to some known sys- 
 tem of rectangular axes, preferably the parallel and meridian 
 at the point chosen as origin of co-ordinates. 0,, 0,, . . . 
 the angles which the most probable positions of/ 3 /",, PP. 2 , , . . 
 make with the axis of x. $,', #/, . . . approximate values 
 of these angles as computed from the co-ordinates, and 
 # #,, . . . corrections to these approximate values, so that 
 
 y,, ( . . . the angles observed at P between the directions 
 /Y', PP., . . . and the initial direction PI\. 
 
 X, Y, . . . the co-ordinates of P referred to the same axes 
 as A' lt Y l : A\, Y.,; . . . Suppose X', Y' to be close 
 approximations to the values of X, F, found graphically, 
 and ,r, y corrections to these values, so that 
 
 X=zX' + x t Y= Y'+y, 
 
 Now, 
 
 Y - - F, 
 tan =
 
 2IO THE ADJUSTMENT OF OBSERVATIONS. 
 
 or 
 
 V Y A- v 
 tan(0/ + *0 = *d 
 
 A - A! -\- * 
 
 Take logs, of both members and expand, then 
 log tan 0/ + dfr = log ( F ' - F,) + o j, - log (A- - A',) - o 
 
 where o,, o 2 , o 3 are tabular differences, as explained in Art. 7. 
 But 
 
 F' F 
 
 tan #/ -T77 ^r 1 , and is, therefore, known. 
 
 A A, 
 
 Hence 
 
 j suppose, 
 
 or 
 
 Similarly 
 
 0= *' 
 
 Now, comparing computed and observed values of the 
 angle P,P1\, 
 
 Calling the quantity 
 
 - ft - (^/ - 0/) = A 
 
 the observation equation may be written 
 
 (a, - a^x + (l\ - b^y = /, 
 
 An observation equation for each of the other angles 
 y v <f> 3 , . . . observed at P may be formed in the same way. 
 
 In the expansions above we have used log. differences 
 as being most convenient and as leading to results close 
 enough in problems of this kind. Five-place tables are 
 sufficient; indeed, in most cases four-place are ample.
 
 INDIRECT OBSERVATIONS. 211 
 
 On this problem consult Bessel, Zach's Monatliche Cor- 
 respondent, vol. xxvii. p. 222; Gerling, Pothenotsche Aufgabe, 
 Marburg, 1840 ; Gauss, Astron. Nachr., vol. i. No. 6 ; Schott, 
 C. S. Report, 1864, App. 13; Petzold, Zeitschr. fiir Vermess., 
 vol. xii. p. 227. 
 
 Ex. Given the rectangular co-ordinates of six known points, and the 
 angles observed at the point P whose position is to be determined, as follows 
 (C. S. Report, 1864, App. 13): 
 
 X, = 1845.0 F, = 5534.0 cp t = 61 12' 10" 
 
 X-, 1485.0 
 
 F, = 2486.7 
 
 cp., = 97 48' 27' 
 
 X 3 = o.o 
 
 F 3 = o.o 
 
 <Pa = 195 55' 56' 
 
 JT 4 = 4418.2 
 
 F 4 = 1416.7 
 
 <pi = 205 35' 04 
 
 ^ = 6163.8 
 
 F 5 = 398.1 
 
 ep 5 = 215 53' 44 
 
 X r , = 5810.6 
 
 F 6 = 1255.7 
 
 
 to find the co-ordinates of/ 3 . 
 
 By plotting the points we may take off the approximate values 
 
 X' = 344S Y' = 2440 
 
 Following the form laid down in the preceding Art., and using five-place 
 log. tables (Art. 7), 
 
 log (3094 r) = 3-49 52 14-0;' 
 log ( J 6o3 x)= 3.20493;* + 27-ur 
 
 .'. log tan 6V + <5i$i = 0.28559;? 27. ix 14.01- 
 
 Now, fc>/ = 117' 23' 19" and 5, = 31 for i' 
 
 Hence 
 
 <r)i = 117 23' 19" + 52.5^ + 27.1;' 
 
 Similarly 6> 2 = 178 38' 14" + 2.5_r + 103. Gy 
 
 W 3 = 215 17' 07" 28.0* 4- 39. 6j 
 
 t) 4 = 313 28' 28' IO3. 2.V 97-8^ 
 
 6 5 = 323" 03' 45" 35.6.V- 47. 3j 
 ^f. = 333" 22' 37" - 35.6.V 71.0;' 
 
 Subtracting <->i from each of the values W 2 , (-) 3 , ... in succession, and com- 
 paring with the measured values <p,, <p a , . . ., we have the observation 
 equations 
 
 50. ar 76.5^' = 165 
 
 So.5.v 1 2. 5^=32 1 
 
 I55-7-*- + 124.97=: 553 
 
 SS.i.v+ 74.47=322 
 
 83.IJT4- 98.1 v 334 
 
 28
 
 212 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hence the normal equations 
 
 48746* + 298 13 y = 177986 
 29813* + 36767^= 109157 
 
 The solution gives 
 
 x = + 3.66 weight 24572 
 y= 0.02 weight 18533 
 
 which added to the approximate values above give the final values 
 
 X= 3451. 66 'Y= 2439. 98 
 Substitute for x, y in the observation equations, and we have the residuals 
 
 Vi = + 19.5, z/2 = 27.8, v a + 8.1, v 4 = + i.o, 7/5 = 13. 5 
 
 Hence, since \vv\ = 1402, 
 
 ,. _ . / 1402 
 
 f* 4/ = 22 
 
 Y 5-2 
 
 22 
 
 and J Ll x == ai 4 
 
 ^24572 
 
 22 
 
 H Y = 0.16 
 
 ^18533
 
 CHAPTER V. 
 
 ADJUSTMENT OF CONDITION OBSERVATIONS. 
 
 108. We now take up the third division of the subject as 
 laid down in Art. 39. So far the quantities we have dealt 
 with, whether directly observed or functions of the quan- 
 tities observed, have been independent of one another ; but 
 if they are not independent of one another that is, if they 
 must satisfy exactly certain relations that exist a priori and 
 are entirely separate from any relations demanded by ob- 
 servation they are said to be conditioned by these relations. 
 
 All problems relating to condition observations may be 
 solved by the rules laid down in the preceding chapters. 
 
 Let, with the usual notation, V V . . . V n denote the 
 most probable values of n directly observed quantities 
 M lf M v . . . M n whose weights are A A? P respectively. 
 Let the n c conditions to be satisfied exactly by the most 
 probable values, when expressed by equations reduced to 
 the linear form, be 
 
 a 'V a"l . . . -L' =o 
 
 where a', a", . . . ; b' , />", .../...; L' , L ', . . . are known 
 constants. 
 
 If v^ v v . . . v n denote the most probable corrections to 
 the observed values, so that 
 
 V^-M^v, 
 V, - M, = v, 
 
 .... (2)
 
 214 THE ADJUSTMENT OF OBSERVATIONS. 
 
 we have the reduced condition equations 
 
 a'v^ -f- a"v^ -f- . . . /' =: o 
 b'v, -f 'V 2 + . . . - I" = o 
 
 or [av\ I' = o 
 
 L J V.J/ 
 
 where I' = L' - \_aM~\, I" = L" - \bM\ . . ., and are, there- 
 fore, known quantities. 
 
 The most probable system of corrections is that which 
 makes 
 
 \_pvv~\ = a minimum, GO suppose. 
 
 The problem is to solve this minimum function when the 
 corrections v are subject to the above n c conditions. 
 
 Direct Solution Method of Independent Unknowns. 
 
 109. It is plain that n c of the corrections can, by means 
 of the condition equations, be expressed in terms of the 
 remaining n n c corrections, and that by substituting these 
 n c values in the minimum function we should have a 
 reduced minimum function containing n n c independent 
 unknowns. This function can be found in the usual way 
 by equating to zero its differential coefficients with respect 
 to each unknown in succession. The n n c resulting equa- 
 tions, taken in connection with the n c condition equations, 
 determine the n corrections i\, i\, . . . v n . Thence \_pvv~] 
 is found. 
 
 The solution of the n n c equations can be carried 
 through by any of the methods of Chapter IV. The pre- 
 cision of the adjusted values, or of any function of them, 
 can also be found as in Chapter IV.
 
 CONDITION OBSERVATIONS. 21 5 
 
 Ex. i. Take that already solved in Ex. 4, Art. 83. 
 
 Let Vi, vi, v 3 , v^, z> be the most probable corrections to the measured 
 angles, then the conditions to be satisfied are 
 
 PSB + z/ 4 = FSB + v* FSP vi 
 OSB 4- z>5 = FSB + v 3 FSO vi 
 
 Substituting for PSB, FSB, etc., their measured values, the condition equa- 
 tions may be written 
 
 v\ v 3 + vt = 0.76 
 Vi v 3 + z/ s = 1.66 
 wiih 
 
 $v i~ + 7V* 1 + 4^3 2 + 7Vt- + 4rv = a min. 
 
 Substitute for z/ 4 , ?' B in the minimum equation, and 
 
 SVi a - + 7~v + 4^a 2 + 7(z'i Va + 0.76) 2 + 4(1/2 v 3 + i.66) s = a min. 
 
 Hence, differentiating with respect to TI, r% v 3 as independent variables, we 
 have the normal equations 
 
 I2-c', - 7<' 3 = 5.32 
 
 iir-a 47' 3 = - 6.64 
 
 7^1 4c/ 2 + 157-3 = 11.96 
 
 whence 
 
 and from the condition equations 
 
 T' 4 = 0".03 I/ 8 = 0".62 
 
 These results are the same as those already found on p. 149. 
 
 Ex. 2. The angles A, J5, C of a spherical triangle are equally well 
 measured ; required the adjusted values and their weights. 
 The condition equation to be satisfied is 
 
 A +B + C- 180 + e (i) 
 
 where e is the spherical excess of the triangle. 
 
 Putting yl/, + :,, J/ 3 + -', M 3 + 7- 3 for A, B, C, the condition equation 
 becomes 
 
 ''i + ''i + r- 3 = 1 80 + [,V] 
 
 = / suppose (2) 
 
 Also 
 
 "V + 7' 2 ! + TV = a min. 
 
 Substitute for 7' 3 from (2) in the minimum function, and 
 
 7V + 7' 3 S + (?, + 7', /) 3 = a min.
 
 2l6 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Differentiating with respect to the independent variables vi, z> a , and 
 
 V\ + 2V* = l (3) 
 
 which give 
 
 Also from Eq. 2, 
 
 Hence the correction to each angle is one-third of the difference of the theoretical 
 and measured sums of the three angles. 
 
 To find the weight of the adjusted value of an angle, as A. 
 
 The function is 
 
 dF ' = v\ 
 
 Hence, following the method of Art. 101 (b), 
 
 where G\ = i, and Qi, @ 2 are found from 
 
 2(?i + (? 2 = I 
 
 @i + 2<2-, = o 
 that is, weight of A = f if weight of measured value is unity. 
 
 Check. Weight of direct measure of A = i 
 
 Wt. of indirect meas. (=180+ B C) of A = % 
 
 Weight of mean = i 
 as already found. 
 
 Ex. 3. To find the weight of a side, a, in a triangle all of whose angles 
 have been equally well measured, the base, b, being free from error. 
 
 sin A 
 Here F-=a = br- 
 
 sin B 
 
 .'. dF a sin i" (cot A v cot r-j) 
 The weight is found from 
 
 u f = a sin i" cot A Qi a sin i" cot B Q 2 
 
 where Qi, Qi satisfy the equations (Art. 101) 
 
 2 Qi + (?a a s ' n T cot -^ 
 Qi + "2.Q.I. = a sin i" cot B 
 Hence 
 
 Up ^a'' sin 2 i" (cot 2 A + cot 2 B + cot A cot B)
 
 CONDITION OBSERVATIONS. 
 
 Ex. 4. The measured values of the angles of a triangle have the same 
 weight. Show that if the corrections to the angles are expressed in terms of 
 the corrections to the log. sines of the angles, and the corrections to these 
 log. sines found by treating them as observed quantities, the same results 
 will be obtained as in Ex. 2. 
 
 For example take 50, 60, 70" oo' 30". 
 
 Indirect Solution MctJiod of Correlates. 
 
 1 10. If the unknowns in the condition equations are much 
 entangled the direct solution would be very laborious. It 
 is in general, therefore, advisable, instead of eliminating the 
 n c unknowns directly, to do so indirectly by means of unde- 
 termined multipliers, or correlates, as they are called. 
 
 If we multiply the condition equations 3, Art. 108, in 
 order by the correlates k' , k", . . ., we may write 
 
 GO = k' (|>;] - /') -f- "(O] - O + + O'"] = a min. (i) 
 
 and determine k', k" , . . . accordingly. 
 By differentiation, 
 
 d<*> = (a'k 1 + b'k" -f . . . -f 2p,v l )dv l 
 
 + (a"k' + b"k" + . . . + 2A^Kz' 2 + ... (2) 
 
 If we place equal to zero the coefficients of n c of the dif- 
 ferentials dv^ dv v . . . we shall have n c equations from which 
 to find k', k", . . . Substitute these n c values in the ex- 
 pression lor </cy, and there will remain n ;/,. differentials 
 which are independent of one another. In order that the 
 function may satisfy the condition of a minimum, the co- 
 efficients of each of these differentials must be equal to zero. 
 This gives ;/ ;/,. equations, which equations, taken in con- 
 nection with the n c condition equations, give the n unknowns 
 
 The practical solution would, therefore, be : Form ;/ 
 equations by placing equal to zero the differential coefficients 
 of the minimum function with respect to each of the 
 quantities c\, 7'.,, . . . TV From these ;/ equations and the 
 n c condition equations determine the n-\-n c unknowns 
 k' , k", . . ., v lf :>, . . ., and thence the function [/t/t/].
 
 218 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 In carrying this out the form of the differential equation 
 2 shows that it would be advantageous to multiply the 
 minimum equation by |, and so write (i) in the form 
 
 >] = a min. (3) 
 Differentiating, we have the n correlate equations 
 
 a"k'+b"k"+ . . . =/X (4) 
 
 Substituting for z/,, ZA,, . . . in the condition equations their 
 values derived from these equations, and the normal equa- 
 tions result. Thev are 
 
 (5) 
 
 . . from (4), 
 
 Solving, we obtain k ', k" , . . ., and thence z\, ' v . 
 and F,, F a , . . . from (2), Art. 108. 
 
 The normal equations may be written 
 
 \uaa\k' -f \uab~\k" + . . . = I' 
 
 where ?/ z^ 2 , . . . denote the reciprocals of the weights 
 / / 2 , ... The form of these equations shows that the 
 coefficients \_uaai], \_uab~], . . . may be computed as in Art. 
 85, the corresponding scheme being 
 
 k'
 
 CONDITION OBSERVATIONS. 2 19 
 
 If the elimination of the normal equations is performed 
 by the method of substitution (Art 86), we have, by col- 
 lecting the first equations of the successive groups, 
 
 \tiaa\k' -f \iiab~\k" -f- \iiac\k'" -{-...=/' 
 + \iibb.\\k + \iibc.i\k'" + . . . r.\ 
 
 + \ucc.2\k'" -f . . . =/'".2 (7) 
 
 where /', /".i, I'". 2, . . . correspond to [#/], [bl.i], \cl.2\ . . . 
 respectively. 
 
 These equations being precisely similar in form to Eq. 8, 
 Art. 86, the elimination gives (see Art. 96) 
 
 I' - l I l '' 1 R> I l '"' 2 R" I 
 
 ~ \uaaY \ubb.i\ \_ucc. 2\ 
 
 ' , Z^e , (8) 
 
 ] r \_ucc. 2] 
 
 where 
 
 ._ 
 
 ] (9) 
 
 /". I = I'R + /" 
 
 i" f .2 = i'R" +rs n + /'" (10) 
 
 29
 
 22O THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. i. Take that solved in Ex. i, Art. 109. 
 The condition equations are 
 
 Vi vs + v\ = o. 76 
 
 v-t v 3 + 7' 5 = 1.66 (i) 
 
 The correlate equations consequently are 
 
 k' = 5^1 
 
 k" = iv* 
 
 k' k" = 47/3 (2) 
 
 k' = 7^4 
 
 k" = 4v 6 
 
 To form the normal equations we may substitute for v\, Va, . . , from (2) 
 in (i), or proceed by means of the tabular form on p. 218. We find 
 
 0.5929^' + o.25/" =0.76 
 o. 25/6' + 0.6429^" = 1.66 
 
 The solution of these equations gives 
 
 &' = 0.230 k" = 2.492 
 whence, from the correlate equations, 
 
 vi = o".os, v-i = o".36, v a = + o".6S, z/ 4 = o".O3, V& = o".62 
 Check. The results satisfy the condition equations. 
 
 Ex. 2. The angles, A, B, C, of a spherical triangle are measured with their 
 weights, /i, /a,/a ; required their adjusted values. 
 
 The condition equation may be written (see Ex. 2, Art. 109) 
 
 "<J\ + v? + v s / 
 
 with t/z' 2 ] = a min - 
 
 The correlate equations are 
 
 and the normal equation 
 
 Hence the adjusted values are known. 
 When the weights are equal, then 
 
 the same results as in Ex. 2, Art. 109.
 
 CONDITION OBSERVATIONS. 221 
 
 Note. If a condition equation is of the form 
 
 the weights of the measured values being /,, / a , . . . /. then, proceeding as 
 in the above, we have 
 
 This result is very important and will be often referred to. 
 
 Ex. 3. At U. S. Coast Survey station Pine Mt. the following were the 
 angles observed between the surrounding stations in order of azimuth : 
 
 Jocelyne-Deepwater, 65 n' 52". 500 weight 3 
 
 Deepwater-Deakyne, 66 24' 15". 553 " 3 
 
 Deakyne-Burden, 87 02' 24". 703 " 3 
 
 Burden-Jocelyne, 141" 21' 21". 757 " i 
 
 required their most probable values. 
 
 The condition to be satisfied is that the sum of the angles should be 360'. 
 Now, 
 
 sum of measured values = 359 59' 54". 513 
 theoretical sum = 360 oo' oo".ooo 
 
 .'. residual error 5 ".487 
 
 Hence, as in the preceding example, 
 
 i 
 
 correction to each of first three angles = - - 3 -- X 5 ".487 
 
 a + a + s + J 
 
 correction to fourth angle X 5 .487 
 
 7? + 8 + -3 : + l 
 
 - 2". 744 
 
 Ex. 4. Given in a triangle, of which the longest side is 4 miles, the three 
 angles measured with equal care and with the values i", 5", 179 59' 58" ; 
 adjust the triangle. 
 
 [Angles as small as single seconds occur in practice. In the primary 
 triangulation (1877) of the Great Lakes, carried out by the U. S. Engineers, in 
 the neighborhood of the Chicago base two angles were measured with values 
 o".8is and i".i8s respectively.] 
 
 Ex. 5. "In order to find the content of a piece of ground, I measured 
 with a common circumferentor and chain the bearings and lengths of its 
 several sides. But upon casting up the difference of latitude and departure
 
 222 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 I discovered that some error had been contracted in taking the dimensions. 
 Now, it is required to compute the area of this enclosure on the most probable 
 supposition of this error. 
 
 " Let ABCDE be a survey accurately protracted according to the 
 measured lengths and bearings of the sides AB, BC, . . . A the place of be- 
 ginning, E of ending, A G a meridian, AF, FE 
 the errors in latitude and departure. Now, 
 the problem requires us to make such changes 
 in the positions of the points B, C, . . . that 
 " we may remove the errors AF, FE in other 
 words, that E may coincide with A ; and 
 these changes must be made in the most 
 We have, therefore, to fulfil the three following con- 
 
 Fig.ll 
 
 probable manner. 
 ditions : 
 
 "All the changes in departure must remove the error in departure EF. 
 
 " All the changes in latitude must remove the error in latitude AF. 
 
 " The probability of these changes must be a maximum." 
 
 Let Oi, a?, a 3 , . . . ; 3i, 3 2 , $3, . . . ; denote the measured lengths and 
 bearings of the sides AB, BC, . . ., and xi, x t , x 3 , . . .;y\,yi,y*, . . . their 
 most probable corrections. 
 
 Now, since the corrected latitudes must balance, and the corrected de- 
 partures must also balance, we have the conditions 
 
 (ai + Xi) cos (Si + yi) + (a? + jr 2 ) cos (3 2 + y-i) + . . . = o 
 (rtj + JTi) sin (3i + ;'i) + (an + * 2 ) sin (3 2 + y-t) + . . . = o 
 
 or, reducing to the linear form, 
 
 cos3i xi a\ sin3i y\ + cos3 2 
 sin3i Xi + ai cos3j yi + sin3 2 
 
 with 
 
 a a sin3 2 
 + a a cos3 a 
 
 . + [a cos 3] = o 
 
 . + [a sin 3] = o ' 
 
 [A* 2 ] + [gy 2 ] = a minimum, 
 
 the weights of x t , x z , . . . ; y\, yi, . . . being / 1( / 2 , 
 spectively. 
 
 Hence the correlate equations 
 
 , k' + sinSi k" p l x l 
 
 i k' + a t cos3i k" q\y\ 
 
 q it 
 
 (2) 
 
 and the normal equations 
 
 sin 3 cos 3 
 
 _ f aii si n ^ cos 3"] ) 
 
 sin 3 cos 31 P* 2 sin 3 cos 3 
 
 ( Tsin 3- cos 31 T 
 
 1L"~7""J~L 
 
 = [a cos 3] 
 
 = [a sin 3]
 
 CONDITION OBSERVATIONS. 223 
 
 Now if we assume 
 
 as " this seems best to agree with the imperfections of the common instru- 
 ments used in surveying," the normal equations reduce to 
 
 K [a] [a cos 3] 
 
 k"[a] = [a sin 3] 
 
 from which k' , k" are known. 
 
 The corrections .TI, x?, . . . ; y\ _r 2 . . ., are known from (2). 
 The errors in latitude (see Eq. i) now reduce to 
 
 [a cos 31 
 
 cosS, x t a, sin3i r, = a^ r ^ 
 
 [J 
 
 cos 3] 
 
 cos3 2 .r 2 rt.> sin3 2 )'_> = a 2 
 
 w 
 
 and the errors in departure to 
 
 sin3, xi + , cos3i n = j r , ~ 
 
 w 
 
 [a sin 3] 
 stn3 2 jr 2 + <z 2 sm3 2 v 2 = a 2 
 
 Hence Bowditch's rule for balancing a survey: " Say as the sum of all the 
 distances is to each particular distance, so is the whole error in departure 
 to the correction of the corresponding departure, each correction being so 
 applied as to diminish the whole error in departure. Proceed in same way 
 for the correction in latitude." 
 
 This problem was proposed as a prize question by Robert Patterson, of 
 Philadelphia, in vol. i. No. 3 of the Analyst or Mathematical Museum, 
 edited by Dr. Adrain. of Reading, Pa., and published in 1807. In vol. i. 
 No. 4 are two solutions one by Bowditch, to whom the prize was awarded, 
 and the other by Dr. Adrain. Adrain's mode of solution is nearly the same 
 as by the ordinary Gaussian method. He employs undetermined multipliers 
 or correlates, exactly as Gauss subsequently did. To Adrain, therefore, is 
 due not only the first derivation of the exponential law of error, but its first 
 application to geodetic work. See Appendix I.
 
 224 THE ADJUSTMENT OF OBSERVATIONS. 
 
 To Find the Precision of the Adjusted Values, or of any Func- 
 
 tion of them. 
 
 in. The method of proceeding is the same as in Art. 
 101. 
 
 The first step is to find //, the m. s. e. of a single observa- 
 tion, and next the weight, /v, of the function, whence the 
 m. s. e. of the function is given by 
 
 Up being the reciprocal of the weight. 
 
 (a) To find //. 
 
 In Art. 99 it was shown that in a system of observation 
 equations the m. s. e. fj. of an observation of the unit of 
 weight is found from 
 
 vv\ 
 
 where \_pvv\ is the sum of the weighted squares of the 
 residuals z>, n is the number of observation equations, and 
 tif the number of independent unknowns. 
 
 Hence in a system of condition equations, n being the 
 number of observed quantities and n c the number of con- 
 ditions, the number of independent unknowns is n ;/,., 
 and 
 
 n(rin c ) 
 
 /\pvv\ 
 
 Y (i) 
 
 Liiroth's formula (Art. 99) may be used as a check on the 
 value of//. 
 
 Checks of \_pvv\. When the number of residuals is large, 
 in order to guard against mistakes \_pvv\ should be com- 
 puted in at least two different ways. The following check 
 methods will be found useful:
 
 CONDITION OBSERVATIONS. 225 
 
 () The correlate equations 4, Art. no, may be written 
 
 I X A 7-, =: l/^ a' k' + i/w, V k" + . . . 
 l 7 / a 7' 2 =: Vu, a"k' -f Vn. 2 b"k" + . . . 
 
 Square and add, and 
 
 [/>7'7'] = \_uaa~\k' V -\- 2\uab\k k" -f 2\uac\k'k'" -f- . . . 
 + \ubb~\k" k" -f 2J>&r]"/r' + . . . 
 
 + \iicc]k'"k'" + ... (2) 
 
 + . . . 
 = [/'/] from (6), Art. 1 10. 
 
 - /'y&' -f /"yfe" 
 
 R 
 
 1 \ubb.i\ 
 
 I 1 f , /, /, T H 1 r , , n~\ I ' ' 
 
 _(/7_ (/".I)' (/-.2)' 
 
 ^ 
 
 [ttbb.l'] \_UCC.2] 
 
 by addition attending to Eq. 10, Art. no. 
 
 This expression is very readily computed from the solu- 
 tion of the correlate normal equations, as shown in Ex. 2 
 following. Compare the computation of [77'] from the 
 scheme in Art. 100. 
 
 The sum [/TT] can in general be computed more rapidly 
 by these methods than by the direct process of summing 
 the weighted squares of the residuals. 
 
 Ex. i. The three angles of a triangle are measured with the weights 
 /i, /a, /a / required the mean-square error of a single observation. 
 Using the values of r^, ?, 7-3 found in Ex. 2, Art. no, we have 
 
 r. ...... _"' /2 , "* r ' . " /a 
 
 - M * + [] + M . 
 
 -11 
 
 "M
 
 226 
 Hence 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 / 
 
 // = since n c = i 
 Check (i). [pvv\ = [/t/] 
 
 as before. 
 
 , 
 ~M 
 
 Check (2). \_pvv\ = pr-y directly from Eq. 3, since [uaa] i. 
 [u] 
 
 Ex. 2. To find the m. s. e. of a single observation in Ex. i, Art. no. 
 The first step is to find the value of \_pvv\. Three methods are given 
 
 p 
 
 1! 
 
 pvv 
 
 5 
 
 0.05 
 
 .0125 
 
 7 
 
 0.36 
 
 .9072 
 
 4 
 
 + 0.68 
 
 1.8496 
 
 7 
 
 0.03 
 
 .0063 
 
 4 
 
 0.62 
 
 1.5376 
 
 
 
 4.3132 = [/ZT>] 
 
 (2) 
 
 k 
 
 < 
 
 / 
 
 O.23O 
 - 2.493 
 
 0.76 
 
 - 1.66 
 
 0.1748 
 4-1384 
 
 4.3132 = [/?'-'] 
 
 (3) From the solution of the correlate normal equations: 
 
 -6' 
 
 k" 
 
 
 + 0.5929 
 + 0.2500 
 
 + 0.2500 
 + 0.6429 
 
 0.76 
 - 1.66 
 
 = r } 
 
 r 
 
 + i. 
 
 + 0.4217 
 
 - 1.2818 
 
 = J 
 
 [u.aai 
 
 
 + 0.5375 
 + i. 
 
 - 1-3395 
 
 = /".i 1 
 
 
 [ul>/>.i\
 
 CONDITION OBSERVATIONS. 
 ' [/"''''] =o.j() X 1.2819 + 1.3395 X 2.492 
 
 = 4.3136 
 
 Hence, the number of conditions being two, 
 
 (b) To find ii F . 
 
 Let the function whose weight is to be found be 
 
 F=f(V lt F 2 , . . . r) (4) 
 
 and let it be conditioned by the n c equations 
 
 /,(r,, r ft . . . r u )= 
 
 Expressing F m terms of the observed values, M^M. r . . . J\f, n 
 which are independent of one another, and reducing to the 
 linear form, we have 
 
 f)F f)F 
 
 ' +v * + - 
 
 Hence as in Art. 101, 
 
 where ?/,, ?/.,, . . . are the reciprocals of the weights of the 
 observed values. 
 
 As it usually requires a long elimination to express F 
 in terms of J/,, M, . . . M M direct!} 7 , it is better to compute 
 
 :, . . . trom the torms 
 
 6M 
 
 HL *_ F i^_. ()F : )F ' i 
 o J/, ~" o I \ oM, "" o F 2 r) J/, ~" 
 
 rtF oF f) F
 
 228 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. 3. To find the m. s. e. of a side, a, in a triangle wliose angles have 
 been measured with the weights />,, f^, f> 3 , the base, />, being free from error. 
 The function equation is 
 
 sin A 
 
 F=a = b - 
 sin B 
 
 and the condition equation 
 
 A + B + C= 180 + 
 Hence from Ex. 2, Art. no, expressing A, B in terms of the observed values, 
 
 B = M, + j-? j 180 + - (Aft + M, + M,) 
 Now, 
 
 = a sin i" / < ( i r n ) cot A + ~. cot B\-v\ 
 \\\ []/ [] i 
 
 I 1 / S\ ) I #1 W-j 
 
 + ^ - j^j COt ^ - ( I - - } COt B -7' 2 + -] - f COt /i +j : 
 I M V M/ i < M L] 
 
 Therefore 
 
 / \ / i \ I ~ 
 
 II F = a' J sin- i I - I i p 1 cot A + j -= cot j9 v ?/, 
 
 = rt 2 sin- i" -, ( , "-: | cot' 2 ^ + ( a r^r ) 
 
 ( V L] / \ H / 
 
 2 cot ^ cot 
 
 and jiif = jii \'up 
 
 where // is the m. s. e. of a single observation. 
 
 If the weights/i, p*, /> 3 are each equal to unit\ r , this reduces to 
 
 tip' 1 = | a" sin 2 1" /< 2 (cot 2 ^ + cot 2 .5 + cot A cot B) 
 and if the triangle is equilateral, 
 
 [<F- = 3 a* sin 2 1" // 2 
 
 Also, if the base, instead of being considered exact, had the m. s. e. ///,, 
 
 2 
 the expressions for /ip- would be increased by in" and //*'- respectively.
 
 CONDITION OBSERVATIONS. 229 
 
 It is, however, usually much more convenient in practice 
 to use the method of correlates. 
 
 Let the function, reduced to the linear form, be written 
 
 dfe/X +/',+ . . . (9) 
 
 This is conditioned by the n c equations, also in the linear 
 form, 
 
 a'i\ -f- a "" i '-. ~\- /' = o 
 
 /A' 1 -{-V 2 + . . . -/"=o (10) 
 
 with 
 
 \ pi>i'\ = a minimum. 
 
 Referring to the principle of Art. 1 10, we see that by 
 using correlates /', k" , . . ., and determining them properly, 
 we can express the function in terms of the quantities 
 ?',, i>. 2 , . . . v n as it independent; that is, 
 
 dF= (/' - a'k' - b'k" - . . .) r, 
 
 + (/" -a'k'-b'k*- . . >,+ . . . (II) 
 
 and, therefore, 
 
 HF= (f a'k' b'k" -- . . .)'"'//, 
 
 + (/" -a!'k'-b"k" - . . .)X+- (12) 
 
 It remains to determine k' ', k' ', . . . Now, when the most 
 probable values of the corrections z/,, v^ . . . i' n are sub- 
 stituted in the value of the function dF, this function must 
 have its most probable value, and, therefore, its maximum 
 weight. We may, therefore, determine the correlates k 
 from the condition that the weight of dF is a maximum ; 
 that is, that U F is a minimum. Differentiate, then, U F with 
 respect to k' , k", ... as independent variables, and we have 
 the equations 
 
 \uaa\k' -f- \iiab\k" -f- . . . = \naf~\ 
 [uaW+[uMW+ =\&rt (13) 
 
 from which k ', k" , . . . are found.
 
 230 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 These equations being precisely of the form of ordinary 
 normal equations, it follows, as in (c) and (d), Art. 100, 
 that 
 
 F = Wf\-\.f\k' - \ubf\k" - (14) 
 
 or 
 
 it r H ff~\ ( T c* \ 
 
 tip **JJ \ - -p -, . 7 -. ... v ! b / 
 
 [uaa] \ubb. i J 
 
 The form of the last expression for u p shows that it ma)' 
 be found by means of the following scheme, in which 
 [#/], \_ubf~], . . . are added as an extra column in the 
 solution of the correlate normal equations (13), in the man- 
 ner shown in Art. 100. For three correlates the scheme 
 would be 
 
 V 
 
 k" 
 
 k'"- 
 
 
 \}iad\ 
 
 ' \jiaU\ 
 
 \jiac\ 
 
 \_naf\ 
 
 
 \iibU\ 
 
 \_ubc] 
 
 w\ 
 
 
 
 [we] 
 
 \_rnf -\ 
 
 
 \ubb,\\ 
 
 \ubc.i] 
 
 \_ucc.i] 
 
 \UCf.l\ 
 
 
 < 
 
 [HCC.2] 
 
 w^ 
 
 
 
 
 = Up
 
 CONDITION OBSERVATIONS. 
 
 231 
 
 Ex. 4. To find the weight of the angle PSB in Ex. i, Art. 109. 
 Here 
 
 dF = vi + 7' 3 
 .-./,= -!, f-2 =o, / 3 - + i 
 
 From the condition equations 
 
 a = + i 
 a"'' = i 
 a"" = + I 
 
 b" = + i 
 //" = - i 
 //"" = + i 
 
 .-. [uaf] = J X - i + i X 
 [*/]=-* 
 [</] = + 0.45 
 
 - 0.45 
 
 The correlate normal equations with the extra column for finding U F : 
 
 k' 
 
 k" 
 
 ' 
 
 
 
 + 0.5929 
 + 0.2500 
 
 + I 
 
 + 0.2500 
 4- 0.6429 
 
 + 0.4217 
 + 0.5375 
 
 o. 7600 
 I. 6600 
 
 1.2818 
 
 - 1-3395 
 
 0.4500 [>?/l 
 + 0.2500 = [itl>f~\ 
 
 -0.7590 
 c.o6o2 [ul>/.i] 
 
 + 0.4500 = [//] 
 + 0.3416 
 
 + 0.1084 =[//!] 
 
 
 + I 
 
 2.492 
 
 0. 1 1 2O 
 
 4- 0.0067 
 
 + O.IOI7 =[tlff.2\ 
 
 Also 
 
 as before. 
 
 = 1.47 \ 0.1017 from Ex. 
 = o".47 
 
 Ex. 5. To find the %veight and m. s. e. of the adjusted value of an angle 
 of a triangle when all three angles are measured, their weights being/,, / 2 ,/ 3 
 respectively. 
 
 The function is 
 
 dF = r, 
 
 and the condition equation
 
 232 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hence from (15) 
 
 M 
 
 Also /<-= /' Vu 
 
 / / I \ 
 
 ~~^\V r -\uT < SeeEx - 1 - 
 
 The weight of an angle before adjustment is to the weight after adjustment, as 
 
 1. M 
 
 Ml ' #l(2 + U 3 ) 
 
 If^i, =/ =/> 3 = i, the weights are as 2 : 3. This result is independent of 
 the magnitude of the angle. It therefore applies to any problem in which 
 the condition to be satisfied is that the sum of two quantities shall be equal to 
 a third, or in which the sum of all three is equal to a constant. For other 
 solutions see Ex. 2, Art. 109. 
 
 Ex. 6. If n angles measured at a station close the horizon, find the weight 
 of the adjusted value of any one of them. 
 
 [The solution is exactly as in the preceding example. 
 The weight of V lt for instance, is found from 
 
 1 1 the weights /i, / a> . . . are all equal to one another, the weight of an angle 
 after adjustment is to its weight before adjustment as 
 
 n : n i ] 
 
 Ex. j. Show that the weight of the sum of the adjusted angles of a triangle 
 is infinite. 
 
 [Sum = 180 + e , a fixed quantity, 
 
 . ". m. s. e. =o, and weight = co 
 or otherwise
 
 CONDITION OBSERVATIONS. 233 
 
 Ex. 8. In the "longitude triangle" Brest, Greenwich, Paris, as de- 
 termined by the U. S. Coast Survey in 1872, the observed values were 
 
 m. s. 
 
 Brest-Greenwich, 17 57.154 weight 10 
 Greenwich-Paris, 9 21.120 " 7 
 Brest-Paris, 27 18.190 " 9 
 
 Show that the most probable values are 
 
 m. s. 
 
 T 7 S7- 1 3 weight 14 
 
 9 21.086 " 12 
 
 27 18.216 " 13 
 
 Ex. 9. To find the weight of a side in a chain of triangles, all of the angles 
 of each triangle having been equally well measured and the base being free 
 from error. 
 
 Let A be the measured value of the base, and let - j- ^ 
 
 Hi, a-i, . . . <r n be the sides of continuation in order as \ / \ / \ ft 
 
 computed from b ; a n being the side whose weight is /A, B\/ \/ 
 required. Fig. 12 
 
 If A:, Bi, A^, B-i, . . . are the measured values 
 
 of the angles used in computing a n from />, the angles A,, A- 2 , . . . being 
 opposite to the sides of continuation, then 
 
 Hi _ sin A i a-i _ sin AI a n _ sin A n 
 
 /> sin Bi c/i sin B? ' ' n n - \ sin fi n 
 
 Hence by multiplying these expressions together, 
 
 _ sin Ai sin Ai sin A n 
 
 sin Bi sin - 2 sin B n (r) 
 
 We may now proceed in two ways. 
 (a) Differentiating directly, 
 
 da* a n sin i" [cot A (A) cot B (B)] 
 where (A), (/?), . . . denote the corrections to A, B, . . . 
 
 [In a chain of triangles it is convenient to use the notation (A), (B) . . . 
 
 for T'i, v-i the parentheses indicating corrections.] 
 
 The condition equations, from the closure of the triangles, are 
 
 / f \ i / /? A i / /"* \ ' ' 
 
 v' i / T 1-^-2 i ) ~r \L* i ) i 
 
 Substituting in Eq, 15, 
 
 tia n = I <?- sin- i" [cot- A + cot- B + cot A cot B\ (3) 
 
 the result required. 
 
 If the triangles are equilateral this reduces to 
 
 Ma>i = * n/>- sin- i" (4)
 
 234 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hence in a chain of equilateral triangles the weights of the sides decrease as 
 we proceed from the base, />, through the successive triangles, inversely as the 
 number of triangles passed over ; that is, are as the fractions 
 
 1, i, i i, 
 (b) Taking logs, of both members of Eq. I and differentiating, 
 
 /log a n -^ log sin Ai (Ai) - log sin J5 t (B t ) + . . . 
 a A i (Mi i 
 
 = [8A(A)-8 B (B)\ (5) 
 
 or expanding the first member, 
 
 da n da H =\8A(A)-8 B (B)~\ (6) 
 
 where S a is the tabular difference for one unit for the number a n , and 8 A, SB 
 are the logarithmic differences corresponding to i" for the angles A, B in a 
 table of log. sines. (See Art. 7.) 
 
 Hence attending to the condition equations 2, we have from (15) 
 for Eq. 5, 
 
 U loga n = It^V + S A $B + S *~] 
 
 and for Eq. 6, 
 
 /r + 8 A SB + SB"] 
 
 as giving the weight of the logarithm of the side and the weight of the side 
 respectively. 
 
 Of the two forms (a) and (b), the logarithmic is in general the most con- 
 venient in practice. 
 
 Ex. 10. From a baseAB(=/>) proceeds a chain of equilateral triangles, 
 B all of the angles being equally 
 
 ~7\t^< BA / well measured, and the sides BC, 
 
 > / \ / \ / \ / C^' being in the same straight 
 
 VA B A/ V line. Find the in. s. e. of the line 
 
 BN, which is n times the base. 
 Take first the simple case of n = 2. 
 
 sin C\ sin Ai sin A? sin C* 
 
 F = BN =b h l> 
 
 sin Bi sin Bi sin -B-i sin B 3 
 
 cot 3(83) + cot C 3 (C 3 )\ 6 sin i" 
 Also, we have the condition equations
 
 CONDITION OBSERVATIONS. 235 
 
 Hence 
 
 1<H = 3 [/J =o 
 [W.i] = 3 0/.i]=o 
 
 O.2] = 3 [</.2]=0 
 
 [//J = (cot- .-/ 1 + 4 cot- j9, 4- cot 2 C, + cot'- .-/.j 4- cot- , + cot- /^ 4- cot- G)/'- sin* i " 
 '.j /''- sin'- i" since cot'-' Go" = .\- 
 
 Substituting in Eq. 15, 
 and therefore 
 
 where /i is the m. s. e. of an observed angle. 
 Generally, 
 
 dF=(n i) cot,-/, (A,) n cot A (#,) + cot C, (C,) 
 4- (w - i) cot A-, (//,.) ( i) cot j9 2 (B.S) 
 
 ( i) cot B* (/>',) 4- cot C s (C 3 ) 
 + .......... 
 
 and 
 
 If the cliain proceeds in the opposite direction until At\"=BA', then 
 since n . (iV J =// A , V '-', and ^V.\ T ' = 2/> approximately, we have 
 
 / A sn i4 
 If NN' is times the base | putting ;/ = - j 
 
 ,. /2/r yt 
 jUtfX> =n AN sin i 4/ - 
 
 + io 
 iSw 
 
 Hence it follows th.it in a chain of ccjuilateral triangles whete one base only 
 is measured, it is better to place the base at the centre of the chain rather than 
 at either end. 
 
 .r. ii. If a chain of equilateral triangles proceeds from the base AB, as 
 the chain in Ex. io, but in the opposite direction, show that the m. s. e. of 
 BN' , which is ;/ times the base, is 
 
 and if the chain proceeds also in the opposite direction until AX-BX, then 
 if XX' be taken ;; times the base, 
 
 // VA ., = // AW sin i"4 /2// '' 
 31
 
 236 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. 12. If a chain of equilateral triangles proceeds from the base AB, 
 
 which is in the same straight line as 
 
 r A B N the derived side BN, show that the 
 
 in. s. e. of AN, which is (n + i) times 
 
 the base b, is 
 
 V V 
 
 Fig.14 
 
 sin i " V4 :i + gn- + 5 
 
 tsin^] sin Ai sin C 3 sinAi sinA? sin^4 3 sinA* sin C 5 "1 
 
 F = b + b h b - + 
 
 sini sin/y 2 sinB 3 sin BI s\nB- 2 sin B 3 sin/? 4 sinj9 5 J 
 
 If also a chain of equilateral triangles proceeds from AB in the opposite 
 direction to JV', then if NN' is times the base, show that the m. s. e. of 
 
 NN' is 
 
 f.i NN 
 
 ' sin I"A/[ 
 
 n 1 + 3 4 
 gn 
 
 Hence show that in computing a line NN' , equal to n times the base AB, 
 through a chain of equilateral triangles, the least loss of precision is with the 
 form of Fig. 12. 
 
 Ex. 13. To find the m. s. e. of the altitude of a triangle, the base, l>, being 
 supposed free from error, and the reciprocal weights of the angles being 
 7/1, 7/2, u a respectively. 
 
 The function is 
 
 , sin^4 sinC 
 F=b - : 
 sin j? 
 
 .'. dF=Fs\n i"|cot^ (A) co\.B (B) + colC(C)\ 
 
 Also the condition 
 
 (A) + (B) + (C) = I 
 Substituting in (15) 
 
 U F = F^ \ ?/i cot 5 A + 7/2 cot' 2 B + 7/ s cot 2 C 
 
 __ (T/I cot A 7/2 cot B + 7/3 cot C) 2 
 
 If Z A = Z C, and u^ = 7/2 = 7/3=- 
 
 / 
 then 
 
 2 F- 
 HF-- sin- i cosec 2 ./? 
 
 and 
 
 Hb sin i" B 
 
 /IF- ., - cosec 2 - 
 
 where// is the m. s. e. of the angle corresponding to the unit of weight.
 
 CONDITION OBSERVATIONS. 
 
 237 
 
 E.\. 14. If two similar isosceles triangles on opposite sides of the base 
 ACarc measured independently, thus forming a rhombus 
 (vertices B,B'), then, taking the weight of each angle unity, 
 
 f.tb sin i" B 
 
 /<= f^ ~ cosec- 
 
 r> 
 
 and if KB' is n times the base /', then, since cot = n, 
 
 2 
 
 _ sin i 
 
 2 V3 
 
 Caution. If we solved for the rhombus directly it 
 would not do to take 
 
 D 
 
 BB' = b cot - 
 
 2 
 
 and then form HBB>. The result would be V 2 times too great. For as the 
 triangles are measured independently, each half of BB' must be considered 
 separately, so that we must use the form 
 
 BB' = - ( cot + cot ) 
 
 2 \ 2 2 / 
 
 with the condition equations 
 
 (A) + (B) + (C) = /, 
 (A 1 ) + (B') + (C) = /, 
 
 corresponding to the angles of the two triangles. 
 
 Ex. 15. If on a b.ise, b, as diagonal two similar isosceles triangles are 
 described, forming a rhombus, and on the other diagonal of this rhombus two 
 triangles similar to the former are described, forming a second rhombus, and 
 so on / times, required th'j m. s. e. of the last diagonal, all of the angles 
 being equally well measured. 
 
 For the i tk diagonal </ 
 
 cot 4- c 
 
 y? 2W 
 ot I 
 
 2 / \ 
 
 cot h cot 
 
 Fig.16 
 
 are the vertical angles in 
 
 where B\, />'..., . 
 order. 
 
 Now, as the triangles are all similar, 
 
 /,', = />., =r />':, . . . = A'.j,,, = B suppose. 
 Hence 
 
 b B Ji . , 
 
 f =. f =...= - cot" 1 " 1 cosec- sin i 
 422
 
 238 THE ADJUSTMENT OF OBSERVATIONS. 
 
 and 
 
 ( {/>" /> 
 
 tid = 2m-\ I cot 2 '"- 2 
 
 j Vi6 2 
 
 inb~- />' . II . ., 
 
 -- cot 2 '"" 2 cosec sin- i 
 
 But 
 
 db cot'" 
 
 If d is times the base, 
 
 For further development ol this subject consult Helmert, Stiuiicn fiber 
 rationelle Vennessnngtn. Leipzig, 1868. 
 
 Solution in Two Groups. 
 
 112. In geodetic work it often happens that the observed 
 quantities are subject to a simple set of conditions which 
 may be readily solved as observation equations by the 
 method of independent unknowns, and are also subject to 
 other conditions which are best solved by the method of 
 correlates. The equations are thus divided into two groups 
 for solution, and the complete solution, therefore, consists 
 of two parts. The observation equations forming the first 
 group are solved by themselves and give approximations to 
 the final values of the unknowns. The corrections to these 
 approximate values due to the second group are next found 
 by solving this second group by the method of correlates.* 
 
 The merit of the method consists in utilizing the work 
 expended in the solution of the first group in determining 
 the additional corrections due to the second group. The 
 
 * The first exposition of this method was given by ]>esscl in the Gradmcssiingin Ostfrcusscn. 
 The method of finding the precision of the adjusted values is due to Andrae, Den Danskc Graii- 
 Htaaling, vol. i. Very complete statements will lie found in the introduction to Die frcussischf 
 T.andestriangiiliitiitii) vol. i., I!erlin, 1874; Ferrero, Exposizionedcl metoda dci minimi quaii- 
 rati, Florence, 1876; Jordan, Hamibuch der I'crmcssungskundc, Stuttgart, 1878.
 
 CONDITION OBSERVATIONS. 239 
 
 solution is rigorous, and, being broken into two purls, is 
 more eusily managed than if all of" the equations had been 
 solved simultaneously. 
 
 Let the first group of equations be ihe observation equa- 
 tions, u in number and containing ;/ unknowns (;/ > //), 
 
 ",-i' + b,y -f . . . - /, = c\ weight /, 
 
 /+. -l* v, " p., ID 
 
 and the second group the condition equations, n c in number, 
 involving the same unknowns (u c < //), 
 
 a x -j- a"y -]-... /' = o 
 
 b'x-{-b"y-\- . . . I" - o (3) 
 
 The most probable values of the unknowns .r, j', . . . are 
 those which are given by the relation 
 
 [/TV] = a minimum. (^3) 
 
 It is required to find them. 
 
 The value of an unknown is found in two parts, the first, 
 (.r), (j'), . . ., arising from the observation equations, and 
 the second, (i), (2), . . ., arising from the condition equa- 
 tions, thus : 
 
 Now, overlooking for the present the condition equations 
 and taking the observation equations only, (.r), (j-i, . . 
 would be found by solving these equations in the usual 
 wav. \Ve have, therefore, reducing all to weight unitv for 
 convenience in writing, the normal equations 
 
 l> |C.r) -f [bb ](;')+ --l/'/i (5)
 
 240 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The solution of these equations gives (see Art. 97) 
 
 Hence (x), (j'), . . . are known. 
 
 To find the condition corrections (i), (2), . . . , eliminate 
 T'J, 7'.,, . . . i' n by substituting" in the minimum equation, 
 which then becomes 
 
 \ad\xx-\-2\ab~\xy-\- . . . 2\_al~\x 
 
 + Ww+. - - -2[/> (7) 
 
 + [//] = a rain. 
 
 This equation is conditioned by equations 2. Thus the 
 solution is reduced to that already carried out in Art. no. 
 
 Calling /, //,... the correlates of equations 2, we 
 have the correlate equations 
 
 | \ni\x -f- \aU\ y + . . . [al~\ = a'I+ 
 ]x + \bb\y +. . . -[l>r\ = a"I+ 
 
 These equations, taken with (4) and (5), give the relations 
 
 aa 
 
 suppose 
 
 2 i (8) 
 
 which being of the same form as (5), their solution gives 
 
 or substituting for | i |, \2\, . . . their values from (8), 
 
 (i) A'/+B'//-|-C'/// + . . . 
 
 CV//+ ... (10)
 
 CONDITION OBSERVATIONS. 241 
 
 where 
 
 A' = \aa~\a' -\- \a,i\a" -)- . . . 
 
 15 ' = \tuL\b' -f [>V |//' + . . (\ i ) 
 
 and arc known quantities. 
 
 We have, therefore, expressed the corrections (i), (2), . . . 
 in terms of the unknown correlates /, //,... It remains 
 now to find these correlates. 
 
 Substituting for x, y, . . . their values from (4) in the 
 condition equations, and 
 
 '(!)+" (2)+. . .=/.' 
 
 //(!) + " (2)+. . . =// (12) 
 
 where 
 
 and are, therefore, known quantities, since (x], (j')> are 
 known. 
 
 Substitute the values of (i), (2), . . ., from (10) in (12), 
 and we have the correlate normal equations 
 
 [rtA] I -\-\_a~\i\ //+... =/.' 
 ^T//4-. ~- 
 
 where 
 
 [rt A] = 
 
 etc. = etc. 
 
 (15) 
 
 The solution of these equations gives the correlates 
 /, //, . . . Hence the corrections (i), (2), . . . are known. 
 Also, since (V), (j), . . . have been found from (6), the total 
 corrections ,r, j, . . . are known.
 
 242 THE ADJUSTMENT OF OBSERVATIONS. 
 
 113. In carrying the preceding solution into practice 
 the following order of procedure will be found convenient: 
 
 (a) The formation and solution of the observation equa- 
 tions (i). 
 
 The partially adjusted resulting values (V), (y\ . . . are 
 now to be used. 
 
 (b) The formation of the condition equations (12). 
 
 *''(!) + ' (2)+. . . =/' 
 
 //(!) + // (2) + . . .=/: 
 
 (c) The formation of the weight equations (9). They 
 are at once written down irom the general solution of the 
 observation equations in (a), and are 
 
 (d) The formation of the correlate equations (8). 
 
 |Tj = *'/+'//+ . . . 
 IT a"I+b" 11+ . . . 
 
 (e) The expression of the corrections in terms of the 
 correlates by substituting from (d) in (c). 
 
 (2) = AV+B*//+. . . 
 
 (f ) The formation of the normal equations by substitut- 
 ing from (e) in (b). They are, 
 
 (g) The determination of the corrections by substituting 
 the values of the correlates in (e).
 
 CONDITION OBSERVATIONS. 243 
 
 1 14. To Find the Precision of the Adjusted 
 Values or of any Function of them. 
 
 (a) First find //, the m. s. e. of an observation of weight 
 unity. 
 
 We have (Art. m) 
 
 number ot conditions 
 
 since ;/ ;/ is the number of conditions in the observation 
 equations, and //,. the number in the condition equations. 
 To find [vv~\. From the first observation equation 
 
 Similarl 
 
 where 
 
 that is, T',", z' 2 , . . . are the residuals arising from taking 
 the observation equations only. 
 
 Attending to Eq. 5, p. 239, it follows evidently that 
 
 [*7'] = O [^'] = O, . . . 
 
 Square the residuals 7',, TV,, . . . and add, then 
 
 = [''"''"] + [wzt'J suppose. 
 
 The total sum [7-7-] may therefore be found in two parts, one 
 from squaring the residuals of the observation equations, 
 and the other from the corrections (i), (2), . . .
 
 244 THE ADJUSTMENT OF OBSERVATIONS. 
 
 We proceed to put \ww\ in a more convenient shape for 
 computation. 
 
 from Eq. 8, p. 240. 
 
 Substitute for (i), | 1 1 , (2), . . . their values from equa- 
 
 tions S and 10, and expand ; then 
 
 which may be transformed, by means of Eq. 14, into the 
 form 
 
 [aw] = /'/+/."//+. . . 
 
 or, as in Art. m, into the form 
 
 (/')* (//.I)' (^.2)' 
 
 [aw] = p^ + ==, + r -- 4 + 
 
 These forms may be readily computed as in Art. 100. 
 
 (b) Next find the weight of the given function of the 
 adjusted values. 
 
 Let the function, reduced to the linear form, be 
 
 dF=gs+g^+. . . (16) 
 
 where ,,, . . . are known quantities. 
 
 Put for x, }>, . . . their values (V)-|-(i), (j') + ( 2 ) 
 and 
 
 Put for (i), (2), . . . their values from (10), and 
 
 ... (17)
 
 CONDITION OBSERVATIONS. 245 
 
 where /, //,... are found from the equations 
 
 Using the multipliers j, 4, ... in order to eliminate 
 /, //, . . ., we have, as in Art. in, 
 
 . . . +/'*, + O&.+ . . . 
 
 (18) 
 
 We may determine ,, 2 , ... so as to cause the co- 
 efficients of/, //,... to vanish ; that is, so as to satisfy the 
 equations 
 
 [diX]^ -f DzB^yf . . . = [A] 
 
 and then we shall have 
 
 JF =&(*)+&( 
 
 Substitute for /', / u ", . . . from (13), and 
 
 G&) + ^(7) + (20) 
 
 where 
 
 G l =g l a'k l b'k^ . . . 
 G ^g^-a"k,-b"k,-. . . (21) 
 
 We have thus expressed the function in terms of (-r), (f), . . . 
 and known quantities. 
 
 Now, since (,r), (y), . . . are not independent, but are 
 connected by the equations 
 
 the problem is reduced to that i^ready solved in Art. 101.
 
 246 THE ADJUSTMENT OF OBSERVATIONS. 
 
 If, therefore, n F is the reciprocal of the required weight, 
 
 *= [GQ] (22) 
 
 where 
 
 Q l =[aa-]G l + [ap]G t +. . . 
 
 . - . (23) 
 
 the quantities [], [/3], being as in the weight equa- 
 tions 9. 
 
 Putting for G lt G 9 , . . . their values from (21) in these 
 equations, and attending to (11), we find 
 
 Si^i-A'^-B'/fc,- . . . 
 
 0, = ft -A^ 1 -B^;-. . . (24) 
 
 where 
 
 ?i = [Ui + []&,+ 
 
 (25) 
 
 Substituting in (22) for C,, G" 9 , . . . <2n (2 a their values 
 from (21) and (24), 
 
 But from (11) and (25) 
 
 Hence, attending to (19), the above expression reduces to 
 
 \GG\ 
 
 or to 
 
 \cc.2\
 
 CONDITION OBSERVATIONS. 247 
 
 To compute [^7]. Multiply each of equations 25, in order, 
 by g g v and add, and 
 
 where [], [/5J, . . . may be taken from the weight equa- 
 tions. 
 
 The remaining terms of the second form of \GQ\ may be 
 found from the solution of the normal equations, as shown 
 in Art. in. 
 
 Solution by Successive Approximation. 
 
 115. This method of solution (due to Gauss) is of the 
 greatest importance in adjustments involving many con- 
 ditions. It may be stated as follows : 
 
 The condition equations may be divided into groups, 
 and the groups solved in any order we please. Each suc- 
 cessive group will give corrections to the values furnished 
 by the preceding groups, and the corrected values will be 
 closer and closer approximations to the most probable 
 values which would be found from the simultaneous solu- 
 tion of all the groups. 
 
 For suppose we have the condition equations 
 
 a'v. + d'v^ . . . = 4 
 
 y Vl +y v , + . . .=i b 
 
 h'v,+h"v t +. . .=/ 
 k'v, + k"v, + . . . 4 
 
 with 
 
 v* =a min.
 
 248 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Let z//, v,', ... be the values of v lt v,, . . . obtained from 
 solving the first group alone ; that is, from 
 
 = a mn. 
 
 If now (?>/), (z//), . . . are the corrections to these values 
 resulting from the remaining equations, then since 
 
 the condition equations are reduced to 
 
 with 
 
 and the values of (z/) found from the simultaneous solution 
 of these equations, added to the values of z/ found from the 
 solution of the first set, would be equal to the value of v 
 found directly. 
 
 Similarly if v" , v" , ... be the values of (v'} obtained by 
 solving the second set alone, and (v"), (?'/) be the cor- 
 rections to these values resulting from the remaining equa- 
 tions, then since 
 
 O 2 ] = O' 2 ] + O-] + [X*") 1 ] 
 
 the condition equations are reduced to 
 ') +
 
 CONDITION OBSERVATIONS. 249 
 
 with 
 
 [/(>")'] = a min. 
 
 The quantities [/>"], \_pv nv \, being positive, the mini- 
 mum equation is reduced with the solution of each set, and 
 thus we gradually approach the most probable set of values. 
 Beginning with the first set a second time, and solving 
 through again, we should reduce the minimum equation 
 still farther, and by continuing the process we shall finally 
 reach the same result as that obtained from the rigorous 
 solution. In practice the first approximation is in general 
 close enough. 
 
 It is plain that the most probable values can be found 
 after any approximation by solving simultaneously the 
 whole of the groups, using the values already found as 
 approximations to these most probable values. 
 
 Examples will be found in the next chapter.
 
 CHAPTER VI. 
 
 APPLICATION TO THE ADJUSTMENT OF A TRIANGULATION. 
 
 116. The adjustment of the measured angles of a tri- 
 angulation net is a special case of the problem discussed 
 in the preceding chapters. We assume the reader to be 
 acquainted with the construction and method of handling 
 of instruments used in measuring horizontal angles, and 
 shall confine ourselves to the methods of adjusting the 
 measured values of the angles. 
 
 117. For clearness we will explain in some detail the 
 preliminary work necessary for the formation of the con- 
 dition equations. In a triangulation there must be one 
 measured base at least, as AB. Starting from this base and 
 
 measuring the angles CAB, ABC, we may 
 compute the sides AC, BCbj the ordinary 
 rules of trigonometry. In plotting the 
 figure the point C can be located in but 
 one way, as only the measurements neces- 
 F ' 9 ' 17 sary for this purpose have been made. 
 
 Similarly, by measuring the angles CBD, DCB we may 
 plot the position of the point D, and this can.be done in but 
 one way. If, however, the observer, while at A, had also 
 read the angle DAB, then the point D could have been 
 plotted in two ways, and we should find in almost all cases 
 that the lines AD, BD, CD would not intersect in the same 
 point. In other words, in computing the length of a side 
 from the base we should find different values, according to 
 the triangles through which we passed. Thus the value of 
 CD computed from AB would not, in general, be the same 
 if found from the triangles ABC, BCD, and from ABC, CAD.
 
 APPLICATION TO TRIANGULATION. 251 
 
 If the blunt angle ABD had also been measured we 
 should have another contradiction, arising from the non- 
 satisfaction of the relation 
 
 DBC+ CBA + ABD = 360 
 
 And not these contradictions only. For \ve have considered 
 so far that in a triangle only two of the angles are meas- 
 ured. If in the first triangle, ABC, the third angle, BCA, 
 were also measured, we know from spherical geometry that 
 the three angles should satisfy the relation 
 
 CA + ABC+BCA = i8o + sph. excess of triangle 
 
 which the measured values will not do in general. A 
 similar discrepancy may be expected in the other triangles. 
 In a triangulation net, then, with a single measured 
 base, in which the sides are to be computed from this base 
 through the intervening triangles, we conclude that the 
 contradictions among the measured angles may be removed 
 and a consistent figure obtained if the angles are adjusted 
 so as to satisfy the two classes of conditions: 
 
 (1) Those arising at each station from the relations of 
 the angles to one another at that station. 
 
 These are known as local conditions. 
 
 (2) Those arising from the geometrical relations neces- 
 sary to form a closed figure. 
 
 (a) That the sum of the angles of each triangle in the 
 figure should be equal to 180 increased by the spherical 
 excess of the triangle. 
 
 (b) That the length of any side, as computed from the 
 base, should be the same whatever route is chosen. 
 
 These are known as general conditions. 
 
 118. The number of conditions to be satisfied will de- 
 pend on the measurements made. Each condition can be 
 stated in the form of an equation in which the most prob- 
 able values of the measured quantities are the unknowns. 
 The number of equations being less than the number of 
 
 33
 
 252 THE ADJUSTMENT OF OBSERVATIONS. 
 
 unknowns, an infinite number of solutions is possible. The 
 problem before us is to select the most probable values 
 from this infinite number of possible values. 
 
 The general statement of the method of solution is this: 
 Adjust the angles so as to satisfy simultaneously the local 
 and general conditions ; that is, of all possible systems of 
 corrections to the observed quantities which satisfy these 
 conditions, to find that system which makes the sum of the 
 squares of the corrections a minimum. 
 
 The form of the reduction depends on the methods em- 
 ployed in making the observations. These 
 methods, in general terms, are as follows: 
 Let O in the figure be the station occupied, 
 * and A, B, C signals sighted at. The angles 
 AOB, BOC are required. By pointing at A 
 and then at B we find the angle AOB. 
 Point now at B and next at C, and we have 
 the angle BOC. These two angles are independent of one 
 another. 
 
 If, however, we had pointed at A, B, Cm succession we 
 should also have found the angles AOB, BOC, but they 
 would not be independent of one another, as the reading to 
 B enters into each. 
 
 In the first method of measurement, which is known as 
 the method of independent angles, either a repeating or a 
 non-repeating theodolite may be used ; in the second, or 
 method of directions, a non-repeating theodolite only. 
 
 The Method of Independent Angles. 
 
 119. As the case of independent angles is the simplest to 
 reduce, we shall begin with it. 
 
 A distinction must be made between angles that are in- 
 dependently observed and angles which are independent in 
 the sense that no condition exists between them. Thus at 
 the station O, above, the angles AOB, BOC, AOC might be
 
 APPLICATION TO TRIANGULATION. 253 
 
 observed independently of one another, but we should not 
 call them independent angles, since the condition 
 
 AOC=AOB -\-BOC 
 
 must be satisfied between them. By independent angles, 
 therefore, in the reduction, we mean those measured angles 
 in terms of which all the measured angles can be expressed 
 by means of the conditions connecting them. In the pre- 
 sent case any two of the three angles AOB, BOC, AOC 
 may be taken as independent, and the third angle would be 
 dependent. 
 
 Angles may be measured independently either with a 
 repeating or with a non-repeating theodolite. In primary 
 work a non-repeating theodolite in which the graduated 
 limb is read by microscopes furnished with micrometers is 
 to be preferred. The method of reading an angle is as fol- 
 lows: The instrument, having been carefully adjusted, is 
 directed to the left-hand signal and the micrometers read. 
 It is then directed to the other signal and the micrometers 
 again read. The difference between these readings is called 
 a positive single result. The whole operation is repeated in 
 reverse order; that is, beginning with the second signal 
 and ending with the first, giving a negative single result. 
 The mean of these two results is called a combined result, 
 and is free from the error arising from uniform twisting: of 
 
 cy o 
 
 the post or tripod on which the instrument is placed, or 
 from " twist of station," as it is called. 
 
 The telescope is next turned 180 in azimuth and then 
 1 80 in altitude, leaving the same pivots in the same wyes, 
 and another combined result obtained. The mean of the 
 two combined results is free from errors of the instrument 
 arising from imperfect adjustments for collimation, from in- 
 equality in the heights of the wyes, and from inequality of 
 the pivots. 
 
 The distinction between these two combined results is 
 noted in the record by " telescope direct " and " telescope 
 reverse."
 
 254 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Besides those mentioned there are two kinds of system- 
 atic error in measuring angles that deserve special atten- 
 tion. They are the errors arising from the regular or 
 " periodic " errors of graduation of the horizontal limb of 
 the instrument, and the error from the inclination of the 
 limb itself to the horizon. The effects of the first may be 
 got rid of by the method of observation, as follows : The 
 reading of the limb on the first signal is changed (usually 
 after each pair of combined results) by some aliquot part 
 of the distance, or half-distance, between consecutive micro- 
 scopes in case of two-microscope and three-microscope in- 
 struments respectively. Thus if n is the number of pairs of 
 
 combined results desired, the changes would be and - 
 
 n n 
 
 respectively with the instruments mentioned. The opera- 
 tion of reversal in case of a three-micro- 
 scope instrument causes each microscope 
 to fall at the middle of the opposite 120 
 space, the limb remaining unchanged. 
 Thus if the full lines in Fig. 18 represent 
 the positions of the microscopes with tele- 
 scope direct, the dotted lines show their 
 positions with telescope reverse. In this 
 lies the greatest advantage of three micro- 
 scopes over two, since with the latter, in reversing, the 
 microscopes simply change places with each other, without 
 reading on new portions of the limb. 
 
 The error arising from want of level of the horizontal 
 limb cannot be eliminated by the method of observation, but 
 with the levels which accompany a good instrument, and 
 with ordinary care, it will usually be less than o".i. In case, 
 however, of a signal having a high altitude above the hori- 
 zon, the error from this source may be greater, and then 
 special care should be taken in levelling. For an expres- 
 sion for its influence in any case see Chauvenet's Astronomy, 
 Vol. II. Art. 211. 
 
 The observations should be made on at least two days
 
 APPLICATION TO TRIANGULATION. 
 
 255 
 
 when conditions are favorable. Results obtained at differ- 
 ent hours of a day are of more value than the same number 
 of results obtained on different days at the same hour of the 
 day. This is on account of variation in external conditions 
 (direction of light, phase, distinctness, refraction, etc.) 
 
 120. We shall for illustration take the following example, making use of 
 such parts of it from time to time as may belong to the subject in hand, and 
 finally, after explaining the method of forming the condition equations, 
 solve it in full. 
 
 In the triangulation of Lake Superior executed by the U. S. Engineers 
 the following angles were measured in the quadrilateral N. Base, S. Base, 
 Lester, Oneota. 
 
 LNO = 
 
 124 
 
 09' 
 
 40". 
 
 69 wei: 
 
 SNL = 
 
 "3 
 
 39' 
 
 05". 
 
 07 ' 
 
 ONS = 
 
 122 
 
 li' 
 
 15" 
 
 .61 
 
 NSO = 
 
 23 
 
 08' 
 
 05", 
 
 .26 
 
 LSN = 
 
 47 
 
 3l' 
 
 20". 
 
 4i 
 
 LSO = 
 
 70 
 
 39' 
 
 24", 
 
 ,60 ' 
 
 SON = 
 
 34 
 
 40' 
 
 39" 
 
 ,66 
 
 NOL - 
 
 43 
 
 46' 
 
 26" 
 
 .40 
 
 OLS = 
 
 30 
 
 53' 
 
 30" 
 
 ,81 
 
 These angles we 
 
 shall 
 
 denote by Mi, 
 
 Fig.19 
 
 14 
 
 23 
 
 6 
 
 7 
 
 3i 
 i 
 8 
 
 .. M respectively. 
 
 The length of the line N. Base S. Base (Minnesota Point) is 6o56"'.6, 
 and the latitudes of the four stations are approximately 
 
 N. Base, 46 45' 
 S. Base, 46 43' 
 
 Lester, 46 52' 
 Oneota, 46 45' 
 
 121. The Local Adjustment. When in a system of 
 triangulation the horizontal angles read at a station are 
 adjusted for all of the conditions existing among them, then 
 these angles are said to be locally adjusted. 
 
 From the considerations set forth in Art. 1 17 it is readily 
 seen that at a station only two kinds of conditions are 
 possible 
 
 (a) that an angle can be formed from two or more 
 others, and 
 
 (b) that the sum of the angles round the horizon should 
 be equal to 360.
 
 256 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The second of these is included in the first, and the 
 method of adjustment may be stated in general terms as 
 follows : 
 
 An inspection of the figure representing the angles at 
 the station will show how all of the measured angles can be 
 expressed in terms of a certain number of them which are 
 independent of one another. These relations will give rise 
 to condition equations, or local equations, as they are called, 
 which may be solved as in Chapters IV. or V. 
 
 Thus, if M,, M v . . . M n denote the single measured 
 angles, and ?',, v v . . . v n their most probable corrections, 
 then if any of the angles M h , M k can be formed from others, 
 we have, by equating the measured and computed values, 
 the local condition equations, 
 
 or 
 
 <', + ?' + *'* = 4 suppose 
 
 with 
 
 ' = a minimum 
 
 where A> A> A denote the weights of the angles. 
 
 The solution ma) ? be in general best carried out by the 
 method of correlates, as in Chap. V. 
 
 The following special cases are of 
 frequent occurrence : 
 
 (i) At a station O the n i single 
 angles AOB, BOC, . . . are measured, 
 and also the sum angle AOL, to find the 
 adjusted values of the separate angles, 
 all of the measured values being of the 
 same weight.
 
 APPLICATION TO TRIANGULATION. 257 
 
 The condition equation is 
 
 or 
 
 = / suppose, 
 with 
 
 [V*] = a minimum. 
 
 The solution gives (Art. 109 or no) 
 
 that is, the correction to each angle is - of the excess of the sum 
 
 angle over the sum of the single angles, and the sign of the cor- 
 rection to the sum angle is opposite to that of the single angles. 
 
 (2) At a station O the n single angles AOB, BOC, . . . 
 LOA are measured, thus closing the horizon, to find the 
 adjusted values of the angles. 
 
 The condition equation is 
 
 = I suppose, 
 with [/# a ] = a minimum. 
 
 The solution gives 
 
 where , = -,, = -,. . ., and [//] = I . 
 
 A A L/J 
 
 If the weights are equal, then 
 
 l_ 
 
 ^ v, . . . ?' - 
 
 that is, the correction to each angle is - of the excess of 360 
 over the sum of the measured angles.
 
 258 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. i. The angles at station N. Base close the horizon ; required to 
 adjust them. 
 
 We have (Art. 120) 
 
 M-i + ?'i = 124 09' 40". 69 + v-i weight 2 
 Mi + v t = 113 39' 05". 07 + v-t " 2 
 
 M 3 + z' 3 = i22 n' 15". 61 + z> 3 " 14 
 
 Sum = 360 oo' oi".37 + v\ + v? + v s 
 Theoretical sum = 360 oo' oo".oo 
 
 .'. Local equation is o = i".37 + v\ + v* + v a 
 
 Hence (Ex. 2, Art. no) 
 
 = o".64 
 z>z = o".64 
 Z'a = o".O9 
 and the adjusted angles are 
 
 124 09' 40". 05 
 113 39' 04". 43 
 
 122 II' 15". 52 
 
 Check-sum = 360 oo' oo".oo 
 
 Ex. 2. Precisely as in the preceding we may deduce at station South Base 
 the local equation 
 
 o = i".o7 + v t + v 6 v 6 
 and the adjusted angles 
 
 23 08' 05". 13 
 47 31' 19". 91 
 70 39' 2 5 ".04 
 
 122. Number of Local Equations at a Station. If s sta- 
 tions are sighted at from a station that is occupied, the 
 number of angles necessary to be measured to determine all 
 of the angles that can be formed at the station occupied is 
 si. If, therefore, an additional angle were measured, 
 its value could be determined in two ways : from the direct 
 measurement and from the s i measures. The contradic- 
 tion in these two values would give rise to a local (condition) 
 equation. If, therefore, n is the total number of angles 
 measured at a station, the number of local equations, as 
 indicated by the number of superfluous angles, is 
 
 n s-\- I.
 
 APPLICATION TO TRIANGULATION. 259 
 
 123. The General Adjustment. With a single 
 measured base the number of conditions arising from the 
 geometrical relations existing among the different parts of 
 a triangulation net can be readily estimated. For if the net 
 contains s stations, two are known, being the end points of 
 the base, and s 2 are to be found. 
 
 Now, two angles observed at the end points of the base 
 will determine a third point ; two more observed at the end 
 points of a line joining any two of these points will de- 
 termine a fourth point, and. so on. Hence to determine 
 the s 2 points, 2 (s 2) angles are necessary. If, there- 
 fore, ;/ is the total number of angles measured, the number 
 of superfluous angles, that is, the number of conditions to 
 be satisfied, is 
 
 n 2(s 2) 
 
 Ex. In a chain of triangles, if s is the number of stations, show that the 
 number of conditions to be satisfied is s 2 ; and in a chain of quadrilaterals, 
 with both diagonals drawn, the number of conditions is 2s 4. 
 
 The equations arising from these conditions are divided 
 into two classes, angle equations and side equations. 
 
 124. The Angle Equations. -The sum of the angles of a 
 triangle drawn on a plane surface is equal to 180. The 
 sum of the angles of a spherical triangle exceeds 180 by the 
 spherical excess (e) of the triangle, which latter is found 
 from the relation 
 
 _area of triangle 
 R sin \" 
 
 R being the radius of the sphere. 
 
 From surveys carried on during the past two centuries 
 the earth has been found to be spheroidal in form, and its 
 dimensions have been determined within small limits. Now, 
 a spheroidal triangle of moderate size may be computed as 
 a spherical triangle on a tangent sphere whose radius is 
 Vp l ft t , where /> p y are the radii of curvature of the meridian 
 and of the normal section to the meridian respectively at 
 
 34
 
 260 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 the point corresponding- to the mean of the latitudes tp of 
 the triangle vertices. 
 
 Hence we may wrap our triangulation on the spheroid 
 in question by conforming it to the spherical excess com- 
 puted from the formula 
 
 rt^sin C 
 
 (in seconds) = : 7 . 
 
 2,o 1 /> 2 sin i 
 
 where a, b are two sides and C is the included angle of the 
 triangle. 
 
 For convenience of computation we may write 
 
 = A ab sin C 
 
 when log A may be tabulated for the argument <p. The fol- 
 lowing table is computed with Clarke's values of the ele- 
 ments of the terrestrial spheroid corresponding to latitudes 
 from 10 to 70. The metre is the unit of length to be 
 used. 
 
 V 
 
 log A 
 
 <P 
 
 log A 
 
 9 
 
 log X 
 
 10 
 
 1.40675 
 
 30 
 
 1.40547 
 
 50 
 
 1.40349 
 
 11 
 
 672 
 
 3i 
 
 537 
 
 5i 
 
 339 
 
 12 
 
 668 
 
 32 
 
 528 
 
 52 
 
 329 
 
 13 
 
 663 
 
 33 
 
 519 
 
 53 
 
 319 
 
 14 
 
 659 
 
 34 
 
 509 
 
 54 
 
 309 
 
 15 
 
 1.40654 
 
 35 
 
 1.40500 
 
 55 
 
 1.40299 
 
 16 
 
 649 
 
 36 
 
 49 1 
 
 56 
 
 289 
 
 17 
 
 643 
 
 37 
 
 481 
 
 57 
 
 280 
 
 18 
 
 637 
 
 38 
 
 47i 
 
 58 
 
 271 
 
 19 
 
 631 
 
 39 
 
 461 
 
 59 
 
 262 
 
 20 
 
 1.40625 
 
 40 
 
 1.40451 
 
 60 
 
 1.40253 
 
 21 
 
 618 
 
 4i 
 
 441 
 
 61 
 
 244 
 
 22 
 
 611 
 
 42 
 
 43i 
 
 62 
 
 235 
 
 23 
 
 604 
 
 43 
 
 420 
 
 63 
 
 226 
 
 24 
 
 597 
 
 44 
 
 410 
 
 64 
 
 218 
 
 25 
 
 1.40589 
 
 45 
 
 1.40400 
 
 65 
 
 1.40210 
 
 26 
 
 58i 
 
 46 
 
 39 
 
 66 
 
 202 
 
 27 
 
 573 
 
 47 
 
 380 
 
 67 
 
 195 
 
 28 
 
 564 
 
 48 
 
 369 
 
 68 
 
 188 
 
 29 
 
 555 
 
 49 
 
 359 
 
 69 
 
 181 
 
 30 
 
 1-40547 
 
 50 
 
 1.40349 
 
 70 
 
 1.40174
 
 APPLICATION TO TRIANGULATION. 26 1 
 
 To find a, b, tp, a preliminary geodetic computation must 
 first be made of the triangulation to be adjusted, starting 
 from a base or from a known side. The values found from 
 using the unadjusted angles will be close enough for this 
 purpose. The latitudes need only be computed to the 
 nearest minute. 
 
 A useful check of the excess results from the principle 
 that the sums of the excesses of triangles that cover the 
 same area should be equal. In our example the spherical 
 excesses of the triangles ONS, LSO will be found to be 
 o".o5 and o".37 respectively. 
 
 In each single triangle, then, the condition required to 
 wrap it on the spheroid, that is, that the sum of the three 
 measured angles shall be equal to 180, together with the 
 spherical excess, gives a condition equation.* This is 
 called an angle equation, or by some a triangle equation. 
 
 Ex. In the triangle N. Base, S. Base, Oneota, if 573, #4, v* denote the cor- 
 rections to the three angles, we have for the most probable values 
 
 ONS=. 122 n' 15". 61 + z> 3 
 NSO= 23 08' 05". 26 + z> 4 
 SON = 34 40' 39". 66 + z>7 
 
 Sum = 180 oo' oo".53 + z 3 + z' 4 + z- 7 
 Theoretical sum = 180 oo' oo".o5 = 180+ s 
 
 and the angle equation is formed by equating these sums. The result is 
 
 <^3 + v\ + vi + o".4S = o 
 Similarly from the triangle Lester, S. Base, Oneota, the angle equation is 
 
 Ve + Z'7 + V 6 + "'a + I*. 10 = 
 
 125. Number of Angle Equations in a Net. It is to be 
 expected that in a triangulation net some of the lines will 
 be sighted over in both directions, and some in only one 
 direction. If these latter lines are omitted the number of 
 angle equations will remain unaltered. Thus in our Lake 
 Superior quadrilateral (Fig. 19) the line NL has been 
 
 * We confine ourselves throughout to triangles to which Legendre's theorem is applicable. 
 For very large triangles other formulas for spherical excess must be used. See, for example, 
 Helmert, Theorieen d. lioheren Gcoddsie, vol. i. p. 362.
 
 262 THE ADJUSTMENT OF OBSERVATIONS. 
 
 sighted over from N, but not from L, so that we have only 
 
 two angle equations: namely, those 
 resulting from the triangles ONS, 
 OLS, just as if the figure had been 
 O f the form of Fig. 21, in which the 
 line NL is omitted. 
 
 Generally, if s is the number of 
 stations occupied, the polygon form- 
 ing the outline of the net will give 
 rise to one angle equation. Each 
 diagonal that is drawn will form a 
 figure, giving rise to an additional 
 angle equation. Hence if in the net there are / t lines 
 sighted over in both directions, the number of diagonals 
 will be /, s, and the number of angle equations 
 
 Z-t+l 
 
 If / 2 of the lines are sighted over in one direction only, 
 and / is the total number of lines in the figure, then since 
 /,=/ / 2 , the number of angle equations would be ex- 
 pressed by 
 
 l-l-s+i 
 
 Thus in the figure the polygon LONS gives an angle 
 equation, and the line OS gives rise to a second angle equa- 
 tion from either the triangle ONS or OLS. We might, 
 therefore, form the equations from either of the three sets 
 of figures, 
 
 LONS LONS ONS 
 
 ONS OLS OLS 
 
 and should have respectively 
 
 ! + '. + '. + ' + + 3-06=0 
 
 ^3 + ^ + ^+0.48 = 
 
 ^ + v* + ^ + v s + v t + 3.06 = o 
 
 V K + V 7 -\-V s + V,+ 1-10=0 
 
 ^3 + v< + v, + 0.48 = o 
 
 W * *0 = O
 
 APPLICATION TO TRIANGULATIOX. 
 
 263 
 
 which pairs of equations, by means of the relations already 
 found (pp. 258, 261), 
 
 *'! + *' + *'+ 1-37 = 
 
 W4 + V. V.+ 1.07=0 
 
 reduce to the same two equations for each set of polygons. 
 
 Ex. In the quadrilateral ABCD, in which all of the 8 angles are measured, 
 show that there are three independent angle equations, 
 and that these equations may be found from the follow- A 
 ing 8 sets of figures : 
 
 ABD, ABC, A CD; ABD, ABC, ABCD; 
 ABD, A CD, ABCD; 
 
 BDA, BCD, BCA ; BCA, BCD, BCD A ; 
 CDB, CAB, CD A ; CDB, CD A, CDBA; 
 DAB, DBC, DAC. 
 
 Fig. 22 
 
 Fig.23 
 
 126. The Side Equations. In a single triangle, or in a 
 simple chain of triangles, the length of any assigned side 
 can be computed from a given side in but one way. When 
 the triangles are interlaced this is not the case. 
 
 Thus in Fig. 21 any side can be computed from A T S in 
 but one way. The only condition equations apart from the 
 
 , Li local equations would be the two 
 angle equations. But in Fig. 19, 
 in which the line NL is sighted 
 over from N, we have the further 
 condition that the lines OL, NL, 
 SL intersect in the same point, L. 
 The figure plotted from the meas- 
 ured values would be of the form 
 of Fig. 23. 
 
 To express in the form of an 
 equation the condition that the 
 three points L,, L.,, 7, 3 must coincide, we proceed as follows: 
 Starting from the base A T S, we may compute SL, directly
 
 264 THE ADJUSTMENT OF OBSERVATIONS. 
 
 from the triangle SNL l and SL S from the triangles SON, 
 SOL y This gives 
 
 sin SN _ sin SL,N 
 
 sin SL 1 sin 
 
 sin SN _ sin SON sin SL 3 O 
 sin SL 3 ~~ sin SNO sin SOL 3 
 
 But SL, must be equal to S 3 
 
 Hence the condition equation is 
 
 sin SLN sin SNO sin 5(97. 
 
 sTn~52VX sin SON sin 
 which is called a .rzV& equation or 5//^ equation. 
 
 Ex. In the figure ABCD 3 Di A , the three angle equations, 
 
 BD^A = 180 
 
 3 " "^ + -ffC/4 + CAB = 180 + 
 
 Fi 9- 24 .#CZ> 3 + CZ> 3J 5 + DzBC = 1 80 + 3 
 
 D, 
 
 given by the triangles D^AB, ABC, BCD*, may be satis- 
 fied and yet the figure not be a perfect quadrilateral. 
 Show by equating the values of BDi and BD$ that the 
 further condition necessary is 
 
 sin DAB sin BCA sin CDB _ 
 sin BDA sin CAB sin BCD 
 
 The side equation 
 
 sin SLN sin SOL sin .S7V0 _ 
 
 sin SNL sin SZ<9 sin S6W 
 gives the identical relation 
 
 sin SN sin SL sin 50 
 
 sin >SX sin SO sin .S/V 
 
 _ 
 
 Hence in forming a side equation we may proceed mechani- 
 cally in this way. Write down the scheme 
 
 SN SL 50_ 
 SL SO SN~ l
 
 APPLICATION TO TRIANGULATION. 265 
 
 the numerator and denominator each. being formed by the 
 lines radiating from the point S in order of azimuth, and the 
 first denominator being the second numerator. The side 
 equation results from replacing the sides by the sines of the 
 angles opposite to them. 
 
 The point 5 is called the/<?<fc of the quadrilateral for this 
 equation. 
 
 127. Position of Pole. It is easily seen that in forming the 
 side equation any vertex may be taken as pole. For plot- 
 ting the figure from the angles of the triangles ONS, OLS, 
 the side equation with pole at S means that the points L t 
 and L 3 must coincide. The side equation with pole at N 
 means that Z-,, L^ coincide, and with pole at O that L^ L 3 
 coincide. If any one of these conditions is satisfied the 
 others are also satisfied, as each amounts to the same con- 
 dition that L is not three points but one point. 
 
 Similar reasoning will show that by plotting the figure 
 from LONS, ONS, the side equations formed by taking the 
 poles at N, L, S, mean that O is not three points but one 
 point, and so on. Hence the side equation formed from 
 any vertex as pole in connection with the angle equations 
 fixes each point of the figure definitely and removes all 
 contradictions from it. 
 
 It will be noticed that the reasoning is in no way affected 
 by the line NL being sighted over in only one direction. 
 
 Ex. i. In a quadrilateral ABCD, in which all of the 8 angles are measured, 
 show that of the 15 side equations that may be formed, 7 only are different in 
 form, and that by taking the angle equations into account all of them may 
 be reduced to a single form. 
 
 Also show that there are 56 ways of expressing the three angle and one 
 side equations necessary to determine the quadrilateral. 
 
 Ex. 2. Examine the truth of the following statement. In a quadrilateral 
 an angle equation may be replaced by a side equation, so that the quadri- 
 lateral may be determined by 3 angle equations and one side equation, 2 
 angle equations and 2 side equations, one angle equation and 3 side equa- 
 tions, the number of conditions remaining four, and the four not being all of 
 one kind.
 
 266 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 128. If the triangulation net, instead of involving quad- 
 rilaterals only, involves central polygons, such that, in com- 
 puting the lengths of the sides, 
 we can pass from one side to 
 any other through a chain of 
 triangles, the same process is 
 followed in forming the side 
 equations as in a quadrilateral. 
 Thus in the figure which rep- 
 resents part of the triangula- 
 tion of Lake Erie west of Buf- 
 falo Base there are side equa- 
 tions from 
 
 Fifl.25 
 
 the quadrilaterals CDHG, GHFA 
 the pentagons GABCH, HGDEF 
 
 The scheme for the pentagonal side equation GABCH, for 
 example, would be just as in the case of a central quadri- 
 lateral, taking G as pole, 
 
 GA 
 GB 
 
 GB_ 
 ~GC 
 
 and the side equation 
 
 sin GBA sin GCB 
 
 GC GH _ 
 ~GH "GA ~ 
 
 sin GHC sin GAH 
 
 sin GAB sin GBC sin GCH sin GHA 
 
 = i 
 
 129. Reduction to the Linear Form. Thus far we have 
 considered the side equations in their rigorous form. But 
 in order to carry through the solution by combining them 
 with the other condition equations they must be reduced to 
 the linear form. We proceed to show how this may be 
 done. (See Art. 7.) 
 
 Let the side equation be 
 
 sin F, sin F 3 
 ~ 
 
 where F,, F 2 , . 
 angles. Let 
 
 . denote the most probable values of the 
 M v . . . denote the measured values, and
 
 APPLICATION TO TRIANGULATION. 267 
 
 v t , v v . . . the most probable corrections to these values ; 
 then the equation may be written 
 
 sin (M l -\- Q sin (M t -f v t ) 
 sin (M, + z/ a ) sin (M t + v 4 ) ' 
 
 Taking the log. of each side of this equation and expanding 
 by Taylor's theorem, we have, retaining the first powers of 
 the corrections only, 
 
 log sin M, + -JM (log sin M,) v, 
 
 f . . .=o (3) 
 
 which may be written in two forms for computation : 
 
 First, if the corrections to the angles are expressed in 
 seconds, we may put 
 
 where d' is the tabular difference for \" for the angle M 1 in a 
 table of log. sines. Then we have 
 
 d'v, o'v, -f . . . + log sin M, log sin M, -f . . . = o 
 
 that is, 
 
 [oV] = / (4) 
 
 where / is a known quantity. 
 Secondly, we may replace 
 
 -r^r (log sin^f,) by mod sin \" cot J/, 
 
 where mod denotes the modulus of the common system of 
 logarithms. Eq. 3 may then be arranged 
 
 Cot M, v, cot M, v 9 + . . . 
 
 = , - ^ - - - (log sin M. log sin M. + . . .) (5) 
 10' mod sin i ' 
 
 if the seventh place of decimals is chosen as the unit. 
 
 35
 
 268 THE ADJUSTMENT OF OBSERVATIONS. 
 
 For convenience of computation there is not much to 
 choose between the two forms. The second is perhaps, on 
 the whole, to be preferred in ordinary triangulation work 
 with well-shaped triangles. 
 
 For the method of computing log. sines and log. differ- 
 ences for small angles or for angles near 180, and also if a 
 ten-place table is used, see Art. 7. 
 
 130. Check Computation. The side equation deduced 
 from spherical triangles must also follow from the corre- 
 sponding plane triangles, the angles of each spherical tri- 
 angle being transformed according to Legendre's theorem ; 
 that is, for example, we should obtain the same constant 
 term / by reducing to the linear form the equation 
 
 sin SLN sin SOL sin SNO __ 
 sin SNL sin SLO sin SON~ 
 
 or the equation 
 
 s'm(SLN-^) s'in(SOL-^) sin (SNO - % 
 
 sn 
 
 _ 
 ~ 
 
 where s 19 2 , 3 denote the spherical excesses of the triangles 
 SNL, SOL, and SON respectively. 
 
 It affords a check of the accuracy of the numerical work 
 to compute the side equation with both the spherical 
 angles and the plane angles. It is evidently simpler to use 
 the spherical angles, so that if but a single computation 
 is to be made they should be chosen. 
 
 For a check of the coefficients of the corrections we 
 have, by expanding the second equation by Taylor's theorem, 
 the relations 
 
 etf' - <T) -f f a (d'" - <T") -f 3 (d"'" - <5""") = o 
 or 
 
 f,(cot SLN cot SNL) -f f a (cot SOL - cot SLO) 
 
 + 3 (cot SNO - cot SON) = o 
 
 for the first and second forms of reduction respectively. 
 This useful check is given by Andrse.
 
 APPLICATION TO TRIANGULATION. 269 
 
 Ex. The quadrilateral N. Base, S. Base, Oneota, Lester (Fig. 19). 
 Take the pole at Lester. 
 We have the scheme 
 
 LS LN LO _ 
 
 LN LO~LS~ 
 
 from which we write down the side equation 
 
 sin LNS sin LON sin LSO _ 
 sin LSN sin LNO sin LOS ~ 
 that is, 
 
 sin (Mi + 7- a ) sin (M a + 7' fi ) sin (M e + V K ) 
 
 sin (Hf., + z-s,) sin (A/i + 7-1) sin (M-, + M s + 7-, + 7- 6 ) 
 
 First Form of Reduction. 
 
 log sin (113 39' 05". 07 + 7- 2 ) =9.9618969,7 9,227-2 
 
 log sin ( 43 46' 26". 40 + 7',) = 9.8399903,4 + 21,98 7- g 
 
 log sin ( 70 39' 24". 60 + 7- 6 ) =9.9747656,9+ 7,397-6 
 
 530,0 
 
 log sin ( 47 3 31' 20". 41 + 7- 5 ) =9.8677859,5 + 19,287-5 
 
 log sin (124 09' 40". 69 + 7-,) =9.9177470,2 14,297-! 
 
 log sin ( 78" 27' 06". 06 + 7- 7 + z- b ) = 9.991 1 180,3 + 4,30(7-7 + rv) 
 
 510,0 
 Hence the side equation in the linear form is 
 
 14.297-, 9.227-.J 19.287-5 + 7.397-6 4.307-7 + 17-687-8 + 20.0 = 
 the unit being the seventh place of decimals. 
 
 Check of the constant term by computing the log. sines after deducting 
 from each angle \ of the spherical excess of the triangle to which it belongs. 
 
 Angle. 
 
 113^ 39' 05". oo 
 43' 46' 26 ".36 
 "o 39' -4"- 4S 
 
 Log Sin. 
 
 9.9618970,3 
 9.8399902,5 
 
 Angle. 
 
 47^ 31' 2o".34 
 124' 09' 40". 65 
 78 27' 05 ".93 
 
 Log Sin. 
 9.8677858,2 
 9.9177470,7 
 9.9911179,8 
 
 528,8 
 
 508,7 
 
 + 20,1 
 
 agreeing closely with the value found from the spherical angles. 
 Check of the coefficients. 
 o.i9( 9.22 19.28) + 0.12(21.98 + 14.29) + 0.37(7.39 4-3) = + o-oS 
 
 This check would have been closer had the spherical excesses been carried 
 out to three places of decimals. We have taken o'.i9, o'.os, and o".i2 for the
 
 2/O THE ADJUSTMENT OF OBSERVATIONS. 
 
 excesses of the single triangles LNS, ONS, LNO, and o".37 for the sum 
 triangle LSO. The first two are more nearly o".ig5 and o".o55. 
 
 Second Form of Reduction. 
 
 Log Sin. Log Sin. 
 
 9.9618969,7 0.43802/2 9-8677859,5 + 0.91562/5 
 
 9.8399903,4+ 1.04372/8 9.9177470,2 0.67862/1 
 
 9.9747656,9 + 0.35102/6 9.9911180,3 + 0.2043(777 + t') 
 
 530,0 
 510,0 
 
 20,0 log I.30IO3 
 
 j log 8.67664 
 io 7 mod sin i 
 
 9.97767 0.950 
 and the side equation is 
 0.67862/1 0.43802/2 0.91562/5 + 0.35102/0 0.20432/7 + 0.83942/8 + 0.950 = o 
 
 This result may be checked in the same way as in the first form. 
 
 In reducing a side equation to the linear form the coefficients of the cor- 
 rections should be carried out to one place of decimals farther than the abso- 
 lute term. This for a short computation would be unnecessary, but in the 
 reduction of an extensive triangulation net it is rendered necessary by the 
 accumulation of errors from the dropping of the last figures in products and 
 quotients. 
 
 It will be noticed that in the preceding example we have 
 carried out the log. sines to 8 places of decimals, the seventh 
 place being the unit. This is amply sufficient, in primary 
 work, for our present methods of measurement, as already 
 indicated in Art. 34. Indeed, it is in general sufficient to 
 carry them to 7 places of decimals only, an ordinary 7-place 
 table being used. As there is no 8-place table published, 
 the labor of forming the log. sines with a lo-place table, and 
 then cutting down the results to 8 places, is, in very many 
 cases, hardly justified by the extra precision attained. 
 
 For a secondary triangulation 6 places of decimals, and 
 for a tertiary triangulation 5 places, may be used.* 
 
 * On the Coast Survey and Lake Survey the practice in primary triangulation has been to 
 carry out the log. sines to io places of decimals. On the English Ordnance Survey they were also 
 carried out to io places, but on the more modern Great Trigonometrical Survey of India to 7 
 places only. In the triangulation of Denmark, Andra: used 8 places ; and in this he has been fol- 
 lowed by the Prussian, Italian, and other modern European surveys.
 
 APPLICATION TO TRIANGULATION. 2/1 
 
 131. We have seen that the coefficients of the correc- 
 tions in a side equation are given by the differences for 
 i" of the log. sines of the angles, or by the cotangents of the 
 angles that enter. Now, since an equable distribution of 
 errors arising from the approximate computation is best 
 attained by securing the greatest possible equality of co- 
 efficients throughout the condition equations, and since the 
 coefficients of the corrections in the angle equations are 
 -f- 1 or i, it follows that it would be most convenient to 
 put the side equations on the same footing as the angle 
 equations. To do this we may divide the side equation by 
 such a number as will make the average value of the co- 
 efficients equal to unity. This, for angles ordinarily met 
 with in triangulation, would be effected by taking the sixth 
 place of decimals as the unit in the side equation. Thus in 
 our example, dividing by 10, which is approximately the 
 mean of the coefficients, and which amounts to the same 
 thing as expressing the log. differences in units of the sixth 
 place of decimals, the equation may be written 
 
 1.43^-0.92?',- 1.93^ + 0.74^-0.43^ + i.777/ 8 +2.00 = 
 
 It would have been equally correct to have multiplied 
 each of the angle equations by 10, and so have put them on 
 the same footing as the side equations. Dividing the side 
 equations is, however, to be preferred, as the coefficients 
 are made smaller throughout, and the formation and solu- 
 tion of the normal equations is consequently easier. 
 
 A striking difference between condition equations and 
 observation equations is here brought out. As a condition 
 equation expresses a rigorous relation among the observed 
 quantities altogether independent of observation, it may be 
 multiplied or divided by any number without affecting that 
 relation ; with an observation equation, on the other hand, 
 the effect would be to increase or diminish its weight. 
 (Compare Art. 58.) 
 
 132. Position of Pole. In a quadrilateral, taking any of the 
 vertices as pole, the conclusion was reached in Art. 127 that
 
 2/2 THE ADJUSTMENT OF OBSERVATIONS. 
 
 any one of the resulting forms of side equation was as 
 good as any other in satisfying the conditions imposed. 
 But when a side equation is reduced to the linear form 
 and is no longer rigorous the question deserves farther 
 notice. 
 
 Two points are to be considered precision of results and 
 ease of computation. As regards the first, since the differ- 
 ences in a table of log. sines are more sharply defined for 
 small angles, and these differences are the coefficients of 
 the unknowns in the side equation, it follows that in general 
 that vertex should be chosen which allows the introduction 
 of the acutest angles into the side equation. 
 
 Labor of computation will be saved by choosing the pole 
 so that as few sine terms as possible enter. Thus by choos- 
 ing the pole at O, the intersection of the diagonals (Fig. 22), 
 the side equation would contain 8 terms, whereas if taken 
 at any of the vertices only 6 terms would enter. Also, 
 other things being equal, we should choose that pole which 
 introduces the smallest number of unknowns into the equa- 
 tion, for then the normal equations would be more easily 
 formed. 
 
 If the approximate form of solution in Art. 115 is em- 
 ployed it is advantageous to choose the pole at the inter- 
 section, O, of the diagonals, as will be seen in the sequel. 
 
 133. Number of Side Equations in a Net. A line being 
 taken as a base, its extremities are known. To fix a third 
 point we must know the other two sides of the triangle of 
 which this point is to be the vertex. Hence if we have a 
 net of triangles connecting s stations, two of the stations 
 being the ends of the base, we must have, in order to plot 
 the figure, 2(5 2) lines besides the base ; that is, 2^3 
 lines in all. 
 
 Starting from the base, each line in this figure can be 
 computed in but one way, but any additional line, whether 
 observed over in one or both directions, can be computed 
 in two ways, and therefore gives rise to a side equation. 
 If, then, the total number of lines in the figure is /, the num-
 
 APPLICATION TO TRIANGULATION. 273 
 
 her of side equations, as indicated by the number of super- 
 fluous lines, is 
 
 /-2.J+3. 
 
 134. Check of the Total Number of Conditions. Leaving 
 local equations out of account, if/ is the number of lines in 
 a figure sighted over in both directions, and s the number 
 of stations, the total number of angles in the figure is 2/ s. 
 If 4 of these lines have been sighted over in one direction 
 only, the number of angles is reduced to 2/ / 2 s. 
 
 Now, the number of angle equations in the figure is 
 
 and the number of side equations is 
 
 /-2J+3 
 
 .*. the total number of condition equations is 
 
 2/ / 2 35 -f- 4, that is, 11 2s -{- 4 
 
 where ;/ is the number of angles in the figure. 
 
 But we have seen in Art. 123 that the total number of 
 conditions to be satisfied among the same n angles is 
 
 n 2s -f- 4 
 
 We conclude, therefore, that the conditions are completely 
 covered by the angle and side equations. 
 
 135. Manner of Selecting the Angle and Side 
 Equations. In the selection of the angle and side equa- 
 tions in a triangulation net we have two dangers to guard 
 against: first, that we omit no necessary conditions, and, 
 second, that we admit no unnecessary ones. The rule 
 usually followed is to start from some line as base, and plot, 
 the figure proceeding from station to station, 
 writing down the conditions that express 
 the connections of each station to the net as 
 the net grows. 
 
 For example, let Fig. 27 represent a tri- Fi s- 26 
 angulation net. Taking AB as base (Fig. 26), a third station, 
 C, gives an angle equation from the triangle ABC, A
 
 274 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 fourth station, D, gives in addition angle equations from 
 ABD,ACD; side equation from ABCD. A fifth point, , 
 gives in addition (Fig. 27) angle equations 
 from ABE, ADE ; side equation from 
 ABDE. 
 
 We have thus in the complete figure 
 (Fig. 27) 5 angle equations and 2 side equa- 
 tions. 
 
 Fig.27 
 
 Check. Number of stations = 5 
 Number of lines = 9 
 .'. Number of angle equations 
 and Number of side equations 
 
 As an illustration of the dif- 
 ficulties which may arise in se- 
 lecting the angle and side equa- 
 tions, let us take the triangulation 
 around the Chicago Base (1877). 
 It is represented in the figure. 
 
 From the rules laid down in 
 Arts. 125, 133 it follows that there 
 are in the adjustment 10 angle 
 equations and 8 side equations. 
 
 The peculiarity of the system 
 is that the station F is very close 
 to the base line DE. Thus the 
 angle equation from the triangle 
 DEF 
 
 EDF 00 oo' oo".8i5+^ 
 FED = 00 oo' i". 1 854-1;, 
 DFE=i 79 59' 5 6".733 + *, + 
 
 =9 2x5+3 = 
 
 Fig.28 
 
 179 59' 5* 
 
 1 80 oo' oo".ooo 
 
 o = - i".267 4- v t 4- v a 4- v^ 4- v, 
 
 In the selection of the side equations it is advisable to 
 avoid those quadrilaterals in the figure which are entangled
 
 APPLICATION TO TRIANGULATION. 275 
 
 with the above triangle; that is, the quadrilaterals ADF 9 
 BDFE, GDFE. For example, if we take the quadrilateral 
 GDFE we have, in units of the seventh place of decimals, 
 pole at G, the side equation 
 
 54.772 i?, o.oooSv, -f- o.ooi 5^ 4 4o.7o66(? a -f~ ^) 
 
 -f- 26. 1 803*', 0.0003 ( v + *0 = 40. 1 3 
 
 Now, since the coefficients of ?, v 4 , ?, ? are each less than 
 2 in the third place, this equation is nearly the same as 
 
 54.772^ - 40.707(X + v.) + 26. iSo? 7 = 40. 1 3 
 or dividing by 40 and replacing ? & -j- ? by v 3 v t , 
 1-369*'! + i.oi8(? s + v<) + 0.654^ = i 
 
 which is nearly the same as the angle equation from the 
 triangle DEF. 
 
 Similarly the quadrilaterals AEDF, BDEF give respec- 
 tively, neglecting coefficients less than 5 in the fifth place, 
 
 1.2192;, + o.2o8f 7 + 0.722(2/3 -f- ^ 4 ) = 0.790 
 0.256?, + 2.466?, + 1.3430s + O = 0.836 
 
 both of which express approximately the same relations 
 among the angles of the small triangle DEF as the angle 
 equation formed from this triangle. We therefore conclude 
 that in the formation of the condition equations other 
 quadrilaterals than these should be chosen. 
 
 136. Again, in selecting the side equations in a net care 
 must be taken that only independent conditions are chosen. 
 Thus in Fig. 28 we might have chosen the following eight 
 quadrilaterals from which to form the side equations: 
 
 AGBD, AGDE, AGBE, BDFG, 
 BFEG, AGDF, ACBE, BDEG. 
 
 A careful examination will show that these figures are not 
 independent. For, taking the four, AGBD, AGDE, AGBE, 
 
 36
 
 276 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 BDEG, and choosing G as pole, which is common to all, we 
 have the side equations 
 
 sin ADG sin DBG sin BAG _ i 
 
 ~sm DAG sirHWG sin ABG 
 
 sin DAG sin EDG sin AEG 
 
 sin ADG sin DEG sin -ZL46P 
 sin y^C sin ^^^ sin 
 
 i 
 
 sn 
 sin 
 
 sin EBG sin AEG 
 
 sn 
 
 sn 
 
 
 sin DBG sin EDG sin BEG 
 which equations multiplied together reduce to the identical 
 form 
 
 i = i 
 
 showing that from any three the fourth may be found. 
 
 The entrance of mutually dependent conditions would, 
 however, be detected in the course of the solution of the 
 normal equations, as we should arrive at two identical equa- 
 tions ; or, in other words, one of the correlates would be- 
 come indeterminate. 
 
 If the rule given on p. 273 is followed closely this 
 repetition of conditions will hardly occur. 
 
 Fig.30 
 
 Fig. 32 
 
 Fig.33 
 
 Fig. 34
 
 APPLICATION TO TRIANGULATION. 
 
 277 
 
 137. In the example chosen (Fig. 28) the selection of the 
 angle and side equations may be made in the following 
 order, proceeding from the line AB, and adding station to 
 station and line to line till the complete figure is reached : 
 
 From Fig. 29, 
 
 From Fig. 30 in addition, 
 
 From Fig. 31 in addition, 
 From Fig. 32 in addition, 
 
 From Fig. 33 in addition, 
 From Fig. 34 in addition, 
 
 Angle equations. 
 
 Triang. AGB 
 AEB 
 BEG 
 
 DAG 
 BDG 
 
 DAE 
 
 DAF 
 
 BDF 
 
 BFE 
 
 BAG 
 
 Side equations. 
 
 Quad.AGSE 
 
 " AGBD 
 
 " AGDE 
 
 " AGDF 
 
 " BDFG 
 
 " KFEG 
 
 " ACBE 
 " CBDE 
 
 In all 10 angle equations and 8 side equations, as should 
 be (p. 274). 
 
 138. We finally notice the arrangement for solution of the 
 condition equations in the net adjustment. On first thoughts 
 it might seem that it would be well to arrange the angle 
 equations and side equations in two separate sets, and so 
 carry them forward for solution. This was done in some of 
 the older work.* The only objection to doing it is that the 
 process of finding the corrections is more troublesome. Ex- 
 
 * See for an example Verification and Extension of La Cailles Arc of ' Meridian, by Sir 
 Thomas Maclear, vol. i. pp. 496 seq.
 
 2 7 8 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 perience shows that the solution of a series of normal equa- 
 tions is much facilitated if the coefficients are arranged as 
 the steps of a stair rather than irregularly. Thus 
 
 \ad\x-\-\aU\y = [al~\ 
 
 \ab~\x + \bU\y -f \bc~\s = \bl] 
 
 \bc\y +\cc-\z=\cl} 
 
 is a more convenient form for solution than 
 
 \aa~\x -f- [ab] y = [/] 
 
 Fig. 35 
 
 The condition equations should therefore be so arranged 
 that, as far as possible (and it cannot always be done at the 
 first trial), the normal equations will fall in the first of the 
 above forms rather than the second. A good rule is to 
 begin with an angle equation, proceeding from triangle to 
 triangle until the points gone over are covered by a side 
 equation, and then introduce it. Continue the process with 
 the remaining sets of triangles. 
 
 As to the angle equations themselves, it matters but 
 little which triangles are taken to form them. It is better, 
 
 however, to avoid triangles with 
 very small angles, such as DEF 
 in Fig. 28, and, where angles so 
 very small occur, to avoid tri- 
 angles involving angles imme- 
 diately contiguous to these small 
 angles. 
 
 139. In the explanation of the 
 formation of the side equations 
 we have assumed that, the com- 
 putation of one side from another 
 assumed as base, by different 
 routes we proceed through chains of triangles. The 
 omission of certain lines in the measurement may make it 
 necessary to proceed through polygons, and then the for- 
 mation of the side equations becomes more complicated.
 
 APPLICATION TO TRIANGULATION. 279 
 
 A good illustration occurs in the triangulation of Lake 
 Superior* (1871) near Keweenaw Base, as shown in Fig. 35. 
 
 Here there are 18 lines and 8 stations, requiring 5 side 
 equations. Proceeding from the line DG and adding point 
 to point, we form the first four side equations from the 
 quadrilaterals 
 
 HGDE, HGEF, HEFC, CEFB 
 
 With these there is no difficulty. 
 
 The fifth side equation is furnished by the pentagon 
 ABCED. We cannot compute directly any specified side 
 of this pentagon from another side through triangles only. 
 A little artifice, however, will enable us to do this. Sup- 
 pose the line CD drawn. Call the angle GDC=,r and 
 ECD=}>. This new line CD gives an additional side 
 equation. We take the two pentagons ABCED, DEGFC. 
 
 From the first pentagon, ABCED, 
 
 CE CD CA CB__ 
 ~CD CA CB CE~ 
 or 
 
 sin (EDG -f- -*") sin CAD sin CBA sin CEB _ 
 
 sin CED sin (ADG -\- x) sm CAB sin CBE = 
 
 and from the second pentagon, DEGFC, pole at E, 
 
 EG EF EC 
 
 ^ 
 EG EF EC ED 
 
 or 
 
 sin EGD sin EFG sin ECF sin (EDG-\-x] _ 
 sin EDG sin EGF sin EFC sin,)' 
 
 Now,j can be expressed in terms of x from the triangle 
 CDE; thus 
 
 (3) 
 where f is the spherical excess of the triangle CDE. 
 
 * Report of Chief of Engineers, 1872.
 
 280 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Eliminating x from equations i, 2, 3, and the required 
 side equation results. 
 
 To find it write (i) in the form 
 
 cot (EDO -f x) 
 
 i sin CAD sin ABC sin EEC 
 
 T-, cot ADE 
 
 sin ADE sin CED sin BAC sin 
 
 from which the value of x follows at once. 
 
 Hence since y is known from (3), all the angles in (2) 
 are known, and the solution is finished in the usual way. 
 
 If chains of triangles intersect so as to form a closed 
 polygon, this method may be employed in simple cases. 
 Such forms are, however, in general better treated by 
 other methods, which will be found in works on the higher 
 geodesy. 
 
 140. Ex. Adjustment of the Quadrilateral WSOLi (Fig. 19). 
 
 The method of forming the condition equations having now been ex- 
 plained, we are ready to adjust the quadrilateral NSOL, as promised in 
 Art. 120. 
 
 The condition equations have all been formed in the preceding sections. 
 Collecting them, we have : 
 
 Local equations (Ex. i, 2, Art. 121) 
 
 v\ + v?. + z> 3 = 1-37 
 w 4 4- v*, v 6 = 1.07 
 
 Angle equations (Ex. Art. 124) 
 
 Vs + V4 + V^ 0.48 
 
 V 6 + VT + V S + Vu = 1. 10 
 
 Side equation, the unit being the sixth place of decimals (Ex. Art. 130), 
 1.43^1 0.92^2 i. 93^6 + 0.747/0 0.43^7 + i.>7z/ 8 = 2.00 
 
 The methods of solution have been explained in Chap. V., and we shall pro- 
 ceed in the order there given for the four forms.
 
 APPLICATION TO TRIANGULATION. 
 
 281 
 
 FIRST SOLUTION METHOD OF INDEPENDENT UNKNOWNS. 
 
 There being 9 unknowns and 5 condition equations connecting them, 
 there must be 4 independent unknowns. We shall choose v t , 7^, v t , 7/0. 
 Expressing all of the unknowns in terms of these four, we write the equations 
 in the form of observation equations, as follows (see Art. 109): 
 
 7'3= + 
 
 Va = - Vi 
 
 7'4 
 
 7-B = 0.5657', + 
 7-3 = 0.435T-1 
 
 -i-37 
 
 i- 
 
 7/4 
 
 weight 2 
 " 2 
 " 14 
 " 23 
 
 6 
 
 + 7-4 + 7-o + 1.07 
 
 7-2 7' 4 + 0.89 
 
 37' 2 0.6617-4 + 0.672775 I.36l 
 37/2 + O.66l7' 4 r.6727'5 1.699 
 
 Hence the normal equations 
 
 7-1 
 
 
 
 4 
 
 V& 
 
 Const. 
 
 
 + 48.83 
 
 + 50.70 
 
 - 3 2 -93 
 
 + 5-44 
 
 - 53-45 
 
 
 
 + 72.45 
 
 - 40.33 
 
 + 24.09 
 
 - 69.70 
 
 
 
 
 + 64.93 
 
 - 2.29 
 
 + 28.18 
 
 
 
 
 
 + 35-82 
 
 29.30 
 
 
 
 
 
 
 + 83.79 
 
 = w 
 
 Solving these equations (page 284), we have the values of the cor- 
 rections 
 
 Vi = O".S2 7/4= O".22 
 
 v-i = o".36 5-0= o"_47 
 
 and thence from the condition equations 
 
 7^3= o".ig Z' K i".33 
 
 z/o = +0^.38 v.j= o".oS 
 
 z/7 = o".o7 
 
 These corrections applied to the measured values of the angles give the most 
 probable values as follows : 
 
 M l - 124 09' 39". 87 
 
 ^/ 2 = ii3 39' 04". 71 
 
 -d/3 = i22 n' 15". 42 
 
 Jlf t = 23 08' 05 ".04 
 
 M,, 47 31' 19". 94 
 
 M a = 70 39' 24". 98 
 
 -rt/7=34 40' 39"- 59 
 
 ^/e = 43 46' 25". 07 
 
 M v ^o 53' 30". 73
 
 282 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The Precision of the Adjusted Values. (a) To find the m. s. e. of an obser- 
 vation of the unit of weight (Arts. 99, 101). 
 From the above values of the residuals v 
 
 [pw] = 7- 53 
 
 Check of [/TO]. Carrying through the solution of the normal equations the 
 extra column required by the sum [///], we find (page 284) 
 
 Hence 
 
 9-4 
 = i".23 
 
 (b) To find the weight and m. s. e. of the adjusted value of an angle. 
 Take the angle NLS. Proceeding as in Art. 101, we have 
 
 F=NLS 
 
 = 180 + (M-i + v* + Mt, + v 6 ) 
 
 .'. C/f= 772 7^6 
 
 Hence from the extra column, the sixth, carried through the solution of the 
 normal equations (page 284), 
 
 U F = 0.053 
 and therefore 
 
 Up- 1.23 1/0.053 
 
 (c) To find the weight and m. s. e. of the adjusted value of a side, the 
 base, NS, being supposed to be free from error. 
 Let us take the side OL. We have 
 
 F= OL 
 
 _ S'JL^^ sin Lso 
 
 ~ sin OLS 
 
 sin {Mi + VT) sin (M + v) 
 For check we shall proceed in two ways.
 
 APPLICATION TO TRIANGULATION. 283 
 
 (i) Expand /'directly; then 
 
 / 6F 6F 6F 6F \ 
 
 dF= ( TTT *' + nrr v + TUT ''> + ^nrr T ' ) sin l 
 \dMs dM* ^M^ dM a ) 
 
 0.05057/3 + 0.02827'e O.Il607-7 0.13427-9 
 
 = 0.0077/1 +0.1717-2 + 0.0567/4 +0.2537/5 
 
 by substituting for 7-3, v a , ?'-, ''a their values from equations i. 
 
 Carry through the solution of the normal equations the extra column re- 
 quired by these coefficients, and (see page 284) 
 
 U F = 0.0019 
 Hence 
 
 I.I F =1.23 V 0.0019 
 = o'".o5 
 
 (2) Take logs, of both members of the equation ; then 
 
 log /'log NS + log sin (M 3 + v 3 ) + log sin (M + z> 6 ) 
 
 log sin (A/, + v-,) log sin (M* + v 
 
 But since JVS is constant, we have, in units of the sixth place of decimals, 
 
 f/log F= 1.337-3 + 0.747-8 3-04^7 3-527'a 
 
 = 0.187-1 + 4.5O7' 2 + 1.457/4 4- 6.637-5 from equations r. 
 
 Hence from the last column addejd to the solution of the normal equations, 
 
 u , o F = 1.50 in units of the sixth place of decimals. 
 Also, 
 
 ^togF =I - 2 3 V ^ 
 
 = 1.5 in units of the sixth place of decimals. 
 Now, since (p. 23) 
 
 dF 
 
 (/log F= mod 
 t 1 
 
 and (p. 255) 
 
 F= 16556^, 
 .-. f.i p o m .o6. 
 
 37
 
 284 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The solution of the normal equations, with the extra columns required by 
 the weight determinations, is as follows : 
 
 -k 
 
 V, 
 
 *4 
 
 * 
 
 I 
 
 /(angle). 
 
 /(side). 
 
 /(side). 
 
 + 48.83 
 
 + 50.70 
 + 72.45 
 
 - 32.Q3 
 - 40.83 
 + 04.93 
 
 + 5-44 
 + 24.09 
 - 2.29 
 + 35.82 
 [///] = 
 
 - 53-45 
 - 69.70 
 + 28.18 
 - 29.30 
 + 83.79 
 
 - i. 
 
 - 0.007 
 + 0.171 
 + 0.056 
 + 0.253 
 
 - 0.18 
 + 4-50 
 + 1-45 
 + 6.03 
 
 + I. 
 
 + 1.0383 
 
 - 0.6743 
 
 + 0.1114 
 
 - 1.0946 
 
 
 - O.OOOI 
 
 - 0.0037 
 
 
 + 19.8082 
 
 - 6.6430 
 + 42.7253 
 
 + 18.4420 
 + 1-3784 
 + 35.2140 
 
 + 14.2038 
 + 7.8652 
 + 23.3454 
 + 25.2836 
 
 - i. 
 o. 
 
 + 0.1761 
 + 0.0527 
 
 + 0.2535 
 
 0. 
 
 + 4.6871 
 + 1.3285 
 + 6.6501 
 + 0.0007 
 
 
 + I. 
 
 - 0.3354 
 
 + 0.9310 
 
 + 0.7176 
 
 + 0.0505 
 
 + 0.0089 
 
 + 0.2366 
 
 
 
 + 40.4972 
 
 + 7.5630 
 + 18.0445 
 
 + 12.6322 
 + 10.1114 
 + 15.0910 
 
 + 0.0505 
 
 + o.uiS 
 + 0.0894 
 
 + O.OOIO 
 
 + 2.9002 
 + 2.2867 
 + 1.1089 
 
 
 + I. 
 
 + 0.1868 
 
 + 0.3119 
 
 - 0.0083 
 
 + 0.0027 
 
 + 0.0716 
 
 
 
 + 16.6317 
 
 + 11.1516 
 
 - 0.0057 
 + 0.0028 
 
 + 0.0690 
 + 0.0003 
 
 + 1.7452 
 + 0.2076 
 
 
 + i. 
 
 + 0.4661 
 
 - 0.0004 
 
 + 0.0003 
 
 + 0.1049 
 
 
 [pvv\ = 
 
 + 7-5376 
 
 o. 
 
 o.oooo 
 0.0505 
 0.0028 
 o.oooo 
 
 o. 
 
 o.oooo 
 0.0016 
 0.0003 
 o.oooo 
 
 + 0.1832 
 
 0.0007 
 1.1089 
 0.2076 
 0.1832 
 
 
 0.0533 
 
 J f 
 
 1.5004 
 
 The solution has been carried through to four places of decimals, on account 
 of loss of accuracy arising from dropping figures in multiplications. The re- 
 sulting values of the corrections have been cut down to two places of deci- 
 mals. The work was done with a machine, as explained on p. 161, the recip- 
 rocals of the diagonal terms being used so as to avoid divisions. Thus the 
 first reciprocal is 0.02048. 
 
 SECOND SOLUTION METHOD OF CORRELATES. 
 Arranging the condition equations in tabular form, we have 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 weights 2 
 
 2 
 
 J4 
 
 23 
 
 6 
 
 7 
 
 31 
 
 I 
 
 8 
 
 
 i 1-43 
 
 - 0.92 
 
 
 
 - i-93 
 
 + 0.74 
 
 - 0.43 
 
 + 1.77 
 
 
 - 2.00 
 
 (- i. 
 
 + I. 
 
 + I. 
 
 
 
 
 
 
 
 - 1.37 
 
 
 
 + I. 
 
 + I. 
 
 
 
 + I. 
 
 
 
 - 0.48 
 
 
 
 
 + I. 
 
 + i. 
 
 - i. 
 
 
 
 
 - 1.07 
 
 
 
 
 
 
 + i. 
 
 + I. 
 
 + I. 
 
 + i. 
 
 - 1. 10
 
 APPLICATION TO TRIANGULATION. 
 
 The Correlate Equations. 
 
 232/4 = 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 + 1-43 
 
 + i. 
 
 
 
 
 
 -0.92 
 
 + i. 
 
 
 
 
 
 + i. 
 
 + i. 
 
 
 
 
 
 + i. 
 
 + I. 
 
 
 -i-93 
 
 
 
 + I. 
 
 
 + 0.74 
 
 
 
 I. 
 
 + I. 
 
 -0.43 
 
 
 4- i. 
 
 
 + I. 
 
 + 1-77 
 
 
 
 
 + I. 
 
 
 
 
 
 + I. 
 
 The Normal Equations. 
 
 I. 
 
 II. 
 
 III. 
 
 - IV. 
 
 V. 
 
 / 
 
 + 5.284 
 
 + 0.255 
 
 0.014 
 
 -0.427 
 
 + 1.862 
 
 2.OO 
 
 
 + 1.071 
 
 + 0.071 
 
 
 
 -1-37 
 
 
 + 0.147 
 
 + 0.043 
 
 + 0.032 
 
 0.48 
 
 
 
 
 + 0.353 
 
 -0.143 
 
 -1.07 
 
 
 
 
 
 + 1.300 
 
 I.IO 
 
 The solution of these equations gives (see page 287) 
 
 I. = 0.3973 
 II. = - 1.0749 
 
 III. = - i. 6006 
 
 IV. = -3.5/21 
 V. = 0.6301 
 
 Substituting these values in the correlate equations, the same values of the 
 corrections result as before. Also, 
 
 = 7- 53
 
 286 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The Precision. (a) To find the m. s. e. /< of an observation of weight unity. 
 From the values of v we find directly 
 
 [pvv] = 7-53 
 
 Checks of [pf'v~\. These are worked out in the sol ution of the normal equa- 
 tions on page 287, according to the formulas of Art. in, and give 7.54 and 7.55 
 respectively. 
 
 Hence taking the mean, [/w] = 7-54, and the number of conditions 
 being 5, 
 
 = i ".23 as before. 
 
 Compare Ex. 2, Art. in. 
 
 (b) To find the weight and m. s. e. of the adjusted value of an angie. 
 Take the angle NLS. 
 
 From the values of u, a, !>,... in the condition equations in connection with 
 the values of f given by this function, we have 
 
 [naf] = + 0.782 [<(/"] = 0.167 
 
 \ubf~\ 0.500 [ U /] = o. 
 
 [ucf] = o. [uff] = +0.667 
 
 Hence from the seventh column in the solution of the normal equations 
 (page 287), 
 
 u F = 0.053 
 and 
 
 fi p = 1.23 o.o53 
 
 = 0".2S 
 
 Compare Ex. 4, Art. in. 
 
 (c) To find the weight and mean-square error of the adjusted value of a 
 side, the base being free from error. 
 
 Take the side Oneota-Lester. 
 
 As in (c), page 282, we have 
 
 dF~= 0.05057/3 + 0.02822/6 0.1160277 0.13422/9 
 Also from the condition equations 
 
 [uaf] = + 0.0046 ["/] = 0.0040 
 
 [/] = 0.0036 [*/] = 0.0165 
 
 [ucf] = 0.0073 [//] = + 0.0030
 
 APPLICATION TO TRIANGULATION. 287 
 
 Hence from the eighth column in the solution of the normal equations, 
 
 UP 0.0023 
 and finally, 
 
 HP 1.23 V.OO23 
 = o".o6 
 
 Solution of the Normal Equations. 
 
 I. 
 
 ii. 
 
 III. 
 
 IV. 
 
 V. 
 
 I 
 
 /(angle). 
 
 /(side). 
 
 4 5-284 
 
 
 
 
 
 
 
 
 + 1.071 
 
 4 O.C7I 
 + 0.147 
 
 + 0.043 
 + 0.353 
 
 4 O.O32 
 - 0.143 
 4- I.30O 
 
 - 1-37 
 - 0.48 
 - 1.07 
 
 - I. 10 
 
 - 0.500 
 - 0.167 
 
 4 0.667 
 
 - 0.0036 
 - 0.0073 
 - 0.0040 
 - 0.0165 
 
 4 0.0030 
 
 + I. 
 
 4 0.0483 
 4 1.0587 
 
 - O.OO26 
 
 4 0.0717 
 4 0.1470 
 
 - 0.0808 
 
 4 O.O2O6 
 4 0.0419 
 4 0.3185 
 
 4- 0.3524 
 
 - O.OQ02 
 4 0.0369 
 
 4 0.0075 
 4- 0.6438 
 
 - 0.3785 
 
 - 0.4853 
 - 1.2316 
 - 0.3952 
 + 0.7570 
 
 4 0.1480 
 
 - 0.5377 
 4 O.002I 
 - 0.1035 
 - 0.2756 
 4 O.55IO 
 
 + O.OOO9 
 
 - 0.0038 
 - 0.0073 
 - 0.0036 
 - 0.0182 
 
 
 4 I. 
 
 4 0.0677 
 4 O.I42I 
 
 4 0.0204 
 
 4 0.0404 
 4 0.3l8t 
 
 - 0.0851 
 
 4 0.0434 
 4 O.O093 
 + 0.6361 
 
 - 1.2025 
 
 - 0.3991 
 - 1.2068 
 - 0.5037 
 
 4 I . 5309 
 
 - o 5078 
 
 4 0.0385 
 - 0.0930 
 - 0.3214 
 4 0.2780 
 
 - o 0036 
 
 - O.OO/O 
 - 0.0035 
 - 0.0185 
 4 O.O030 
 
 Values of the Unknowns 
 
 I. = - 0.39 
 II. = - 1.07 
 
 m.= - 1.60. 
 
 l l~ = ~ 3 'E 
 \ . = - 0.631 
 
 4 1. 
 
 4 0.2843 
 4 0.3066 
 
 + 0.3054 
 
 - 0.0030 
 + 0.6228 
 
 - 2.8o86 
 - 1.0933 
 
 - o 3818 
 
 4 I.I209 
 
 4- 0.2709 
 
 - 0.1039 
 - 0.3332 
 4 0.2676 
 
 - 0.0493 
 
 - O.OOI5 
 - 0.0164 
 
 + o 0027 
 
 73 
 
 % 
 il 
 
 31 
 
 4 I. 
 
 - 0.0098 
 4 0.6228 
 
 - 3.5659 
 
 - 0.^924 
 4 3.8986 
 
 - 0.3389 
 
 - 0.3342 
 + 0.2324 
 
 - O.0049 
 
 - 0.0164 
 4 0.0027 
 
 
 4 I. 
 
 - o 6301 
 
 + 0.2472 
 
 - 0.5366 
 
 4 0.0531 
 
 - 0.0264 
 4- 0.0023 
 
 
 I. 1' = - o 3973 x - 2.00 = 0.79 
 II. /" - 1.0749 x - 1.37= 1.47 
 III. 1'" = - 1.6006 x -0.48 = 0.77 
 IV. /"" - 3.^721 x _ 1.07 = 3.82 
 V. /"'"=- 0.6301 x - 1.10 = 0.69 
 
 0.7^70 
 1.5309 
 1 .12Oq 
 3.8986 
 0.2472 
 
 
 
 
 7-54 
 
 7.5546 
 
 The values of [/>fz p ] are found from equations 2, 3, Art. in.
 
 288 THE ADJUSTMENT OF OBSERVATIONS. 
 
 THIRD SOLUTION SOLUTION IN Two GROUPS. 
 The form given in Art. 113 is followed. 
 
 The Local Adjustment. 
 
 (a) At North Base. 
 The Observation Equations. 
 
 p 
 
 c*o 
 
 (* a ) 
 
 / 
 
 2 
 
 + I 
 
 
 O. 
 
 2 
 
 
 + I 
 
 0. 
 
 14 
 
 I 
 
 I 
 
 -1-37 
 
 The Normal Equations. 
 
 (*i) (**) 
 
 16 + 14 19.18 = [pal] suppose 
 14 + 16 19.18 = 
 
 Solving in general terms. 
 
 ( Xl ) = + 0.267 [pal] 0.233 
 
 Hence 
 
 (jr 2 ) = - 0.233 [pal] + 0.267 [pbl] 
 
 (x t ) = o".64 
 (jT 2 ) = o".64 
 
 (x 3 ) = + o".64 + 0^.64 i".37 
 = o".og 
 
 and 
 
 Local Angles. 
 124 09' 40". 05 
 113 39' 04". 43 
 
 122 II' 15". 52 
 
 To find the m. s. e. of a single observation. 
 The value of [pvv] = [pxx] = 1.75. 
 Hence for this station, the number of conditions being 32, 
 
 =1 /I 
 
 r o 
 
 = i -3
 
 APPLICATION TO TRIANGULATION. 
 
 (b) At South Base. 
 The Observation Equations. 
 
 / 
 
 (*0 
 
 (*) 
 
 / 
 
 23 
 
 + I 
 
 
 o. 
 
 6 
 
 
 +.1 
 
 o. 
 
 7 
 
 + I 
 
 + I 
 
 -1.07 
 
 The Normal Equations. 
 
 289 
 
 Hence 
 
 Also, 
 
 30 + 7 = - 7.49 
 7 + 13 = -7-49 
 
 (jr 6 ) = o".so 
 
 (.*) = o".i3 o".5O + i ".07 
 = + o"-44 
 
 Local Angles. 
 
 23 08' 05". 13 
 
 47 3i' IQ"^ 1 
 70 39' 25". 04 
 
 = 3.24 
 
 The General Adjustment. 
 Most Probable Angles. 
 
 At N. Base, 124" 09' 40". 05 + (i) 
 113 39' 04". 43 + (2) 
 122 n' 15". 52 (i) (2) 
 
 At S. Base, 23 08' 05". 13 + (4) 
 
 47 3i' i9"-<)i + (5) 
 
 70 39' 25".04 + (4) + (5) 
 
 At Oneota, 34 40' 39". 66 + (7) 
 43 46' 26".40 + (8) 
 
 At Lester, 30 53' 30". Si + (9)
 
 290 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The Angle and Side Equations. 
 
 t 
 
 (a) Triangle, N. Base, S. Base, Oneota. 
 
 Angle SNO 122 n' 15". 52 (i) (2) 
 " NSO 23 08' 05". 13 + (4) 
 " NOS 34 40' 39". 66 + (7) 
 Sum = 180 oo' oo".3i 
 180 + 180 oo' oo'.os 
 
 o=o".26-(i)-( 2 ) + ( 4 ) + (7) 
 
 (b) Triangle Lester, Oneota, S. Base. 
 
 Angle NSO 70 39' 25". 04 + (4) + (5) 
 SOL 78 27' 06". 06 + (7) + (8) 
 OLS 30 53' 30". 81 + (9) 
 
 180 oo' oi".9i 
 180 oo' oo".37 
 
 0=1 ".54 + (4) + (5) + (7) + (8) + (9) 
 
 (c) Quadrilateral N. Base, S. Base, Oneota, Lester. 
 
 sin LNS sinLSO sin LOW _ 
 
 sin LNO sin NSL sin LOS ~ 
 
 LNS = 113 39' 04". 43 + (2) LNO 124 09' 40". 05 + (i) 
 
 LSO 70 39' 25". 04 + (4) + (5) NSL = 47 31' 19". 91 + (5) 
 
 LON= 43 46' 26". 40 + (8) LOS = 78 27' 06". 06 + (7) + (8) 
 
 9.9618975,6 9,22 (2) 9.9177479,3 14,29(1) 
 
 9.9747660,1+ 7,391(4) + (S)} 9.8677849,8 + 19,28(5) 
 
 9.8399903,4 + 21,98(8) 9.9911180,3+ 4,3o|(7) + (8)[ 
 
 539.1 509.4 
 
 509,4 
 
 29,7
 
 APPLICATION TO TRIANGULATION. 29! 
 
 Check by deducting J- of the spherical excesses of the triangles from the 
 angles. 
 
 113 39' 04". 36 124 09' 40". 01 
 
 70 39' 24". 92 47 31' 19". 84 
 
 43 46' 26". 36 78 27' 05". 93 
 
 9.9618976,2 9.9177479,9 
 
 9-9747659.3 9.8677848,6 
 
 9.8399902,5 9.9911179,8 
 
 38,0 8,3 
 8,3 
 
 29,7 
 The two methods agree well. 
 
 A glance at the log. differences for i" shows that by expressing them in 
 units of the sixth place of decimals their average value is unity nearly. We 
 have, then, for the side equation, 
 
 i. 43(1) -0.92(2) + 0.74(4)- 1. 19(5) -o.43(7) + 1-77(8) + 2.97 = 
 
 The Weight Equations. 
 (0 = 0.233 |jj +0.267 |~2~| 
 (2) = +0.267 I i | 0.233 I 2 | 
 
 (4) = +0.038 | 4 | 0.021 | j>J 
 
 (5)= 0.021 j~4~j +0.088 |~5~j 
 
 (7)= +0.032 |~7~| 
 
 (8)= +i.ooo|T] 
 
 (9)= +0.125 | 9 | 
 
 The Correlate Equations. 
 
 I. 
 
 ii. 
 
 in. 
 
 Check. 
 
 Qj= -' 
 
 
 + 1.43 
 
 - 0-43 
 
 |_2J = - I 
 
 
 0.92 
 
 + 1.92 
 
 |Tj= +i 
 
 +- 1 
 
 + 0.74 
 
 - 2.74 
 
 ULJ = 
 
 + 1 
 
 - 1.19 
 
 + 0.19 
 
 IT) = + i 
 
 + 1 
 
 0.43 
 
 1-57 
 
 |_8J = 
 
 + 1 
 
 + 1.77 
 
 - 2.77 
 
 nn = 
 
 + 1 
 
 
 1. 00 
 
 The check is formed by adding each horizontal row (Art. 84). 
 38
 
 292 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Expression of the Corrections in Terms of the Correlates. 
 
 
 I. 
 
 + 0.233 
 0.267 
 
 II. 
 
 III. 
 ~ 0.333 
 0.246 
 
 Check. 
 4- O. IOO 
 
 + 0.513 
 
 - 0.034 
 
 - 0.579 
 
 + 0.613 
 
 (2) = 
 
 0.267 
 + 0.233 
 
 
 + 0.382 
 + 0.214 
 
 O.II5 
 - 0.447 
 
 - 0.034 
 
 + 0.596 
 
 0.562 
 
 (4) = 
 
 + 0.038 
 
 + 0.038 
 O.02I 
 
 + 0.028 
 + O.O24 
 
 O.IO4 
 0.004 
 
 + 0.038 
 
 + 0.017 
 
 + 0.052 
 
 O.IO8 
 
 (5) = 
 
 O.O2I 
 
 O.02I 
 + 0.088 
 
 0.016 
 0.105 
 
 + 0.058 
 + 0.017 
 
 
 + 0.067 
 
 O.I2I 
 
 + 0.075 
 
 (7) = 
 
 + 0.032 
 
 + O.O32 
 
 0.014 
 
 0.050 
 
 (8) = 
 
 
 + I. 
 
 + 1.770 
 
 - 2.770 
 
 (9) = 
 
 
 + 0.125 
 
 
 O.I25 
 
 The Corrections in Terms of the Correlates (collected}. 
 
 (2) = 
 
 (4) = 
 
 (5) = 
 
 (7) = 
 
 (8) = 
 
 (9) = 
 
 I. 
 
 - 0.034 
 
 - 0.034 
 + 0.038 
 
 O.O2I 
 + 0.032 
 
 II. 
 
 + 0.017 
 + 0.067 
 + 0.032 
 + I.OOO 
 
 + 0.125 
 
 III. 
 
 - 0.579 
 + 0.596 
 + 0.052 
 
 0. 121 
 
 O.OI4 
 + 1.770 
 
 Formation of the Normal Equations. 
 
 -(2) 
 
 + (4) 
 + (7) 
 
 I. 
 
 + 0.034 
 + 0.034 
 + 0.038 
 + 0.032 
 
 [+ 0.138 
 
 + 0.017 
 + 0.032 
 
 + 0.049 
 
 III. 
 + 0.579 
 
 0.596 
 4- 0.052 
 
 0.014 
 
 O.O2I 
 
 Check. 
 
 O.6I3 
 + 0.562 
 
 o. 108 
 
 0.050 
 
 + o. 209 
 
 1(4) 
 (5) 
 i<7) 
 (8) 
 (9) 
 
 + 0.017 
 + 0.067 
 + 0.032 
 + I. 
 + 0.125 
 
 + O.O52 
 
 0. 121 
 
 0.014 
 
 + 1.770 
 
 0.108 
 + 0.075 
 
 0.050 
 - 2.770 
 
 0.125 
 
 + 0.049 
 
 + 1.241 
 
 1.687 
 
 - 2.978
 
 APPLICATION TO TKIANGULATION. 
 
 293 
 
 II. 
 
 - 0.92(2) 
 + 0.74(4) 
 
 - i.i'X5) 
 
 - 0.43(7) 
 + 1-77(8) 
 
 1.687 
 
 III. 
 
 + 0.852 
 + 0.533 
 -f- 0.038 
 + 0.144 
 + 0.006 
 + 3-133 
 
 + 4.706 
 
 Check. 
 
 0.804 
 
 - 0.564 
 
 o.oSo 
 
 0.089 
 
 + O.O22 
 
 - 4-93 
 
 - 6.418 
 
 The Normal Equations (collected}. 
 
 I. II. III. 
 
 + 0.138 + 0.049 + 0.021 = 0.260 
 
 + 1.241 + 1.687 = 1.540 
 
 + 4.706= 2.970 
 
 The solution of these equations gives (page 295) 
 
 I. = - 1.597 
 
 II. = -0.642 
 
 III. = 0.394 
 
 Substitute for I., II., III. their values in (4), and we have the general cor- 
 rections. 
 
 Adding the local corrections and general corrections together, the total 
 corrections to the measured angles result and are as follows : 
 
 Local. 
 
 General. 
 
 Total. 
 
 p 
 
 pvv 
 
 Final Angles. 
 
 -fi = 
 
 o" 
 
 64 
 
 o" 
 
 18 
 
 = o" 
 
 82 
 
 2 
 
 i-34 
 
 124 
 
 09' 
 
 39". 7 
 
 -'2 = 
 
 o" 
 
 64 
 
 + o" 
 
 28 
 
 = o" 
 
 36 
 
 2 
 
 .26 
 
 H3 
 
 39' 
 
 4"-7i 
 
 JCa = 
 
 o" 
 
 09 
 
 o" 
 
 10 
 
 = o" 
 
 19 
 
 14 
 
 50 
 
 122 
 
 II 
 
 i5"-42 
 
 -T4 = 
 
 o" 
 
 13 
 
 o" 
 
 09 
 
 = o" 
 
 22 
 
 23 
 
 I.IO 
 
 23 
 
 08' 
 
 5"- 04 
 
 -V 5 = 
 
 o" 
 
 50 
 
 + o" 
 
 04 
 
 = o" 
 
 4 6 
 
 6 
 
 1.27 
 
 47 
 
 3i' 
 
 !9"-95 
 
 X* = 
 
 + o" 
 
 44 
 
 o" 
 
 05 
 
 = + o" 
 
 39 
 
 7 
 
 i. 06 
 
 70 
 
 39' 
 
 24 ".99 
 
 X 7 = 
 
 
 
 o" 
 
 07 
 
 = o" 
 
 o? 
 
 3i 
 
 15 
 
 34 J 
 
 40' 
 
 39"- 59 
 
 X f = 
 
 
 
 l" 
 
 33 
 
 I " 
 
 33 
 
 i 
 
 i-77 
 
 43" 
 
 46' 
 
 25"-07 
 
 -r a = 
 
 
 
 o" 
 
 08 
 
 = o" 
 
 08 
 
 8 
 
 05 
 
 3 ' 
 
 53' 
 
 30"- 73 
 
 
 
 
 
 
 
 
 [/H 
 
 = 7-50 
 
 
 
 
 Number of local conditions = 2 
 Number of general conditions = 3 
 
 Total = 5 
 
 The method of solution just given is substantially the same as that em- 
 ployed on the survey of the Great Lakes between Canada and the United 
 States by the U. S. Engineers.
 
 294 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The Precision of the Adjusted Values. 
 
 (a) To find the m. s. e. of an observation of weight unity. 
 
 Computation of [/r^]. 
 
 (r) From the preceding table [/w] has been found directly ; thus 
 
 [>] = 7. 50 
 
 (2) Check (Art. 114). From the station adjustments find [vv] 
 
 N. Base gives (p. 288) 1.75 
 S. Base gives (p. 289) 3.24 
 
 4.99 = [z/V]. 
 
 From the general adjustment find [ww]. 
 
 (<r] /' X I. = -0.26 X - 1-597= +0.42 
 /" X II. = 1-54 X 0.642= + 0.99 
 /"' X III. = 2.97 X 0.394= + 1. 16 
 
 + 2.57 
 
 (/5)/ ' X f===f=- 0.26 X -1.885= +0.49 
 
 LAj 
 
 /o"l 
 
 / .1 X F =z: = - 1.45 X 1.183= + 1.72 
 OB.I] 
 
 /o'"2 
 
 / .2 Xr--=^ = -94 X 0.394 = + 0.37 
 [fC.2] 
 
 + 2.58 
 
 .'. [ww] 2.58 
 
 and [p-cn>] =4.99+ 2.58 
 
 = 7-57 
 Hence taking the mean of the values of \_pvv\, 
 
 there being 2 local conditions and 3 net conditions. 
 
 (b) To find the m. s. e. of an angle in the adjusted figure. 
 Angle = NLS 
 
 ... ^=_( 2 )_( 5 ) 
 
 = + O.O55 I- 0.067 II- + O.7OO III. 
 
 from the weight equations. 
 
 From equations 25, Art. 114,
 
 APPLICATION TO TRIANGULATION. 295 
 
 The values of [anr], [a/if] . . . are given in the weight equations. Hence 
 
 q\ =+0.267 X 0.233 X i =+0.233 
 q-i = o. 233 X o + o. 267 X I = o. 267 
 q 3 = +0.038 X o 0.021 X I = +0.021 
 q^ = 0.021 X + 0.088 X I = 0.088 
 
 g 
 
 1 
 
 gq 
 
 o 
 
 + 0.233 
 
 o. 
 
 I 
 
 0.267 
 
 0.267 
 
 o 
 
 + O.O2I 
 
 o. 
 
 I 
 
 0.088 
 
 0.088 
 
 l> 
 
 = O.O22. 
 
 (See the solution of the 
 normal equations.) 
 
 [VC.2] 
 
 = 0.006 
 
 = 0.274 
 
 0.302 
 
 = -355 
 0.302 
 
 u F = + 0.053 
 
 .'. /v = i".23 Vo.053 
 
 = 0".28 
 
 (c) To find the m. s. e. of a side in the adjusted figure. 
 Side = Oneota-Lesier. 
 
 i " OSL 
 
 --- 
 
 sin SON sin OLS 
 
 Therefore 
 
 dF = 1.33(1) + L33(2) + 0.74(4) + o.74(5) - 3-04(7) - 3-52(9) 
 in units of the sixth place of decimals, 
 
 0.174 I - 0.475 II- + - OI 5 HI- 
 
 from the weight equations. 
 
 The solution is carried through exactly as in the preceding case. We find 
 
 [gq~\ 2.01 r and U F 1.49 
 Hence H VU F = 1.23 V 1.49 
 
 = 1.5 in units of the sixth place of decimals. 
 
 Now log 0Z = 4.2189699 OL- 16556"* 
 
 16556 
 
 . '. m. s. e. of side = - X 0.0000015 
 o.434 
 
 = o w .o6
 
 296 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Solution of the Normal Equations. 
 
 I. 
 
 II. 
 
 III. 
 
 / 
 
 /(angle) 
 
 + 0.138 
 
 + 0.049 
 + 1.241 
 
 + O.O2I 
 + 1.687 
 
 + 4- 706 
 
 0.260 
 - 1-540 
 - 2.970 
 
 + 0.055 
 0.067 
 + 0.700 
 
 + 1. 
 
 + 0.355 
 
 + 0.152 
 
 - 1.885 
 
 + 0.399 
 
 
 + 1.224 
 
 + I. 680 
 + 4.703 
 
 - 1.448 
 2.930 
 
 0.087 
 + 0.692 
 
 
 
 
 
 + O.O22 
 
 
 + I. 
 
 + 1-373 
 
 -1.183 
 
 O.O7I 
 
 
 
 + 2.396 
 
 - 0.945 
 
 + 0.8II 
 + O.OO6 
 
 
 + I. 
 
 -0.394 
 
 + 0.338 
 
 
 
 
 + 0.274 
 
 Ex. i. Adjust the observed differences of longitude* given in the follow- 
 ing table : 
 
 FOILHOMMERUM 
 CONTENT 
 
 Fig.36 
 
 Dates. 
 1851 
 
 h. 
 
 Cambridge-Bangor, o 
 
 Observed Differences. 
 . s. . 
 9 23.080 0.043 
 
 1857 
 
 Bangor-Calais, 
 
 6 
 
 00.316 0.015 
 
 1866 
 
 Calais-Heart's Content, 
 
 55 
 
 37-973 0.066 
 
 1866 
 
 Heart's Content-Foilhommerum, 2 
 
 5i 
 
 56.356 0.029 
 
 1866 
 
 Foilhommerum-Greenwich, 
 
 4i 
 
 33.336 0.049 
 
 1872 
 
 Brest Greenwich, 
 
 17 
 
 57.598 O.O22 
 
 1872 
 
 Brest-Paris, 
 
 27 
 
 18.512 0.027 
 
 1872 
 
 Greenwich-Paris, 
 
 9 
 
 2I.OOO 0.038 
 
 1872 
 
 St. Pierre-Brest, 3 
 
 26 
 
 44.810 0.027 
 
 1872 
 
 Cambridge-St. Pierre, 
 
 59 
 
 48.608 O.O2I 
 
 1869-1870 
 
 Cambridge-Duxbury, 
 
 i 
 
 5O.I9I O.O22 
 
 1870 
 
 Duxbury-Brest, 4 
 
 2 4 
 
 43.276 0.047 
 
 (1867 
 <I8 7 2 
 
 Washington-Cambridge, 
 
 23 
 
 41.041 o.oiS 
 
 1872 
 
 Washington-St. Pierre, i 
 
 23 
 
 29-553 0.027 
 
 
 * Coast and Geodetic Survey Report, 
 
 1880, 
 
 app. No. 6. 
 
 Corrections.
 
 APPLICATION TO TRIANGULATION. 
 
 297 
 
 [Number of conditions = w s + i, where n is number of observed dif- 
 ferences of longitude, and s is number of longitude stations. 
 
 The condition equations are 
 
 i. 
 
 V +V^ 7'g = + 0.086 
 
 7'i T'a 7> 3 V 4 Z>5 + Vt + Vg + Vio = + 0.045 
 
 Vg V\n + V\\ + Vu 0.049 
 
 + Vio + 7'13 Vn = 0.096 
 
 The weights are taken inversely as the squares of the p. e. 
 Solution by method of correlates, as in Art. no.] 
 
 Ex. z. The system of triangulaiion shown in the figure was executed by 
 Koppe in the deter- 
 mination of the axis 
 (Airolo - Goschenen) 
 of the St. Gothard 
 tunnel.* In the fol- 
 lowing table the ad- 
 justed values are giv- 
 en side by side with 
 the measured values. 
 It is proposed as a QOSCH 
 problem of adjust- 
 
 lent. 
 At Goschenen. 
 
 Measured. 
 
 VII VI 
 
 Adjusted. 
 
 At IV. 
 
 Measured. Adjusted. 
 
 II. 
 
 
 
 oo' 
 
 oo". 
 
 OO 
 
 oo". 
 
 oo 
 
 V. 
 
 O 
 
 oo' 
 
 oo".oo 
 
 oo".oo 
 
 III. 
 
 44 
 
 33' 
 
 10". 
 
 88 
 
 10". 
 
 03 
 
 VI. 
 
 15 
 
 41' 
 
 3"-57 
 
 6". 29 
 
 IV. 
 
 69 
 
 30' 
 
 12" 
 
 5i 
 
 n". 
 
 62 
 
 VII. 
 
 74 
 
 1 2 
 
 20". 55 
 
 19". 86 
 
 V. 
 
 124" 
 
 58' 
 
 4" 
 
 23 
 
 5". 
 
 13 
 
 Goschenen 
 
 80 
 
 32' 
 
 48". 99 
 
 SO". 12 
 
 
 
 
 
 
 
 
 II. 
 
 135 
 
 44' 
 
 49"-77 
 
 50". 91 
 
 At II. 
 
 
 
 
 
 
 
 III. 
 
 199 
 
 24' 
 
 ii".56 
 
 io".73 
 
 III. 
 
 o" 
 
 oo' 
 
 oo". 
 
 oo 
 
 oo". 
 
 00 
 
 A * 17 
 
 
 
 
 
 
 
 
 
 
 
 
 At V. 
 
 
 
 
 
 IV. 
 
 37 
 
 53' 
 
 54" 
 
 33 
 
 52". 
 
 97 
 
 IV. 
 
 o 
 
 oo' 
 
 oo".oo 
 
 oo'.oo 
 
 V. 
 
 60 
 
 29' 
 
 33" 
 
 13 
 
 33" 
 
 .82 
 
 VIII. 
 
 78 
 
 40' 
 
 s'-g 1 
 
 6". 72 
 
 VI. 
 Goschenen 
 
 77" 
 93 
 
 4' 
 
 n' 
 
 5" 
 4i" 
 
 .67 
 .69 
 
 8". 17 
 40". 5 7 
 
 IX. 
 VI. 
 
 140 
 215 
 
 44' 
 
 32' 
 
 43"-5i 
 45"- 4i 
 
 44"-45 
 
 43"- 45 
 
 VII. 
 
 124" 
 
 16' 
 
 33" 
 
 .98 
 
 33". 27 
 
 VII. 
 
 286 
 
 19' 
 
 25". 30 
 
 27". 21 
 
 At III. 
 
 
 
 
 
 
 
 Goschenen 
 
 316 
 
 oo' 
 
 44". 92 
 
 43"- 61 
 
 VIII. 
 
 o 
 
 oo' 
 
 oo" 
 
 .00 
 
 oo".oo 
 
 II. 
 
 338 
 
 20' 
 
 33"-53 
 
 3i"-74 
 
 IX. 
 
 53 
 
 58' 
 
 14" 
 
 .48 
 
 15" 
 
 49 
 
 At VII 
 
 
 
 
 
 VI. 
 
 99 
 
 47' 
 
 5o" 
 
 .21 
 
 50" 
 
 .86 
 
 II. 
 
 o" 
 
 oo' 
 
 oo".oo 
 
 oo'.oo 
 
 IV. 
 
 102" 
 
 32' 
 
 5i" 
 
 36 
 
 51" 
 
 .90 
 
 III. 
 
 T 9 
 
 n' 
 
 58". 44 
 
 59"-3 
 
 Goschenen 
 
 138 
 
 44' 
 
 28" 
 
 .Si 
 
 29" 
 
 .70 
 
 IV. 
 
 32 
 
 4' 
 
 49". 32 
 
 4S".6S 
 
 VII. 
 
 I44~ 
 
 28' 
 
 12" 
 
 47 
 
 Jl" 
 
 .40 
 
 V. 
 
 64 
 
 n' 
 
 54"-oS 
 
 56". 05 
 
 II. 
 
 i So 
 
 59' 
 
 38" 
 
 94 
 
 39" 
 
 .n 
 
 VI. 
 
 90 
 
 05' 
 
 39"-47 
 
 37". oo 
 
 * Zcitst.hr. fur I 
 
 'eriess.< vol. iv. 
 
 
 

 
 298 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 At VIII. 
 
 At XI. 
 
 
 
 
 
 XI. 
 
 o 
 
 oo' 
 
 oo" 
 
 .00 
 
 oo" 
 
 ,00 
 
 XII. 
 
 o 
 
 oo' 
 
 oo".oo 
 
 oo".oo 
 
 XII. 
 
 18 
 
 56' 
 
 17" 
 
 43 
 
 17" 
 
 54 
 
 Airolo 
 
 16 
 
 55' 
 
 55".o6 
 
 54"-38 
 
 X. 
 
 43 
 
 50' 
 
 24" 
 
 03 
 
 24", 
 
 70 
 
 IX. 
 
 37 
 
 13' 
 
 59"- 79 
 
 58".43 
 
 IX. 
 
 50 
 
 18' 
 
 22" 
 
 52 
 
 20", 
 
 27 
 
 VIII. 
 
 152 
 
 26' 
 
 30". 24 
 
 30". 44 
 
 VI. 
 
 106 
 
 3o' 
 
 15" 
 
 .04 
 
 15" 
 
 37 
 
 
 
 
 
 
 V. 
 
 112 
 
 
 28" 
 
 .72 
 
 29", 
 
 24 
 
 
 
 
 
 
 III. 
 
 130 
 
 n' 
 
 30" 
 
 .81 
 
 4i". 
 
 54 
 
 At XII 
 
 
 
 
 
 At IX. 
 
 
 
 
 
 
 
 IX. 
 
 o 
 
 oo' 
 
 oo".oo 
 
 oo".oo 
 
 VI. 
 
 o 
 
 oo' 
 
 oo" 
 
 .00 
 
 oo", 
 
 ,00 
 
 Airolo 
 
 30 
 
 31' 
 
 2". 30 
 
 3"- 39 
 
 V. 
 
 8 
 
 28' 
 
 17" 
 
 13 
 
 15". 
 
 ,06 
 
 X. 
 
 42 
 
 13' 
 
 20". 53 
 
 2i"-33 
 
 III. 
 
 18 
 
 33' 
 
 3" 
 
 27 
 
 5" 
 
 .00 
 
 VIII. 
 
 90 
 
 3' 
 
 2". 22 
 
 i ".74 
 
 VIII. 
 
 63 
 
 4i' 
 
 28" 
 
 .63 
 
 28". 
 
 55 
 
 XI. 
 
 98 
 
 40' 
 
 i4"-95 
 
 13". 72 
 
 X. 
 
 76 
 
 59' 
 
 5o" 
 
 .89 
 
 5i. 
 
 48 
 
 
 
 
 
 
 XL 
 
 79 
 
 10' 
 
 36" 
 
 33 
 
 36". 
 
 34 
 
 At Airolo. 
 
 Airolo 
 
 109 
 
 45' 
 
 39" 
 
 23 
 
 39"' 
 
 33 
 
 
 
 
 
 
 XII. 
 
 123 
 
 16' 
 
 23" 
 
 .76 
 
 24". 
 
 23 
 
 XI. 
 
 o 
 
 oo' 
 
 oo".oo 
 
 oo".oo 
 
 
 
 
 
 
 
 
 XII. 
 
 94 
 
 54' 
 
 56". 06 
 
 55". 26 
 
 At X. 
 
 
 
 
 
 
 
 IX. 
 
 230 
 
 53' 
 
 7". 5i 
 
 6". 98 
 
 XII. 
 
 o 
 
 oo' 
 
 oo" 
 
 .00 
 
 oo". 
 
 oo 
 
 X. 
 
 296 
 
 26' 
 
 49"-43 
 
 51". n 
 
 Airolo 
 
 9 
 
 49' 
 
 30" 
 
 .02 
 
 37". 
 
 92 
 
 
 
 
 
 
 IX. 
 
 9i 
 
 30' 
 
 5" 
 
 .16 
 
 5" 
 
 .96 
 
 
 
 
 
 
 VIII. 
 
 252 
 
 43' 
 
 46" 
 
 -75 
 
 47" 
 
 49 
 
 
 
 
 
 
 XI. 
 
 275 
 
 12' 
 
 8" 
 
 44 
 
 9" 
 
 74 
 
 
 
 
 
 
 The distance X-XII. is 44i6.8. 
 
 There are 19 angle equations and 15 side equations in the adjustment. 
 
 SOLUTION BY SUCCESSIVE APPROXIMATION. 
 
 141. The rigorous forms of solution which have been 
 given are suitable for a primary triangulation where the 
 greatest precision is required. In secondary or tertiary 
 work it would not be advisable to spend so much labor in 
 the reduction, since the systematic errors remaining would 
 probably largely outweigh the accidental errors eliminated. 
 For work of this kind the method of solution by successive 
 approximation is to be preferred. 
 
 The principle underlying the process of solution is that 
 explained in Art. 115. Each condition or set of conditions
 
 APPLICATION TO TRIANGULATION. 299 
 
 is adjusted for independently in succession, the values of 
 the corrections found at each adjustment being closer and 
 closer approximations to the final values. Should the 
 values found after going through all of the conditions not 
 satisfy the first, second, . . . groups of condition equations 
 closely enough, the process must be repeated till the re- 
 quired accuracy is attained. 
 
 To make the operation as simple as possible let us take 
 but a single condition at a time. 
 
 (i) Local equation at N. Base, 
 
 The solution is given in Ex. i, Art. 121, 
 
 "i\ = o".64, ?', = - o".64, 7' 3 = o".O9 
 
 (2) Local equation at S. Base, 
 
 ''4 + ?'.- 7', + 1.07=0 
 
 The solution is given in Ex. 2, Art. 121, 
 
 v 4 = - o".i3, v, = - o".so, TV, = : + o"-44 
 
 (3) Angle equation, 
 
 ?' 3 + ?'4 + *', + 0.48 = 
 
 Using the values of v 3 , v 4 already found as first approxi- 
 mations, the equation reduces to 
 
 < ; 3 + *'< + ?' T + 0.26 = 
 
 The method of solution is given in Ex. 2, Art. no, 
 
 *', = o".i3, 7' 4 = - o".o8, r. = - o".os 
 39
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The successive approximations found so far, when 
 added, give 
 
 v,= -o".6 4 v 6 = -C/.55 
 
 V 3 = O".22 V 7 = C/.05 
 
 V t =-O ff .2I 
 
 Proceed similarly with the remaining two condition equa- 
 tions. The resulting values will agree closely with the 
 rigorous values already found. 
 
 142. In order to bring out still more clearly the advan- 
 tages of solving in this way, let us take a more extended 
 example. A good one is furnished by the triangulation 
 (1874-1878) of the east end of Lake Ontario, omitting the 
 system around the Sandy Creek base. 
 
 .38 
 
 The measured values of the angles are given in the 
 following table. Each angle is taken to be of the same 
 weight. In the last column are given the locally corrected 
 angles found by the rigorous methods of solution.
 
 APPLICATION TO TRIANGULATION. 
 
 301 
 
 Station 
 occupied. 
 
 Angle as measured between 
 
 Locally 
 corr. 
 angles. 
 
 Sir John, 
 
 Carlton and Kingston, 
 Wolfe and Kingston, 
 
 90 17' 44". 91 
 56" 24' 09". 77 
 
 Carlton, 
 
 Wolfe and Sir John, 
 Kingston and Sir John, 
 
 120 48' 06". 54 
 62 03' 27". 56 
 
 
 Kingston, 
 
 Sir John and Wolfe, 
 Carlton and Wolfe, 
 Wolfe and Amherst, 
 
 64 40' 50". 9 1 
 37 02' 04". 43 
 88 19' 14". 70 
 
 
 Wolfe, 
 
 Duck and Carlton, 
 Amherst and Carlton, 
 Kingston and Carlton, 
 Sir John and Carlton, 
 
 188 07' iH".54 
 140 12' 34". 44 
 84 13' 14". 34 
 25 18' i6".So 
 
 
 Amherst, 
 
 Kingston and Wolfe, 
 Kingston and Duck, 
 Wolfe and Duck, 
 Grenadier and Duck, 
 Duck and Vanderlip, 
 Vanderlip and Kingston, 
 
 35 41' 23". 02 
 in 45' 28". 46 
 76 04' 06". 32 
 54 38' oo".34 
 71 15' 25".43 
 176 59' 06". 1 1 
 
 22". 69 
 28 ".68 
 05 "-99 
 
 25\32 
 
 06". oo 
 
 Duck, 
 
 Oswego and Vanderlip, 
 Vanderlip and Amherst, 
 Amherst and Wolfe, 
 Wolfe and Grenadier, 
 Grenadier and Stony Pt., 
 Stony Pt. and Oswego, 
 
 104 08' 58". 93 
 70 26' 3 1 ".99 
 56 01' 12". 47 
 18 45' 43"-36 
 49 53' 12". 77 
 60 44' 19". 46 
 
 59". 10 
 32". 16 
 
 12". 64 
 
 43". 53 
 12". 94 
 19". 63 
 
 Grenadier, 
 
 Stony Pt. and Duck, 
 Duck and Amherst, 
 Duck and Stony Pt., 
 Amherst and Stony Pt., 
 
 78 13' 33". 64 
 50 35' 04". 28 
 2SI 3 46' 25". 89 
 231" n' 22". 04 
 
 33".8 4 
 04". 19 
 26". 16 
 21". 97 
 
 Stony Pt., 
 
 Oswego and Duck, 
 Duck and Grenadier, 
 Grenadier and Duck, 
 
 88 22' oo".S6 
 51 53' 12". 60 
 308' 06' 47 ".2 1 
 
 12". 70 
 47"-30 
 
 Oswego, 
 
 Sodus and Vanderlip, 
 Sodus and Duck, 
 Sodus and Stony Pt., 
 Vanderlip and Duck, 
 Vanderlip and Stony Pt., 
 Duck and Stony Pt., 
 
 80 29' 46".io 
 107" 19' 03". 28 
 138' 12' 49"-44 
 26 49' i6".6i 
 57 43' oi ".96 
 30 53' 42".SS 
 
 46"- 59 
 03". 96 
 
 48'. 28 
 
 17". 37 
 oi ".69 
 
 44" -32 
 
 Vamlerlip, 
 
 Amherst and Duck, 
 Amherst and Oswego, 
 Duck and Oswego, 
 Duck and Sodus, 
 Oswego and Sod us. 
 Sodus and Amherst, 
 
 38 18' 07. 12 
 87" 19' 53"-47 
 49 oi ' 45 ".54 
 87 59' i2\55 
 38" 57' 26". 55 
 233' 42' 40". 41 
 
 07 "-30 
 53". 16 
 45"- 86 
 12". 42 
 26". 56 
 40". 28 
 
 Sod us, 
 
 Vanderlip and Oswego, 60 32' 57". 55 

 
 3O2 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The local and general equations are formed as usual (see 
 Arts. 117-140). The general rule in the solution is to 
 adjust for one condition at a time. Instead, however, of 
 following out this rule strictly, it is often better to adjust for 
 a group of conditions simultaneously. Often a group is 
 almost as easily managed as a single condition. No rule 
 can be given to cover all cases, and much must be left to 
 the judgment and ingenuity of the computer. 
 
 143. The Local Adjustment tit Each Station. 
 
 (a) Adjust for each sum angle separately. 
 Rule and example in Art. 121. 
 
 (b) Adjust for closure of the horizon. 
 Rule and example in Art. 121. 
 
 At stations Sir John, Carlton, Kingston, Wolfe there are 
 no local conditions, and at each of the stations Amherst, 
 Stony Point, Sodus there is one angle independent of the 
 others, and therefore not locally adjusted. 
 
 The angles at station Amherst may be rigorously ad- 
 justed, as in Art. 121. The resulting values are given in 
 the table. If we break the adjustment into two parts, as in 
 
 (a) and (b), we have : 
 
 
 
 (a) Sum Angle. 
 
 Adjusted. 
 
 22". 73 
 06 ".03 
 
 28". 76 
 
 28". 75 check. 
 
 22 ".65 
 05 ".96 
 
 25"- 35 
 06". 04 
 
 
 Mea 
 
 sured values. 
 
 Kingston-Wolfe, 
 
 35 
 
 41' 
 
 23". 
 
 02 
 
 o" 
 
 .29 
 
 Wolfe-Duck, 
 
 76 
 
 04' 
 
 06", 
 
 32- 
 
 o' 
 
 .29 
 
 
 iu 
 
 45' 
 
 29". 
 
 34 
 
 
 
 Kingston-Duck, 
 
 111 
 
 45' 
 
 28". 
 
 46 + 
 
 o" 
 
 .29 
 
 
 
 
 3)o" 
 
 .88 
 
 
 
 
 
 
 o" 
 
 .29 
 
 
 
 (b) Closure of Horizon. 
 
 
 
 
 
 
 
 Kingston- Wolfe, 
 
 35 
 
 4i' 
 
 22". 
 
 73- 
 
 o" 
 
 .08 
 
 Wolfe-Duck, 
 
 76 
 
 04' 
 
 06". 
 
 03- 
 
 o" 
 
 .07 
 
 Duck-Vanderlip, 
 
 7i 
 
 15' 
 
 25"- 
 
 43- 
 
 o" 
 
 .0& 
 
 Vanderlip-K ings ton, 
 
 176 
 
 59' 
 
 06", 
 
 ii 
 
 o" 
 
 .07 
 
 
 
 
 4)00" 
 
 30 
 
 
 
 
 
 
 oo" 
 
 075 
 
 
 
 oo".oo check. 
 
 The adjusted values agree closely with those from the simultaneous 
 solution, as given in the table.
 
 APPLICATION TO TRIANGULATION. 303 
 
 At station Duck the angles close the horizon. Hence 
 the correction to each angle is one-sixth of the difference of 
 their sum from 360. (See Art. 121.) 
 
 144. The General Adjustment. The local adjust- 
 ment being finished, we shall consider the adjusted angles to 
 be independent of one another and to be of the same 
 weight. We are therefore at liberty to break up the net 
 into its simplest parts. We have in our figure, first a 
 quadrilateral SCWK, next two single triangles KWA, 
 AWD, next a central polygon DAGSOV, and, lastly, a 
 single triangle VOS. These three figures include most 
 cases that arise in any triangulation net. 
 
 (a i) Adjustment of a Quadrilateral. In the quadrilateral 
 SCKW all of the eight angles i, 2, ... 8 are supposed to 
 be equally well measured. 
 
 (i) The Angle Equations. 
 
 The angle equations from the triangles SCIV, CWK, 
 WKS may be written in general terms 
 
 ^ + V, + ^3 + V* = A 
 ^3 + ^4 + ^ + *'* = 4 
 
 v* + *' + * ; 7 + v* = l* 
 
 As these equations are entangled, if we 
 adjusted for each in succession a g-eat 
 many repetitions of the adjustment would 
 be necessary to obtain values that would 
 
 satisfy the equations simultaneously. It is, therefore, better 
 
 to adjust simultaneously, and it happens that a very simple 
 
 rule for doing this can be found. 
 
 Call /(',, &,, &,, /' 4 the correlates of the equations in order; 
 
 then the^ correlate equations are 
 
 k, = v, 6, + 3 = v, 
 
 k l = v, , -f- /-, = v t
 
 304 THE ADJUSTMENT OF OBSERVATIONS. 
 
 and the normal equations 
 
 4*. + 2*. =4 
 
 2k, -f 4 a + 2, = 4 
 
 2/C- 2 + 4^ 3 = / 3 
 Solving these equations, there result 
 
 Substitute these values in the correlate equations, and 
 , = . = *(+ 34 -*4 + 
 
 . = '4 = i(+ 4 + 2/,- /,) 
 . = . = *(- 4 + 2/ a + /,) 
 
 , = . = *(+ A-2/ a -f- 3 /,) 
 which ma be written 
 
 whence follows at once the convenient rule for adjusting 
 the quadrilateral, so far as the angle equations are con- 
 cerned : 
 
 () Write the measured angles in order of azimuth in two 
 sets of four each, the first set being the angles of SCW, and the 
 second those of WKS. 
 
 (,9) Adjust the angles of each set by one- fourth of the dif- 
 ference of this sum from 180 -f- excess of triangle, arranging 
 the adjusted angles in two columns, so that the first column ivill 
 show the angles of SCK, and the second those of CWK. 
 
 (y) Adjust the first column by one-fourth of the difference of 
 its sum from 1 80 -f- excess of triangle, and apply the same cor- 
 rection, with the sign changed, to the second column.
 
 APPLICATION TO TRIANGULATION. 
 
 305 
 
 The spherical excesses of the triangles SCW, CWK, WKS being o"i6, 
 o".35, and o"47 respectively, the adjustment of (he quadrilateral may be ar- 
 ranged as follows : 
 
 Measured angles. 
 
 
 Adjusted angles. 
 
 33 53' 35"-i4 
 62 03' 27". 56 
 58 44' 38". 98 
 25 18' i6".8o 
 
 35". 56 
 27"- 98 
 39 "-40 
 17". 22 
 
 35"- 4 
 27". 82 
 39"- 56 
 I?"- 38 
 
 179 59' 58".4S 
 180 + F. = 180 oo' oo".r6 
 
 
 oo".i6 check 
 
 4)1". 68 
 
 58 54' 57"-54 
 37 02' 04".43 
 27 38' 46". 48 
 56 24' 9". 77 
 
 58". 1 1 
 04". 99 
 47". 04 
 io".33 
 
 58". 27 
 
 05 "-1 5 
 46". 88 
 
 179 59' 58". 22 
 
 180 + = 180 oo' oo".47 
 
 oo".gi 
 oo".2S 
 
 oo".47 check 
 
 4)2". 25 
 
 4)0". 63 
 
 
 o". 5 6 
 
 o".i6 
 
 
 (2) The Side Equation. 
 
 Using the values of the angles just found, we next form 
 the side equation with pole at O. It is 
 
 sin OSC sin OCW sin OWK sin OKS 
 sin CWO sin WKO s'mKSO 
 
 = i 
 
 or writing it in general terms when reduced to the linear 
 form (see Art. 129) 
 
 art + a.pj + art + art + art + art + art + art = /< 
 
 where ?>/, ?'/, . . . are the corrections resulting from the 
 side equation. 
 
 Solving as in Ex. 2, Art. no, we have the corrections 
 
 - ^'., 
 
 \ad\ 
 
 p ^' t i 
 
 [aa]
 
 306 THE ADJUSTMENT OF OBSERVATIONS. 
 
 These corrections may be found still more rapidly as 
 follows : Since the side equation may be so transformed 
 that the coefficients a iy a. 2 , . . . are approximately equal to 
 unity numerically (see Art. 131), we may take each of them 
 to be unity, and then 
 
 that is, the corrections to the angles are numerically equal, but 
 are alternately -f- and . 
 
 This plan has the additional advantage of not disturbing 
 the angle equations. 
 
 Returning to our numerical example, we first reduce the side equation to 
 the linear form. 
 
 OSC = 33 53' 35"-40 + 7/1 SCO = 62 03' 27". 82 + v, 
 
 OCW = 58 44' 39". 56 + 7/3 CWO = 25 18' 17". 38 + 7/4 
 
 OWK=*$ 54' 58". 27 + 7/5 WKO ^ 02' O5".i5 + z/g 
 
 OA'S =27 38' 46". 88 + 2' 7 A'SO = 56 24' io".i7 + 7/ 8 
 
 9.7463587 + 3I-3 i 
 9.9318952 + 12.87/3 
 9.9326832 + 12.72/5 
 9.6665301 + 40.27/7 
 
 72 
 70 
 
 9.9461673 + 1 1. 2 7/ a 
 9.630869! + 44.52/4 
 9.7797125 + 27.97' e 
 9.92O6l8l + I4.O7/e 
 
 70 
 
 2 
 
 Dividing by 20, which will reduce the coefficients to unity approximately, and 
 1.567/1' 0.567/2' + 0.647/3' 2.222/4' + 0.642/5' 
 
 1.407/d' + 2.OI7/7' O.7O.7/ e ' + O.IO O 
 
 Hence 
 
 [aa] = 15 
 and 
 
 zV = p".oi, W = o".oo, 7/3' = o".oo, z/ 4 ' = + o".oi, etc. 
 
 By the second rule the corrections would be T ~, that is, To".oi alternately, 
 
 8 
 
 which values differ but little from the preceding. 
 
 The total corrections to the angles are the sums of the two sets of cor- 
 rections from the angle and side equations.
 
 APPLICATION TO TRIANGULATION. 307 
 
 (a 2) Adjustment of a Quadrilateral. By the following 
 artifice the quadrilateral may be rigorously adjusted for the 
 side equation without disturbing the angle equation adjust- 
 ment, which amounts to the same thing as the simultaneous 
 adjustment of the angle and side equations. 
 
 Suppose that the angle equations have been adjusted as 
 already explained in (a i). If v/, z'/, . . . ?'/ denote the 
 corrections arising from the side equation, the condition 
 equations may be written 
 
 ",*',' + <w' + " 3 7 V + <w' + <*?* -f <***' 
 
 B writin the corrections in the form 
 
 the first three condition equations become = identi- 
 cally, and we have therefore to deal only with the single 
 condition equation 
 
 ("i + <* 9 + a + "n ^ : , ^ a. a r )v -f (rt-, rt-.j7'' 
 
 + (", <><'" + (^5 B )f'"' + (", ^)''"" = /,' 
 
 with 
 
 (T' + T'7+(z'-70 2 +(-7- + 7'"/ + (-7'-7'"r+. . . = a mill. 
 
 The correlate equations are 
 
 40
 
 308 THK ADJUSTMKNT OF OBSERVATIONS. 
 
 Substitute in the condition equation, and 
 
 k\\(a, -f a., -f a :> -f a, a y a, rf. a^ + (>, 
 
 from which / can be found. 
 
 Hence the corrections are known. 
 
 The complete adjustment of our quadrilateral is contained in the follow- 
 ing table : 
 
 Meas. Angles. 
 
 
 Local 
 Angles. 
 
 l."R. Sines. 
 
 LOR. Diff. 
 
 1 
 
 o* 
 
 
 
 
 
 
 
 33 53' 35"-M 
 
 3 ? ",6 
 
 ^s"-4 9-74^3587 
 
 + 31-3 
 
 
 
 t>2 03' 27". 56 
 
 2 7 ".u8 
 
 9.9461673. 
 
 -f- I 1. 2 
 
 42-5 
 
 1806 
 
 58 44' 38" 98 
 
 39'-4 
 
 39". 56 ; 9.9318952 
 
 + 12.8 
 
 
 
 25 18' i6".8o 
 
 17". zz 
 
 i7"-33 
 
 9.6308691 
 
 + 44-5 
 
 57-3 
 
 3283 
 
 58". 48 
 
 
 
 
 
 
 
 oo".i6 
 
 
 
 
 
 
 
 i".68 
 
 
 
 
 
 
 
 58 54' 57"-54 
 
 58". II 
 
 58"-2 7 
 
 9.9326832 
 
 + '2.7 
 
 
 
 37 02' 04". 43 
 
 04 ".99 
 
 05"-'S 
 
 9.7798125 
 
 + 27-9 
 
 40.6 
 
 1648 
 
 27 38' 4 0". 48 
 
 47' -4 
 
 46". 88 
 
 9.6665301 
 
 + 40.2 
 
 
 
 56 24' 09". 77 
 
 i"-33 
 
 io".i7 
 
 9.9206181 
 
 + M-o 
 
 54-2 
 
 2938 
 
 58". 22 
 
 ','' 9 o 
 
 
 
 
 
 
 
 
 oo '.47 
 
 oo .28 
 
 
 ^2 7 o 
 
 I,!2 5 92.1 
 
 
 
 2". 25 
 
 o".63 
 
 
 
 
 
 
 
 
 
 
 2 
 
 4)10.4 
 
 
 
 
 
 
 
 2.6 
 
 5-2 
 
 27 
 
 
 
 
 
 
 
 9702 
 
 4 2 -5 57-3 
 
 2 
 9702 
 
 Hence the corrections are kno-.vn. These corrections, applied to the local 
 angles, give the final angles required. 
 
 (b) Adjustment of a Single Triangle. 
 
 Rule and example in Kx. 2, Art. 109, and Ex. 2, Art. 1 10. 
 
 The single triangles in our figure are Kingston, Wolfe, 
 Amlierst; Wolfe, Duck, Amherst ; Vanderlip, Oswego, 
 Sodtis.
 
 APPLICATION TO TRIANGULATION. 
 
 309 
 
 For example, take the first (.' = n".~2) 
 
 Measured. Adjusted. 
 
 Kingston, 88 19' 14". 70 15". 78 
 
 Wolfe, 55 5<y 20". io 21 ".18 
 
 Amherst, 35 41' 22". 69 23". 77 
 
 I7<) 5'V 57 '-4'J o". 73 check. 
 180 + c = I So" oo' oo .72 
 
 3)3". 23 
 
 r.os 
 
 (c) Adjustment of a Central Polygon. In the central poly- 
 gon Duck, Amherst, Grenadier, Stonv Point, Oswego, Yan- 
 derlip tlie condition equations in general terms are: 
 
 Fig.40 
 
 Local equation (hori/on equation), 
 
 'u + ' + ' + '' + '' = /, 
 Angle equations, 
 
 ' P, + < ', + v, = 1, 
 ' *> * =/ 
 
 1 1:1 "l ' 11 "l ' )r> ' 6 
 
 Side equation (jole at Duck), 
 
 We may adjust tor these equations in order, Hist the hori- 
 zon equation, then the angle equations separately, as they 
 are not entangled, and next the sii.le equation. 
 
 A rigorous adjustment may, however, be carried out at 
 once with very little additional labor. Adjust first each 
 angle equation by itself, and let (:,}, I; 1 ..), ... be the values 
 that result. Let (l), (2), . . . denote the farther correc- 
 tions to the measured angles in order arising from the local 
 and side equations, so that
 
 310 THE ADJUSTMENT OF OBSERVATIONS. 
 
 If we substitute these values in the above equations we 
 have the new condition equations 
 
 (i) + (2)4- ( 3 )=o 
 (4) + (5)+ (6)=o 
 
 (13) + (14) -|- (i 5) = o 
 
 from which to find (i), (2), (3), . . . (15). 
 
 Calling k lt ,, /, //, . . . the correlates of the condition 
 equations in order, we have the correlate equations 
 
 (1) = ,*, + 7 ( 4 ) 
 
 (2) = ,*> + 7 (5) 
 
 (3)= * + 7 (6)= , + 77 
 
 Eliminate now the angle equation correlates. By addition 
 (, + ",X'i + 37 
 
 Hence 
 
 (2)= -Htf.-w.)^.-i 
 
 (3) =-!(,+ ,)*, + I 
 
 Substitute these values of (i), (2), ... in the condition 
 equations, and we have the normal equations 
 
 2 1 [aa] a,a t a<a b . . . JX-, [ii\k 9 = 3/' 
 
 - [a]*, -f io/', = 3 /" 
 
 Solving, we find /,, &,, and thence (i), (2), . . . are known.
 
 APPLICATION TO TRIANGULATION. 311 
 
 The normal equations may be written down directly in 
 every case, as the law of their formation is as evident as 
 that of ordinary normal equations. They involve only two 
 unknowns, /',, /, no matter how many sides the polygon 
 has. 
 
 It is evident that the elimination of the angle equation 
 correlates /, //,... from the correlate equations before 
 forming the normal equations amounts to the same thing as 
 first forming the normal equations in the usual way and 
 then eliminating /, //,... For the normal equations from 
 the correlate equations are 
 
 \aa\k, + (a, + ,)/ + (a t + *.) //+ . . . = /' 
 
 + 5*. + / + //+...=/" 
 
 (, + #,+ *, + 3/ =0 
 
 (a 4 + a>) l + *, +3/7 =o 
 
 Substitute for /, //,... their values from the third, 
 fourth, . . . equations in the first two, and we find the two 
 normal equations as before. 
 
 The introduction of the method of eliminating the corre- 
 lates arising from one set of condition equations before 
 forming the normal equations is due to Schleiermacher, of 
 the Hessian survey. For a fuller account of it see Fischer's 
 Geodiisie, part iii. p. 93 ; Huge I in General Bt'richt der euro- 
 pa isc hen Gradinessung, 1867, pp. 106 seq. ; Nell in ZcitscJir. 
 fiir Venncss., vol. x. pp. I seq. ; vol. xii. pp. 313 seq. 
 
 The process is not of any special advantage except in 
 such problems as that under discussion, and then if. is 
 better to use the final formula for the normal equations 
 directly. 
 
 We shall now proceed with our numerical example. 
 
 At station Duck the measured values of the angles 
 
 o 
 
 are taken, at the other stations the locally adjusted 
 angles.
 
 312 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Given Angles. 
 
 l,og. Sines. 
 
 Did. i". 
 
 Squares. 
 
 Products. 
 
 Sums. 
 
 54 38' oo".34 oo".62 
 
 9.911 4060 
 
 4- 14-9 
 
 222. 
 
 257-8 
 
 - 2.4 
 
 50 35' 04". 19 04". 47 
 
 9 8879338 
 
 + '7-3 
 
 299 . 3 
 
 
 
 74 46' 55"-83 56". 12 
 
 
 
 
 
 
 180 oo' oo"-36 
 
 
 
 
 
 
 180 oo' oi".zi =i8o-(- e 
 
 
 
 
 
 
 3)"-8 5 
 
 
 
 
 
 
 0".28 
 
 
 
 
 
 
 78 13' 33". 84 34". 49 
 
 9:9907654 
 
 + 4.4 
 
 19.4 
 
 72.6 
 
 12. 1 
 
 51 53' 12". 70 13". 35 
 
 9.8958618 
 
 + 16.5 
 
 272.2 
 
 
 
 49 53' 12". 77 13" 42 
 
 
 
 
 
 
 i"-95 
 
 
 
 
 
 
 o .65 
 
 
 
 
 
 
 88 22' oo".86 oo". 4 7 
 
 9.9998235 
 
 + 0.6 
 
 4 
 
 21 . I 
 
 -34-6 
 
 3" 53' 44"-32 43"-93 
 
 9.7105188 
 
 + 35-2 
 
 1239.0 
 
 
 
 60 44 it)". 46 iy".o7 
 
 
 
 
 
 
 o 4 ".64 
 
 
 
 
 
 
 3 '-47 
 
 
 
 
 
 
 "-39 
 
 
 
 
 
 
 26 49' 17" 37 i8".is 
 
 9.6543842 
 
 + 41.6 
 
 1730.6 
 
 761.3 
 
 + 23.3 
 
 49 01 ' 45". 86 46". 64 
 
 9.8779750 
 
 + '8.3 
 
 334-9 
 
 
 
 104 08' 58". 93 59"-7 
 
 
 
 
 
 
 2".l6 
 
 
 
 
 
 
 4". 49 
 
 
 
 
 
 
 a"-33 
 
 
 
 
 
 
 o". 7 8 
 
 
 
 
 
 
 38 18' 07". 30 06". 33 
 
 9.7922537 
 
 + 26.7 
 
 712.9 
 
 189.6 
 
 + 19.6 
 
 71 15' 25' .32 24". 36 
 
 9 .976^353 
 
 + 7-i 
 
 50-4 
 
 
 
 70 26' 31". 99 3i".o2 
 
 
 
 
 
 
 04". 61 
 
 6328 6247 
 
 
 4881.1 
 
 1302.4 
 
 - 6.2 
 
 i .71 
 
 6247 
 
 
 1303.4 
 
 
 
 2". 90 
 
 81 
 
 
 6183.5 
 
 
 
 ""97 
 
 3 
 
 
 2 
 
 
 
 
 243 . 
 
 
 12367.0 
 
 
 
 The Noimal Equations. 
 12367^, + 6.2/t, = 243 
 
 6.2^i + I0^a = 2. 01 
 
 . '. /'i = O.O20 
 
 k* + 0.213 
 
 Local Equation at Station Duck. 
 
 74 ' 4' 5^". 12 
 49 53' 13"- 42 
 60 44' 19". 07 
 104" 08' 59". 70 
 70 26' 3 1 ".02 
 
 359" 59' 59"-33 
 360' oo' oo".oo 
 
 oo".f>7 
 3
 
 APPLICATION TO TRIANGULATION. 313 
 
 Corrections. Adjusted Angles. 
 
 (!) = 
 
 -o .39 
 
 54 
 
 3 s 
 
 oo .23 
 
 (2) = 
 
 + o".2(> 
 
 5" 
 
 35' 
 
 04"- 73 
 
 (3) = 
 
 + o".i3 
 
 74 " 
 
 46' 
 
 5 f >"-25 
 
 (4) = 
 
 -o".24 
 
 7 s " 
 
 13' 
 
 34". 25 
 
 (5) = 
 
 + o'.iS 
 
 51 
 
 53' 
 
 13". 53 
 
 (6) = 
 
 + o".o6 
 
 49 
 
 53' 
 
 I3'48 
 
 (7) = 
 
 o".3i 
 
 88 
 
 22' 
 
 (IT)", if) 
 
 (8) = 
 
 -)- o".4O 
 
 30 
 
 53' 
 
 44"-33 
 
 (9) = 
 
 o".O9 
 
 60 
 
 44' 
 
 iS".(>S 
 
 (10) = 
 
 -o".;5 
 
 26 
 
 49' 
 
 17". 40 
 
 (") = 
 
 + o".45 
 
 49" 
 
 01' 
 
 47"- 09 
 
 (12) = 
 
 4- o".3O 
 
 104 
 
 08' 
 
 60". oo 
 
 (13) = 
 
 - o".47 
 
 38 C 
 
 1 8' 
 
 05 ".86 
 
 (M) = 
 
 + O".20 
 
 71 
 
 15' 
 
 24". 56 
 
 (5) = 
 
 4- 0".27 
 
 70 
 
 26' 
 
 31 ".29 
 
 APPROXIMATE METHOD OF FINDING THE PRECISION. 
 
 145. An adjustment may be carried out rigorously so 
 far as finding the values of the unknowns is concerned, but 
 only an approximate value of the rn. s. e. of the angles or 
 sides may be thought necessary. 
 
 In good work the following method will give results 
 nearly the same as those found by the rigorous process. 
 
 The average value // of the m. s. e. of an angle in a 
 triangulation net after adjustment is easily seen from Art. 
 102 to be 
 
 n 
 
 where 
 
 ;/ m number of angles observed. 
 
 n c = number of local and general conditions. 
 
 ft m. s. e. of a measured angle of weight unity. 
 
 The value of// is, by the usual formula,
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 To find the m. s. e. of a side of a triangle a single chain 
 of the best-shaped triangles between the base rind the side 
 is selected, all tie lines being rejected. Then, assuming the 
 base to be exact and the m. s. e. of eacli adjusted angle to 
 be //, we have from Ex. 9, p. 234, 
 
 where O A , O B are the log. differences corresponding to i" for 
 the angles A, B in a table of log. sines. 
 
 A form still more approximate was used on the U. S. 
 Lake Survey in the determination of the precision of a side 
 of the primary triangulation. The angles ot each triangle 
 were taken to be independent of one another. In this case 
 evidently (Ex. 5, p. 109) 
 
 The earlier work of the Coast Survey was computed from 
 this same formula. 
 
 Ex. To find the m. s. e. of the side OL as derived from the base NS in 
 the figure ONSL (Fig. 19). 
 
 Number of angles measured = 9. 
 
 Number of conditions, local and general, = 5. 
 
 From the adjustment (Art. 140) [/?>?'] =r 7.54. 
 
 5
 
 APPLICATION TO TRIANGULATION. 
 The chain of triangles is OA r S, OLS. 
 
 315 
 
 Station. 
 
 Angles. 
 
 5 Sqs. 
 
 Prods. 
 
 N. Base, 
 
 122 II' 15" 
 
 - 13-2 174.2 
 
 
 S. Base, 
 
 23^ oS' 05" 
 
 
 - 401.3 
 
 Oneota, 
 
 34 40' 40" 
 
 + 30-4 924-2 
 
 
 S. Base, 
 
 70 39' 25" 
 
 + 7-4 54- 
 
 
 Oneota, 
 
 78" 27' 05" 
 
 260. 5 
 
 Lester, 30 53' 30" -+- 35.2 1239.0 
 
 
 
 
 2251.4 
 
 = 31.6 in units of the seventli decimal place. 
 Also OL 16556 metres. 
 
 Hence /i nr is known. 
 
 The RIctJiod of Directions. 
 
 146. This method is clue to Bessel. Various modifica- 
 tions of Bessel's plan of making the observations are used 
 on different surveys. The following is that used on the 
 U. S. Coast Survey. . 
 
 " In any set, after the objects have been observed in the 
 order of graduation they are re-observed with instrument 
 reversed in the opposite order: the mean of the two obser- 
 vations upon each object is then taken. The number of 
 such sets and the number of positions made depends on the 
 accuracy required and upon the perfection of the instru- 
 ment." A single series of means is called an "arc." 
 
 "The direction instrument requires that it should be 
 turned on its stand or changed in position, in order that the 
 direction of any one line, and consequently of all, should 
 fall upon different parts of the circle as the only security 
 against errors of graduation. The number of positions 
 41
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 varies from five to twenty-one of nearly equal arcs; and in 
 each position the circuit of the horizon is made, giving the 
 direction of each line by two observations, one in the direct 
 and the other in the reversed position of the telescope. 
 These circuits or series are repeated in each position until 
 two to five values of each direction are obtained. Each 
 angle is therefore determined by from 35 to 63 measure- 
 ments in the direct and a like number in the reversed posi- 
 tion of the telescope." 
 
 147. The Local Adjustment. Let O be the station 
 occupied, and i, 2, 3, . . . the stations sighted at in order 
 of azimuth. Let some one direction, 
 as Oi, be selected as the zero direc- 
 tion, and let A, B, . . . denote the 
 most probable values of the angles 
 which the directions of the different 
 signals make with this direction. 
 
 In the first arc let X l denote the 
 most probable value of the angle be- 
 tween the zero of the limb of the 
 instrument and the direction of the 
 
 signal taken as the zero direction ; then if J//, J//', . . . 
 denote the readings of the limb for the different signals, 
 and z//, v" , . . . the most probable corrections to these 
 readings, we have the observation equations 
 
 Fig.4l 
 
 The zero of the limb being changed in the next arc, we have 
 in like manner 
 
 X - M = v' 
 
 and so on for the remaining arcs.
 
 APPLICATION TO TRIANGULATION. 317 
 
 If now />,', /,", . . . ; //, /,", .../... denote the weights 
 of the measured directions of the several series of arcs, the 
 normal equations follow at once. They are 
 
 [>,] x, +P: A +pi n B+ . . . = [/vi/j 
 
 p 
 
 from which the unknowns may be found. 
 
 In order to shorten the numerical work a course similar 
 to that of Art. 41 may be followed. Let 
 
 where J//, Ml, . . . A', B ', . . . are approximate values of 
 X lt X, ... A, ,..., and .f,, x v . . . (A), (B), . . . denote 
 their most probable corrections. 
 
 Also, for convenience in writing, put 
 
 ml' = Ml' - Ml - A' ml' = J/ a " - Ml - A' 
 
 ml" = Ml" - Ml - B' ml" = Ml" - Ml - B' 
 
 The normal equations now become 
 
 -h
 
 3l8 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The quantities ,f lf x. t , . . . being merely auxiliary quanti- 
 ties, we eliminate them by substituting their values as found 
 from the first group of normal equations in the second 
 group. We have then 
 
 
 
 which may be solved as usual. 
 
 These equations may be written 
 
 where [], [ rf ^]> are to be looked on as mere 
 symbols. 
 
 In the cases that occur in ordinary work the computa- 
 tion may be still farther shortened. If we arrange the 
 observations in groups containing readings on the same 
 series of signals, then, these readings being of equal value, 
 we have for the first group 
 
 A / =A // = ./ suppose 
 
 A'=A" =,/ " 
 
 . 
 
 and therefore 
 
 ;// being the number of signals sighted at in this group. 
 Similarly for the other groups.
 
 APPLICATION TO TRIANGULATION. 
 
 319 
 
 Hence if ///, // rt ", . . . denote the number of arcs in the 
 several groups, the coefficients of the normal equations 
 become 
 
 // ' /' '/ 
 
 [<W] = L/J- -> / -*T,*P 
 
 >'s s 
 
 
 where ,/>', ,/>" . . . denote any of the equal weights in 
 the several columns of the first group, and />', 2 />", . / 
 3 p', 3 p", .../.. . denote corresponding quantities in the 
 second, third, . . . groups. These weights for the signals 
 sighted at may be taken to be each equal to unity, and, for 
 the signals not sighted at, zero. 
 
 After having found the quantities ///, w", ... by taking 
 the differences between the approximate values of the 
 angles and the several measured values, it is convenient to 
 arrange the formation of the normal equations according to 
 the following scheme: 
 
 No. of 
 Group. 
 
 /"'" 
 
 [P\ [/'"'"I 
 
 The coefficients of the normal equations may now be 
 written down at sight.
 
 320 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Special Case. If every arc is full, and every direction 
 is equally well measured in every arc, then 
 
 and the normal equations become 
 
 By addition, the number of arcs being // s i, 
 
 w 
 
 )==? 
 
 Hence 
 
 () = [;'"] 
 
 as is evident a priori. 
 
 148. Checks of the Normal Equations. 
 i. The sum 
 
 [ cc _] + . . . 
 
 is equal to half the number of observations, less half the 
 number of arcs.
 
 APPLICATION TO TRIANGULATION. 321 
 
 For substituting for [#<?], [^], their values we 
 find the above sum equal to 
 
 pi + fi" + Pi'" + . . . _ /i" 2 +//j ! +/ > i"/i"' 4- . . . 
 
 2 [/i] 
 
 pi" -$- pi" -t- ft" + . . . /'-' 2 4 A'" ! 4 />2 'A'" 4 ... 
 
 [/] 
 
 4- - ... 
 
 = 1 1 [/] + [/"] + [/"] + - I ~ ^Jjj - ^| - 
 
 Now,// // == . . . = i, and therefore 
 
 [/'] -f- [/>"] -j- \ p'"] -}-... = the number of observations 
 
 r/, 2 ] r/, 2 ] ry, a i 
 
 -^ -f . . . = the number of arcs. 
 
 [AJ [A] [A] 
 2. The sum 
 
 [f/] + [*/] + L^] + - -[ 
 
 where ['/] is formed in the same way as [^ 
 For 
 
 
 since [/"w/"J + [/'"''"] + 
 
 - [A'.l - [A W J -...== [ 
 
 ^= o 
 which proves the proposition.
 
 322 THE ADJUSTMENT OF OBSERVATIONS. 
 
 149. The Precision of the Adjusted Va/ucs. The first step 
 is to find //, the m. s. e. of a single direction. 
 We have generally 
 
 . 
 
 No. ol obs. quan. No. of indep. unknowns. 
 
 Now, from the observation equations, Art. 147, it is evident 
 that the number of independent unknowns is equal to the 
 number of arcs //, together with the number of signals ;/, 
 sighted at, less one. Hence 
 
 \j>w\ 
 
 ft - : - L/ 
 
 n (tr a -\-u s I ) 
 
 n being the number of directions. 
 
 To compute the value of | pvi<\. Eliminate .r n x^, . . . 
 from the normal equations, Art, 147, and we have the re- 
 duced normal equations 
 
 + A V) +//"(#) + =[A< 
 
 Hence, as in Art. 100, 
 
 A AJ 
 
 = [p,nm\ - Iil-I 
 
 The quantity \pnnn\ should always, if possible, be found 
 from the original observations. It will in general be quite 
 different if found from the means forming the different 
 groups of arcs taken as single observations. (See Ex., 
 Art. 62.)
 
 APPLICATION TO TRIANGULATION. 
 
 323 
 
 Takinof the weight of each observed direction to be o 
 
 o o 
 
 the same value unity, the expressions for[z/z'] may be written 
 
 \vv] = \jnni\ 
 
 [',]' 
 
 1^1 
 
 -[aa\ 
 
 150. The General Adjustment. The general ad- 
 justment is carried out as in the case of independent angles. 
 The angle and side equations are formed as in Arts. 1 17-139, 
 and the solution effected according to the programme of 
 Art. 113. 
 
 151. Ex. At station Clark Mt., in the triangulation of the Blue Ridge, Va,, 
 readings were made with a non-repeating theodolite in the method of arcs. 
 The following, taken from these readings, will be sufficient to illustrate the 
 method of reduction. 
 
 The original observations are arranged in sets containing readings on 
 the same groups of signals, and the quantities given in the table beiow are 
 the remainders found by subtracting the reading of the first direction in earh 
 series from the readings of the other directions; that is, MI" Mi, 
 Mi" -Mi, . . . 
 
 Ai Claik Mt. 
 
 Spear. Humpback 
 
 Fork. 
 
 oo" oo' 00". oo 
 
 .OO 
 .CO 
 .OO 
 
 24 09' 35^-70 
 
 33"-55 
 
 78' 26' oS".55 
 09 ".60 
 
 09". 33 
 io".45 
 
 oo' oo' 00". oo 
 
 .OO 
 
 
 78"' 26' IO".20 
 1 1 ".03 
 
 oo oo' oo'.oo 
 
 .OO 
 .OO 
 .OO 
 .00 
 
 24 og' 36". 10 
 37"-4Q 
 
 3S".I5 
 39".oo 
 
 
 
 oo oo' oo'.oo 
 .00 
 .00 
 .00 
 
 54 1 6' 31". 85 
 3i" 94 
 
 36'iig 
 
 42
 
 324 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The Local Adjustment. 
 Assume the most probable values of the angles 
 
 Spear, 00 oo' oo".oo 
 
 Humpback, 24 09' 36". 90 + (A) 
 Fork, 78 26' 9". 90 + () 
 
 Take the differences between the approximate values of the angles and 
 the several measured values. We then have 
 
 /';' 
 
 p"m" 
 
 p'" m'" 
 
 Sums. 
 
 o.oo 
 
 n a ' = 4 .00 
 .00 
 / = 3 .00 
 
 o.oo 
 
 1. 2O 
 i/' = 1 + 2.50 
 0.67 
 
 - 3-35 
 
 - 1-35 
 
 i/"r=I 0.30 
 -0.57 
 + 0.55 
 
 -4-39 
 
 2.72 
 
 -1.6 7 
 
 1l a "=2 O.OO 
 
 n s "2 .00 
 
 if' 1 - 
 
 + 0.30 
 tp'" = I + I.I3 
 
 + 1-43 
 
 o.oo 
 
 + i-43 
 
 o.oo 
 
 0.80 
 
 
 
 .00 
 
 .00 
 
 .00 
 .00 
 
 o.oo 
 
 + 0.50 
 0.37 
 
 + 1.25 
 
 + 2.IO 
 
 
 + 2.68 
 
 + 2.68 
 
 
 o.oo 
 .00 
 
 - 1.15 
 i. 06 
 
 
 
 .00 
 .00 
 
 -0.97 
 + 3-13 
 
 0.05 
 
 0.00 
 
 0.05 
 
 [/'/'] = o.oo 
 
 \_p"m"] = - 0.04 
 
 [/'"w'"]=: 0.29 
 
 [//] = -0.33 
 
 [/] = II 
 
 [/'] = J 3 
 
 [/"] = 10 

 
 APPLICATION TO TRIANGULATION. 
 Next form the table 
 
 325 
 
 No. of 
 group. 
 
 "a 
 "s 
 
 "a 
 
 p' in' 
 
 /";" 
 
 /'";'" 
 
 Sum. 
 
 Sum 
 n s 
 
 I 
 
 1 
 
 4 
 
 
 
 2.72 
 
 -1.6 7 
 
 ~4-39 
 
 ~ I-463 
 
 2 
 
 a 
 
 2 
 
 o 
 
 
 + 1-43 
 
 + 1.43 
 
 + 0.715 
 
 3 
 
 f 
 
 5 
 
 
 
 + 2.68 
 
 
 + 2.68 
 
 + 1-340 
 
 4 
 
 ^ 
 
 4 
 
 o 
 
 .00 
 
 0.05 
 
 0.05 
 
 0.025 
 
 Sums, 
 
 15 
 
 
 0.04 
 
 0.29 
 
 - 0.33 
 
 Check 
 
 The coefficients of the normal equations 
 
 [jzfl ] = 13 -!-f-! = + 7i 
 [_**_] = - I - 1 = - 34 
 [_W_] = IO -f-f-f= + 5f 
 
 r ,r/ ] = 0.04 + 1.463 1.34 + 0.025 = +0.108 
 r bl i = 0.29 + 1.463 0.715 +0.025 = + 0.483 
 [^/] =-1.463 + 0.715 + 1-34 =+0.592 
 
 Check (i) [art] + [jitfj + [_WJ = gj 
 
 = i(34-i5) 
 
 as it should. 
 
 (2) [ 
 
 as it should. 
 
 The normal equations are 
 
 = 0.5QI 
 = [,/] 
 
 =+ 0.483 = 
 
 The general solution of these equations gives 
 C<4) = + 0.1921 [a/] + o. 
 (B)= + O.I 1 30 [a/] +o.242g[W] 
 
 Substituting for [/], [^/] their values, there result 
 
 (A) + o".075 
 (B)= + o".i30
 
 326 THE ADJUSTMENT OF OBSERVATIONS. 
 
 and hence the local directions 
 
 Spear o oo' oo".ooo 
 
 Humpback, 24 09' 36". 975 
 Fork, 78 26' io".O3O 
 
 To find the m. s. e. of a single observation. 
 
 The value of [?'?'], computed according to Art. 150, is found to be 25.7. 
 Therefore 
 
 the divisor in this case being 15. 
 
 This completes the local adjustment at this station. Proceed similarly 
 at the remaining three stations. 
 
 The General Adjustment. 
 
 At Clark. 
 Most probable directions. 
 
 Spear, o oo' oo".ooo 
 
 Humpback, 24 09' 36".975+(i) 
 Fork, 78 26' io".o3o 4- (2) 
 
 Weight Equations. 
 
 (l)= + O.I92I | I | + O.II30 | 2 | 
 (2)= +O.II30 | 1 | + 0.2429 | 2 | 
 
 [w]=25.7 Divisor, 17 
 
 At Spear. 
 
 Humpback, o" oo' oo".ooo 
 
 Fork, 32" 08' n".793+(3) 
 
 Clark, 54 06' 29". 197 + (4) 
 
 Weight Equations. 
 
 (3)= + 0.2061 jjj +0.0485 fjj 
 (4)= + 0.0485 |Tj +0.1879 IT? 
 
 \vv\ = 58.7 Divisor, 17
 
 APPLICATION TO* TRIANGULATION. 327 
 
 At Humpback. 
 
 Clark, o" oo' oo".ooo 
 Spear, 101 44' 3". 123 + (5) 
 Fork, 332 58' n".i57+(&) 
 
 Weight Equations. 
 
 (5)= +0.1333 I 5 I +0.0667 I f > I 
 
 (6)= +0.0667 | 5 | +0.1833 | 6 | 
 
 [rT'] 106.0 Divisor, 23 
 
 At Fork. 
 
 Clark, o oo' oo".ooo 
 
 Spear, 79 35' 42". 479 + (7) 
 
 Humpback, 98 41' 43". 926 + (8) 
 
 Weight Equations. 
 (7)= +0.2970 | 7 I +- o.i 394 I' 8 | 
 (8)= +0.1394 Qj +0.1879 |JJ 
 [<vz/]=47.5 Divisor, 17 
 
 The angle and side equations are formed as already explained. The 
 angle equations from the triangles SFC (E = io".773), HFC (f. 7". 386), 
 SI/C(f. 9". 789), and the side equation from the quadrilateral CSHF (pole 
 at C), will be (ound to be 
 
 (2) -(3) + (4) + (7) -0.860 
 
 -(i) + (2) -(6) + (8) = 1.562 
 
 (i) + (4) + (5) =0.494 
 
 2.609(3) - 1.847(4) + 0.2187(5) - 2.0635(6) + 0.0193(7) + 0.1611(8) = 0.0424 
 
 From this point the solution is carried through exactly as in Art. 140. 
 The finally adjusted directions will be found to be 
 
 At Clark, Spear, o" oo' oo".ooo At Humpback, Clark, o' oo' oo".ooo 
 
 Humpb-.ck, 24* 09' 36". 844 Spear, ioT 44 03". 279 
 
 Foik, 78 26' io".47S Fork, 332' 58' io".7S4 
 
 At Spear, Humpback, o oo' oo".ooo At Fork, Clark, o oo' oo'.ooo 
 
 Fork, 32 08' n".799 Spear, 79 35' 42". 428 
 
 Clark, 54" 06' 29". 666 Humpback, 98 41' 44". 536 
 
 The m. s. e. of an observation of weight unity is i'-77. 
 
 The m. s. e. of the adjusted value of the angle CSF = o".46.
 
 328 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The form of reduction used on the U. S. Coast Survey 
 for the adjustment of the primary triangulation is essentially 
 the same as that just explained, so far as finding the val- 
 ues of the local corrections is concerned. The method of 
 weighting employed in the general adjustment is not given, 
 as it is not so satisfactory as that in the text. It will be 
 found in Report, 1864, app. 14. 
 
 152. Approximate Method of Reduction. A very 
 convenient method of approximation may be derived from 
 the normal equations, Art. 147. It depends on the theorem 
 of Art. 115, and hence, if the process is repeated often 
 enough, leads to the same result as the rigorous solution. 
 
 In the second group of normal equations, Art. 147, assume 
 
 X, = M,', X, = M,', . . . 
 then we have as approximate values of A, B, . . . 
 
 _ \p"(M" - _ 
 
 [/'J [/"] 
 
 Let x l denote the correction to X^ ,t' 2 to X^ . . . Then the 
 values A', />', ... of A, />, . . . substituted in the first 
 group of normal equations, give as approximate values 'Of 
 
 _ 
 
 [A] 
 
 which values substituted in the second group give as second 
 approximations to the values of A, B, . . . 
 
 A _ A'W: - M: - /) +P:(M:- - M: - Q + . . . 
 
 T7T 
 
 W'-M';- Q + . . . 
 
 and so on.
 
 APPLICATION TO TRIANGULATION. 329 
 
 This approximate form of reduction was used on the 
 Ordnance Trigonometrical Survey of Great Britain in the 
 reduction of the principal triangulation. The approxima- 
 tions were carried out only as far as A", B\ . . . This is 
 in general sufficient in good work. 
 
 Instead of finding A', Z>', . . . as above, it is often more 
 convenient to follow the Ordnance Survey plan, and, in- 
 stead of deducting J//, AJ t ", . . . from the readings of the 
 signals in the different arcs in order, to add to the readings 
 of the signals in the different arcs quantities which will 
 make the column M' constant throughout. We should 
 then have 
 
 The corrections x and the other approximations are made 
 as before. 
 
 Should the observation of the zero direction be want- 
 ing in any arc as, say, the third the quantity to be added 
 to each reading in this arc is given approximately by the 
 value of X 3 in the first set of normal equations. Thus, given 
 the readings prepared as explained above, 
 
 M', MS', M,'" 
 M', M^', M t f " 
 
 o, M,'" - M," 
 to find M: - M 3 f . 
 
 If A', B' have been found from the first two arcs, then 
 from the normal equations 
 
 and 
 
 
 M " - M '- 
 
 = A r approx.
 
 330 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hence the complete form would be 
 
 M', M,", M,'" 
 
 M', M:, M,'" 
 
 A' + M', M,'" - M 3 "+A f + M' 
 
 or 
 
 Ex. 
 
 o.MS- M f , M/" - M' 
 o, M," -M', M n '" -M' 
 
 A', M,'" -M t "-\-A' 
 
 At Clark Mt. 
 
 p 
 
 
 M" - M' 
 
 M'" - M' 
 
 
 4 
 
 2 
 
 00 oo' oo".oo 
 .00 
 
 24 09' 36". 22 
 
 78 26' 09". 48 
 io".62 
 
 5 
 4 
 
 .00 
 
 37"-44 
 00 oo' oo" 
 
 54 1 6' 32. "99 
 
 
 
 A' = 24 09' 36". 90 
 
 
 4 
 
 2 
 
 00 oo' oo".oo 
 .00 
 
 24 Og' 36". 22 
 
 78 26' 09". 48 
 
 !0".62 
 
 5 
 4 
 
 .00 
 
 37"- 44 
 36". 90 
 
 9". 89 
 
 
 Means 
 
 36 ".90 
 
 9". 88 
 
 Corrections to the arcs: 
 
 
 
 M" - M 1 A' 
 
 M'" M' B' 
 
 X 
 
 
 o".oo 
 
 -o".6S 
 
 o".4O 
 
 o".36 
 
 
 .00 
 
 
 + o".74 
 
 + o".37 
 
 
 .00 
 
 + o".54 
 
 
 + 0".27 
 
 
 
 o".oo 
 
 + o".oi 
 
 + o".oo 
 
 
 M'M'x 
 
 M'" M' x 
 
 4 
 
 2 
 
 o"-36 
 59"- 6 3 
 
 24" 09' 36". 58 
 
 78 26' 09". 84 
 
 I0".25 
 
 5 
 4 
 
 59". 73 
 
 37"- 17 
 36". 90 
 
 9". 89 
 
 
 Means, 59"-94 
 Final values, oo".oo 
 
 3f>" -90 
 36". 96 
 
 9"-94 
 9". 98
 
 APPLICATION TO TRIANGULATION. 331 
 
 Modified Rigorous Solution. 
 
 153. The forms that have been given in the preceding 
 articles for the rigorous adjustment of a triangulation, 
 though analytically very elegant, are somewhat compli- 
 cated. A method which shall give a marked diminution 
 of work in the reduction without increasing the field work 
 materially is a desideratum. 
 
 In the reduction of a long net of triangulation we have 
 seen that labor is saved by breaking the work into two 
 parts, first adjusting at each of the stations for the local 
 conditions, and then using this work in the further adjust- 
 ment arising from the angle and side equations. Now, if 
 the measurements were made on a uniform plan the local 
 adjustment would be simplified, as we should have a similar 
 problem to solve at each station. This would lead to a 
 great saving of labor, since, if the measurements are made 
 at hap-hazard, the local adjustment may be quite compli- 
 cated. 
 
 If we decide, then, that the observer must work in ac- 
 cordance with some regular form, our next inquiry is, What 
 shall that form be ? First, shall the angles be measured in- 
 dependently or in arcs ? The point to be aimed at in this 
 as in all work of precision is to get rid of systematic error. 
 The accidental errors are trifling in comparison. When 
 we consider twist of triangulation station from the action 
 of the sun's rays; the influence on distinctness of vision for 
 the same focus for different lengths of lines sighted over; 
 the interruptions that may occur in the course of reading a 
 long arc; the more uniform light that may always be had 
 when the number of signals in use at one time is small, etc., 
 we cannot but conclude that greater precision is to be 
 attained by measuring the angles independently. The 
 errors are more likely to mutually balance. Even Andras, 
 the author of the most important contributions to the 
 method of directions since Bessel, and who used this 
 method in the triangulation of Denmark, acknowledges 
 43
 
 332 THE ADJUSTMENT OF OBSERVATIONS. 
 
 that " in place of observations of directions in arcs it is pre- 
 ferable to return to the old method of Gauss in measuring 
 angles." * 
 
 As regards the cost, it must be acknowledged that for 
 an equal number of results, leaving quality out of account, 
 the method of arcs has the advantage. Nowadays, how- 
 ever, when facilities exist for measuring angles by night as 
 well as by day, there is less delay in waiting for suitable 
 conditions than when day work alone had to be depended 
 on.f Taking this into account, the difference in cost would 
 not be great in any case, more especially as a triangulation 
 party is never a very large one. 
 
 Having decided that angles should be measured inde- 
 pendently, it is in accordance with general experience that 
 instead of spending all of the time of observation in meas- 
 uring the single angles themselves better results would be 
 obtained by spending part of it in measuring combinations 
 of the angles. A simple form that at once suggests itself 
 would be to close the horizon at each station ; that is, to 
 measure all of the angles AOB, BOC, . . . LOA in order 
 round the horizon (see Fig. 20). The local corrections 
 
 would each be - of the discrepancy of the sum of the 
 
 angles from 360 (Art. 121), and the reduction is thus 
 simple and uniform. However, as measuring the closing 
 angle LOA is the same as measuring the sum angle AOL, it 
 would seem that if we measure one sum angle we ought to 
 measure all possible sum angles.:}: Though the form of 
 adjustment for this combination of measures is a special 
 case of that already given, I shall, at the risk of a little 
 repetition, sketch it in full. 
 
 * VerhaniUungen der europtiischen Gradmcssung, 1878, p. 47. 
 
 t Sec C. S. Report 1880, App. No. 8. Experiments made at Sugar Loaf Mountain, Georgia, 
 have shown that an apparatus cheap and easily operated can be used ; that night observations are 
 a little more accurate than those by day, and <that the average time of observing in clear weather 
 can be more than doubled by observing at night. 
 
 t This form of combination was introduced by Gauss in the triangulation of Hanover. 
 A similar form is employed on the New York State Survey. See also C. S. Report 1876, 
 App. 20.
 
 APPLICATION TO TRIANGULATION. 
 
 333 
 
 154- The Local Adjustment (Angles). Let O be the 
 
 station occupied, and 1,2,3,4 the sta- 
 tions sighted at in order of azimuth ; 
 then the angles to be measured would 
 be 
 
 1 02 
 
 103 203 
 
 104 204 304 
 
 Take the first three as independent 
 
 unknowns, and let A, B, C denote their 
 
 most probable values. Also let / ia , / 13 , . . . denote the 
 
 several measured values. 
 
 The observation equations are 
 
 Fig. 43 
 
 -A + B 
 -A 
 
 and the normal equations 
 
 (0 
 
 Solve these equations, and 
 
 (3) 
 
 It is useful to notice as a check that 
 
 A+JS+C=l lt + i tt + / lt (4) 
 
 For practical computation arrange in tabular form as 
 follows. Sum first the horizontal rows, next form the sum
 
 334 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 differences, and then place these quantities in the proper 
 columns. Add each column and divide the sums by 4 : . 
 
 A, 
 
 /,. 
 
 / 
 
 Sum, 
 
 
 
 4s 
 
 4,4 
 
 Sum, 
 
 Sum, Sum 2 
 
 
 
 A 4 
 
 Sum 3 
 
 Sum, Sum 3 
 
 Sum, Sum, 
 
 Sum, Sum 3 
 
 Sum, 
 
 
 
 A 
 
 
 
 C 
 
 Check 
 
 
 Check Solution. The form of the expressions found as 
 the values of A, B, C suggests a method of finding these 
 values in accordance with the fundamental principle of the 
 mean. For, writing the value of A in the form 
 
 A = 
 
 2+1 + 1 
 
 (5) 
 
 we see that it is the weighted mean of the values 
 
 weight i 
 
 - 
 
 that is, the value of A is found by taking the weighted 
 mean of the measured value of A and of all the values of A 
 that can be found by combining two other measures. 
 
 Similarly for the remaining angles. 
 
 The Precision. From Eq. 3 or from Eq. 5 it is evident 
 that each of the angles A,B,C has the weight 2. The weight 
 Pot the angle A -\- B is given by (see Art. 101) 
 
 
 and P=^
 
 APPLICATION TO TRIANGULATION. 
 
 335 
 
 The same result will be found for the remaining angles. 
 Hence after the station adjustment each angle has the 
 same weight, being double the weight of a single measured 
 angle in our case of 4 stations sighted at. With n stations 
 
 sighted at the weight of an adjusted angle would be ' 
 times the weight of a single measured angle. 
 
 Ex. i. Given at station Oswego, in the triangulation of Lake Ontario 
 (Fig. 38), 
 
 /n = 80 29' 46". IO 
 
 /13 = 107 ig' 03". 28 
 
 / 14 = I38 12' 49"-44 
 
 / as == 26 49' l6".6l 
 4u= 57 43' oi".g6 
 Au= 30 53' 42". 88 
 required the adjusted values. 
 
 Solution. 
 
 46". 10 
 
 63". 28 
 
 109". 44 
 
 218". 82 
 
 
 i6".6i 
 
 6i".g6 
 
 78". 57 140". 25 
 
 
 
 42". 88 
 
 42". 88 i75"-94 
 
 140". 25 
 
 i75"-94 
 
 218". 82 
 
 
 iS6". 3 5 
 
 255"-83 
 
 433"- 10 
 
 
 46". 59 
 
 63". 96 
 
 108". 28 
 
 218". 83 check. 
 
 A 
 
 B 
 
 C 
 
 
 and the adjusted angles 
 
 
 
 80 
 
 29' 
 
 46" 
 
 59 
 
 
 
 
 107 
 
 19' 
 
 03* 
 
 .96 
 
 
 
 
 138 
 
 12' 
 
 48" 
 
 .28 
 
 
 
 
 26 
 
 49' 
 
 I?" 
 
 37 
 
 
 
 
 57 
 
 43' 
 
 01", 
 
 69 
 
 
 
 
 30 
 
 53' 
 
 44" 
 
 32 
 
 
 Check Solution. 
 
 
 Angle A. 
 
 
 Angle 
 
 A'. 
 
 
 Angle C. 
 
 46". 10 wt. i 
 
 3 
 
 '.28 wt. 
 
 i 
 
 49". 44 wt. i 
 
 46 
 
 '.67 ' \ 
 
 2 
 
 '71 
 
 " 
 
 i 
 
 46". 16 " ^ 
 
 . 47 
 
 "48 " \ 
 
 6 
 
 "56 
 
 " 
 
 i 
 
 48". 06 " ^ 
 
 Mean, 46'. 59 
 
 48". 28 
 
 Similarly for the remaining angles. 
 
 The weight of each adjusted angle is twice that of each measured angle. 
 
 Ex. 2. The angles at station Vanderlip (Art. 142) may be adjusted in the 
 same way as the above.
 
 336 THE ADJUSTMENT OF OBSERVATIONS. 
 
 155. The General Adjustment (Angles). The angle 
 and side equations are formed as explained in Arts. 117- 
 139. The connection between the local and general adjust- 
 ment is through the weight equations. In our example of 
 4 stations, in which every angle between every two direc- 
 tions is measured, the weight equations are in three groups, 
 of which the first is (see Eq. 2, p. 333) 
 
 which equations are moderately simple in form. 
 
 If we had simply closed the horizon the first group of 
 weight equations would have been 
 
 which are as complicated as the preceding. 
 
 Hence, in a net in which the angles at each station are 
 measured in either of the ways indicated, the solution 
 would 'be simplified so far as the local adjustment is con- 
 cerned, in that the adjustment at each station is of a fixed 
 form, but in the general adjustment arising from the angle 
 and side equations the gain is comparatively little. 
 
 If, however, instead of finding the corrections to the 
 angles we had found the corrections to the directions of the 
 arms of the angles, the weight equations become much 
 simplified, and therefore also the whole reduction. This 
 idea, which, I believe, is due to Hansen, will now be 
 developed.* 
 
 156. The Local Adjustment (Directions). If X, 
 A, B, C denote the readings of the 4 directions i, 2, 3, 4 
 from O, then since A X, B X, C X correspond to the 
 
 * See Die f>renss!schc Landestriangulation. Berlin, 1874, scq.
 
 APPLICATION TO TRIANGULATION. 337 
 
 A, B, C in the angle adjustment, the observation equations 
 may be written 
 
 -X+A -/ = *,. 
 
 -X + B -/,. = 
 
 -JT + c-4 = * I4 
 
 -.* + /? -/ = *' (0 
 
 -A + C-/, t = ^ t 
 
 -JB + C-l. t = v, t 
 
 and the normal equations 
 
 3 x- A- B- <:=-/-/-/ 
 -,r^- 5- c= /-/-/ 2 
 
 Adding these equations, there results 
 
 = 
 
 and therefore the unknowns cannot be found without some 
 further relation connecting them. The reason of the inde- 
 terminate form is that directions are nothing but the angles 
 which the rays Oi, O2, . . . make with some common zero 
 ray whose position is not fixed, and which may therefore be 
 taken arbitrarily. To carry out the solution it will be most 
 convenient to fix the zero ray by making the arbitrary 
 assumption 
 
 X+A+B+C=o (3) 
 
 By adding this to each of the normal equations they re- 
 duce to 
 
 4'Y" A 1 /I 3 ~ A 4 
 
 *A-- /_/-/ 
 
 48= /,, + /,,-/,, (4) 
 
 4C /,4 + /,4 + '., 
 
 which give the values of X, A, B, C directly.
 
 338 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The computation may be rendered quite mechanical by 
 arranging in tabular form : 
 
 I 
 
 2 
 
 3 
 
 4 
 
 Sums. 
 
 
 4, 
 
 4s 
 
 4, 
 
 Sum, 
 
 
 
 4s 
 
 4, 
 
 Sum 2 
 
 
 
 
 4 4 
 
 Sum 3 
 
 - Sum, 
 
 - Sum, 
 
 - Sum 3 
 
 
 
 4X 
 
 4 A 
 
 AB 
 
 4C 
 
 
 X 
 
 A 
 
 B 
 
 C 
 
 Check 
 
 The transformation of the normal equations (2) into (4) 
 by means of the arbitrary relation (3) is allowable. For 
 since the sum of the coefficients of the unknowns in each of 
 equations 2 is zero, whatever values of X, A, B, C satisfy 
 those equations X-\-a, A -\-a, B-\-a, C -\- a, where a is any 
 constant, will also satisfy them. Hence whatever set of 
 values is taken to satisfy the equations, the differences 
 A X, B X, CX will be the same in value. There- 
 fore by arbitrarily fixing the zero direction we find deter- 
 minate values for the corrections to the other directions. 
 
 Ex. Take that of Ex. i, Art. 154. 
 seconds only, is 
 
 The tabular scheme, writing down the 
 
 
 46". i o 
 
 63". 28 
 
 109". 44 
 
 218". 82 
 
 
 
 i6".6i 
 
 6i".g6 
 
 78"-57 
 
 
 
 
 42 ".88 
 
 42". 88 
 
 218". 82 
 
 - 78". 57 
 
 42". 88 
 
 
 
 218". 82 
 
 - 32". 47 
 
 37". 01 
 
 214". 28 
 
 
 - 54" -70 
 
 8". 12 
 
 9". 25 
 
 53"- 57 
 
 
 X 
 
 A 
 
 B 
 
 C 
 
 
 and A X, B X, CX give the same values of the angles as found before. 
 Hence, whether we find the local corrections to the angles or to the directions
 
 APPLICATION TO TRIANGULATION. 
 
 339 
 
 of the arms of the angles, the result is the same. One method or the other 
 may, therefore, be used, as is most convenient. 
 
 To avoid the use of large numbers certain approximate values may be 
 assumed for the directions, and the corrections to these approximate values 
 found. (Compare Art. Si.) Thus if (X), (A), (B), (C) denote the corrections 
 to assumed values of X, A, B, C, we may proceed as follows : 
 
 Assumed approximate angles 
 
 Sodus, o oo' oo* + (A') 
 
 Vanderlip, 80 29' 46" + (A) 
 
 Duck, 107 19' 03" + (B) 
 
 Stony Pt., 138 12' 4 
 
 Tabular Form. 
 
 I 
 
 2 
 
 3 
 
 4 
 
 Sums. 
 
 
 + o".io 
 
 + 0".2S 
 
 + o'.44 
 
 + 0".S2 
 
 
 
 - o"-39 
 
 - i ".04 
 
 - i -43 
 
 
 
 
 3". 12 
 
 3'. 12 
 
 0".S2 
 
 + i"-43 
 
 + 3"- 12 
 
 
 
 0".S2 
 
 + i "-53 
 
 + 3". oi 
 
 3"-?6 
 
 
 o" . 20 
 
 + o".38 
 
 + "-75 
 
 o".g4 
 
 
 (X) 
 
 (A) 
 
 (S) 
 
 (0 
 
 
 and the adjusted angles are as before. 
 
 157. The General Adjustment (Directions). The 
 
 form of the normal equations 4, Art. 156, shows that at 
 each station the quantities X, A, . . . are determined inde- 
 pendently of one another. The weight of each direction 
 at a station, as shown by the weight equations, is repre- 
 sented by the number of stations sighted at. If, therefore, 
 (i), (2), . . . denote the corrections to the values A', A, . . . 
 locally adjusted, which arise from the angle and side equa- 
 tions, we may consider X, A, ... as independent quanti- 
 ties, all of equal weight and subject to certain rigorous con- 
 ditions, and proceed to carry out the solution according to 
 the simple form of Art. no. Hence, as the solution breaks 
 into two simple problems of adjusting quantities as if inde- 
 pendently observed, we see the advantage of adjusting the 
 directions instead of the angles. 
 
 44
 
 340 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 ' Ex. The quadrilateral Buchanan, Brule, Aminicon, Lester, in the tri- 
 angulation of Lake Superior. 
 
 Direction. 
 
 At Buchanan, 2-1 
 
 At Brule, 
 
 3-2 
 
 At Aminicon, 8-7 
 
 At Lester, 
 
 Measured Angle. Local Adj . Angle. 
 
 47 57' 36". 25 36". 21 
 
 97 26' 41". 29 4i"-32 
 
 49 29' 05". 15 5". 1 1 
 
 5-4 
 
 40 
 
 25' 
 
 47"-49 
 
 47"-36 
 
 6-4 
 
 71 
 
 58' 
 
 27". 31 
 
 27". 45 
 
 6-5 
 
 31 
 
 32' 
 
 40". 22 
 
 40". 09 
 
 8-7 
 
 37 
 
 47' 
 
 05". 51 
 
 5"- 1? 
 
 9-7 
 
 97 
 
 51' 
 
 03". 1 1 
 
 3"-45 
 
 9-8 
 
 60 
 
 03' 
 
 58". 62 
 
 58". 28 
 
 II-IO 
 
 5i 
 
 oo' 
 
 39"- 77 
 
 40". oo 
 
 12-10 
 
 92 
 
 43' 
 
 49". 42 
 
 49". 20 
 
 1 2-1 1 
 
 41 
 
 43' 
 
 o8". 9 7 
 
 9". 2O 
 
 The local adjustment is carried through as in Art. 154, it being more con- 
 venient to adjust the angles directly father than the directions. 
 
 Taking the locally adjusted angles as independent, we next find the cor- 
 rections to the directions arising from the angle and side equations. 
 
 The angle equations, formed in the usual way from the triangles Buchanan, 
 Brule, Aminicon (e = i"-37) ; Brule, Aminicon, Lester (e = i".i9); Lester, 
 Buchanan, Brule (s = i". 19) ; and the side equation, from the quadrilateral 
 itself (pole at Lester), are 
 
 -(i) + (2) -(4) + (6)- (8)+ ( 9 )=-o'. 5 7 
 - (i) + (3) - (5) + (6) - (10) + (ii) = - o".22 
 -(4) + (5) -(7) + (9) -() + (") = + i".i8 
 
 -O.I4(l) -0.90(2)+ 1.04(3) + I- 24 (4) -2. 95(5) 
 
 + i . 72(6) + i . 50(7) - i . 36(8) - o. 14(9) = - o" . 80 
 
 The number of stations pointed at from' each station occupied being 3, the 
 weight of each locally adjusted angle is the same throughout the net. 
 Hence we have the correlate normal equations 
 
 I. II. III. IV. 
 
 + 6.0 +2.0 +2.0 + 0.94 =0.57 
 
 + 2.O + 6.O 2.O + 5.84 = O.22 
 
 + 2.0 2.0 +6.0 - 5.84 = + 1.18 
 + 0.94 +5.84 5.84 +19.22 = 0.80 
 
 I. = 0.273 
 II. = +0.136 
 
 III. = +0.377 
 
 IV. = + 0.045
 
 APPLICATION TO TRIANGULATION. 341 
 
 and the corrections 
 
 C 1 ) +o". 13 (7) = o".3i 
 
 (2)= 0".3I (8)=r+o".2I 
 
 (3)=s+o".i8 (9)=+o".io 
 
 (4) = o'.os (10) = o".i4 
 
 (5)=+o'.n (ii) = o".24 
 
 (6)= o".o6 (i2)=: + o".3S 
 
 On the Breaking of a Net of Triangulation into Sections for 
 Convenience of Solution. 
 
 158. In a long chain of triangulation or in a complicated 
 net the simultaneous solution of the condition equations 
 would be very troublesome, not from any principle in- 
 volved, but from its very umvieldiness. Accordingly it is 
 necessary to break the work into sections and solve each 
 section by itself. As this breaking into sections causes 
 more or less disturbance of the local conditions at the lines 
 of breaking off, each section should be as large as can be 
 conveniently managed, and the lines of breaking off should 
 be so chosen as to disturb as few conditions as possible. 
 By the method of adjustment explained in Arts. 153-157 
 much larger sections can be taken than by any other. 
 This is a strong argument in favor of its use. 
 
 The contradictions that occur at the lines of breaking off 
 of the several sections are most conveniently bridged over 
 by means of the principle of Art. 115. As an example we 
 may cite the reduction of the principal triangulation of the 
 British Ordnance Survey. There were 920 condition equa- 
 tions to be satisfied in the net. The following was the 
 method of solution employed:* 
 
 " The triangulation was divided into a number of parts 
 or figures, each affording a not unmanageable number of 
 equations of condition. One of these being corrected or 
 computed independently of all the rest, the corrections so 
 obtained were substituted (so far as the} 7 entered) in the 
 
 * Account of the Principal Triangulation, p. 272.
 
 342 THE ADJUSTMENT OF OBSERVATIONS. 
 
 equations of condition of the next figure, and the sum of 
 the squares of the remaining corrections in that figure made 
 a minimum. The corrections thus obtained for the second 
 figure were substituted in the third, and so on." 
 
 In the triangulation of Mecklenburg* this method was 
 carried out even more systematically. The adjustment 
 was divided into five groups of 22, 22, 22, 21, 22 condition 
 equations respectively. The corrections resulting from the 
 solution of group 1. were carried, so far as they entered, into 
 group II., and this group solved, and so on through groups 
 III., IV., V. The whole operation was repealed four 
 times, and the small contradictions still remaining were 
 distributed empirically. 
 
 For an interesting conference on the whole question 
 see Comptes Rendus de r Association Gcode'sique Internationale, 
 1877. 
 
 Adjustment of a Triangulation for Closure of Circuit. 
 
 159. In the adjustment of a triangulation we have so far 
 considered it only with reference to a single measured base. 
 We have seen how at each station the discrepancy arising 
 from sum angles and from closure of the horizon can be got 
 rid of, and also how in a net joining several stations the 
 conditions arising from closure of triangles and from 
 equality of lengths of sides computed by different routes 
 can be satisfied. There remains the question as to the 
 mode of procedure when several bases enter whose lengths 
 are known and whose positions have been fixed astronomi- 
 cally. Special cases would be where a circuit of triangula- 
 tion closed on the initial line, and where a secondary system 
 is to be made to conform to two lines in a primary system, 
 the primary lines being assumed to be known in length and 
 position. 
 
 The conditions to be satisfied in the adjustment are four 
 in number- that the value of a base computed from another 
 
 * Grosshcrzoglich mecklcnburgi&che Landesi'i:ri>iessung. Schwerin, 1882.
 
 APPLICATION TO TRIANGULATION. 343 
 
 should agree with the measured value in azimuth, in length, 
 and in latitude and longitude of one of the end points. 
 
 The measured angles of the triangles connecting the 
 bases having been already adjusted with reference to one 
 base for the local and general conditions, the additional cor- 
 rections necessary to satisfy the closure of the circuit will 
 be small. But little difference in the results will therefore 
 be found by making a simultaneous solution of all of the 
 condition equations of closure and a solution by successive 
 approximation according to the method of Art. 115. 
 
 We shall consider only a single chain of triangles, all tie 
 lines of the system being rejected. This on account of sim- 
 plicity, and also for the reason stated above, that, in good 
 work the corrections being small, it gives results practically 
 close enough nearly the same, in fact, as a rigorous solu- 
 tion. If thought necessary in any special case a rigorous 
 solution of the condition equations can be carried out by 
 the method of correlates. 
 
 1 60. Adjustment for Discrepancy in Azimuth. 
 Let 1-2 and 5-6 be two bases connected through inter- 
 mediate stations 3, 4 by a single chain 
 of the best-shaped triangles that can 
 be selected from the net. 
 
 In computing the base 5-6 from 1-2 
 the sides 1-4, 3-4, 3-6 are at once sides 
 of continuation and bases, according 
 to the triangles considered. For example, in the triangle 
 i 2 4 the side 1-4 is a side of continuation from the base 1-2, 
 but in 134 the side 1-4 is a base with reference to 3-4 as a 
 side of continuation. 
 
 In the chain of triangles let 
 
 A lt A^ ... be the angles opposite to the sides of con- 
 tinuation. 
 
 /?,, B^, . . . the angles opposite to the bases in order of 
 computation. 
 
 C }} *,... the angles opposite to the flank sides.
 
 344 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Let z, z l denote the values of the measured azimuth of the 
 bas.es 1-2, 5-6 respectively. These values are assumed to 
 be correct, and receive no change in the adjustment. 
 
 A geodetic computation* of the azimuth of the base 5-6 
 from the base 1-2 is now made, using the values of the 
 angles of the intervening triangles resulting from the adjust- 
 ment for local and general conditions. Call the value of 
 the azimuth of 5-6 computed in this way z' . 
 
 Now, reckoning azimuth in the usual way from the 
 south as origin, and the direction of increase from south to 
 west, it is easily seen, by passing from 1-2 along the sides 
 1-4, 4-3, 3-6, that the excess 4 of the observed value z l over 
 the computed value z' of 5-6 is given by 
 
 -(Q + (Q-(Q + (Q = 4 (i) 
 
 where (7,), (C y ), . . . denote the corrections to the angles 
 * ,,... This is the azimuth condition equation. 
 
 In order that the correction arising from the azimuth 
 equation may not disturb the conditions of closure existing 
 among the angles of the triangles, it is necessary that the 
 corrections to the angles should satisfy the conditions 
 
 The unknowns in equations I, 2 are subject to the re- 
 lation 
 
 ' - . . = a rain. 
 
 Call ,, k^ k^ / 4 the correlates of equations (2), and k the cor 
 relate of equation (i), and we have the correlate equations 
 
 *. = 
 
 \ k (Q k, + k = (Q k. - k = (Q 4 + ^ = (Q 
 
 * For methods of doing this see Lee's Tables and Formulas, Washington, 1873 ; Coast Sur- 
 vey Report, 1875.
 
 APPLICATION TO TRIANGULATION. 345 
 
 whence the normal equations 
 
 3^ k = o 
 
 3^ 3 -f k = o 
 
 If we had ;/ triangles instead of four, the normal equations 
 would be of similar form, and, solving, we should find 
 
 *= 
 
 2n 
 
 and therefore the corrections 
 
 ~ 
 
 
 
 Hence the rule : Divide the excess of the observed over the 
 computed azimuth by the number of triangles, and apply one-half 
 of this quantity to each of the angles adjacent to the flank sides 
 on one side of the chain, and the total quantity, with the sign 
 changed, to the third angle. The signs are reversed for the 
 angles on the other flank. 
 
 If the azimuth mark is not on a triangulation side the 
 line of azimuth may be swung on to such a side by adding 
 the angle between the mark and the side.
 
 34^ 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. The sketch represents the secondary triangulation of Long Island 
 Sound. The sides 1-2, 14-15 are the lines of junction with the primary sys- 
 
 tem. These lines are assumed to remain unchanged in azimuth in the adjust- 
 ment, and the secondary system is to be made to conform to them. 
 
 The main chain of triangles joining the two primary lines is indicated in 
 the figure by heavy lines. The number of these triangles is n. The system 
 has been adjusted for local and geometrical conditions, and the resulting 
 angles of these triangles are as follows : 
 
 Angle. 
 
 log sin 
 diff. i" 
 
 Sph. 
 excess. 
 
 Angle. 
 
 log sin 
 diff. i" 
 
 Sph. 
 excess. 
 
 A, 
 
 82 49' 19". 25 
 
 0.27 
 
 
 A, 
 
 58 25' 08". 08 
 
 1.29 
 
 
 B, 
 
 64 55' 40". 32 
 
 0.99 
 
 3"-66 
 
 B-, 
 
 82 16' i6".i6 
 
 0.29 
 
 3"-63 
 
 c, 
 
 32 15' 04". 09 
 
 
 
 C-, 
 
 39 18' 39"-39 
 
 
 
 A* 
 
 52 37' 47"-9 8 
 
 1.61 
 
 
 A & 
 
 69 12' 06". 02 
 
 0.80 
 
 
 B* 
 
 74 10' 50". 56 
 
 0.60 
 
 4". 96 
 
 * 
 
 71 58' 54"-i8 
 
 0.68 
 
 3"-3 
 
 c, 
 
 53 n' 26". 42 
 
 
 
 c s 
 
 38 49' 02".83 
 
 
 
 A 3 
 
 69 20' 1 6". 03 
 
 0.79 
 
 
 A, 
 
 59 18' 26". 33 
 
 1.25 
 
 
 B* 
 
 67 43' 43"- 33 
 
 0.86 
 
 3". 53 
 
 B 
 
 102 08' 26". 94 
 
 -0.45 
 
 i "-33 
 
 C 3 
 
 42 56' 04". 16 
 
 
 
 c a 
 
 18 33' 08". 05 
 
 
 
 A, 
 
 39 31' 29". 66 
 
 2-55 
 
 
 A^ 
 
 67 38' 52". 77 
 
 0.87 
 
 
 B, 
 
 120 36' 36". 1 1 
 
 -1-25 
 
 i"-3i 
 
 Bio 
 
 70 26' 23". 26 
 
 0.75 
 
 2". 41 
 
 c< 
 
 19 5i' 55"-54 
 
 
 
 Go 
 
 41 54' 46". 38 
 
 
 
 A, 
 
 83 19' 3i"-o8 
 
 0.25 
 
 
 An 
 
 29 43' 55"- 99 
 
 3.69 
 
 
 B & 
 
 68 57' 47"- 75 
 
 o.Si 
 
 i ".42 
 
 n 
 
 55 23' 09". 36 
 
 1-45 
 
 2". 1 3 
 
 c, 
 
 27 42' 42". 58 
 
 
 
 CM 
 
 94 52' 56". 78 
 
 
 
 A t 
 
 87' 25' 26". 66 
 
 0.09 
 
 
 
 
 
 
 B, 
 
 44 10' 45". 25 
 
 2.17 
 
 3". 47 
 
 
 
 
 
 Ct 
 
 48 23' 5 1 ".56 
 
 
 
 
 

 
 APPLICATION TO TRIANGULATION. 347 
 
 A geodetic computation for latitude, longitude, and 
 azimuth was carried through from the line 1-2 to 14-15, 
 using the above angles. It was found that approximately 
 
 observed az. of 14-15 computed az. of do. = 2". 93 
 
 Hence the corrections to the angles of the triangles for this 
 discrepancy in azimuth are for the 
 
 first triangle (/!,) o". 13 second triangle (A^) = -\-o". 13 
 
 (/>',) =-o". 1 3 W = + o*.is 
 
 (Q = -f ".26 (Q = - o".26 
 
 and so on. 
 
 161. Adjustment for Discrepancy in Bases. This 
 is fully explained in Arts. 168-170. 
 
 Using the last form given in Art. 170, we may write the 
 base-line equation 
 
 \d A (A)-d B (B)-\ = l 
 
 from which, using the values of A^ />,, . . . found in the 
 azimuth adjustment as first approximations, further correc- 
 tions to the angles are found, as in equations 5, Art. 170. 
 Since the angles C do not enter into this adjustment, the 
 corrections resulting will not disturb the adjustment for 
 azimuth already made. 
 
 The advantage of this method of proceeding is that the 
 subsequent work does not disturb any adjustment already 
 made and thus render it necessary to make a new approxi- 
 mation. The labor is reduced to a minimum, and the re- 
 sults obtained are practically close enough. 
 
 For an example see Ex. i, Art. 170. 
 
 162. Adjustment for Discrepancy in Latitude 
 and Longitude. The corrections to the angles arising 
 from discrepancy in azimuth and in bases having been ap- 
 plied, the value of one base computed from another will 
 a<;ree with the measured value in direction and length. 
 
 o * 
 
 45
 
 348 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The discrepancy in position, as shown by the differences be- 
 tween observed and computed latitudes and longitudes, 
 alone remains. This discrepancy, being small, may be 
 eliminated closely enough by distributing it proportionally 
 from one end of the chain of triangles to the other, accord- 
 ing to Bowditch's rule as given in Ex. 5, Art. no that is, 
 the error in latitude in proportion to the longitudes, and 
 the error in longitude in proportion to the latitudes of the 
 several stations. Each station, being thus made slightly 
 eccentric, is next reduced to centre, when the whole net 
 will be consistent.
 
 CHAPTER VII. 
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 
 
 163. During the present century two forms of apparatus 
 have been used in the measurement of primary bases, the 
 compensation bars and the metallic-thermometer appa- 
 ratus. On the English Ordnance Survey the two principal 
 lines, the Lough Foyle and Salisbury Plain bases, were 
 measured with the Colby compensation bars. Most of the 
 bases of the U. S. Coast Survey and five of the eight bases 
 of the U. S. Lake Survey were measured with the Bache- 
 Wiirdemann compensation apparatus. On the Continent 
 of Europe the Bessel metallic-thermometer apparatus is 
 very generally used. 
 
 Indications are not wanting that both forms will be sup- 
 planted before long by an apparatus consisting of simply a 
 single metallic bar.* 
 
 The essential part of any form of base apparatus consists 
 of one or two bars of metal, usually from 4 to 6 metres in 
 length and of about 40 X 15 mm. cross-section. If the ex- 
 treme points of a bar are the limits of measure, so that a 
 measurement is made by contact [end-measures], two bars 
 are necessary. If, however, the length of a bar is con- 
 sidered to lie between two marks made on it [line-meas- 
 ures], so that in a measurement the transition from one bar 
 to the next depends, not on the stability of the bar, but on 
 some outside appliance, only one bar is necessary. 
 
 Descriptions of the various forms of apparatus used 
 on different surveys will be found in reports of those 
 surveys. 
 
 * See Ibanez, Zfitschr. fiir Instrumentenkiinrif, 1881. Also Art. 166.
 
 350 THE ADJUSTMENT OF OBSERVATIONS. 
 
 164. Precision of a Base-Line Measurement. For 
 
 clearness it will be necessary to outline the principles on 
 which the measurement is made. 
 
 First, we must find the length of the measuring bar in 
 terms of some standard of length ; and as the measurements 
 of the line itself are made at various temperatures, the co- 
 efficients of expansion of the metals in the measuring appa- 
 ratus must also be known. Comparisons must, therefore, 
 be made with the standard during wide ranges of tempera- 
 ture ; and as these comparisons are fallible, the results 
 found for length and expansion will be more or less er- 
 roneous. 
 
 The principle involved in the measurement is exactly 
 the same as in common chaining with chain and pins. 
 There are, indeed, various contrivances for getting a pre- 
 cision not looked for in chaining, such as for aligning the 
 measuring bar, for finding the inclination of each position 
 of the bar, and for establishing fixed points for stopping at 
 and starting from in measurement. But these make no 
 change in the essential principle. 
 
 The errors in the value of a base line may, therefore, be 
 considered to arise from two principal sources, comparisons 
 and measurement. Experience has shown that the main 
 error arises from the comparisons, and that, even it our 
 modes of measuring the base line itself were perfect, the 
 precision of the final value would be but little increased so 
 long as the methods of comparison are in their present 
 state. Thus in the Lake Survey primary bases, if the field 
 work had been without error, the total p. e. of the bases 
 would have been diminished only about ^ part. 
 
 These errors differ essentially in character. An error 
 arising from the comparisons, being the same for each bar 
 measurement, is cumulative for the whole base, while errors 
 arising in the measurement of the base itself, were the 
 measurements repeated often enough and the conditions 
 sufficiently varied, would tend to mutually balance, and 
 could, therefore, be treated by the strict principles of least
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 351 
 
 squares. But as the number of measurements is not often 
 more than 2 or 3, and as these are made usually at about 
 the same season of the year, only a comparatively rough 
 estimate of the precision is to be looked for. 
 
 As a check on the field work a base is usually divided 
 into sections by setting stones firmly in the ground at 
 approximately equal intervals along the line, so that in- 
 stead of being able to compare results at the end points 
 only, \ve may compare results just as well at 6 or 8 points. 
 In this way a better idea of the precision of the work is 
 obtained, as we have 6 or 8 short bases to deal with instead 
 of a single long one. 
 
 We proceed now with the problem of determining the 
 precision of measurement. Ft may be stated as follows: 
 A base is measured in sections with a bar of a certain 
 length, each section being measured ;/, times. By the first 
 measurement the first section contains J// bars, the second 
 M" bars, . . . ; by the second measurement the first sec- 
 tion contains MJ bars, the second Mf bars, . . . ; and so 
 on. The weights of the measurements in order being 
 A' A' / P" > A"> / respectively, required the 
 m. s. e. of the most probable value of the base. 
 
 Let F, most prob. value of first section 
 
 F a = most prob. value of second section 
 
 then we have the observation equations 
 First section, V, M{ = ?>/ wt. />,' 
 
 V,-M{'=,v;' wt. A" 
 
 Second section, F, J// : = r>/ wt. // 
 V, - M," = v," wt. A" 
 
 and so on. 
 
 Now, either of two assumptions may be made.
 
 352 THE ADJUSTMENT OF OBSERVATIONS. 
 
 (a) In the first place, that the precision of the measure- 
 ment of each bar is the same throughout the different 
 sections. 
 
 We have, then, nn l equations containing n unknowns, 
 and the normal equations are 
 
 whence F,, F 2 , . . . are known, and therefore the whole 
 line V= V, -f F a + . . . -f V n is known. 
 
 The mean-square error /j. of an observation of weight 
 unity that is, of a single measurement of a bar is given 
 by (see Art. 99) 
 
 [fivv] 
 
 No. of obs. No. of indep. unknowns. 
 
 Now, the length of the measuring bar being taken as the 
 unit of measurement, the weight of a section, as depending 
 on the measurement, may be expressed in terms of the 
 number of bars measured. 
 
 For since ft is the m. s. e. of a measurement of a single 
 bar, the 'm. s. e. of the measurement "of a length of M bars 
 
 is n VM. Hence =^ is the weight of a measurement of 
 
 length M when the weight of a measurement of the unit of 
 length is unity. 
 
 Writing, therefore, for the weights / their values in 
 terms of M, 
 
 / I Vvi 
 
 y (,- i)\Jf 
 
 vv~\ 
 M\ 
 
 In the case usually occurring in practice, where the line 
 is measured twice, we may put this formula in a form more
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 
 
 353 
 
 convenient for computation. For if the first measurement 
 of the w, sections gave lengths M lt M t , . . ., and the second 
 measurements gave lengths J/,-)-^, M^-\-d v . . . for the 
 same sections in order, then, since 
 
 we have for the m. s. e. of one measurement of a bar and for 
 the mean of two measurements respectively 
 
 Hence the m. s. e. of a single measurement and of the mean 
 of the two values of the whole base are 
 
 the number of bars in the line being [M]. 
 
 Ex. The Bonn Base, measured in 1847, near Bonn, Germany, with the 
 original Bessel metallic-thermometer apparatus. The base was a broken 
 one, the two parts making an angle of 179 23'. Each part was measured 
 twice, as follows:* 
 
 
 Differences. 
 
 No. of bars. 
 
 Northern Part, Sec. i 
 
 L 
 0.183 
 
 116 
 
 Sec. 2 
 
 + o.cn)4 
 
 87 
 
 Sec. 3 
 
 0.013 
 
 6 1 
 - 264 
 
 Southern Part, Sec. i 
 
 0.007 
 
 92 
 
 Sec. 2 
 
 + o.(xj5 
 
 60 
 
 Sec. 3 
 
 + 0.757 
 
 131 
 
 283 
 
 * Das rhcinischf Drciecksnct:. Berlin, 1876.
 
 354 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hence the m. s. e. of the northern part, arising from errors of measurement 
 only, is 
 
 y_ .094' 
 
 6 "" 87 
 and the m. s. e. of the southern part is 
 
 L 
 -- = 0.093 
 
 283/.007 2 .cos' 2 -757' 2 
 
 -+- - + - L 
 
 92 60 131 
 
 \ L 
 
 = - 
 
 / 
 
 The other two main sources of error are : 
 
 1. Error in comparison of the measuring bars with one another. 
 
 2. Error in the determination of their length. 
 
 The m. s. e. arising from these sources are respectively 
 
 L L 
 
 0.386, 0.313 for the northern part 
 
 0.391, 0.335 f r the southern part 
 Remembering that these latter errors are systematic, we have, finally, 
 
 m. s. e. of base = Kogs^ .327'^+ (.386 + .391)'- + (.313 + .335)- 
 
 L 
 
 = 1.07 
 
 (b) In the second place, if we assume that the law of 
 precision of the measurements of the different sections is 
 unknown, and that these sections are independent, we have 
 for the mean of the values of the several sections and their 
 m. s. e. 
 
 v _ 
 
 J&wL 
 
 since p , ,,= m . i 
 
 v _ 
 
 n 2 - a a . , . 
 
 / K r . -,/ v - r Since A =ff=. . . = 
 
 [AJ<X - i) ,(, - i)
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 355 
 
 If V denotes the whole line, so that 
 
 r=* r ,+ * r .+ . +Vn 
 
 then, since the measurements are independent, 
 
 and the (m. s. e.) 2 of a single measurement of the line 
 
 
 The number of bars in the line being [M], we have for the 
 average value of the (m. s. e.) 2 of a single measurement of -a 
 bar 
 
 If, for example, the line has been measured twice, and 
 d^ */,... ^4 are the differences of the measurements ot the 
 several sections, then 
 
 and therefore 
 
 r 2 i 
 
 and the (in. s. e.) 2 of a single measurement of a bar is i FTT 
 
 4 6
 
 356 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. The Chicago Base, measured in 1877 with the Repsold metallic- 
 thermometer apparatus belonging to the U. S. Engineers. The base was 
 divided into 8 sections and was measured twice. 
 
 Section. 
 
 No. of Bars. 
 
 Diff. of Measures. 
 
 I. 
 
 227.25 
 
 mm 
 - 1-3 
 
 II. 
 
 230.25 
 
 + 2.5 
 
 III. 
 
 234.50 
 
 + 2.3 
 
 IV. 
 
 232. 
 
 + 0.7 
 
 V. 
 
 231. 
 
 + i.5 
 
 VI. 
 
 225. 
 
 + i.i 
 
 VII. 
 
 300.50 
 
 + 1.3 
 
 VIII. 
 
 196.80 
 
 O.2 
 
 Taking the errors of the different sections as independent, the p. e. of the 
 mean of the two measures of the base is 
 
 :. 4 6 
 
 0.6745 \ 
 
 4 
 
 The p. e. arising from the other sources of error were 
 
 mm 
 
 (1) measuring bar, 6.38 
 
 (2) metallic thermometer, 2.82 
 
 (3) elevation above mean tide, N. Y., 0.36 
 
 Assuming these to be independent, the p. e. of the Chicago Base at sea level is 
 
 mm 
 
 Vi.46' 2 + 6.38- + 2.82 s + 0.36- = 7.14 
 
 Of the two assumptions (a) and (b), the first is due to 
 Bessel and the second to Andrse. For a more elaborate 
 discussion of the points involved see Astron. Nac/ir., Nos. 
 1924, 1935, and V. J. Sclir. d. Astr. Ges., 1878, p. 69. 
 
 165. Length of Base and Number of Bases neces- 
 sary in a Triangulation. In computing a side of a 
 triangle from another and shorter side as base, the loss of 
 precision arising from the influence of the acute angle 
 opposite to the base is the greater the more acute that 
 angle is. To guard against this, if a measured base were a 
 side of a triangle it would be necessary that its length be
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 
 
 about the average length of the triangle sides of the sys- 
 tem. In the older work the practice was to measure long 
 bases, even as long as 20 kilometres and over. The modern 
 practice was introduced by Prof. Schwerd, of Speyer, Ger- 
 many, in 1819. By measuring a short base of 860** and 
 checking on to a line computed from a base of 15 kilo- 
 metres, he found a result agreeing within o m . i. His con- 
 elusion was that " with a small expenditure of time, trouble, 
 and expense the base of a large triangulation can be de- 
 termined by a small exactly measured line." 
 
 A similar conclusion was reached by Gen. Ibanez as the 
 result of measuring the base of Madridejos, Spain, in 1859. 
 He says: " The question, so much disputed between French 
 and German geometers, as to whether it is necessary to 
 measure long bases, or if short bases are sufficient, had 
 occupied the attention of the observers. Taking advantage, 
 therefore, of the favorable opportunity which presented 
 itself to them, they proposed to compare the results ob- 
 tained from the direct measure of the whole base with those 
 calculated from a special triangulation depending on the 
 central section of the base."f 
 
 The adjustment of this triangulation net gave rise to 36 
 angle equations and 28 side equations containing 90 unknown 
 quantities. The net is shown in the figure. 
 
 " We give in the following table the results of the direct 
 measures reduced to the sea level, and their comparison 
 with the values found trigonometrically : 
 
 Measured. Triangulation. 
 
 Fi 9- 47 
 
 3077459 3077-462 
 
 2216.397 2216.399 
 2766.604 
 
 2/23.425 2723.422 
 
 3879.000 3879.002 
 
 14662.885 14662.889 
 
 * Die kleinc Sfi-yercr Flasis. Speyer, i8?z. t Astron. A'ac/ir., No. 1462.
 
 358 THE ADJUSTMENT OF OBSERVATIONS. 
 
 " The remarkable agreement which the two operations 
 present is sufficient authority to limit the length of bases, 
 and to be satisfied with those of 2 or 3 kilometres in length, 
 always on the condition that they are joined to the sides of 
 the main triangulation by means of a system of lines ar- 
 ranged so as to be able to apply to it the proper method of 
 adjustment." 
 
 If errors arising from the angles of the triangles are 
 neglected, it is easy to show that if a short base M is 
 measured n times, and the arithmetic mean of the n meas- 
 urements taken as base from which a line equal to nM is 
 derived by triangulation, the precision of this line is the 
 same as if it had been measured once directly.* 
 
 For if [JL M = the m. s. e. of a length M, then f* M V7i is the 
 m. s. e. of a length nM. Also, the m. s. e. of the arithmetic 
 
 mean of n measurements of the line M is 7^, and when 
 
 Vn 
 
 from this line the line nM is derived trigonometrically its 
 m. s. e. is .-- n = H M Vn t agreeing with the preceding. 
 
 V Jl 
 
 Since, however, the principal errors arise from the tri- 
 angulation, the advantage is with the longer base. 
 
 We may estimate the relative amount of influence of 
 errors in the base and of errors in the angles in a chain of 
 triangles, on the value of any side computed from the base, 
 as follows : 
 
 Assuming the triangles to be approximately equilateral, 
 we have (Ex. 9, Art. in) 
 
 H a * /V -J- f // sin 2 \" Vn 
 
 where a H is the computed side, b the base, n the number of 
 triangles intervening, and // the m. s. e. of a measured 
 angle. 
 
 Suppose, for the sake of fixing our ideas, that b 10000, 
 then 
 
 u a ^ = fj-b +-1600 nfjf millimetres. 
 
 * Gradmessung in Ostpreussen, p. 36.
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 359 
 
 Now, with a primary base apparatus a precision of a 
 m. s. e. of 2 mm. in icoo m. can be easily reached.* Hence 
 from the above formula we see that even in a short chain 
 of triangles the error of a side arising from the error of the 
 base is small in comparison with that arising from the errors 
 in the angles measured. 
 
 Also, since the m. s. e. of a side arising from errors in 
 the angles measured increases as the square root of the 
 number of triangles from the base (Ex. 9, Art. in), we con- 
 clude that it is better to measure bases frequently with 
 moderate precision rather than to measure a few at long 
 intervals in the net, but with great precision. 
 
 But little is gained in precision by repeating the meas- 
 urement of the base many times. We have seen in Art. 
 164 that the main sources of error to be leared arise from 
 the comparisons with the standards and are independent of 
 the measurement of the line. Accordingly, though bases 
 have in the present century been measured from 6 to 8 
 times, it is the general custom now to do so only twice. 
 In this way any gross error is checked and a sufficiently 
 close precision determination of the measurement can be 
 found. As a compromise between the error arising from 
 the connection with the triangulation and the systematic 
 error introduced by the base apparatus itself, the general 
 practice is to measure bases of from 4 to 8 kilometres f in 
 length, or about one-sixth of the triangle sides. Besides, 
 long bases are not always to be had in positions .just where 
 wanted as triangle sides, as the configuration of most coun- 
 tries will not allow of it. 
 
 * The p. e. of the Madridejos Base (1858), Spain, is part ; of the Grossenhain Base 
 
 5865800 
 
 1872), Saxony, part ; of the Atlanta Base (1872-1873), U. S., part ; of the Chicago 
 
 Base (1877), U. S., part ; of the Sanduskv Base (1878), U. S., part. 
 
 1052200 1148600 
 
 t For example, Salisbury Plain (1849), England, u.i kil. ; Halland (1863), Sweden, 7.3 kil. ; 
 Oran (1867), Algeria, 9.4 kil. ; Harlem (1868-1869!, Holland, 60 kil. ; Atlanta (1872-1873), U . S., 
 9.3 kil. ; Radautz (1874), Austria, 4.6 kil. ; Udine (1874), Italy, 3.2 kil. ; Chicago (1877), V. S., 7.5 
 kil. ; Vich (1877), Spain, 2.5 kil. ; Gottingen (1880), Germany, 5.2 kil. ; Meppen (1883), Germany, 
 7.0 kil.
 
 360 THE ADJUSTMENT OF OBSERVATIONS. 
 
 166. Measuring a base line is not necessarily a difficult 
 operation. The great trouble with all forms of apparatus 
 hitherto employed arises from temperature changes. But 
 with the apparatus proposed by Mr. E. S. Wheeler, in 
 which the measuring bar is simply a bar of metal packed in 
 melting ice, this difficulty would be altogether overcome. 
 The length of the bar would remain unchanged throughout 
 the measurement, as its temperature is kept constant, being 
 that of melting ice. The same temperature could at any 
 time be had at which to find the length of the bar in terms 
 of the official standard of length. 
 
 The apparatus might be constructed as follows: The 
 measuring bar, a bar of steel 25 mm. in diameter and 6m. in 
 length, placed in a circular cast-steel tube I m. in diameter, 
 made stiff by bracing, but as light as possible. Along the 
 top of this tube slots of about 75 mm. in width would be cut 
 to allow the introduction of ice around the bar. The hole 
 for drainage would be at the centre of the tube on the 
 under side. For supports during the measurement two 
 trestles placed i-I m. from the ends would be best. Effects 
 of flexure would be got rid of by having the graduation 
 marks showing the length of the bar placed on the neutral 
 axis of the bar. The reading microscopes, alignment appa- 
 ratus, sector and level for determining the inclination of the 
 bar during measurement, such as those made by Repsold 
 for the U. S. Engineers. The mode of measurement the 
 same as with the Repsold apparatus.* The amount of 
 computation necessary to reduce the measurements made 
 in this way would be small in comparison with that 
 required with the forms of apparatus at present in 
 use. 
 
 167. Connection of a Base with the Main Tri- 
 aiigulatioii. We shall now consider the best method of 
 connecting a base line with a side of the main triangulation 
 with the least possible loss of precision. This, like most 
 
 * See Professional Papers Corps of Engineers U. S. A., No. 24, for description of the Rep- 
 sold base apparatus.
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 361 
 
 geodetic questions, is not to be decided by least-squares' 
 methods alone. 
 
 There are two points to be considered as little loss of 
 precision as possible, combined with economy of work in the 
 measurement and reduction of the intervening triangula- 
 tion. To secure the first we must use only well shaped 
 triangles, avoiding in particular very acute angles opposite 
 the bases. To secure the second as few stations as possible 
 should be occupied. 
 
 The solution is a tentative one. Various forms of con- 
 nection may be tried, and the m. s. e. of the triangle sides 
 computed from the bases by the methods of Chapter V., on 
 the hypothesis of a regularity of figure which conforms 
 more or less closely to the case in hand. Many of the re- 
 sults will be found in Ex. 10-15, Art. in. These results we 
 shall now make use of. 
 
 Taking the length of a triangulation side to be from 15 
 to 30 miles, as giving the best results in angle measurements, 
 the proper length of the base would be from 3 to 5 miles 
 that is, about one-sixth of a triangle side. Now, if the con- 
 nection between the base AB and a triangle side 6 times 
 the base is through a chain of equilateral triangles of the 
 form of Fig. 12 ; or of the rhomboidal form of Fig. 15, com- 
 posed of two similar triangles, in which the first diagonal 
 BB' is 6 times the base ; or of the rhomboidal form of Fig. 16, 
 in which the second diagonal .is 6 times the base, the re- 
 spective weights of the derived side are, roughly, as the 
 numbers 16, 3, 7. 
 
 Hence, so far as the precision is concerned, the advantage 
 is with the first form. When, however, we consider that 
 this form requires 12 stations to be occupied, while the third 
 needs only 6, we see that when amount of labor as well as 
 degree of precision is taken into account the advantage is 
 with the latter form. The second form, though the most 
 economical, is condemned by the acute angles that occur in 
 it opposite to the base. 
 
 Similar comparisons of other forms of connection will
 
 362 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 lead us to the conclusion that the rhomboidal form (Fig. 16), 
 in which a side equal in length to about 6 times the base 
 can be found by passing through two sets of similar tri- 
 angles from the base, is the normal form, and that the forms 
 used in practice should approximate to it as closely as the 
 nature of the country will allow. As examples take the 
 connections of the Coast Survey Atlanta Base (1872-1873), 
 the Lake Survey Chicago Base (1877), and the Prussian 
 Gottingen Base (1880). 
 
 Fig.50 
 
 Fig.49 
 
 Fig.48 
 
 ATLANTA 
 
 QOTTINQEN 
 
 The form of the triangulation net to connect the base 
 with the main triangulation being decided on, it becomes a 
 question (but only a secondary one) to decide with what 
 care the several angles in this net should be measured in 
 order to attain the greatest precision with a given expendi- 
 ture of labor. This will be found fully discussed in ZeitscJir. 
 fiir Vermess., 1882, pp. 122 seq. 
 
 1 68. Adjustment of a Triaiigulatioii when more 
 than one Base is considered. In all discussions so far 
 we have considered only one measured base. But in a net 
 of triangles, bases must be measured at intervals, for reasons 
 already assigned. In computing an intermediate side from 
 different bases discrepancies will be found. How shall 
 these discrepancies be treated? 
 
 The triangulation adjustment could have been made in
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 363 
 
 such a way that no discrepancy would show itself in pass- 
 ing from one measured base to another. It was only neces- 
 sary to introduce equations connecting the bases in the 
 form of an ordinary side equation, thus (Fig. 12), 
 
 a sin A sin A 
 
 b sin B^ sin B^ 
 
 where a, b are two bases, and A lt B^ . . . are the angles of 
 continuation. 
 
 There is no objection to doing this so long as the bases 
 have been measured with the same apparatus and the inter- 
 vening triangulation is first-rate. But as the discrepancy 
 between the measured value of the base and its value as 
 computed from another base through the triangulation 
 affords a good test of the quality of the work, it is better 
 not to introduce an equation connecting the bases into the 
 first adjustment. 
 
 169. Suppose that the triangulation has been adjusted in 
 sections, each with reference to a single measured base, as 
 explained in Chapter VI., and we wish to adjust for the 
 discrepancy arising from computing one base from another 
 through a chain of the best-shaped triangles in the system. 
 It will be sufficient to confine our attention to this chain of 
 triangles, all tie lines being rejected. 
 
 (i) Rigorous Solution. The condition equation to be 
 satisfied, arising from the connection of the bases, may be 
 written 
 
 <* + (*) sin \A, + (Aft sin \A t +(A t )\ 
 b + (b] sin {/?,+(#,)} sin {*, 
 
 where a, b, A t , Z?,, . . . are measured values, and (a), (/;), 
 (/?,), (-5,), . . . are their most probable corrections. 
 
 Taking logs, and reducing to the linear form (Ex. 4, 
 Art. 64), 
 
 - d.(a) + W) + [3 A (A) - J^)] = / 
 
 47
 
 364 THE ADJUSTMENT OF OBSERVATIONS. 
 
 where / is the excess of the observed over the computed 
 value of log a, and d a , o b , d A , d B are the log. differences as 
 usual. 
 
 Also, since the angles of each triangle in the chain must 
 satisfy the condition of closure, we have the conditions 
 
 with 
 
 = a mn - 
 
 \ ] 
 
 where /^ a , /Jt b are the m. s. e. of the bases, and //,, // 2 , . . . the 
 m. s. e. of the three angles of each of the triangles in the 
 chain. 
 
 The solution may be carried out by the method of cor- 
 relates. 
 
 If the bases have been measured with different apparatus 
 the question of the comparison of the measuring bars with 
 the standards of length is the important one. After that 
 has been satisfactorily settled the connection of the bases 
 may be treated as above. 
 
 Ex. This mode of reduction maybe illustrated by the simple case of a 
 triangle having two sides and the three angles measured. 
 
 In the triangle W. Base, E. Angle, W. Angle (ABC), Sandusky Base, 
 there were measured 
 
 ft. ft. 
 
 BC = 6742.420 o.oio 
 
 AC = 6602.386 o.oio 
 
 From the adjustment of the triangulation with reference to one measured 
 base, and which was carried out by the methods already explained in Chapter 
 VI., the three adjusted angles of the triangle were found to be 
 
 i 07' 51". 35 o".2o 
 ABC= i 06' 26". 74 o".20 
 ACB = 177 45' 41". 91 o".2o 
 
 Required the most probable values of the sides and angles of the triangle 
 from a second adjustment into which the two measured sides enter.
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 365 
 
 If (a), (/>), (A), (B), (C) arc the corrections to the above quantities in order, 
 the condition equations are 
 
 (A) + (B) + (C) = o 
 
 6742.420 + (a) _ sin \ i 07' 51". 35 + (A)\ 
 6602.386 + (/>) ~ sin | r 06' 26". 74 4- (B)\ 
 
 or, reducing the latter to the linear form, 
 
 o.644(<z) + 0.658(6) i.oS()(A) + i.obi(B) 0.044 
 with 
 
 (a)* W_ , W . (JBT- (C) 2 
 (.01)- + (.01)* + (.20)- + (.20)* + (^ - 
 
 that is, 
 
 4oc<rt) 2 + 40o(/') 2 + (A)- + (B)- + (C)- = a min. 
 
 The solution of these equations gives 
 
 ft. 
 
 (a) = + 0.00003 (^) = o".O2 
 
 (t>) = 0.00003 (^) = + o".O2 
 
 (C) = o".oo 
 
 This example is noteworthy as showing the combination of heterogeneous 
 measures in the same minimum equation. 
 
 (2) Approximate Solutions The two of most importance 
 are (a) when the angles alone are changed, (b) when the 
 bases alone are changed. Either of these is practically 
 more important than the rigid solution, as a base line in 
 good work receives a very small correction from the ad- 
 justment. 
 
 (a) When the angles alone are changed. The formulas 
 to be used in this case follow from those of the rigorous 
 method just given by putting the base corrections equal to 
 zero. Thus, if all of the adjusted angles are of the same 
 weight, then, since (a] = o, (//) o, the baseline equation 
 becomes 
 
 with 
 
 The angle equations are as before.
 
 366 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hence if k is the correlate of the base-line equation, we 
 have, by eliminating the angle equation correlates, 
 
 (Q=- (V- d B '}k 
 
 (2) 
 (A,}= (20Y + d B "}k 
 
 = - OY- V)* 
 . . . 
 
 whence, by substituting in the base-line equation, 
 
 Hence the corrections to the angles are known. 
 
 A still further approximation may be made. It is evident 
 that the corrections to the angles C are small compared 
 with those to A and B, and that they vanish when A=B. 
 Hence, as we have assumed the triangles to be well shaped, 
 we may take 
 
 = (Q=. . . =o (3) 
 
 The angle equations then become 
 whence 
 
 .^_ '^B_ k . (5) 
 
 2 
 
 where
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 367 
 
 Ex. i. To find the changes in the angles resulting from the equation 
 connecting the lengths of the lines 1-2 and 14-15 in the triangulation of Long 
 Island Sound (Fig. 46). 
 
 The excess of the log. of the observed v.ilue of 14-15 over the value- 
 computed from 1-2 through the triangulation is 4.00 in units of the sixth 
 decimal place. 
 
 If the lines themselves receive no correction the condition equation, 
 expressed in the linear form, is (see table, p. 346) 
 
 0.27(^1) o.99(j9i) + i.6i(/l a ) o.6o(B-i) + . . . 4.00 
 Now, 
 
 [(8 A + s B y>] = 51.48 
 
 and therefore 
 
 0.27 + o.qq 
 (A l ) = -(B l ) = ^X 4-00=0". 10 
 
 the corrections required. 
 
 Ex. 2. To find the precision of a side in a chain of triangles joining two 
 bases. 
 
 Having found the adjusted angle?, we may find the weight of the log. of 
 any side of continuation as the m th in a chain of triangles joining two bases, 
 all of the angles being of the same weight. 
 
 For simplicity in writing take in =2, n 3. 
 
 The condition equations are 
 
 3 ) - 8 B "(B,) + 8 A '"(A 3 ) - 8 B "\B,) 
 and the function /', whose weight is to be found, 
 
 F=8 A '(Ai) - 8 B '(B^ + 8 A "(A 9 ) - 8 B "(B*) 
 The solution follows at once from Eq. 15, Art. in. The result is 
 
 a 8 A '"* + 8 A " SB" + 8 B "'- 
 
 UF = f [8 A > + 8 A d B + 8 B *] ~ , 
 
 [<>A* + t>A 8 B + 6 B -] t 
 
 the summation of the 8's extending between the limits indicated.
 
 368 THE ADJUSTMENT OF OBSERVATIONS. 
 
 As an example of a different method of treating this 
 special case (a), the Coast Survey connection of three 
 primary bases, the Fire Island, Massachusetts, and Epping, 
 may be cited.* 
 
 The triangulation net connecting the bases is conceived 
 to consist of three branches, one for each base and pro- 
 ceeding therefrom to a line in common. Adjusting each 
 branch, three independent values of the length and p. e. of 
 the line of junction are obtained. The weighted mean of 
 these values is taken to be the most probable value of the 
 line of junction. This value, as well as the length of each 
 base, is considered to be exact. The adjustment of each 
 branch of the triangulation is then repeated with an addi- 
 tional equation fixing the ratio of the length of the base to 
 the line of junction. In making this adjustment the form 
 of solution given in Art. 90 will be found very convenient 
 in handling the extra equation. 
 
 In the final computation of the lengths of the triangle 
 sides "from any one of the measured base lines, we shall, 
 as we recede from it, obtain a proportional amount of the 
 influence of the measure of the other two lines as we ap- 
 proach them, and finally, reaching one or the other, the effect 
 of that base from which we set out is lost. Each distance 
 will have assigned to it its most probable value ; we shall 
 have no discrepancy whatever in the geometrical figure of 
 the triangulation, and the resulting sides and angles will 
 have nearly the same probability as those derived from a 
 theoretically perfect solution." 
 
 (b) By changing the bases alone. The argument for 
 changing the bases only is that as the computed discrep- 
 ancy is always small, and as the triangulation has been 
 already adjusted, it is a great saving of labor to change 
 all the triangles proportionally, and thus get a consistent 
 result by a method which, if not rigorous, is good enough. 
 This is the more evident if we consider with what accuracy 
 geodetic work can now be done. 
 
 * Report, 1865, app. No. 21.
 
 APPLICATION TO BASE-LINE MEASUREMENTS. 369 
 
 Let ,, b. ... be the measured lengths of the bases, and 
 /,, Pv . . . their weights. 
 
 If s is the most probable length of a common side com- 
 puted from the bases, and /,, / 2 , . . . are the ratios of the 
 lengths of the bases to this side, then 
 
 s = y weight p}>?, and weight, of log y 1 is pjb* 
 / , X , 
 
 * = r M' log is /.A* 
 
 ' v <i A 
 
 The weighted mean value of log s is 
 
 With this value of log s, leaving the angles of the triangles 
 as they are, the sides may be recomputed, when, of course, 
 the bases will be changed. 
 
 We have seen in Art. 164 that in base-line measurements 
 the error arising from errors of comparison of the measuring 
 bars is the main one, and that the error arising from the 
 measurement of the line itself is relatively small. Hence 
 we may take the m. s. e. of a base as proportional to the 
 length, and the weight, therefore, inversely as the square 
 of the length. The weight of each computed value of log s 
 will then be the same, and we have now 
 
 As the weights taken are tolerably close, this simple formula 
 will give a result near enough for most work. 
 
 It is evident that if we compute each base from all of 
 the others, and take the mean of the logarithmic values 
 found, consistent values of the sides throughout the tri- 
 angulation will be found by starting from any base.
 
 370 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Thus in the Coast Survey example already referred to, 
 
 
 Epping Base. 
 
 Mass. Base. 
 
 Fire Island Base. 
 
 Measured, 
 
 3.9403143.4 
 
 4.2387077.4 
 
 4-1479535-3 
 
 Computed from Epping, 
 
 
 4.2387115.5 
 
 4.1479556.3 
 
 " " Mass., 
 
 3.9403075.8 
 
 
 4.1479518.4 
 
 " Fire Id., 
 
 3.9403122.4 
 
 4.2387094.6 
 
 
 Means, 
 
 3.9403113.9 
 
 4.2387095.8 
 
 4.1479536.7 
 
 There will now be no contradiction in the values of any 
 triangle side deduced from the different bases.
 
 CHAPTER VIII. 
 
 APPLICATION TO LEVELLING. 
 
 170. The object of levelling is to find the difference of 
 height of two or more points on the surface of the earth. 
 As only differences of height, and not absolute values, are 
 required, some one height may be selected as the standard 
 of reference, and to it any value at will may be assigned. 
 The common custom in geodetic work is to take what is 
 called the mean level of the ocean as the standard height. 
 To be able to refer to it at any time a fixed mark is estab- 
 lished, say at New York, and readings are taken to the sur- 
 face of the ocean at high and low water for a lunation at 
 least. The height of the fixed mark above mean tide is 
 computed from these readings. Then all that is to be done 
 in order to find the elevation of any point above mean tide 
 is to run a line of levels between this point and the fixed 
 mark just described. 
 
 There are two methods of levelling in common use in 
 geodetic work : spirit levelling of precision-and trigonomet- 
 rical levelling. The difference between the two consists 
 mainly in the greater lengths of lines sighted over in the 
 latter case. 
 
 Spirit Levelling of Precision. 
 
 171. For descriptions of instruments employed see AY- 
 vellement de Precision de la Snisse, par Hirsch et Plnnta- 
 mour, Geneve et Bale, 1867, seq. ; Report Chief of Engineers 
 U. S. A., 1880, App. GO; Report U. S. Coast Survey, 18/9 
 App. 15, 1880 App. II ; Professional Papers Corps of Engi- 
 neers U. S. A., No. 24, Chap, xxii 
 
 4 s
 
 372 THE ADJUSTMENT OF OBSERVATIONS. 
 
 The method of observing used on the Coast Survey is 
 as follows: Two levelling rods are used. "Two lines are 
 run simultaneously, with the rods usually at different dis- 
 tances from the instrument ; and to prevent the gradual 
 accumulation of error supposed to be due to running con- 
 stantly in one direction, alternate sections are run in opposite 
 directions. Each station of the instrument, therefore, con- 
 tains two backsights and two foresights. In stations i, 3, 
 5, etc., rod A, backsight, is invariably read first ; then rod B, 
 backsight; then rod A, foresight; and finally rod B, fore- 
 sight. In stations 2, 4, 6, rod B takes precedence of A in 
 both back and fore sights." 
 
 An important source of error in spirit levelling, and one 
 very commonly overlooked, is the change in the length of 
 the levelling rod from variations of temperature. From 
 experiments made by the Prussian Land Survey, in which 
 the rods were compared daily with a steel standard, the 
 following fluctuations in length were found for four rods 
 
 *~* o 
 
 made of seasoned fir : * 
 
 Rod 13, from May 19 to Aug. 18, 0.51 mm. per metre. 
 14, " " 20 " " 15, 0.46 " 
 9, " " 24 " Sept. 6, 0.37 " 
 10, " " 24 " " 6, 0.43 " 
 
 It is quite ppssible that errors from this source may 
 largely exceed the errors arising from the levelling itself. 
 Each field party should therefore be provided with the 
 means of making a daily comparison of the rods used with 
 a standard of length. A steel metre and a micrometer 
 microscope mounted on a stand would be all that would be 
 necessary. 
 
 It is a curious fact in spirit levelling, but one abundantly 
 verified, that a much greater discrepancy is to be expected 
 in a duplicate line of levels if the lines have been levelled 
 over in opposite directions than if both have been run in 
 the same direction. This was noticed long since in a line 
 
 * Nivellements dcr Lamtesati/nafime, vol. v. Berlin, 1883.
 
 APPLICATION TO LEVELLING. 
 
 373 
 
 of levels run under the direction of a committee of the 
 British Association from the Bristol Channel to the English 
 Channel (Portishead to Axraouth), a distance of 74 miles. 
 The increase in the discrepancy of the forward and back- 
 ward levels was continuous. Thus at 12 miles from Portis- 
 head it was 0.35 ft., at 23 miles 0.53 ft., at 37 miles 0.70 ft., 
 at 49 miles 0.82 ft., at 59 miles 0.92 ft., and at 74 miles 
 1.03 ft.* 
 
 In the winter of 1878-1879 a line of levels was run, under 
 the direction of the U. S. Engineers, from Austin, Miss., to 
 Friar's Point, Miss., a distance of about 44 kilometres. 
 " All lines were levelled in duplicate and in opposite direc- 
 tions. When using two rods both were used on the same 
 line, rod No. 2 being used for the first, third, . . . back- 
 sight and for the second, fourth, . . . foresight, and rod 
 No. 3 for the second, fourth, . . . backsight and for the 
 first, third, . . . foresight, always using rod No. 2 on the 
 closing bench mark." The results for the principal bench 
 marks are given in the following table, where it will be 
 seen that the discrepancy in the forward and backward 
 levels increases from one end of the line to the other: f 
 
 Distance. 
 
 Bench Mark. 
 
 Discrepancy. 
 
 
 Austin I. 
 
 
 m 
 
 
 mm 
 
 16505 
 
 Trotter's Landing. 
 
 + 16.8 
 
 4464 
 
 Glendale. 
 
 -f 16.8 
 
 15950 
 
 Delta. 
 
 + 23.6 
 
 6602 
 
 Friar's Point I. 
 
 + 36.5 
 
 Experience on the survey of the great lakes and of the 
 Mississippi River, and also on the survey of India, has 
 shown that the personal peculiarities of the observer enter 
 
 Report lifit ish Association, 
 
 t Re fort Chit, 
 
 Engineers I', S. A., 1879, P- T 944-
 
 374 THE ADJUSTMENT OF OBSERVATIONS. 
 
 largely into levelling work. It would seem that a much 
 more reliable result will be obtained if the line between two 
 points whose difference of height is required has been run 
 over not only in opposite directions, but in opposite direc- 
 tions by the same observer. The personal bias is probably 
 due most largely to the fact that, the ends of the bubble 
 not being sharply defined, different observers estimate the 
 positions of these end points differently. To eliminate its 
 effect completely the line should be levelled in opposite 
 directions by each of a large number of observers. 
 
 Again, as both rod and bubble should be read simul- 
 taneously at each foresight and at each backsight, and as 
 this is physically impossible for one observer, the effect of 
 the unequal heating of the instrument by the sun, even 
 when shaded by an umbrella, has a tendency to cause a 
 change in the position of the bubble in the interval between 
 the readings of the rod and bubble. This error is cumula- 
 tive, and its reduction to a minimum depends upon the skill 
 of the observer. 
 
 The rule adopted by the European Gradmessung for 
 allowable discrepancy in a duplicate line of levels is that 
 the p. e. of the difference in height of two points one kilo- 
 metre apart should in general not exceed 3 mm., and should 
 in no case exceed 5 mm. On the U. S. Coast Survey, for 
 short distances a discrepancy between two levellings of 
 a distance of D kilometres of an amount not exceeding 
 5 \^2D mm. is allowed. 
 
 172. Precision of a Line of Levels. The precision 
 of a line of levels will be given by the m. s. e. of a single 
 levelling of a unit of distance, which we shall take to be one 
 kilometre. The problem is quite analogous to that already 
 discussed in Art. 164, the unit of distance there being the 
 length of a measuring bar. 
 
 If we suppose that in running a single line of levels be- 
 tween two bench marks, A and B, the ground is equally 
 favorable throughout, and that none but accidental errors 
 ma}' be expected to enter, then if // is the m. s. e. of the
 
 APPLICATION TO LKVELUNO. 37$ 
 
 unit of distance (one kilometre), the m. s. e. of a distance of 
 D kilometres would be 
 
 in other words, the weight of a levelling of a distance of D 
 kilometres would be invt rsely proportional to tliat distance. 
 
 Strictly speaking, we should find the weight from the 
 m. s. e. // // . . . arising from all sources of error that 
 enter into the work, such as from the comparisons of the 
 levelling rod with the standard, from the nature ot the 
 country levelled over, from effects of change of length of 
 sight, etc. Then, considering the errors arising from these 
 sources to be independent, we should find finally the m. s. e. 
 of each line levelled over to be vf/?j, and the weight of the 
 
 line would therefore be as T 
 
 O'J 
 
 The uncertainty attendant on estimating these sources 
 of error is so great, and the influence of the distance so 
 largely exceeds the influence of the others, that it is suffi- 
 cient to adopt the rule first given of weighting as the in- 
 verse distance. It follows from this rule that better work 
 is to be looked for if short sights are taken. Even with a 
 first-rate telescope sights should not be taken to exceed 100 
 metres.* As the rod is more easily read by the observer at 
 short distances, the loss of time from the more frequent 
 settings of the instrument is not so great as would at first 
 appear. 
 
 Suppose now that in finding the difference of height of 
 two bench marks, A and B, , lines of levels have been run, 
 and that the results have been compared at intermediate 
 bench marks in succession D lt D v . . . D n kilometres apart; 
 
 * On the precise levelling of the Coast Survey, " where the slope of the ground is steep the 
 distances may be taken as great as possible, and on comparatively level ground they may range 
 from 50 to 150 metres, according to the condition of the weather and atmosphere." 
 
 The U. S. Engineers follow the rule, " The lengths of sight will depend on the condition of 
 the atmosphere, but the rod should always be near enough to be seen distinctly. It will be seldom 
 that lengths of sight greater than 150 metres can be taken." 
 
 In the Prussian Land Survey "the length of sight since 1879 has not beon taken over 50". 
 Only in special cases for example, in crossing streams is it permitted to exceed 50", whereas a 
 shorter sight is frequently taken."
 
 3/6 THE ADJUSTMENT OF OBSERVATIONS. 
 
 then if z//, v", . . . ; v,', v t ", ...;... denote the residual 
 errors of the differences of height in the n sections, and 
 //, p", . . . ; // p", ...;... denote the weights of the 
 observed differences, we have, as in Art. 164 
 
 (i) On the hypothesis that the precision for each unit of 
 distance (one kilometre) is the same throughout the different 
 sections, 
 
 n (n, - i) 
 
 = 
 
 (*,- 
 
 since the weights are inversely as the distances. 
 
 (2) On the hypothesis of the independence of the sec- 
 tions between the intermediate bench marks, the average 
 value of fJL is given by 
 
 When the number of levellings is two, and the ob- 
 served differences of height at the several bench marks 
 Z>,, Z> 2 , . . . D n kilometres apart give discrepancies of 
 d lt d^ . . . d n respectively, then the above formulas reduce 
 to 
 
 and 
 
 The m. s. e. of a single levelling of the whole line would be 
 respectively 
 
 and 
 
 and for the m. s. e. of the mean of the two measurements 
 these values would each be divided by V2.
 
 APPLICATION TO LEVELLING. 
 
 377 
 
 Ex. i. In the precise Levelling (1880) of the U. S. Coast Survey on the 
 Mississippi River two lines were run simultaneously between eveiy two 
 bench marks. The following are the results from B.M. LIV. to B.M. LV. 
 
 B.M. 
 
 Distance. 
 
 Rod A. 
 
 Rodfl. 
 
 Difference. 
 
 
 kil. 
 3-447 
 
 m. 
 O.S7O2 
 
 >. 
 .- 0.8713 
 
 mm. 
 I.I 
 
 
 3.806 
 
 + 0.5805 
 
 + 0.573S 
 
 6.7 
 
 
 1.204 
 
 -0.2154 
 
 0.2204 
 
 5-0 
 
 
 3.038 
 
 + 0.2667 
 
 + 0.2664 
 
 0.6 
 
 It will be found that // = 2 mm nearly. 
 
 Ex. 2. The distance AB has been levelled n different limes. Calling di 
 the difference between the first and second measurements, il 3 the difference 
 between the first and third, and so on, show that the m. s. e. of an observation 
 of weight unity is 
 
 n(n i) 
 
 Ex. 3. The difference of level of two points, A and B, is found by two 
 routes whose lengths are D t , D- 2 kil. respectively. If the discrepancy in the 
 
 results is d, the m. s. e. of one kilometre is - , 
 
 173. Adjustment of a Net of Levels. If a line of 
 levels is to be run between two points a sufficient check ot 
 the accuracy of the work will in general be afforded by 
 running the line over at least twice. Comparisons may be 
 made at intermediate points not too far apart, and if the 
 discrepancies found are within the limits already mentioned 
 the mean of the results may be taken as giving the eleva- 
 tions sought. This method was used by the U. S. Engi- 
 neers in determining the heights of the great lakes between 
 Canada and the United States above mean tide. 
 
 But when a complete topographical survey of a country 
 is made, and a network of levels is necessary, an additional 
 control of the accuracy of the work is afforded by the polyg- 
 onal closing of the level lines forming the net; that is, 
 from the condition that on passing round the polygon and
 
 378 THE ADJUSTMENT OF OBSERVATIONS. 
 
 arriving at the starting-point we should have the same 
 height as at first. This assumes that the points of the net 
 are in the same level surface and that error of closure de- 
 pends upon errors of observation only. In the usual case, 
 where the net is small and the country comparatively level, 
 such an assumption is quite allowable. On the other hand, 
 if the net is very large or the country mountainous, system- 
 atic sources of error, arising principally from the spheroidal 
 form of the earth and the deviation of the plumb-line, would 
 be introduced. The corrections resulting from these causes 
 would have to be computed and applied. A full investiga- 
 tion of this point will be found \v\Astron. Nachr., Vols. 80-84. 
 The adjustment of a net of levels may be carried out in 
 a similar way to the adjustment of a triangulation. Thus 
 suppose that the lines of levels form a closed figure. The 
 conditions to be satisfied among the observed differences 
 of height may be divided into two classes: 
 
 (a) Those arising from non-agreement of repeated meas- 
 urements of differences of height between successive bench 
 marks. The equations expressing these conditions corre- 
 spond to the local equations in a triangulation. 
 
 (b) Those arising from the consideration that on starting 
 from any bench mark and returning to it through a series 
 of bench marks, thus forming a closed figure, we should 
 find the original height. The resulting equations corre- 
 spond to the angle and side equations in a triangulation. 
 
 If the circuit, instead of being a simple one, has tie lines, 
 the number of closure conditions is easily estimated. For 
 if s be the number of bench marks, and / the number of 
 lines levelled over, the number of lines necessary to fix the 
 bench marks is s i, and therefore the number of super- 
 fluous lines that is, the number of closure conditions to be 
 satisfied among the differences of height is 
 
 / j-f i. 
 
 1 74. Approximate Methods of Adjustment. The differences 
 of height between successive bench marks may be found
 
 APPLICATION TO LEVELLING. 379 
 
 by taking the means of the observed values, as explained in 
 Art. 172. Taking these means as independently observed 
 quantities, they may be adjusted for non-satisfaction of 
 closure conditions in various ways : 
 
 (a) The differences of level between the successive bench 
 marks will give rise to / observation equations. Taking 
 the starting point as origin, .$ i differences are required 
 to fix the s stations. If these s i differences are con- 
 sidered independent unknowns the remaining / s-{- i un- 
 knowns may be expressed in terms of them, and the solution 
 completed by the method of Art. 109. 
 
 (b) Since each independent polygon in the net contains 
 one superfluous measurement, it gives a condition equa- 
 tion. With s stations connected by / lines there must be 
 / s-\- i condition equations, as the number of superfluous 
 measurements is / s -\- i. The solution may be completed 
 by the method of correlates, Art. no. 
 
 As to the relative advantages of the two forms of solu- 
 tion, if the adjusted values only are to be found, without 
 their weights, the test will be furnished by the number of 
 normal equations in each case, as the principal part of the 
 labor consists in solving the normal equations. The num- 
 ber by the first form is s i, and by the second / s -\- i ; 
 and therefore, in an extensive net with few tie lines, the 
 method of condition equations is to be preferred, and vice 
 versa. 
 
 (c) Adjust each simple circuit in order by the principle 
 of Art. 115, and repeat the process until the required accu- 
 racy is reached. 
 
 (d) The method employed on the British Ordnance Sur- 
 vey in reducing the principal triangulation. (See Art. 152.) 
 This method is easily applied and gives results practically 
 close enough with the first approximation. It deserves to 
 be employed much more than it is in work of this kind. 
 
 The Ordnance Survey levels, however, were not reduced 
 by this method. " The discrepancies in the levelling along 
 the different lines or routes brought out in closing on 
 
 49
 
 380 THE ADJUSTMENT OF OBSERVATIONS. 
 
 common points have been treated for England and Wales 
 as a whole and rigorously worked out by the method of 
 least squares, involving ultimately the solution of a system 
 of equations with 91 unknown quantities."* 
 
 Ex. In the figure A WMG, A is the initial point, height zero, and the 
 measured heights and distances are as follows : 
 
 A W m 
 
 A 1^=42.65101 Z>i =37.8 kil. 
 
 AM = 54. 74663 Z> 2 = 35.8 " 
 
 AG =58.56223 Z> 3 = 22.6 " 
 
 Fig. 51 
 
 WM = 12. 10530 Z> 4 =44.2 ' 
 
 MG = 3.79892 Z> 5 = 27.9 " 
 
 required the adjusted values oi the heights. 
 
 First Solution. Let the most probable corrections to the five measured 
 differences of height in order be v\, z/ 2 , z>a, z> 4 , 7' 5 . The points IV, M, G are 
 completely determined by A W, AM, AG. We may, therefore, take v\, z>i, v 3 
 as independent unknowns. 
 
 The observation equations are 
 
 v\ -=v\ weight 26.5 
 
 Vy = v<2 " 27.9 
 
 z/ 3 = z/ 3 44-2 
 
 z> 4 = vi + vi 0.00968 " 22.6 
 
 z5 = Vi + z>s + 0.01668 " 35.8 
 
 1000 
 the weights being computed from - . 
 
 The solution is finished as in first solution, Art. 140. 
 
 Second Solution. From the closed circuits A JVM, AMG we have the 
 condition equations 
 
 v\ V1-\-V^=. 0.00968 
 Vi v a + Vi, = + O.OI668 
 with 
 
 J>\ = 
 
 The solution is finished as in second solution, Art. 140. 
 
 * Abstract of Levelling in England and Wales, Introduction, p. vi.
 
 APPLICATION TO LEVELLING. 
 
 381 
 
 ThirJ Solution. Adjust each simple closed circuit in the figure in order. 
 Since in the circuit AMG the algebraic sum of the three differences of height 
 should be zero, we may apply the principles of Arts, no, in. We have 
 
 AM= 54-74663 D* = 35-3 .'./., = 27.9 
 MG = 3-79892 > 5 = 27.9 p b = 35.8 
 
 AC = 58.56223 D 3 22.6 / 3 = 44.2 
 
 0.01668 
 
 86.3 
 
 Correction to AM =. ~ X 0.01668 = -{-0.00692 
 06.3 
 
 T 
 
 Weight of adjusted AM = 27.9 -f- 
 
 
 
 35.8 
 
 = 47-7 
 Hence for the circuit AMW 
 
 ^^=42.65101 
 
 = I2.I053O 
 
 
 
 44.2 
 
 * =47-7 
 
 ! =26.5 
 
 ^ =22.6 
 
 which may be adjusted as above. 
 
 The circuit AMG may be again adjusted, and so on. 
 
 Fourth Solution. 
 
 
 
 
 I. 
 
 
 
 Weight. 
 
 A 
 
 [K 
 
 M 
 
 G 
 
 Means. 
 
 26.5 
 27-9 
 44.2 
 
 22.6 
 358 
 
 o. 
 
 0. 
 
 o. 
 
 42.65101 
 
 o. 
 
 =4.74663 
 12.10530 
 
 =;8. 56223 
 3.79892 
 
 
 
 
 
 II. 
 
 
 
 
 o. 
 
 0. 
 0. 
 
 42.65101 
 42.65101 
 
 M-7-4663 
 
 54-75631 
 54.75096 
 
 58. 56223 
 
 <&M9&8 
 
 
 Means 
 
 o. 
 
 42.65101 
 
 54-75096 
 
 58.55670 
 
 
 
 
 
 III. 
 
 
 
 
 0. 
 
 o 
 o. 
 
 0. 
 
 0. 
 
 + 0.00433 
 
 - o oo=;35 
 o. 
 
 - o.oo;!;3 
 + 0.00682 
 
 0. 
 
 0.00216 
 - 0.00276 
 - 0.00268 
 + 0.00341 
 
 
 
 
 IV. 
 
 
 
 
 o. 
 f o 00216 
 - 0.00276 
 
 42.65101 
 
 42.64833 
 
 54 74879 
 
 M.75363 
 54-75437 
 
 58-55947 
 58.55329 
 
 
 Means 
 Final values 
 
 - o 00063 
 o. 
 
 42.64978 
 42.65041 
 
 54-75237 
 54.75300 
 
 <^. 55670 
 S8-M733 

 
 382 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Trigonometrical Levelling. 
 
 175. In extensive surveys in which a primary triangula- 
 tion is carried on, the heights of the stations occupied may 
 be conveniently found by trigonometrical levelling. The 
 extra labor required to measure the necessary vertical 
 angles while the horizontal angles are being read at the 
 several stations is but slight. 
 
 When a country is hilly, heights can be perhaps as 
 closely determined by trigonometrical levelling as by spirit 
 levelling ; but if the country is flat and the triangulation 
 stations low, experience has shown that much more de- 
 pendence is to be placed on the latter method. See, for 
 comparisons of the relative accuracy of the two methods, 
 Report of U. S. Coast Snrvty, 1876, App. i6and 17 ; Report of 
 G. T. Survey of India, vol. ii. ; Report of New York State 
 Survey, 1882. Sometimes, indeed, the agreement is so close 
 that it must be regarded as accidental. Thus in the de- 
 termination of the axis of the St. Gothard Tunnel Engineer 
 Koppe found the difference of height of the two ends by 
 trigonometrical levelling to be 39"*. 13. The spirit levelling 
 of precision executed by Hirsch and Plantamour gave it 
 39". 05. 
 
 The great cause of inaccurate work in trigonometrical 
 levelling is atmospheric refraction. This is one of those 
 disturbing causes which is so erratic in its character that 
 no method has yet been devised for determining it that is 
 very satisfactory. Hence the plan adopted in trigono- 
 metrical levelling is to observe only at or near the time of 
 minimum refraction. Without this precaution very dis- 
 crepant results may be looked for. For example, in India, 
 where in certain districts the triangulation has been carried 
 for hundreds of miles over a level country with stations 10 
 or 12 miles apart and from 18 to 24 feet high just high 
 enough to be mutually visible at the time of minimum re- 
 fraction " numerous instances are recorded of the vertical 
 angles varying through a range of 6 to 9 minutes, corre-
 
 APPLICATION TO LEVELLING. 
 
 383 
 
 spending to an apparent change in altitude of 100 to 150 
 feet in the course of 24 hours." 
 
 176. To Find the Zenith Distance of a Signal Point the 
 telescope at a mark on the signal and bisect it with t he- 
 horizontal thread. Then turn the telescope 180 on its 
 axis, transit it, and bisect the mark again. At both bisec- 
 tions read the vertical circle and the level parallel to the 
 vernier arm. One-half of the difference of the circle read- 
 ings corrected for level will give a single determination of 
 the value of the zenith angle sought. The result is free 
 from index error of the circle. 
 
 Errors of graduation with good instruments are so small 
 in comparison with the uncertainties arising from refrac- 
 tion that it is unnecessary to eliminate them in the measure 
 of vertical angles. 
 
 177. Let us consider the general case in which the zenith 
 angles measured at two stations whose difference of height 
 is required are not read simul- 
 taneously. In the figure, if 
 
 A^ A y denote the positions of 
 the instruments employed on 
 two lofty stations, the angles 
 that the observer has attempt- 
 ed to measure are Z^A V A^ and 
 Z^A^A^. But on account of the 
 refractive power of the atmos- 
 phere the path of a ray of light 
 from A t to A,, will not be a 
 straight line but a curve more 
 or less irregular, and the direc- 
 tion in which A^ is seen from A, will be that of the tangent 
 A^T to this curve at A,. The line of sight from A. 2 (oA t 
 will not necessarilv be over the same curve. 
 
 The angles between the apparent directions of the rays 
 of light at A^A^ and the real directions are called refraction 
 
 angles. 
 
 Thus TA t A, is the refraction angle at A,. 
 
 Mem. Roy. Astron. Sac., vol. xxxiii. p. 104.
 
 384 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Let s,, denote the measured zenith angles reduced to 
 the heights of the instruments at A lt A v 
 
 D=the distance between the instruments. 
 C ' = the angle at the earth's centre subtended by D. 
 ,,,,= the refraction angles at A : , A y . 
 
 Now, assuming the paths of the rays of light from one sta- 
 tion to the other to be arcs of circles, which is approximately 
 the case when the lines are of moderate length, we have 
 
 _ t path of ray 
 
 C ^~ 7> : 
 R c 
 
 where R c is the radius of the refraction curve. 
 But approximately 
 
 r _ path of ray 
 ~~R~~ 
 
 where R is the mean radius of the earth. 
 Hence we may put 
 
 C. = iV C, = i*,C (0 
 
 where k^ k^ are constants and may be called refraction 
 factors. 
 
 Observations are frequently made so as to be simul- 
 taneous at A t , A^ and the line of sight may then be as- 
 sumed to be the same arc of a circle for both directions. 
 In this case if we put 
 
 r .\-r IC 
 
 (si TSa K ^ 
 
 k is called the coefficient of refraction. 
 
 178. To Find the Refraction Factors. The triangle 
 CA,A^ gives the relation 
 
 But, with sufficient accuracy, C= : Tl , where R is the 
 
 R sin i 
 
 radius of curvature of the line in question.
 
 APPLICATION TO LEVELLING. 385 
 
 Hence, attending to equations I, Art. 177, we may write 
 the above relation in the form 
 
 * 1 + ^ = 2|r- -^-(^+ ~ 2 - 1 80) | (2) 
 
 This equation shows that a single line will not give the 
 refraction factors, and we must, therefore, have a net of 
 lines with the zenith angles read at the ends of each line. 
 If, for simplicity, we consider a quadrilateral ABCD we shall 
 have six equations of the form (2). As Fi g .53 
 
 these equations contain twelve unknowns 
 we may, in order to reduce this num- 
 ber, assume, if the observations at sta- 
 tion A over the lines AB, AC, AD are 
 nearly simultaneous, that the k for each 
 of these lines that is, the k for station A 
 is the same. Similarly at stations B, C, D, so that in the 
 most favorable case we shall have 4 unknowns and 6 equa- 
 tions. 
 
 It may not always be possible to get one k for each sta- 
 tion, but in a net, if all of the lines have been sighted over, 
 a sufficient excess in number of equations over unknowns 
 will in general be found to admit of solution by the method 
 of least squares. As regards the weights to be assigned to 
 these equations, we may proceed in the ordinary way. If 
 ,7, were observed ;/ t times, and z n were observed w 2 times, 
 the weights of .cr, and 2 2 may be taken to be w, and 2 re- 
 spectively, and the first equation would have a weight P 
 given by 
 
 that is, its weight would be proportional to 
 
 
 
 ", + ,
 
 386 THE ADJUSTMENT OF OBSERVATIONS. 
 
 179. To Find the Mean Coefficient of Refraction. 
 
 When the zenith distances are simultaneous, 
 
 k l = , = k suppose 
 
 and the refraction coefficient at the moment of observation 
 is for both stations, 
 
 ,. 
 
 A value of k can thus be found for each line sighted over 
 from both ends simultaneously. To get an average value 
 of k for the whole system the weighted mean of these 
 separate values must be taken. The same method of 
 assigning weights may be used as in the preceding. Bessel 
 argues that errors arising from irregularities in k are of 
 much more importance than the errors in the zenith angles, 
 and proposes the empirical formula* 
 
 for assigning relative weights. In this he is followed by 
 the Coast Survey in their determination of the coefficient 
 of refraction from observations made in Northern Georgia 
 near Atlanta base. 
 
 The mean values of k found by the Coast Survey in N. 
 Georgia (1873) and by the New York State Survey (1882) 
 were 0.143 and 0.146 respectively. 
 
 1 80. To Find the Differences of Height. There 
 are three cases to be considered: 
 
 (i) When the zenith angles at both ends of each line are 
 used. 
 
 (a) When the zenith angles are not simultaneous. 
 
 Let //, known height of first station 
 // = height of next station 
 
 * Gradinessung in Ostfreusscn, p. ig6.
 
 APPLICATION TO LEVELLING. 387 
 
 Krom the triangle CA,A t (Fig. 52) 
 
 H.-ffi CA. - CA, tanKAt-AJ (} 
 
 Substituting for A t , A y their values in terms of z v k v 
 and reducing, 
 
 (b) When the zenith angles are simultaneous k^-=k v 
 and 
 
 which is the form used on the Coast Survey. 
 
 (2) When the angles observed at each end of a line are 
 used separately. 
 
 In the common case of a line sighted over from one end 
 only, we have, since 
 
 by substituting these values in (i) and reducing, 
 
 = Z? cot ^ + Z) 2 + -=& cof ^ (4) 
 
 which is equivalent to the Coast Survey form. 
 
 When the line is sighted over from both ends we have 
 two values of the difference of height, whose weighted mean 
 gives the required result. 
 
 50
 
 388 THE ADJUSTMENT OF OBSERVATIONS. 
 
 181. Precision of the Formulas for Differenced of 
 Height. It // // 2 , // 3 denote the differences of height be- 
 tween two stations found from non-simultaneous readings 
 of angles at the stations, from simultaneous readings, and 
 from angles read at one of the stations only, respectively, 
 then, with sufficient accuracy, 
 
 7z 2 = /Man i<> 2 - O (i) 
 
 , = Z? cot #, + (i -,)! 
 
 Let the m. s. e. of an observed zenith angle be /jt g . De- 
 note by ///&,, /^ 2 , /// /3 the m. s. e. of the differences of height 
 found, and by fi k the m. s. e. of a refraction factor k. Then 
 by differentiation, taking D, r,, 5 2 , k^ / 2 as independent 
 variables, and remembering that z z^ are each equal to 90 
 nearly, and that we may put dD = o, since the distances 
 are well known in comparison with the heights, we shall 
 have (Art. 65) 
 
 (2) 
 
 These results show that differences of height are found 
 with the greatest precision from simultaneous observations. 
 182. Adjustment of a Net of Trigonometric 
 Levels. It will be sufficient to consider a simple closed 
 figure, as a net of levels can be broken up into a number of 
 closed figures, generally triangles. For simplicity take a 
 triangle.
 
 APPLICATION TO LEVELLING. 389 
 
 The same value of k is assumed for all of the lines 
 radiating' from a station. Denote the values of k at the 
 three stations by i\, k v k t respectively. Then we have the 
 observation equations 
 
 ( R sin i" , } 
 
 ^=2 I i - y: (* + #, 180 ) |- = /,, a known quan. 
 
 The condition to be satisfied amon^ the differences of 
 
 o 
 
 height at the vertices of the triangle is that, on starting 
 from any station and returning 1 to it through the other two 
 stations, we should find the original height. Thus proceed- 
 ing round the triangle in order of azimuth, if//,, // 2 , // 3 de- 
 note the differences of height of the stations, and (//,), (// 2 ), 
 (// 3 ) the most probable corrections to these values, we must 
 have 
 
 that is, we must have a condition equation of the form 
 
 ak^ -j- bk^ -j- ck^ / 4 
 
 where a, b, c, / 4 are constants. 
 
 The four equations may be solved by the method of cor- 
 relates, and the differences of height may next be computed 
 from equations i, Art. 181, and will be found consistent. 
 
 If the circuit, instead of being a simple one, has diago- 
 nals, then, as in Art. 173, if / is the number of lines read 
 over, and ^ the number of stations in the circuit, the num- 
 ber of conditions to be satisfied among the differences of 
 height is 
 
 l-s+i
 
 390 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. In the triangulation of Georgia, near Atlanta Base, there were 
 measured zenith angles as follows :* 
 
 Stations occupied. 
 
 Station read to. 
 
 Zenith Angle. 
 
 Time. 
 
 S. W. Base, 
 
 ( Sweat Mt. 
 ( Stone Mt. 
 
 89 37' 53"-7 
 89 26' 2i".6 
 
 2 days. 
 
 N. E. Base, 
 
 ( Sweat Mt. 
 ( Stone Mt. 
 
 89 39' 5 1 "-7 
 89 24' 04". o 
 
 I " 
 
 Stone Mt., 
 
 fN. E. Base. 
 { S. W. Base. 
 
 90 43' 20". 8 
 90 41' 56". 8 
 
 2 " 
 
 
 [Sweat Mt. 
 
 90 08' 58". 9 
 
 
 Sweat Mt., 
 
 (S. W. Base. 
 
 I N. E. Base. 
 I 
 
 9 33' 37"-2 
 9 3i' 43"-4 
 
 I " 
 
 
 [Stone Mt. 
 
 90 09' 37". o 
 
 
 The mean latitude is 34 N. approx. 
 
 Fig.54 
 
 N.E.B. 
 
 Call ki, 
 respectively. 
 
 S.W.B: 
 
 Distances. 
 
 S. W. Base-Sweat Mt., 
 
 N. E. Base-Sweat Mt., 
 
 N. E. Base-Stone Mt., 16321 
 
 Stone Mt.-S. W. Base, 17761 
 
 Sweat Mt. -Stone Mt., 40726 
 
 , k* the refraction factors for all lines at the stations i, 2, 3, 4 
 
 * The measures are taken from C. S. Report, 1876.
 
 APPLICATION TO LEVELLING. 391 
 
 We must form two classes of equations: 
 
 (i) Observation equations. These are computed from the form 
 
 ' = 2Ji --~ 
 
 BD 
 
 where B = : - 
 
 J? sm i 
 
 The weights are computed according to the formula - V D (Art. 179). 
 
 Ij will be close enough to take i, 3 as the number of days of observation. 
 That is the best we can do with our data. We have then the observation 
 
 equations 
 
 1 + 3 =0.249 weight 1.04 
 
 1 + 4 = 0.267 " 1.33 
 
 3 + 4 = 0.308 " 1.35 
 
 a + 3 = 0.297 " 0.80 
 
 2 + 4 =0.317 " 0.85 
 
 (2) Condition equations. 
 
 The number of lines = 5 
 The number of stations = 4 
 .-. The number of condition equations = 5 4 + 1 
 
 2 
 
 These two condition equations arise from the sums of two sets of three equa- 
 tions, each of the form 
 
 H- H' = Z>tan \(z-z') + ^ (-') 
 They are, for the triangles 134, 234, 
 
 o = 1.52 10.92 ki 4 1.79 3 + 52.71 4 
 o= + 1.84 + 14. 47 2 +40.17 3 54.64/^4 
 
 The solution of these equations, subject to the relation 
 
 [] = a minimum, 
 
 gives 
 
 1=0. 1 18 3 = 0.149 
 
 4 = 0.107 4 = 0.171 
 
 whence the adjusted differences of height follow at once : 
 
 m 
 
 1 to 3 + 198.24 
 
 3 to 4 2.30 }- check sum o.oo 
 
 4 to i 195.94 
 
 3 to 2 191.17 
 
 2 to 4 + 188.88 } check sum + o.oi 
 
 4 to 3 + 2.30 
 
 The precision of the adjusted values, or of any function of them, may be 
 found exactly as in Art. 114.
 
 392 THE ADJUSTMENT OF OBSERVATIONS. 
 
 183. Approximate Method of Adjusting a Net. 
 
 On account of the many uncertainties attendant on finding 
 the refraction factors, it is not often that so elaborate a 
 method of adjusting the heights as the preceding is fol- 
 lowed. 
 
 In the ordinary method of observation, where the ob- 
 served zenith angles are simultaneous at every two stations, 
 
 // = H, H l = D tan | (z, - z,} 
 
 and the differences of height may be computed at once with- 
 out any reference to the coefficient of refraction. These 
 differences of height, considered as observed quantities, 
 may be adjusted for conditions of closure in the net, as in 
 spirit levelling, Art. 174. 
 
 The weights P to be assigned to the differences of height 
 in the solution will be found from 
 
 and therefore the weights are inversely proportional to the 
 squares of the distances between the stations. 
 
 When the zenith angles are not simultaneous, after find- 
 ing the refraction factors, as in Art. 182, and computing 
 the differences of height, we should find the weights of 
 these differences of height from 
 
 It would seem safe to assume p z = 2", p. k = 0.02. 
 Now, Z> 2 sin 2 1"/** < | ~R* P* 
 
 as D > 4 miles. 
 
 Hence for distances between the stations up to 4 miles the 
 first term is the important one, and for greater distances
 
 APPLICATION TO LEVELLING. 
 
 393 
 
 the second term. We should, therefore, for distances be- 
 tween stations not greater than 4 miles, weight inversely 
 as the square of the distance, and for distances over that 
 amount inversely as the fourth powers of the distances. 
 
 Ex. i. In the determination of the axis of the St. Gothard Tunnel (Fig. 
 37) the heights of the trigonometrical stations were determined by trigono- 
 metrical levelling. The following were the results unadjusted with their 
 weights. Required the values adjusted for closure of circuits. 
 
 DifT. of height. Wt. 
 
 Diff. of height, 
 
 , Wt. 
 
 
 m 
 
 
 Airolo- XII. 
 
 914.96 
 
 23 
 
 Airolo- X. 
 
 1287.75 
 
 17 
 
 Airolo- XL 
 
 1299.27 
 
 2 
 
 Airolo- IX. 
 
 1553.09 
 
 5 
 
 XII.- X. 
 
 372.73 
 
 5 
 
 XII.- XI. 
 
 384.41 
 
 2 
 
 XII.- IX. 
 
 638.30 
 
 3 
 
 XII.- VIII. 
 
 814.35 
 
 i 
 
 X.- XI. 
 
 II. 60 
 
 3 
 
 X.- IX. 
 
 265.48 
 
 6 
 
 X.-VIII. 
 
 441.10 
 
 2 
 
 XL- IX. 
 
 253.87 
 
 I 
 
 XI.- VIII. 
 
 429.55 
 
 10 
 
 IX.-VIII. 
 
 175-37 
 
 I 
 
 III.- IX. 
 
 216.46 
 
 i 
 
 V.- IX. 
 
 899.87 
 
 i 
 
 V.-VIII. 
 
 1075-77 
 
 i 
 
 III.-VIII. 
 
 391.74 
 
 i 
 
 VII.-VIIL 
 
 901.78 
 
 i 
 
 V.- VII. 
 
 174-45 
 
 7 
 
 IV.- III. 
 
 296.69 
 
 60 
 
 VII.- III. 
 
 509.49 
 
 4 
 
 GOschenen- III. 
 
 1376.19 
 
 14 
 
 V.- IV. 
 
 387-24 
 
 20 
 
 VII.- IV. 
 
 212.75 
 
 7 
 
 Goschenen- IV. 
 
 1079.50 
 
 30 
 
 Goschenen- V. 
 
 692.35 
 
 15 
 
 The heights of Airolo and Goschenen are 1147.12 and 1108.07 respec- 
 tively, as found by spirit levelling, and are unaltered in the adjustment. 
 The adjusted heights of the stations will be found to be 
 
 III. 
 
 IV. 
 
 V. 
 
 VII. 
 
 VIII. 
 
 2484.26 
 
 2187.57 
 
 1800.36 
 
 1974.78 
 
 2876.07 
 
 IX. 2700.35 
 
 X. 2434.87 
 
 XL 2446.49 
 
 XII. 2062.08 
 
 Ex. 2. From 
 
 //a H\ =. D tan \(zi z\] 
 deduce 
 
 
 //, -H^D tan \(^ - =,) + 
 
 **' '
 
 394 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. 3. If a series of points connected by observed zenith angles begin and 
 end with points whose heights are known, then the number of conditions to be 
 satisfied among the differences of height is 
 
 / S + 2 
 
 where / is the number of lines read over and s the number of points in the net. 
 
 Ex. 4. If zenith angles are read to two stations from a station between, 
 show that the difference of height; //, of the two stations will be found from 
 
 2 
 
 h = Di cot 22 D\ cot z\ 
 
 "2.R 
 
 where D\, Z> 2 are the distances from the station occupied. 
 
 Also show that when D\ Z> 2 the precision of the height h is the same as 
 if the observations had been made simultaneously at the two extreme stations 
 themselves.
 
 CHAPTER IX. 
 
 APPLICATION TO ERRORS OF GRADUATION OF LINE MEAS- 
 URES AND TO CALIBRATION OF THERMOMETERS. 
 
 Observations for the determination of errors of gradua- 
 tion of line measures, and observations for the calibration 
 of thermometers, may be discussed together, as there is no 
 essential difference in the method of reducing them. 
 
 184. Line Measures. Let AB be a line measure 
 divided into n equal parts as nearly as may be at the points 
 1,2,. . .111, and let o and n be the initial and terminal 
 marks. Comparisons are sup- Fi 55 
 
 posed to have been made be- * 1 I j ( 
 
 tween AB and a standard of * n 
 
 length, so that the distance between o and n is known. 
 The problem proposed is to rind the corrections to the 
 intermediate graduation marks that is, the amounts by 
 which the positions of the marks should be changed to be 
 in their true relative positions. When the corrections have 
 been applied to the distances 01, i 2, . . . these distances 
 should be all the same proportional part of the entire dis- 
 tance AB. 
 
 In making the necessary observations two methods are 
 in common use. 
 
 (a) Two microscopes furnished with micrometers are 
 firmly mounted on a frame separate from the support on 
 which the graduated line measure rests. The zeros of the 
 micrometers are placed at a distance apart as nearly as 
 possible equal to the distances to be read. The marks 
 o, I ; 1,2; . . . are in succession brought under the micro- 
 scopes and the micrometers read in each position. Each 
 distance is thus compared with the constant distance be- 
 51
 
 396 THE ADJUSTMENT OF OBSERVATIONS. 
 
 tween the micrometer zeros, which differs from the dis- 
 tance of the true positions of the marks by a fixed but un- 
 known amount. 
 
 Let .*, x^ . . . x n denote the corrections to the gradua- 
 tion marks at o, i, . . . . Then for the first interval o I, 
 if M , M l be the readings at o, I and c be the unknown con- 
 stant distance between the micrometer zeros, and d the dis- 
 tance between the corrected positions of the graduation 
 marks, we should have 
 
 This may be written 
 
 #o *i J = 4, (0 
 
 where 
 
 f ol = M,- M a , and y = c-d 
 
 Hence, taking four spaces only, 
 
 x <> x \ y = ' \ 
 
 (2) 
 
 But as the distance between the initial and terminal points 
 is known, the corrections x and x 4 may each be taken equal 
 to zero. Also, since the equations contain four unknowns 
 and are themselves only four in number, we must either 
 reduce the number of unknowns arbitrarily or make addi- 
 tional observations involving other combinations of the- 
 unknowns, in order to apply the method of least squares. 
 It is usually more convenient to solve these equations by 
 computing the corrections to the intervals 01,02,. . . first 
 of all, and then the corrections to the positions of the marks. 
 Writing z t for x^ x a , ,? for ,r 2 x 1} . . ., and eliminating^,
 
 APPLICATION TO ERRORS OF LINE MEASURES. 397 
 
 we have from the above equations, supposing the first space 
 compared with all of the others, 
 
 (3) ' 
 ^l Z i = t* 4 4 1 
 
 But 
 
 = (4) 
 
 Hence 
 
 V 3 I 4 4 3^0 l) 
 
 (5) 
 
 Also -r, = ^,, .*., = .7, -|- -> , and are therefore known. 
 
 Ex, In order to find the corrections to the double decimetre graduation 
 marks needed to change the nominal into exact values in terms of the interval 
 oi' on a Repsold steel metre comparisons were made as follows: Two 
 microscopes were mounted at a distance of o m .2 approx. from each other, and 
 readings made by pointing at the successive double decimetre marks, the 
 microscopes remaining stationary while the metre was run under them. The 
 order of reading was o m .o and o m .2 ; 0^.2 and o".4; . . . o m .S and i m .o ; o.o 
 and o'".2, the interval o".o and o m .2 being read on at the beginning and end 
 of each set of comparisons. Twenty-four sets were made and the following 
 results obtained.* 
 
 m in tit m n in 
 
 Interval, 0.2 to 0.4 = o.o to 0.2 + 2.1 o.i 
 0.4 to 0.6 o.o to 0.2 + 2.6 o.i 
 0.6 to o.S = o.o to 0.2 + 0.7 o. i 
 
 0.8 tO I.O = 0.0 tO 0.2 4- 2.2 0.1 
 
 With the above notation 
 
 l S a 2.6 
 
 zi 34=0.7 
 
 = + 1.5, and xi = 1.5 
 
 = O.6 JT 2 r=O. 
 
 * The value of the interval o i>* was found, by comparing with the German standard, to be 
 -f- JO* 1 . 65 t, where / is the temperature in degrees centigrade.
 
 398 THE ADJUSTMENT OF OBSERVATIONS, 
 
 or in tabular form, 
 
 Graduation 
 
 o m .o 
 
 O.2 
 
 0.4 
 
 0.6 
 
 0.8 
 
 I'".O 
 
 Correction 
 
 OM.O 
 
 -1.5 
 
 -0.9 
 
 + 0.1 
 
 -0.7 
 
 OM.O 
 
 The Precision. Each of the intervals is entangled with the interval o.o 
 and 0.2, and, therefore, the p. e. given are not independent. From the mode 
 of measurement it is evident that the (p. e.) 2 of o.o to 0.2 is half that of the 
 (p. e.) 2 of each of the other intervals. Hence it n is the p. e. of the first in- 
 terval, and r the p. e. of each of the others, 
 
 r, 2 = 2r 2 
 
 Hence 
 and 
 
 .-. r 2 = 0.007 ^i 2 0.003 
 
 r *S = ^5 (4 X 0.007 + 16 X 0.003) 
 r x i = oc*.o6 
 Similarly for the other marks. 
 
 From equations 2 definite values of the corrections may 
 be found if x v and x^ are supposed known. There is, how- 
 ever, a greater probability of eliminating systematic error 
 if, instead of spending all the time of observation in the 
 direct comparison of the single intervals, we spend part of 
 it in comparing combinations of those intervals. Let, then, 
 as the best arrangement (see Art. 153), the single intervals 
 o i, i 2,23, 34, and all possible combinations of intervals, as 
 o 2, i 3, 2 4 ; o 3, i 4, be equally well compared. The micro- 
 scope intervals would, of course, be different in each set of 
 comparisons, being approximately 01, 02, 03. The ob- 
 servation equations for this arrangement are 
 
 y\ 
 x\ y\ 
 
 = / 3 
 
 (6) 
 
 x^ 
 
 X^ 
 
 / 14 
 
 / 4
 
 APPLICATION TO ERRORS OF LINE MEASURES. 399 
 But x and ;r 4 are known. Hence the normal equations 
 
 + 4*1 JT3 *3 + yi + }'3 = la 1 + /I 2 + A 3 + /l 4 
 
 - Xi + 4X3 ^3 = /OS A S + /S + /8 4 
 
 - Xi Xl + 4X3 - I'* }'3 = /() 3 A 3 /2 3 + /3 I 
 
 + 4,1'l A. 1 A 3 /2 3 /3 4 (?) 
 
 + JTl - JT3 + 3^3 = A) 2 A 3 /3 I 
 
 + JTl - JT 3 + 2J 3 = /u 3 A 4 
 
 from which equations the most probable values of the cor- 
 rections may be found. 
 
 The Precision of the Corrections x x v x y The m. s. e. of 
 an observation of the unit of weight is found from the usual 
 formula. We have 
 
 9 -6 
 
 the number of observations being 9, and of independent 
 unknowns 6. 
 
 The weights of x^ x v x^ may be found by the methods of 
 Chapter IV. 
 
 The weight of x^ and of x 3 is V, ar| d the weight of x ti is 
 \-. Hence the m. s. e. of x : , x^ x^ are known. 
 
 Ex. Solve by this method the example in Art. 187. 
 
 (b) The work of reduction is much facilitated by em- 
 ploying an auxiliary scale, CD, divided into spaces approxi- 
 mately equal to those of AB, and whose values have already 
 been found by comparison with some standard. If, as 
 befjpre, we suppose the single spaces compared, and also 
 all possible combinations of spaces, the observation equa- 
 tions would be the same as equations 6. The second scale, 
 however, enables us to find each microscope interval em- 
 ployed. Hence y^ y^ . . . are known, and may, therefore, 
 be transposed in these equations and added to the terms 
 / 01 , / 12 , . . . The observation equations thus involve only
 
 400 THE ADJUSTMENT OF OBSERVATIONS. 
 
 .f u , . . . ;t- 4 as unknowns. Taking- ^ = o, ,r 4 = o, the normal 
 equations are 
 
 X? Xa = /o l + A 2 + A 3 + A 4 
 - JTl + 4^2 *3 = /O 2 /I 2 + /2 3 + /a 4 (8) 
 
 -l"l #2 + 4-* 3 = /OS As <'a 3 + /3 4 
 
 from which ;r,, ;tr 2 , ;tr 3 may be found. 
 
 The reduction may, however, be made more easily by 
 the application of the artifice employed in Art. 156. If we 
 consider x and ;r 4 as yet unknown, the normal equations 
 may be written 
 
 4-*o Xi Xi Xa X\ "= /o i + /o 2 + /o 3 + A> 4 
 
 X + 4x1 x<i x a x t = /o i + A 2 + /i 3 + l\ 4 
 
 X Xi+ 4X3 Xa X.i = ! 2 /I 2 + /2 3 + /2 4 (9) 
 
 JTo fl ~~ -^2 + 4-^3 X l '03 '13 '2 3 + '3 4 
 
 = /O 4 A 4 /2 4 /3 4 
 
 Adding these equations, there results 
 
 This was to be expected, because we have left the initial 
 and terminal points free. Hence we may assume any 
 arbitrary relation between the corrections. Let us assume 
 as most convenient 
 
 X + Xi + Xy + X 3 + X t =O (lO) 
 
 and then by adding this relation to each of the normal 
 equations we have 
 
 5^0 = + /O 1 + /O 2 + /O 3 + /O 4 
 
 5*1 = lo i + A 2 + A s + A 4 
 
 5^2 = /O 2 A 2 + /2 3 + /2 4 ( J J ) 
 
 5^8 '03 As '23+ '34 
 5.3:4 = - / 4 - A 4 - /2 4 - /3 4 
 
 The whole solution may be conveniently arranged in 
 tabular form. The sums of the horizontal rows are first 
 found and then placed in the proper vertical columns, with
 
 APPLICATION TO CALIBRATION OF THERMOMETERS. 401 
 
 signs changed. The vertical columns are next added and 
 divided by 5. 
 
 
 - / , 
 
 /0 
 
 ~/03 
 
 ~/04 
 
 Sum, 
 
 
 
 -/! 
 
 -/li 
 
 ~/14 
 
 Sum 2 
 
 
 
 
 /2 3 
 
 -/24 
 
 Sum 3 
 
 
 
 
 
 -/34 
 
 Sum 4 
 
 Sumi 
 
 Sum 2 
 
 Sum 3 
 
 Sum 4 
 
 
 
 5* 
 
 5*! 
 
 5* 2 
 
 5*3 
 
 5*4 
 
 
 *0 
 
 #i 
 
 JT 2 
 
 *3 
 
 ** 
 
 
 By putting x = o, x t = o in equations 11, it is easily seen 
 that we have the same results for x x x\ as found from 
 equations 8. 
 
 Ex. The intervals 0-5 mm., 5-10 mm., . . . 25-30 mm. on a steel metre 
 were compared with the interval 92-94 hundredths on a standard inch. The 
 whole intervals 0-30 mm. being known from other comparisons, show that the 
 
 p. e. of the interval 0-5 mm., if the p. e. of a comparison is r, is - ^30. 
 [Call metre intervals s,, S?, . . . s t , and inch interval a. Then 
 
 j, a + At 
 s-i a + A.i 
 
 where A\, // 2 , . . . A& are the errors. 
 By addition, 
 
 = -* ~ T 
 
 where [s] is a known quantity. 
 
 Hence eliminate a from the value of SL ] 
 
 185. Calibration of Thermometers. In Fig. 55 let 
 o, i, 2, ... n denote the graduation marks cut on the glass 
 stem, Afi, of a thermometer. The point A we will take to 
 be the freezing point and B the boiling point. These points 
 are known, being first determined bv the maker of the in-
 
 402 THE ADJUSTMENT OF OBSERVATIONS. 
 
 strument, and can be redetermined at any time by special 
 experiments. 
 
 We will suppose that the errors of the freezing and boil- 
 ing" point marks are known, and proceed to consider the 
 corrections to the intermediate marks due to want of 
 uniformity of the bore of the stem, or the calibration correc- 
 tions, as they are called. If the bore between two assigned 
 marks were uniform it could be filled by a certain volume 
 of mercury. The length of this standard volume or column 
 would be indicated by the readings at the two marks. If 
 the column were moved along until it came between two 
 other marks with the same difference of readings as before, 
 and the bore were not uniform with the bore at first posi- 
 tion, the column would not fill the stem between the marks. 
 The amount of difference would be the calibration correc- 
 tion for this interval. 
 
 In explaining the method of determining the calibration 
 corrections let us for simplicity consider 3 intermediate 
 points only between the freezing and boiling points; that 
 is, 4 intervals, o i, I 2, 2 3, 3 4, of 45 each on Fahrenheit's 
 scale. A column of mercury, of volume sufficient to fill o I 
 as nearly as may be, is broken off and the ends read when in 
 the positions o i, I 2, 2 3, 3 4. Another, equal to o 2, is read 
 in the positions 02, i 3, 2 4, and a third, equal to o 3, in the 
 positions 03,1 4.* 
 
 As it is impossible to break off the exact column re- 
 quired in every case, we break off one as nearly equal to it 
 as possible and neglect the error introduced by the small 
 discrepancy, which need not exceed o.2. 
 
 1 86. The following method of breaking off column 
 lengths was that employed by Mr. Charles C. Brown, of the 
 Lake Survey, in determining the calibration corrections of 
 thermometer 5280 Green (New York): 
 
 A column of mercury 200 to 250 in length was easily 
 obtained by making that amount of mercury run into the 
 
 * This method of making the readings is known as Neumann's. See the corresponding form 
 in triangulation Art. 154.
 
 APPLICATION TO CALIBRATION OF THERMOMETERS. 403 
 
 stem, a fe\v slight jars being sufficient to start the column 
 moving; a little manipulation then brought the empty 
 space in the bulb to the junction of the bulb and the stem, 
 when a sudden turn of the thermometer upright broke off 
 the column, and almost as sudden an inversion preserved it. 
 When too large a space was left in the bulb to be filled by 
 heating the thermometer to 140 or 150, a few drops of 
 mercury were allowed to drop off the end of the column in 
 the stem held upright, good care being taken to stop the 
 operation before the column joined the mercury in the bulb. 
 Then, the thermometer being heated until the mercurv from 
 the bulb began to appear in the stem, the column already in 
 the stem was run down carefully, and partially joined to the 
 mercury in the bulb, leaving a small bubble on one side of 
 the column, the thermometer being allowed to cool slowly 
 until the desired length of column above this bubble (which 
 remains very nearly stationary) was obtained. The column 
 was broken off at the bubble by a slight twitch or jar. If 
 there are objections to heating the thermometer above a 
 certain temperature, column lengths above 10 to 20 or 3O D 
 longer than the number of degrees of that temperature, de- 
 pending on the distance of the 32 point from the bulb, can 
 be obtained by jarring off small drops of mercury from a 
 long column into the reservoir at the top of the stem. It 
 requires much more time, care, and patience to obtain a 
 column in this way than in the other. Columns more than 
 about 160 in length were so obtained in this cnse. It is 
 rather difficult to break off short columns 5 to 15 in length 
 in the manner first described, the weight of mercury in the 
 short column not giving momentum enough to move it 
 away from the rest of the column readily. A little patience 
 is all that is necessary, however. To be able to read the 
 shorter columns at 32 a column 10 to 50 long, depending 
 on the temperature at the time, must be broken off and put 
 into the reservoir at the upper*end of the tube, out of the 
 way. 
 
 187. Let x v , -i',, .r 3 , ,r 3 , x^ denote the calibration correc- 
 52
 
 404 THE ADJUSTMENT OF OBSERVATIONS. 
 
 tions at the several graduation marks. Then, approxi- 
 mately, we have for each interval a relation of the form 
 
 column diff. of read ings -(- diff. ofcal. corr. 
 
 As the volumes of the columns broken off correspond to the 
 constant interval between the microscopes in the case of 
 line measures, the observation equations furnished by the 
 thermometer readings will correspond in form to equations 
 6, Art. 184. 
 
 The rigorous solution of these equations is, however, 
 rather complicated, and though in comparisons of stand- 
 ards of length it may be allowable, from the precision with 
 which measurements can be made, to spend the labor de- 
 manded by this form of reduction, yet in thermometric work, 
 where the readings cannot be very close from the nature of 
 the case, an approximate foraris sufficient. By the following 
 artifice, which is quite analogous to that introduced by 
 Hansen in the adjustment of a triangulation (Art. 155), the 
 labor is reduced very materially. 
 
 Instead of finding the corrections to the graduation 
 marks directly, the corrections to the several intervals be- 
 tween the graduation marks may first be found, and thence 
 the corrections to the marks at the ends of the intervals. 
 Thus if ,, , , 4 denote the corrections to the 4 inte/vals 
 in our example, then 
 
 f - Y - Y "~ - Y - Y ( T ^ 
 
 "Z -- ' t 2 *! "4 - -*4 -i-S \ l J 
 
 From the observation equations 6 we have, by subtract 
 ing in pairs, 
 
 _ r- - / _ / 
 
 ""4 - 34 *a 
 
 - / - / 
 
 *i 3 *0 
 
 - s / - / 
 
 -" ' *
 
 APPLICATION TO CALIBRATION OF THERMOMETERS. 405 
 
 Hence, considering / ia / ,, / 33 / )2 , ... as independently 
 observed quantities, we have the normal equations 
 
 -1 22 ; 3 -4 
 Z\ + 3C'l 2 3 24 = 
 2l 2 3 + 3 2 3 2 4 = 
 2, 2a 2j + 3=4 = 
 
 (A a /o i) + (/i 3 /oa) + (A 4 - /o 3) 
 
 - (A a - /o ,) + (A 3 - /, a ) + (A 4 - /, 3) 
 
 - (/a 3 - A a) + (A 4 /a a) (As- /o a) 
 
 - (A 4 - A 3 ) - (A 4 - A 3 ) - (A 4 - /o 3) 
 
 (3) 
 
 which equations when added give o = o identically. The 
 reason is that we have not yet fixed the initial or terminal 
 points. To do this we may, as most convenient, assume the 
 relation 
 
 2l + 2 a + S S + Si = O (4) 
 
 Adding this relation to each of the normal equations, we 
 find the values of the unknowns. Thus 
 
 4=1 = (A , - / ,) + (/, 3 - / ,) + (/,,- / 3 ) 
 423 = - (A , - / + (A 3 - A 2 ) + (A 4 - A 3 ) 
 4 c 3 = - (A 3 - A a) + (A 4 - A 3) - (A 3 - A. ) 
 434 = _(/, 4 _/,,)_ (/., 4 _ /, ,) _ (/, 4 _ / 3 ) 
 
 The computation of the unknowns may be much facili- 
 tated by arranging in tabular form, as follows: 
 
 -1 
 
 *a 
 
 ,3 
 
 -4 
 
 
 
 - (A -2 - /O ,) 
 
 -(A 3 -/o,) 
 
 -(A, -A, 3 ) 
 
 Sumi 
 
 
 
 - (A 3 - A a) 
 
 - (A 4 - A 3) 
 
 Surria 
 
 
 
 
 - (A , - A 3 ) 
 
 Siini 3 
 
 Sum, 
 
 Sum 2 
 
 Sums 
 
 
 
 42l 
 
 422 
 
 4 = 3 
 
 4=i 
 
 
 -i 
 
 -2 
 
 ~3 
 
 -4 
 
 
 The corrections ,r,, x. r x z are then found from the relations
 
 406 
 
 THE ADJUSTMENT OF OBSERVATIONS. 
 
 since the correction x may be assumed to be zero, and then 
 Xi must also be zero from the relations (i) and (4). 
 
 As a check we notice that the values of 4 and x a must be 
 equal but of opposite sign. 
 
 The Precision. If z//, v,', . . . denote the residuals of 
 equations 2, we have for the m. s. e. of a column length 
 
 6-3 
 
 the number of equations being 6 and of independent un- 
 knowns 3. 
 
 Now, since the column length involves the difference of 
 the calibration corrections at its ends, if we assume the 
 m. s. e. of these corrections to be equal we shall have the 
 m. s. e. of a calibration correction by dividing the above 
 
 result by V2 ; thus 
 
 2(6 - 3) 
 
 The weights of the unknowns or of any functions of them 
 may be found by the methods of Chapter IV. 
 
 Ex. The following were the observed values of the lengths of the 45, 90, 
 and 135 columns of thermometer Green 4470, made to determine the calibra- 
 tion corrections at the 77, 122, and 167" points : 
 
 45 col. 
 
 90 col. 
 
 135 col. 
 
 44. 68 
 
 90. 07 
 
 134. 61 
 
 44. 7i 
 
 90. 09 
 
 134. 68 
 
 44 c -72 
 
 90. 1 1 
 
 
 44. 75 
 
 
 
 The corrections at the freezing and boiling points are known.
 
 APPLICATION TO CALIBRATION OF THERMOMETERS. 407 
 Solution. Arranging in tabular form, 
 
 ., 
 
 -2 
 
 23 
 
 .. 
 
 Surns. 
 
 
 -0-.03 
 
 O.O2 
 
 o.oy 
 
 O.I2 
 
 
 
 o".oi 
 
 O.O2 
 
 -- 3 
 
 + O' . 1 2 
 
 + 0-.03 
 
 
 
 
 + 0.I2 
 
 o.oo 
 
 . O.OO 
 
 0.I2 
 
 
 + o.03 
 
 O.OO 
 
 and 
 
 Also 
 
 = o.O3 
 
 188. The following memoirs may be consulted : Sheep- 
 shanks, Monthly Notices Royal Astronomical Society, vol. xi. 
 pp. 233-248 ; Hanscn, Von der Bestimmung dcr TJieilnngs- 
 fcliler eines gradlinigcn Maassstabcs, Leipzig-, 1874; Russell, 
 American Journal of Science, vol. xxi. pp. 373-379; Thorpe, 
 Report /-iritish Association for Advancement of Science, 1882, 
 pp. 145-204; Brown, Van Ncstrand's Engineering Magazine, 
 vol. xxix. pp. 1-7; Benoit, Mesures de dilatation et comparisons 
 des regies mc'triqucs, Paris, 1883.
 
 CHAPTER X. 
 
 APPLICATION TO EMPIRICAL FORMULAS AND INTERPOLATION. 
 
 189. In all discussions hitherto we have considered the 
 observed quantity, whether a function of one or more 
 variables, to be a function whose form was known and 
 which could therefore be developed in terms of that 
 variable. In the physical sciences we meet with a differ- 
 ent problem. From observation we have values of a func- 
 tion corresponding to certain known values of a variable, 
 and are required to determine the form of the function from 
 the observed values; in other words, to find the algebraic 
 formula connecting the function and the variable. The 
 method of least squares will not enable us to find the 
 most probable form of this function. All it will do is to 
 show how to approach more closely to its form after that 
 form has been found approximately by other means. 
 
 For example, the observed height of the tide at a place 
 may be considered a function of the time of day. If obser- 
 vations are made with a staff gauge three or four times a 
 day, and the stage of water at a time not observed is 
 required, we cannot interpolate between the observed 
 values in the most probable manner till the function con- 
 necting the height and the time of day is known. 
 
 Sometimes, indeed, the observations may be so taken as 
 to record themselves. Thus with Saxton's self- registering 
 tide gauge, in common use in the United States, the curve 
 representing the rise and fall of the water is traced continu- 
 ously on a web of paper moving uniformly past a pencil 
 point by means of clock-work. Plotting the time along 
 one edge of the paper as abscissa, the stage of water at
 
 APPLICATION TO EMPIRICAL FORMULAS, ETC. 409 
 
 any time required can be read at once from the sheet. 
 The curve representing the rise and fall of the water is 
 in this case completely known, and no computation is 
 necessary. 
 
 190. From the nature of physical phenomena, in which 
 observations are made only at intervals, and into which sys- 
 tematic error largely enters, we must admit that it is most 
 probable that the function connecting the observed quantity 
 and the variable is different for each observation. We 
 should thus have a series of equations of the form 
 
 M, =/ (x, a, &,c, . 
 
 M, =f t (x, /i, k, I, . 
 
 where J/,, M t , . . . are the observed values, x the variable, 
 and a, b, . . . / //, k, . . . are constants. 
 
 In order to apply the method of least squares to de- 
 termine the constants, the functions must first be reduced 
 to the linear form, and the number of quantities to be de- 
 termined must be made less in number, arbitrarily it may be, 
 than the number of observations. The first point is to 
 determine as nearly as we can the general form of the 
 functions connecting the observed quantities and the vari- 
 able. 
 
 For this no general rule can be given. From theoretic 
 considerations we may hit upon a certain formula as 
 plausible. It must then be found by trial how closely it 
 fits the observed values. It is often very convenient to 
 make a graphical representation, the variable being the 
 abscissa, and the observed values of the function correspond- 
 ing to known values of the variable the ordinates. The 
 points plotted will in general fall so irregularly, here a 
 group and there a vacant space, that a free-hand curve only 
 can be drawn which will conform to the general outline of 
 the plot. The curve will represent the general form of the 
 function we are seeking. This method of eliminating ir- 
 regularity by drawing a mean curve to represent the value
 
 4IO THE ADJUSTMENT OF OBSERVATIONS. 
 
 of the function corresponds to assuming a common form for 
 the algebraic functions f^f v . . . above. 
 
 A good example is afforded by the method employed by 
 Humphreys and Abbot in their Mississippi River work for 
 finding the law of velocities below the surface in a plane 
 parallel to the current.* The observations for velocity 
 were made with floats at different depths, started from 
 boats anchored at vaiious distances from the shore. In the 
 first place, all of the velocities observed from each anchorage 
 were plotted on cross-section paper, the depths of the floats 
 being the ordinates and the velocities the abscissas. The 
 resulting curves showed marked irregularities of form. To 
 eliminate- errors of observation the velocities were grouped 
 in sets corresponding to nearly equal depths of water and 
 to nearly equal velocities of the river, and the means plotted. 
 These curves indicated a law, though not sufficiently clearly 
 to allow of deducing an algebraic expression for it. " It is 
 manifest that some further combination is necessary in 
 order to eliminate the effect of disturbing causes. Since 
 the absolute depths differ, this can only be done by com- 
 bining the velocities at proportional depths." A curve 
 was. therefore, plotted from the mean of all the velocities at 
 proportional instead of at absolute depths. This curve 
 proved to be approximately a parabola with axis horizontal 
 and vertex about o 3 of the depth of the river below the 
 surface. It could, therefore, be expressed by an algebraic 
 formula. 
 
 In many cases we may find the form of the function 
 over which the observations extend by means of a formula 
 of interpolation. Thus if x' is an approximate value of x, 
 then for observations in the vicinity of x we have, by 
 Taylor's theorem, 
 
 /(*)=/{*' + (*-*')| 
 
 =/(*') + &(* - *'} -f c(* - *y + 
 
 where b, c, . . . are unknown constants to be determined. 
 
 * Physics and Hydraulics of the Mississippi River. Washington, 1876.
 
 APPLICATION TO EMPIRICAL FORMULAS, ETC. 41 I 
 
 Having found the form of the function approximately 
 by such means as the preceding, we can, by the method 
 of least squares, determine the most probable values of the 
 constants involved, so far as the observations on hand are 
 concerned. For let the function be reduced to the linear 
 form, and the most probable values of the constants and 
 the precisions of these values may then be found by the 
 ordinary rules for observation equations as laid down in 
 Chapter IV. 
 
 191. The question naturally arises as to the number of 
 terms of a series, such as the above, that should be taken 
 in any special case. Cut and try is our main guide. II the 
 plot of the observations shows, for example, that the phe- 
 nomenon can be very closely represented by a straight line, 
 we should take the first two terms. Thus it has been usual 
 to take the relation between the length, V, of a bar of 
 metal and the temperature, t, to be of the form 
 
 VM-\-b(t-f} 
 
 where M is the observed length at the temperature /', and b 
 is the coefficient of expansion to be found from the observa- 
 tions. More refined methods of observation indicate, how- 
 ever, that this expression is not sufficient to express the 
 relation between length and temperature. The plot of the 
 observed values indicates a curve of higher dimensions than 
 the first, so that we must take additional terms of the series 
 to represent it, thus : 
 
 V M+ b(t - t'} + c(t - tj -f . . . 
 
 A second guide is afforded by the value obtained of the 
 sum of the squares of the residuals of the observation equa- 
 tions. If the nature of the observations is such as to war- 
 rant the expectation of a value much smaller than that 
 obtained, we may take additional terms of the series; and if 
 this is still unsatisfactory we must seek another function. 
 This test may also be used in comparing one form of func- 
 tion with another. 
 53
 
 412 THE ADJUSTMENT OF OBSERVATIONS. 
 
 In anV case, even when all of the tests applied appear 
 satisfactory, we cannot say that our final result is the most 
 probable value of the function that can be found. We have 
 made too many assumptions for that. In the first place, 
 the form of the function assumed as a first approximation is 
 too uncertain; and, again, \ve have arbitrarily assumed the 
 number of constants to be determined, and determined this 
 limited number so as to satisfy a special set of observations 
 only. All that our final formula will enable us to do is to 
 interpolate in the most probable manner within the range of 
 the special set of observed values, but not to extrapolate. 
 If, however, the formula has been tested by many series of 
 observations made under the most diverse conditions, and 
 is found to satisfy them well, we can with confidence apply 
 it in cases where no observations have been made. We 
 may, in fact, consider that we have found the law connecting 
 the function and the variable. 
 
 192. It is evident, too, that different experimenters may 
 derive formulas widely different to represent like phenome- 
 ( iia, and that each formula may satisfy the special set of 
 observations irom which it was derived tolerably well. 
 Each experimenter may have chosen his constants differ- 
 ently, as well as a different form of function to begin with. 
 As an example of this we may cite the formulas proposed * 
 to represent the variation of the elastic forces of vapor at 
 different temperatures /. 
 
 Young proposed 
 
 P (a -f /;/)"' 
 
 a and b being constants to be found from observation. 
 Roche, 
 
 X 
 
 P = a a J f mx 
 
 where x = / -\- 20. 
 Biot and Regnault, 
 
 log P = a -f- bo* + cjf 
 
 * Mousson, Fhysik, vol. ii. Zurich, 1880.
 
 APPLICATION TO EMPIRICAL FORMULAS, ETC. 
 
 413 
 
 Of these formulas the first involves two constants, the 
 second three, and the third five. The first curve, there- 
 fore, requires two observations to determine it, the second 
 three, and the third five. If a group of observations were 
 plotted with temperatures as abscissas and pressures as ordi- 
 nates, then, since the last lormula represents a curve passing 
 through five of these points, it is evident that if the obser- 
 vations are good this curve would in all probability pass 
 near all intermediate points and more closely than a curve 
 fixed by only two or three points. It should, therefore, be 
 chosen. 
 
 Ex. i. The following are the results of the observations m;ide for velocity 
 of current of the Mississippi River by Humphreys and Abbot at Carrollton and 
 Baton Rouge in 1851. Each is the mean of 222 observations and is given at 
 proportional depths, the whole depth being represented by unit}*. 
 
 Propor. depth of float 
 below surface. 
 
 Obs. velocity in feet 
 per second. 
 
 O.O 
 
 3-I950 
 
 O.I 
 
 3.2299 
 
 O.2 
 
 3-2532 
 
 0-3 
 
 3.2611 
 
 0.4 
 
 3-25I6 
 
 0-5 
 
 3.2282 
 
 0.6 
 
 3.1807 
 
 0.7 
 
 3.1266 
 
 0.8 
 
 3-0594 
 
 0.9 
 
 2-9759 
 
 In Art. 191 it has been shown that the curve of velocities is approxi- 
 mately parabolic in form. As a first approximation, therefore, we take three 
 terms of the expansion from Taylor's theorem, 
 
 V = a + bD + fD- 
 
 where V \s the velocity, D the proportional depth, and a, l>, c constants to be 
 determined from the observations.
 
 414 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Hence the observation equations 
 
 3.1950 = - 
 
 3.2299 = a + o.ib + o.oic 
 3.2532 = + .26+ .04<r 
 3.2611=0+ .36 + .ogc 
 3.2516 = + .4^ + . i6c 
 3.2282 = + .53 + .25<r 
 3. 1807 = a + .63 + .36<r 
 3.1266 = + .7*5+ .49^ 
 3.0594 = + .83+ .64<r 
 2.9759 = a + 0.9^ + O.Sl,r 
 
 To solve these equations proceed as in Art. 104. Take the mean and sub- 
 tract from each, and we have 
 
 + 0.0188 = 0.45*5 0.2851: 
 + 0.0537 = -35^ 0-275<: 
 + 0.0770= 0.25^ 0.245^ 
 + 0.0849 = o. 15$ o. 1951: 
 + 0.0754 o.os/> o. i25c 
 + 0.0520 = + o.os/; O.O35C 
 + 0.0045 = + o. 15^ + 0.075C 
 
 0.0496 = +O.25/; + 0.205^ 
 
 0.1168 = + 0.35^4- 0.355^: 
 
 0.2003 = + 0.45^ + 0.5251: 
 
 Hence the normal equations 
 
 0.2031 = 0.8250^ +-0.74251: 
 
 0.2232 =o.7425/> + o. 
 
 and ^ = 0.4424 c= 0.7652 = 3. 1952 
 
 .'. V 3.1952 + 0.4424!) o. 
 
 The residuals r/are found by substituting for a, l>, c in the observation equa- 
 tions, and give
 
 APPLICATION TO EMPIRICAL FORMULAS, ETC. 415 
 
 If we take four terms of the expansion, so that 
 
 V = a + bD + cD n - + <//>< 
 and proceed similarly, we shall find 
 
 a = 3-1935, ^= + 0-4735 c- 0.8563, </ 4- 0.0675 
 so that 
 
 V ' = 3.1935 + 0.4735 D O.S563/) 2 + 0.0675 D* 
 
 and the sum of the squares of the residuals z> is given by 
 
 [H =0.45 
 
 The sum of the squares of the residual errors being less than in the former 
 case, we conclude that the observations are better represented by the formula 
 last obtained. 
 
 It is to be borne in mind that in the application of the method of least 
 squares only one set of measured quantities, if more than one occur in the 
 problem, can be considered variable. The error is thrown into this set. 
 Thus in the problem just solved the depths are supposed to be correctly 
 measured and the error alone to occur in the measured velocities. So in 
 finding the expansion of a body it is usual to consider all of the error as 
 occurring in the observed lengths, and to take the thermometer readings to 
 be correct. The justice of these assumptions may, however, very fairly be 
 questioned. A discussion of the general problem covering such cases will 
 be found in Art. 106. 
 
 Ex. 2. The tension, H, of saturated stearn at temperature f C. is found 
 from the formula* 
 
 
 where a = 0.0036678, and a, b, c, . . . are constants, >/ + i in number, to 
 be determined from observed values of / and H. We have also given that 
 when 
 
 t = ioo c C., H = 760 "" 
 
 * Travaujc ft Mtmoircs <fu Bureau international lies raids ct Mf surfs, Paris, 1881.
 
 416 THE ADJUSTMENT OF OBSERVATIONS. 
 
 To find the constants a, />, c, . . . we proceed as follows: 
 Substituting the values of I and H in equation i, and eliminating a, we 
 have 
 
 = 7OO IO \i + looo i -(- at) \i+iooo i -j- -t) 
 
 = 760 10 "" ^' + c ^2 + ) suppose. 
 
 log 760 log // = bpi + fpv + . . . (2) 
 
 Next compute approximate values of /', t, ... by selecting of the observa- 
 tions and substituting the observed values of t and H in Eq. 2. Let b' , c', . . . 
 denote the values found from the solution of the resulting equations, and let 
 //' denote the corresponding value of//, so that 
 
 If ' = 760 io~^>'+ c >*+- ' > (3) 
 
 Also, let x, y, . . . denote the corrections to be added to the approximate 
 values b' , c , . . . to find the most probable values, so that 
 
 then, from (2), 
 
 log 760 log \ff' + (^ ff')\ = (b' + r )/i + ' 
 or expanding by Taylor's theorem and remembering that, from (3), 
 
 log 760 log //' = b'p l + c'p-i + . . . 
 we have 
 
 the linear form required. The solution may be finished in the usual way, 
 as given in Chapter IV. 
 
 193. The preceding examples show that as soon as it has 
 been decided what formula will apply to a special set ot 
 observations the main difficulty is in reducing this formula 
 to the linear form. When this has been done so that the 
 formula is in the form of an ordinary observation equation, 
 the reduction by the method of least squares is straight- 
 forward.
 
 APPLICATION TO EMPIRICAL FORMULAS, ETC. 417 
 
 In cases where the data are insufficient or contradictory 
 it is useless to waste time on long computations, as the 
 graphical plot will show at a glance all that the observations 
 can show.' With cross-section paper plots can be made 
 rapidly, and by transferring to tracing linen one plot can 
 be placed over another, so that comparisons can be readily 
 effected and a mean value struck. 
 
 In the problem of the law of velocities in rivers, alter 
 having decided from the various plots that the curve of 
 velocities is approximately parabolic in form, it is better to 
 employ the method of least squares to determine the con- 
 stants of the curve rather than to trust to the graphical 
 method throughout, but only for the reason that the data 
 available are very complete. 
 
 194. Periodic Plienoinena. A large class of physical phe- 
 nomena is more or less periodic in character. The daily 
 temperature throughout the year at a given place, the er- 
 ror of graduation of the limb of a theodolite, the magnetic 
 declination with reference to the time, etc., are examples. 
 The phenomenon may not be strictly periodic in that like 
 periods succeed each other in their proper order, or that 
 even any one period is perfect throughout. If a plot of the 
 observed values of the function corresponding to certain 
 values of the variable involved be made, and it indicates the 
 periodic character of the function, we may assume as the 
 form of the function a number of terms of the series fur- 
 nished by Fourier's theorem. Each observation will give 
 an observation equation, and from the observation equations 
 the values of the constants in the formula will be deter- 
 mined. As in the cases already discussed, the final formula 
 is to be looked on as holding only within the limits of the 
 observations, serving as a guide for the future study of the 
 phenomenon in question, and only to be used with great 
 caution outside the limits of the observed values. 
 
 Suppose, then, that n observations give the values 
 yl/,, J/ a , . . . M n corresponding to the values <p, <p-\- 6, . . . 
 <f -j- (n i)# of the variable <f uniformly distributed over
 
 41 8 THE ADJUSTMENT OF OBSERVATIONS. 
 
 the period represented by 360 ; that is, such that the inter- 
 
 i & 360 
 val 6 is -. 
 
 n 
 
 Now, if f(<p] is any arbitrary periodic function of a vari- 
 able <p, we may, according to Fourier's theorem, write 
 
 f(<p) = X+ h, sin (a, -f <p ) -f /t t sin (a, + 2<p) + . . . 
 
 where X, //,, h v . . . 1? 2 , . . . are constants to be deter- 
 mined from the observed values of f(<f>) corresponding to 
 assigned values of the variable <p. Substitute M lt M^, 
 M 3 , . . .M u for/(<p), and <p, <p -f 6, <p + 26, . . . <p + (;/ - i)0 
 for <f>, in the above equation, and we shall have n equations 
 from which to determine a number of constants not exceed- 
 ing n i. 
 
 To lighten the numerical work let us assume the arith- 
 metic mean of the observed values as an approximate value 
 of X, and let x denote the correction to this value, so that 
 
 
 Then if, as usual, is placed 
 
 the n observation equations may be expressed by the 
 general formula 
 
 x -4- /i, sin (,-(- <f> + in &) 
 
 -f- // sin (a, + 2<p+2mff)+ . . . -/ m + I = ,, m + i 
 
 where m assumes all values from o to n i 
 Writing them at full length, 
 
 x -\-}\ cos o -f- s l sin o -f- J 2 cos 2 o -f - 2 sin 2 o + . . . /, = ?;, 
 x -\-y, cos 6 -f s l sin 6 -(- j 2 cos 2 S-\- s, sin 2 S -|- . . . / = v t
 
 APPLICATION TO EMPIRICAL FORMULAS, ETC. 419 
 
 v t , v y , . . . being, as usual, the errors of observation, and 
 
 j, = h, sin (a, -f <p), j, = //, sin (a, + 2<p), 
 z l = / tl cos (a, 4- <p), s, = //, cos (a, + 2<p), 
 
 The normal equations reduce to the simple forms (see 
 Art. 9) 
 
 nx = [/] = o 
 
 -j, I^^co 
 
 ft 
 
 -^ = 2*4 + 1 sin 
 
 -2,= 2/ m + l sin 2m 6 
 
 where m has all values from o to n i. 
 
 Hence//!,,//,, . . . a,, . . . are known, and their values, 
 being substituted in/(y>), give the function required. 
 
 If the initial value of f (<p) corresponds to <p = o, that is, 
 if the observed values M lt M v M 3 , . . . M n of f(<f) corre- 
 spond to o, 6, 26, ...( 1)6, where 116 = 360, it is 
 simpler to write the equation for f(*p) first of all in the 
 form 
 
 f((f>) X-\-y, cos <p -\- z, cos 2<p -f- . . . 
 -f- 5, sin ^ -|~ " sm 2 ^ + 
 where 
 
 y l = h l sin a, j, = //, sin a,, ... 
 
 ^r, = //, COS a, <cr 2 = //, COS 2 , ... 
 
 Then with the value - - as the approximate value of X, we 
 
 54
 
 420 THE ADJUSTMENT OF OBSERVATIONS. 
 
 have the n observation equations and the normal equations 
 of the same forms as before. 
 
 Hence the function is known. 
 
 195. Two cases of frequent occurrence are : 
 
 (a) n - 10, 9 = 3 T <y>- = 3 6 
 The normal equations may be written 
 
 5;-, = /,_/, + !(/,_ / 7 ) - (A. _/,)} cos 36 + | (/, - /) - (A - /.) } cos 72 
 5 *, = \ (A - /,) + (/. - Ao) I sin 36 + {(/- /) + (A - /.) | sin 72 
 
 5 )< 2 = A + / 6 + j (A + A) + (4 + Ao) | cos 72 | (/s + A.) + (A + A) \ cos 36 
 5 s 2 = |(/ a + A)-(A. + /,u)}sin 72 + {(/ + A)-(A + /.)}sin 36 
 
 It will be noticed that the difference of the subscripts of 
 the /'s in each parenthesis is always five, the same as the 
 coefficient of the unknown. 
 
 (b) n=i2, 6=30 
 The normal equations 
 
 6j/, = (A - A) + 1 (/a - /.) - (/. - A i) } sin 30 + {(/,- / 8 ) - (/. - As) } cos 30 
 62,= (/ 4 /,)+ | (/s / 9 ) + (/ 6 A i) ( cos 30 + | (/., 4) + (/ A 2 ) } sin 30 
 6>< 2 = (A + A) - (/ 4 + /,) + { (/ + /) + (/ + /,,) - (/s + /.)-(^+/ii)| sin 30 
 6s 2 = | (A + /) - (/. + /) + (/ 3 + /.)-(/6+/u) }cos 30 
 
 The difference of the subscripts of the / 's in each parenthesis 
 is six in this case. 
 
 196. The Precision. The m. s. e. of the unit of weight is 
 found from the usual formula 
 
 where n t is the number of constants determined.
 
 APPLICATION TO EMPIRICAL FORMULAS, ETC. 
 
 Check of [w]. Generally (Art. 100) 
 
 r/r/i" \hi i~i l r/-/ 1~\* 
 
 M = [//]- 
 
 421 
 
 M 
 
 [<:<:.2] 
 
 Now substitute for [#/], [/.i], [V/.2], . . . their values from 
 the normal equations above, and remembering that these 
 equations contain only one unknown each, 
 
 -jr. i. The mean monthly heights of the water in Lake Michigan at Chi- 
 cago below the mean level of the lake from iS6o to 1875, for the 12 months of 
 the year 1868, were as in column M of the following table: 
 
 
 M 
 
 / 
 
 i 
 
 M 
 
 / 
 
 Jan. 
 
 ft. 
 I.I7 
 
 /'. 
 4- 0.45 
 
 July. 
 
 ft. 
 
 O. II 
 
 ft. 
 
 0.61 
 
 Feb. 
 
 1-25 
 
 + 0.53 
 
 Aug. 
 
 0-43 
 
 0.29 
 
 March. 
 
 0-59 
 
 0.13 
 
 Sept. 
 
 0.68 
 
 0.04 
 
 April. 
 
 o.6S 
 
 0.04 
 
 Oct. 
 
 0.94 
 
 + 0.22 
 
 May. 
 
 0.29 
 
 - 0.43 
 
 Nov. 
 
 1.05 
 
 + o-33 
 
 June. 
 
 0.17 
 
 - 0.55 
 
 Dec. 
 
 Mean, 
 
 1-32 
 
 + 0.60 
 
 0.72 
 
 Required a formula from which the mean daily height may be found. 
 
 The period is one ye.tr, and if we assume that its 12 months correspond to 
 360 and that the months are of equal length, each interval f~) would be 30. 
 
 The values of the coefficients y lt z t , . . . can be at on 7e written down 
 from (b). They are 
 
 j, = + 0.517 y-i = o.oio 
 
 zi = 0.194 z a = + 0.017 
 and therefore 
 
 a, = 110 34' Ai = + 0.552 
 
 nr a = I t 49 32' h?= 0.020 
 
 1 = 0.72 + 0.552 sin (110 34' + (p) + 0.020 sin (149 32' + 2<p) + . . .
 
 422 THE ADJUSTMENT OF OBSERVATIONS. 
 
 Ex. 2. In a micrometer microscope, to find the correction for periodic 
 error of the micrometer screw to the readings of the graduated micrometer 
 head. 
 
 The necessary observations are made by measuring in succession on a 
 giaduated scale the distance corresponding to intervals which ate aliquot 
 parts of a complete revolution of the screw. 
 
 Let A = the value of the space on the scale. 
 
 <p = the division on the micrometer head read on at the initial reading. 
 
 The correction 10 the first reading will be 
 
 h\ sin(j + tp) + //a sin (a 2 + 2<p) + . . . 
 and the correction to the second reading 
 
 hi sin((iri + A + (p) + hi sin (nr 2 + 2A + 2cp) + . . . 
 
 Hence, since the correction to the observed value M of the scale distance is 
 the difference of these corrections, we have 
 
 A A 
 
 A = M + 2/ii cos(o' 1 H --- h q>) sin h 2^ 2 cos (a 2 + A + zcp)sinA + . . . 
 
 Suppose now that the scale has been read on by tt different parts of the 
 screw, so that to the observed values M\, Mi, . . . M n of the graduated dis- 
 tance correspond the values of q>, 
 
 q>, tp + &, . . . q> + (n i)Q 
 
 the amount by which the screw is shifted ea''h time being & = - . 
 
 a 
 We should then have equations of the form 
 
 A A 
 
 A = M + 2/ii cos (cti -\ --- h q> + ;)sin 
 2 2 
 
 + 2^2 cos (nr a + A + 2cp + 2mf-)) sin A + . . . 
 where m assumes all values from o to n i. 
 
 If we take as an approximate value of A, and put 
 
 the n observations may be expressed by tlie general formula 
 
 A . A 
 
 x + 2/1, cos (IT, 4 --- h <p + m&) sin 
 
 2 2 
 
 + 2/i z cos (o -f A + 2<p + 2m@) sin A + . . . l 
 
 where m assumes all values from o to n i 
 
 Expanding the cosines, the solution may be completed as in Art. 194. 
 
 The screw of the filar micrometer of the 26-in. refractor of the U. S. Naval 
 Observatory was examined for periodic error by'Prof. Hall in 1880 (Washing- 
 ton Observations, 1877).
 
 APPLICATION TO EMPIRICAL FORMULAS, ETC. 
 
 423 
 
 The micrometer was placed under the Harkness dividing engine, and the 
 distance corresponding to each ^ of a revolution of the screw was measured 
 by means of the micrometer belonging to the engine. The following are the 
 means for each ' of a revolution : 
 
 Microm. 
 
 At 
 
 / 
 
 o.o to o.i 
 
 d 
 O.622I 
 
 -f 9 
 
 .1 " .2 
 
 .6229 
 
 + 17 
 
 2 " .3 
 
 .6182 
 
 -30 
 
 3 " -4 
 
 .6212 
 
 o 
 
 4 " -5 
 
 .62OI 
 
 ii 
 
 .5 " .6 
 
 .6227 
 
 + 15 
 
 .6 " .7 
 
 .6226 
 
 + 14 
 
 .7 " -8 
 
 .6150 
 
 -62 
 
 .8 " .9 
 
 .6189 
 
 -23 
 
 0.9 " i.o 
 Me 
 
 0.6285 
 
 + 73 
 
 an, 0.6212 = o". 995 
 
 Assuming these residuals to have a periodic form, required the correction to 
 the reading of the head of the micrometer. 
 
 The observation equations are 
 
 x + y l cos o + zi sin o + y t cos 2 o + s 2 sin 2 o := 9 
 x + yi cos 36 + z, sin 36 + v a cos 72" + z* sin 72' =17 
 
 zi sin 324^ + ^2 cos648 + ; 2 sin 648"= 73 
 
 The values of vi, z\, . . . are at once found from (a). 
 The final result is 
 
 f(q>) + o".ooo2 sin <p + o".oo22 cos <p o".oo22 sin 2<p + o".oo47 cos -i<p 
 
 Ex. 3. To find the correction for periodic error of graduation to the value 
 of an angle measured with a theodolite in which the circle is read by two 
 opposite microscopes, i and 2. 
 
 Let A = the value of the angle. 
 
 cp = the reading of the circle with microscope i when the telescope is 
 pointed at the first signal. 
 
 The correction to the reading of microscope i for periodic error will be 
 ki sin(ar! + g>) + h-i sin (a, + 2<p) + . . .
 
 424 THE ADJUSTMENT OF OBSERVATIONS, 
 
 and to the reading of microscope 2, writing 180 + q> for <p, 
 
 hi sin (i + <p) + hi sin (a? + 2q>) ... 
 The correction to the mean reading on the first signal is, therefore, 
 
 h?. sin (a<i + 2<p) + h^ sin (<T 4 + 4<p) + . . . 
 and the correction to the mean reading on the second signal, 
 
 h* sin | a? + 2 (A + <p) + A* sin * <T 4 + 4 (A + <p) | + . . . 
 
 Hence, since the correction to the observed value MI of the angle is the 
 difference of these corrections, we have 
 
 A = MI + 2hi cos (oti + A +- 2(p) sin A + 2/i t cos (a 4 + 2 A + 4^) sin zA + . . . 
 
 Suppose now, sighting at the same two signals, that readings have been made 
 at n different parts of the circle, so that to tlie observed values of the angle 
 Mi, Mi, .... M n correspond the values of tp ; (p, (p + S, . . . tp + (n i)0, 
 the amount by which the circle is shifted each time being C-). Complete the 
 solution as in Ex. 2. 
 
 Given the measures of the angle Falkirk-Gasport-Pekin made with a 
 Troughton and Simms 12-in. theodolite: 
 
 M 
 
 / 
 
 97 22' 36" 
 
 So 
 
 i".O3 
 
 33" 
 
 54 
 
 + 2". 23 
 
 3i" 
 
 75 
 
 + 4". 02 
 
 34" 
 
 06 
 
 -M"-7i 
 
 38" 
 
 67 
 
 2". 90 
 
 39" 
 
 63 
 
 4". 06 
 
 Mean, 97 22' 35" 
 
 77 
 
 
 A = gf 22' 35".7 
 (p 70 27' 
 
 = 6, e= 60
 
 APPLICATION TO EMPIRICAL FORMULAS, ETC. 425 
 
 The observation equations, 
 
 j'a cos o + zi sin o + y* cos o + z\ sin o = + 1.03 
 yi cos 60 -f 2 a sin 60 + y^ cos 120 + z 4 sin 120= 2.23 
 
 y-t cos 300+ z a sin 300+ j4cos6oo + z^ sin 600=: + 4.06 
 The normal equations, 
 
 3>'i = 4-22 
 
 3Z = 11.44 
 
 3^4 = - 1-04 
 
 3^4 = 0.55 
 
 Hence the correction for periodic error of any angle A measured with this 
 instrument is given by 
 
 4".io cos (11 + A 4 2<p)sin,4 i". 54 cos (91 + zA + 49)) sin zA 
 
 197. Consult Bessel, Abhandlungen, vol. ii. p. 364; Chau- 
 venet, Astronomy, vol. ii. ; Briinnow, Astronomy ; Cole, 
 Great Trig. Survey of India, vol. ii. ; Woodward, Report 
 Chief of Engineers U. S. A., 1876, 1879.
 
 APPENDIX I. 
 
 HISTORICAL NOTE. 
 
 198. The first account of the method of least squares as 
 now employed was published by Legendre in 1805, in his 
 Nouvelles mtthodes pour la determination des orbites des comet 'es. 
 The main points on which he insists are the generality and 
 ease of application of the method. He states without proof 
 the principle that the sum of the squares of the errors must 
 be made a minimum, and deduces the arithmetic mean as 
 a special case of this general statement. He was also the 
 first to make use of the term least squares (inoindres quarry's). 
 
 The first demonstration of the exponential law of error 
 was published in 1808 by Dr. R. Adrain, of Reading, Pa., 
 in the Analyst or Mathematical Museum* He gives two 
 demonstrations, the second being essentially the same as 
 that now known as Herschel's proof. After deriving the 
 principle of minimum squares Adrain applied it to the solu- 
 tion of the following four problems: 
 
 (1) Supposing a, b, c, . . . to be the observed measures 
 of any quantity x, the most probable value of .v is required. 
 
 (2) Given the observed positions of a point in space, to 
 find the most probable position of the point. 
 
 (3) To correct the dead reckoning at sea by an observa- 
 tion of the latitude. 
 
 (4) To correct a survey. (Ex. 5, Art. no.) 
 
 Gauss published in 1809, in the Tlieoria motus corporum 
 civlestium, a third proof of the law of error, thence deducing 
 the principle of minimum squares. He so thoroughly de- 
 
 * There is a copy of the Analyst containing Adrain's proofs in the library of the American 
 Philosophical Society, Philadelphia. An account of Adrain's life will be found in the Demo- 
 cratic AT'/<-tt for 1844. 
 
 55
 
 428 APPENDIX. 
 
 veloped the subject in its principles and practical applica- 
 tions that comparatively little has been added by later 
 writers. An English translation of the TJieoria motus by 
 Admiral Davis, U. S. N., was published in 1858, and a 
 French translation of Gauss' memoirs on least squares by 
 Bertrand in 1855. 
 
 The main contributions to the subject, aside from those 
 mentioned, have been made by Laplace in the fundamental 
 principles, and by Bessel, Hansen, and Andrse in its applica- 
 tions to astronomical and geodetic work. 
 
 199. Many proofs of the law of error have been given. 
 The most important are by Adrain, Gauss, Laplace, and 
 Hagen. 
 
 That by Gauss in 1809 assumes that if a series of obser- 
 vations, all equally good, are made of a quantity, the arith- 
 metic mean of the observed values is the most probable 
 value of the quantity. The law of error is then deduced, 
 and the principle of minimum squares follows at once. 
 
 Laplace, in his first proof (1810), deduces the principle 
 that the sum of the squares of the errors must be made a 
 minimum without reference to any law of error, only as- 
 suming that positive and negative errors are equally prob- 
 able and that the number of observations is infinitely great. 
 A clear account of this proof is given by Glaisher, Mem. 
 Roy. Astron. Soc., vol. xxxix., and by Meyer, Wahrscliein- 
 lichkeitsrechnung, pp. 440-473. See also Airy, Theory of 
 Errors of Observations, second edition ; Todhunter, History of 
 the Theory of Probability. 
 
 Hagen's proof (1837) is founded on the hypothesis that 
 an error of observation is the algebraic sum of an infinite 
 number of element errors which are all of equal value and 
 which are as likely to be positive as negative. A modified 
 form of this proof is given in Appendix II. 
 
 Adrain's second proof is shorter than any of the three 
 mentioned, but is not so satisfactory. This proof is also 
 given by Sir John Herschel, Edinburgh Review, vol. xcii. ; 
 Airy, Theory of Errors of Observations, third edition ; Thomson
 
 APPENDIX. 429 
 
 and Tail, Treatise on ^Natural Philosophy, vol. i. ; Clarke, 
 Geodesy. 
 
 For more extended information on this subject the 
 reader is referred to Merri man's very complete memoir 
 entitled List of Writings Relating to the Method of Least 
 Squares, New Haven, 1877. 
 
 APPENDIX II. 
 
 THE LAW OF ERROR. 
 Proof on Hagens \Yoit ng's\ Hypothesis. 
 
 200. It a quantity, T, is to be determined, and J/, is an 
 observed value of T, then, if the observation were perfect, 
 we should have 
 
 But since, if we make a second and a third observation, we 
 may not find the same value as we did at first, and as we 
 can only account for the difference on the supposition that 
 the observations are not perfect that is, that they are 
 affected with certain errors we should rather write 
 
 where M t , M^ M 3 are the observed values, and J,, J a , J 3 
 are the errors of the observations. 
 
 Now, an error of observation does not result from a 
 single cause. Thus in reading an angle with a theodolite 
 the error in the value found is the result of imperfect ad- 
 justment of the instrument, of various atmospheric changes
 
 430 APPENDIX. 
 
 of want of precision in the observer's method of handling 
 the instrument, etc. Each of these influences may be taken 
 as the result of numerous other influences. Thus the first 
 mentioned may include errors of collimation, of level, etc. 
 Each of these in turn may be taken as resulting from other 
 influences, and so on. The final influences, or element 
 errors, as they may be called, must be assumed to be inde- 
 pendent of one another, and each as likely to make the 
 resultant error too large as too small that is, as likely to 
 be positive as negative. The number of these element 
 errors being very great, we may, from the impossibility of 
 assigning the limit, consider it as infinite in any case. Each 
 element error must consequently be an infinitesimal, and 
 for greater simplicity we may take those occurring in any 
 one series as of the same numerical magnitude. Hence we 
 conclude that an error of observation may be assumed to 
 be the algebraic sum of a very great number of independent 
 infinitesimal element errors s, all equal in magnitude, but as 
 likely to be positive as negative. 
 
 Let the number of these element errors be denoted by 
 2n, as the generality of the demonstration will not be affected 
 by supposing this infinitely great number to be even. If 
 all of the element errors are -(-, the error 2ne results, and 
 this can occur in but one way ; if all but one are -{-, the 
 error (2 2]s results, and this can occur in 2n ways; and 
 generally if n -\- m are -f-, and n m are , the error 2we 
 
 ,, i .1- - 2n (211 i) . . . (n -4- 111 -\- i ) 
 
 results, and this can occur in - 
 
 12. . , (n m) 
 
 ways.* Hence the numbers expressing the relative fre- 
 quency of the errors (that is, the number of times they may 
 be expected to occur) are equal to the coefficients in the 
 development of the 2n th power of any binomial. 
 
 The element errors, infinite in number, being infinitely 
 small in comparison with the actual errors of observation, 
 these latter may consequently be assumed to be continuous 
 from o to 2ns, the maximum error. If, therefore, J denotes 
 
 * Sec Todhunter's or Newcomb's Algebra.
 
 APPENDIX. 
 
 431 
 
 the error in which n -(- ;// -f~* 's and ;/ m e's occur, and 
 J-\-dJ denotes the consecutive error in which n-\-m -(- i 
 -f- 's and // m i e's occur, we have 
 
 J = 2ine 
 J + dJ = (2m -f 2)f 
 
 and therefore 
 
 J 
 
 Calling / the relative frequenc) 7 of the error J, ancl/-f- df 
 that of the consecutive error J -J- */J, we have 
 
 
 2n(2n 1} . . . (n -\-m-\-i) 
 
 n ;// 1 
 
 2(2- I) ... (//-f w + 2) 
 
 n m i i 
 Hence, by division, 
 
 /" n -\-m-\-\ 
 
 or 
 
 f//"_ 2W-J- I 
 
 ~ 
 
 Now, since dA is infinitely small in comparison with J, we 
 may write 
 
 df_ 2J 
 
 7 : ~ 
 
 Also, since ^/" is infinitely small in comparison with /, 2J is 
 with respect to ndA-\- J, and we may neglect J in the de- 
 nominator in comparison with ndJ. We have, therefore, 
 
 df_ 2J_ 
 7 : ~
 
 432 APPENDIX. 
 
 And since A is infinitely small in comparison with ndA, and 
 dA is infinitely small in comparison with J, it follows that n 
 must be an infinity of the second order. It is, therefore, of 
 
 a magnitude comparable with rr^i, and hence n(dA)* must be 
 a finite constant. Calling this constant ^, we have 
 
 *--2h*AdA 
 
 Integrating and denoting the value of/, when J o, by/ , 
 
 The errors being separated by the intervals dA, so that 
 o, dA, . . , A, A-\- dA, . . . are the errors in order of magni- 
 tude, we must, in order to make the system consistent with 
 the definition of probability', and therefore continuous, con- 
 sider not so much the relative frequency of the detached 
 errors as the relative frequency of the errors within certain 
 limits. 
 
 Now, by the definition of probability, the probability of 
 an error between the limits A and A-\-dA is represented by 
 a fraction whose numerator is the number of errors which 
 fall between A and A -j- dA, and denominator the total num- 
 ber of errors committed. If we denote this probability by 
 we may write 
 
 where c is a constant, -f being necessarily a constant for the 
 same series of observations. 
 
 201. The principle of least squares now follows readily. 
 
 Suppose that a series of observations has been made of a 
 function T of a certain known form, 
 
 T = f(X, F, . . .)
 
 APPENDIX. 433 
 
 in which the constants that enter are given by theory for 
 each observation, and X, Y, . . . are the unknowns to be 
 found. 
 
 Call MM . . . the observed values of T, and T lt T v . . . 
 the corresponding true values of the function, so that the 
 errors are found from 
 
 If we knew the true values of X, F, . . ., and therefore of 
 7\ we should have for the probabilities, before the first, 
 second, . . . observations are made, that the errors to be 
 expected lie between J, and J, -|- ^4 4 an d J^ -\-rfJ.,, . . . 
 respectively, are 
 
 where c lt c v . . . //,, // 2 , . . . are constants. 
 
 The probability </> of the simultaneous occurrence of all 
 of these errors, which are independent of each other, is 
 given by (Art. 5) 
 
 But as only the observed values M^M V . . . are known, the 
 true values of T, X, Y, . . . , J,, 4,, . . ., and therefore of </>, 
 are unknown. 
 
 If now arbitrary values of X, Y. . . . are assumed, 
 T, 4> J,, . . . will receive values, and therefore a value of 
 </> will be determined. Other assumed values of X, V, . . . 
 will give other values of <J>, which is therefore a function 
 of X, Y, . . . Of all possible values which are given to 
 X, F, . . . there must be some one set of values which is 
 to be chosen in preference to any others. The most prob- 
 able set is naturally the one that would be chosen. 
 
 Let, then, X t , F,, . . . denote the most probable values 
 of X, Y, . . . Substitute them in the function and let
 
 434 APPENDIX. 
 
 V F s , . . . denote the resulting values of T t , T y , . . . Then 
 we have no longer the true errors 7^ M iy T^ M^ . . ., 
 but the errors F, M v F 2 M . . ., which may be called 
 residual errors of observation, being the difference between 
 the most probable and the observed values. They are 
 denoted by the symbols v } , v v . . . 
 
 Now, assuming that c e~ /lV denotes the probability of 
 a residual between v and v-\-dv, the expression for <p be- 
 comes 
 
 and the most probable set of values of X, Y, . . . would be 
 that which corresponds to the maximum value of this ex- 
 pression, which can only happen when 
 
 [/^V] is a minimum 
 
 for the same set of values of X, F, . . . 
 
 This is the principle of least squares. 
 
 202. The following memoirs may be consulted for other 
 presentations of this proof: Young, Philosophical Transac- 
 tions, London, 1819; Hagen, Grundziige der WahrscJieinlicJi- 
 keitsrechmmg, Berlin, 1837; Wittstein, Lehrbiich der Difftr- 
 ential- und Integralrechnung, Hanover, 1849; Price, Infinitesi- 
 mal Calculus, vol. ii., Oxford, 1865 ; Tait, Edinburgh Trans- 
 actions, 1865 ; Kummell, Analyst, vol. iii., Des Moines, 1876; 
 Merriman, Journal Franklin Institute, Philadelphia, 1877.
 
 APPENDIX. 
 
 435 
 
 TABLE I. 
 
 Values of 0/ = 
 
 a 
 
 r 
 
 eoo 
 
 Diff. 
 
 (1 
 
 r 
 
 e(/) 
 
 Diff. 
 
 O,O 
 O, I 
 
 0,000 
 0,054 
 
 54 
 
 C 1 
 
 2,5 
 2,6 
 
 0,908 
 0,921 
 
 '3 
 
 IO 
 
 O,2 
 0,3 
 
 0,107 
 0,1 60 
 
 5J 
 
 53 
 
 r 1 
 
 2,7 
 2,8 
 
 0,931 
 0,941 
 
 10 
 
 (\ 
 
 0,4 
 
 0,213 
 
 5j 
 
 2,9 
 
 0,950 
 
 V 
 
 0,5 
 
 0,264 
 
 5 1 
 
 3.0 
 
 o,957 
 
 7 
 ft 
 
 0,6 
 
 0,314 
 
 5 
 
 3,1 
 
 0,963 
 
 u 
 
 o,7 
 
 0,363 
 
 49 
 48 
 
 3,2 
 
 0,969 
 
 c 
 
 0,8 
 
 0,411 
 
 -T w 
 
 3,3 
 
 o,974 
 
 J 
 
 o,9 
 
 0,456 
 
 45 
 
 3,4 
 
 0,978 
 
 4 
 
 1,0 
 
 0,500 
 
 44 
 
 3-5 
 
 0,982 
 
 4 
 
 1,1 
 
 0,542 
 
 4 2 
 
 3,6 
 
 0,985 
 
 
 1,2 
 
 0,582 
 
 40 
 
 3,7 
 
 0,987 
 
 
 i,3 
 
 0,619 
 
 37 
 16 
 
 3,8 
 
 o, 990 
 
 I 
 
 1,4 
 
 0,655 
 
 j u 
 
 3,9 
 
 0,991 
 
 i 
 
 1,5 
 
 o,68S 
 
 33 
 
 4-0 
 
 o,993 
 
 
 1,6 
 
 0,719 
 
 3 1 
 
 4.1 
 
 o,994 
 
 i 
 
 1,7 
 
 0,748 
 
 2 9 
 
 4,2 
 
 o,995 
 
 
 1,8 
 
 o,775 
 
 27 
 
 4,3 
 
 0,996 
 
 
 1-9 
 
 0,800 
 
 2 5 
 
 4,4 
 
 0,997 
 
 
 2,0 
 
 0,823 
 
 2 3 
 20 
 
 4-5 
 
 0,998 
 
 o 
 
 2,1 
 
 0,843 
 
 
 4,6 
 
 0,998 
 
 Q 
 
 2,2 
 
 0,862 
 
 19 
 
 4,7 
 
 0,998 
 
 1 , 
 
 2,3 
 
 0,879 
 
 i? 
 1 6 
 
 4,8 
 
 0,999 
 
 O 
 
 2,4 
 
 0,895 
 
 T *7 
 
 4.9 
 
 0,999 
 
 (> 
 
 2,5 
 
 0,908 
 
 1 j 
 
 5,o 
 
 0,999 
 
 1
 
 APPENDIX. 
 
 TABLE II. 
 
 Factors for Bessel's Probable-Error Formulas, 
 
 w 
 
 .6745 
 
 6745 
 
 n 
 
 6745 
 
 6745 
 
 
 V,-I 
 
 *X-,) 
 
 
 VH-I 
 
 Vn(n - i) 
 
 
 
 
 40 
 
 0.1080 
 
 0.0171 
 
 
 
 
 41 
 
 .1066 
 
 .0167 
 
 2 
 
 0.6745 
 
 0.4769 
 
 42 
 
 1053 
 
 .0163 
 
 3 
 
 .4769 
 
 2754 
 
 43 
 
 .1041 
 
 .0159 
 
 4 
 
 .3894 
 
 .1947 
 
 44 
 
 . 1029 
 
 0155 
 
 5 
 
 0.3372 
 
 o . i 508 
 
 45 
 
 o. 1017 
 
 0.0152 
 
 6 
 
 .3016 
 
 .1231 
 
 46 
 
 .1005 
 
 .0148 
 
 7 
 
 2754 
 
 .1041 
 
 47 
 
 .0994 
 
 .0145 
 
 8 
 
 -2549 
 
 .0901 
 
 48 
 
 .0984 
 
 .0142 
 
 9 
 
 .2385 
 
 0795 
 
 49 
 
 .0974 
 
 .0139 
 
 10 
 
 0.2248 
 
 0.0711 
 
 50 
 
 o . 0964 
 
 0.0136 
 
 ii 
 
 2133 
 
 .0643 
 
 51 
 
 0954 
 
 .0134 
 
 12 
 
 .2029 
 
 .0587 
 
 52 
 
 .0944 
 
 .0131 
 
 13 
 
 .1947 
 
 0540] 
 
 53 
 
 0935 
 
 .0128 
 
 14 
 
 .1871 
 
 .0500 
 
 54 
 
 .0926 
 
 .0126 
 
 15 
 
 o. 1803 
 
 0.0465 
 
 55 
 
 0.0918 
 
 0.0124 
 
 16 
 
 .1742 
 
 0435 
 
 56 
 
 .0909 
 
 .0122 
 
 17 
 
 .1686 
 
 .0409 
 
 57 
 
 .0901 
 
 .Ollg 
 
 18 
 
 .1636 
 
 .0386 
 
 58 
 
 .0893 
 
 .OII7 
 
 19 
 
 .1590 
 
 0365 
 
 59 
 
 .0886 
 
 .OII5 
 
 20 
 
 0.1547 
 
 o . 0346 
 
 60 
 
 0.0878 
 
 O.OII3 
 
 21 
 
 .1508 
 
 .0329 
 
 61 
 
 .0871 
 
 .OIII 
 
 22 
 
 .1472 
 
 .0314 
 
 62 
 
 .0864 
 
 .OIIO 
 
 23 
 
 .1438 
 
 .0300 
 
 63 
 
 .0857 
 
 .OIOS 
 
 24 
 
 .1406 
 
 .0287 
 
 64 
 
 .0850 
 
 .OIO6 
 
 25 
 
 0.1377 
 
 0.0275 
 
 65 
 
 0.0843 
 
 O.OIO5 
 
 26 
 
 1349 
 
 .0265 
 
 66 
 
 .0837 
 
 .OIO3 
 
 27 
 
 1323 
 
 0255 
 
 67 
 
 .0830 
 
 .OIOI 
 
 28 
 
 .1298 
 
 .0245 
 
 68 
 
 .0824 
 
 .0100 
 
 29 
 
 -1275 
 
 .0237 
 
 69 
 
 .0818 
 
 .0098 
 
 30 
 
 0.1252 
 
 0.0229 
 
 70 
 
 O.OSI2 
 
 0.0097 
 
 31 
 
 .1231 
 
 .0221 
 
 71 
 
 .0806 
 
 .0096 
 
 32 
 
 .1211 
 
 .0214 
 
 7 2 
 
 .0800 
 
 .0094 
 
 33 
 
 .1192 
 
 .O2O8 
 
 73 
 
 0795 
 
 .0093 
 
 34 
 
 "74 
 
 .O2OI 
 
 74 
 
 .0789 
 
 .0092 
 
 35 
 
 0.1157 
 
 0.0196 
 
 75 
 
 0.0784 
 
 0.0091 
 
 36 
 
 . 1140 
 
 .OlgO 
 
 80 
 
 .0759 
 
 .0085 
 
 37 
 
 .1124 
 
 .0185 
 
 85 
 
 .0736 
 
 .0080 
 
 38 
 
 .1109 
 
 .Ol8o 
 
 90 
 
 .0713 
 
 .0075 
 
 39 
 
 .1094 
 
 0175 
 
 TOO 
 
 .0678 
 
 .0068
 
 APPENDIX. 
 
 TABLE III. 
 
 Factors for Peters Probable-Error Formulas. 
 
 437 
 
 n 
 
 .8453 
 Vn(n I) 
 
 .8453 
 
 n 
 
 .8453 
 
 .8453 
 
 nV7^-~i 
 
 n ^n I 
 
 *'( - i) 
 
 
 
 
 40 
 
 0.0214 
 
 0.0034 
 
 
 
 
 4i 
 
 .0209 
 
 .0033 
 
 2 
 
 0.5978 
 
 0.4227 
 
 42 
 
 .0204 
 
 .0031 
 
 3 
 
 3451 
 
 .1993 
 
 43 
 
 .0199 
 
 .0030 
 
 4 
 
 .2440 
 
 .1220 
 
 44 
 
 .0194 
 
 .0029 
 
 5 
 
 0.1890 
 
 0.0845 
 
 45 
 
 0.0190 
 
 0.0028 
 
 6 
 
 1543 
 
 .0630 
 
 46 
 
 .0186 
 
 .0027 
 
 7 
 
 .1304 
 
 0493 
 
 47 
 
 .0182 
 
 .0027 
 
 8 
 
 1130 
 
 0399 
 
 48 
 
 .0178 
 
 .0026 
 
 9 
 
 .0996 
 
 .0332 
 
 49 
 
 .0174 
 
 .0025 
 
 10 
 
 0.0891 
 
 0.0282 
 
 50 
 
 0.0171 
 
 0.0024 
 
 ii 
 
 .0806 
 
 .0243 
 
 5i 
 
 .0167 
 
 .0023 
 
 12 
 
 .0736 
 
 .O2I2 
 
 52 
 
 .0164 
 
 .0023 
 
 13 
 
 .0677 
 
 .0188 
 
 53 
 
 .0161 
 
 .0022 
 
 14 
 
 .0627 
 
 .0167 
 
 54 
 
 .0158 
 
 .O022 
 
 15 
 
 0.0583 
 
 O.OI5I 
 
 55 
 
 0.0155 
 
 O.OO2I 
 
 16 
 
 .0546 
 
 .0136 
 
 56 
 
 .0152 
 
 .OO2O 
 
 17 
 
 0513 
 
 .0124 
 
 57 
 
 .0150 
 
 .OO2O 
 
 IS 
 
 .0483 
 
 .OII4 
 
 58 
 
 .0147 
 
 .0019 
 
 19 
 
 0457 
 
 .OIO5 
 
 59 
 
 .0145 
 
 .0019 
 
 20 
 
 0.0434 
 
 0.0097 
 
 60 
 
 0.0142 
 
 0.0018 
 
 21 
 
 .0412 
 
 .0090 
 
 61 
 
 .0140 
 
 .0018 
 
 22 
 
 0393 
 
 .0084 
 
 62 
 
 .0137 
 
 .OOI7 
 
 23 
 
 .0376 
 
 .0078 
 
 63 
 
 0135 
 
 .OO17 
 
 24 
 
 .0360 
 
 .0073 
 
 64 
 
 .0133 
 
 .OOI7 
 
 25 
 
 0.0345 
 
 0.0069 
 
 65 
 
 0.0131 
 
 O.OOI6 
 
 26 
 
 .0332 
 
 .0065 
 
 66 
 
 .0129 
 
 .OOl6 
 
 27 
 
 .0319 
 
 .006l 
 
 67 
 
 .0127 
 
 .O0l6 
 
 28 
 
 .0307 
 
 .0058 
 
 68 
 
 .0125 
 
 .OOI5 
 
 29 
 
 .0297 
 
 0055 
 
 69 
 
 .0123 
 
 .OOI5 
 
 30 
 
 0.0287 
 
 O.OO52 
 
 70 
 
 O.OI22 
 
 O.OOI5 
 
 31 
 
 .0277 
 
 .0050 
 
 7i 
 
 .0120 
 
 .OOI4 
 
 32 
 
 .0268 
 
 .0047 
 
 72 
 
 .0118 
 
 .0014 
 
 33 
 
 .0260 
 
 .0045 
 
 73 
 
 .0117 
 
 .0014 
 
 34 
 
 .0252 
 
 .0043 
 
 74 
 
 .0115 
 
 .0013 
 
 35 
 
 0.0245 
 
 0.0041 
 
 75 
 
 O.OII3 
 
 0.0013 
 
 36 
 
 .0238 
 
 .0040 
 
 80 
 
 .OIOO 
 
 .0012 
 
 37 
 
 .0232 
 
 .0038 
 
 85 
 
 .OIOO 
 
 .0011 
 
 38 
 
 .0225 
 
 .0037 
 
 9 
 
 0095 
 
 .OOIO 
 
 39 
 
 .0220 
 
 .0035 
 
 IOO 
 
 .0085 
 
 .oooS
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 
 DC 9 ^J.6C 
 
 * TMPSook i 
 
 Los Angeles 
 
 k is DUE on the last date stamped below. 
 
 IBJECT TO FINE IF M 
 
 EDUCATION 
 
 RECEIVED 
 
 SEP 2 2 1970 
 
 SP 291972 
 
 Form L9-116i-8,'62(D1237s8)444
 
 EDUCATION 
 LIBRARY 
 
 QA 
 
 275 
 W93t 
 
 UCU-ED/PSYCH Library 
 
 QA 275 W93t 
 
 L 005 647 1 72 5 
 
 A 001 086 189