jo-v S^ ^ -<.OFCAllF0fi>(A Law > ^ "' '.S'G[lf "^/JajAINH ]Wv :? o ^vlOSANCflfj-;^ c>^ <\SUIBRARYQr^ ^• ^WE■u^T '^(l/UJllVj jO '^/AiiJAIN.iJW^ ^ ^ ^^\^F•u^T vr ^6'Aavaaii-^^^ c -< ■'oajAi.Mi j^'^ .^ .VlOSANCEl5j^> o M-^^ m^' -n <_J O u- •]\m}'^ ij^ ^ >i ^f5! N^ILIBRARYQ^. '^OJITVDJO'^ F0% >- AMEUNIVERV/, lOSANCElfj> , , ^ J o ^•^ •'jaj.'Mixii J^^^ ^WEyNIVER% .vlOSA ;^^ .\\\EUNIVER% %K\^ ^^^t•LIBRARYQ<^ s 1 inr i '^ .^,OF-CAl!F0/?,t'. •;< "g 1 1 .OF-CA y(j AX^ \\^EUNIVFRV/>, ^NStllBRARY<9/ ^^£HBRARYq<- ^ l:< ^ ,^\\F UNIVERS-/^ ^VlOSANCFlfj> -f^^ f 'dO' ^^OFCALIF' >- Precision of the aritlimetic mean ....,,,..,. 3S V vl TABLE OF CONTENTS PAGE Bessel's formula 3° Distinction between residuals and errors 40 Peters' formula 4° Control of [7/=] 42 Approximate method of finding precision 42 The law of error tested by experience 44 Cautions as to tests of precision 45 Systematic errors 5 1 Observed values of different quality 52 The most probable value — The weighted mean 54 Combining weights 55 Reduction of observations to a common standard 56 Control of weighted mean 57 Precision of weighted mean 58 Observed values multiples of the unknown 60 Precision of a linear function of independently observed values 62 Miscellaneous examples 67 Weighting of observations 74 An approximate method 76 Weighting when constant error is present 77 Assignment of weight arbitrarily 82 Combination of good and inferior work 83 The weight a function of our knowledge 84 General remarks 87 Rejection of observations 87 CHAPTER IV AdjustDient of Indirect Observations Determination of the most probable values 93 Formation of the normal equations 96 Control of the formation 100 Forms of computing the normal equations loi With multiplication tables or a machine 10 1 With a table of logarithms 102 With a table of squares 103 Solution of the normal equations 105 The method of substitution 106 Controls of the solution 107 Forms of solution 109 Solution without logarithms 109 Solution with logarithms 112 The method of indirect elimination 114 Doolittle method of solution 114 Precision of the most probable (adjusted) values 121 TABLE OF CONTENTS vii PAGE First method of finding the weights 122 Special case of two and three unknowns 135 Modifications of the general method 127 Second method of finding the weights 129 The probable error of a single observation 132 Methods of computing [v-] i33 Precision of any function of the adjusted values (three methods) 137 Average value of the ratio of the weight of an observed value to its ad- justed value '-^3 Two special artifices '44 CHAPTER V AdJHstmetit of Condition Observatiofis General statement ^49 Direct solution— Method of independent unknowns 150 Indirect solution — Method of correlates 152 Precision of the adjusted values or of any function of them 158 The probable error of an observation of weight unity 158 Weight of the function '^^ Solution in two groups ^°3 Program of solution '7° Precision of the adjusted values or of any function of them 173 Solution by successive approximation ^77 CHAPTER VI Application to the Adjustment of a Triangulation — Method of Angles General statement '^° The method of independent angles 182 The local adjustment '^5 Number of local equations ^^^ The general adjustment '^° The angle equations '''9 Number of angle equations '9' The side equations '93 Position of pole '95 Reduction to the linear form '97 Check computation '9° Position of pole "°' Number of side equations ^°- Check of the total number of conditions 202 Manner of selecting the angle and side equations 203 Adjustment of a quadrilateral ^°° Solution by independent unknowns -°7 Precision of the adjusted values 208 viii TABLE OF CONTENTS PAGE Solution by correlates 210 Precision of the adjusted values 211 Solution in two groups 213^ Precision of the adjusted values 218 Solution by groups 223 The local adjustment 227 The general adjustment 227 Adjustment of a quadrilateral — Approximate method 228 Adjustment of a quadrilateral — Rigorous method 231 Adjustment of a single triangle 233 Adjustment of a central polygon 234 Approximate method of finding the precision 237 CHAPTER VII Application to the Adjustvie/it of a Triangulation — Method of Directio/is General statement 239 Local adjustment 243 Figure adjustment 243 Condition equations 244 Correlate equations 246 Normal equations 246 The best side equations 247 Length, azimuth, latitude and longitude condition equations 250 Length condition equations 252 Azimuth condition equations 253 Latitude and longitude condition equations 255 Breaking a net into sections 259 CHAPTER VIII Application to Base-Line Measurement and to Leveling Precision of base-line measurement 260 Precision of each bar length the same 263 Precision of each bar length not necessarily the same 265 Application to leveling 267 Method of indirect observations 268 Method of conditioned observations 268 Assignment of weights 270 CHAPTER IX Application to Selection of Methods of Observation General statement 272 TABLE OF CONTENTS ix PAGE Distinction between accidental, systematic, and constant errors 273 More accurate definition of probable error 274 , Detection of systematic and constant errors 276 Zenith telescope latitude observations 278 Telegraphic longitude observations 283 Other illustrations 286 APPENDIX Table I, Values of 6I (/) 291 Table II, Factors for Bessel's probable error formula 292 Table III, Factors for Peters' probable error formula 293 THE ADJUSTMENT OF OBSERVATIONS CHAPTER I INTRODUCTION The factors that enter into the measurement of a quantity are, the observer, the instrument employed, and the conditions under which the measurement is made. I. The Instrument. — If the measure of a quantity is deter- mined 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 descrip- tions 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 astronomers 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 instru- ment confined the attention of the observer to the point at issue, and helped the judgment in arriving at conclusions. 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 instru- mental determination was considered sufficient to give the value of the quantity measured. The next advance was in the questioning of the instrument and in showing that a result better on the whole than a single direct measurement could be found. This opened the way for better in.struments and better methods of observation. For example, Gascoigne's introduction of cross-hairs into the focus of 2 THE ADJUSTMENT OF OBSERVATIONS the telescope led to better graduated circles and to better methods of reading them, resulting finally in the reading micro- scopes now almost universally used. The culminating point was reached by Bessel, who, by his systematic and thorough investi- gation of instrumental 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 noti- cing that there are discrepancies in repeated measurements of a masfnitude with the same instrument, or in measures 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 determined, and the two measure- ments differed widely, we should suspect the chain to be in error, and proceed to examine it before further measuring. So, dis- crepancies 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 applying 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 instru- ment are seen to be in great measure capable of elimination. From geometrical considerations the observer can tell the arrangement of parts demanded by a perfect instrument. He can compute the errors that would be introduced by certain sup- posed irregularities in form and changes of condition. The instrument -maker cannot, it is true, fulfill the conditions neces- sary for a perfect instrument, but he has been gradually approaching them more and more closely. It is to be remem- bered that, even if an 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 instru- INTRODUCTION 3 ment was made adjustable in most of its parts, so that the relative positions of the parts are mider the control of the observer. Tliis is getting to be more and more the case with the better class of instruments. 2. Not only is error diminished by the improved construction 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 graduation 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 circle 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 ordinary leveling, 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 ought to work perfectly is often a great disappointment in practice. A striking example is the compensating base apparatus which has been abandoned on all the leading surveys. The mechanical and thermal difficulties have proved to be insurmountable, and the compensating bars have been replaced by others of much simi)lcr construction. 4 THE ADJUSTMENT OF OBSERVATIONS Similarly, the repeating theodolite has fallen far short of the expectations of its first advocates, who hoped that with it the errors of measurement of an angle could be reduced almost indefinitely. The mechanical difficulties have proved insur- mountable, and the repeating theodolite is now known to be capable of no greater accuracy than the direction instrument. Such is the perfection at present attained in the construction of mathematical instruments, and the skill with which they can be manipulated, that comparatively little trouble in making ob- servations arises from the instrument itself. 3. External Conditions. — The great obstacles to accurate work arise from the influence of external conditions — condi- tions wholly beyond the observer's or instrument-maker's con- trol, and whose effect can, in general, neither be satisfactorily computed nor certainly eliminated by the method of observa- tion. 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 is reading horizontal angles on a high tower a strong wind arises, it may be necessary for him to stop work. If the air commences to "boil " with extreme violence, it may also be best to stop work. If the sun shines on one side of his instru- ment, 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 vari- ations 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 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 elim- inated, in part at least, by the method of observation. In the measurement of horizontal angles where the instrument is placed on a high tower, the influence of the sun causes the INTRODUCTION 5 center post or tripod of the station to twist in one direction during the day. When this influence is removed at night, the twist is in tlie opposite direction. Assuming 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 observ^- ing for time with a sextant, the effect of refraction is often eliminated when the highest degree of accuracy is required, by taking two sets of observations of the sun at about the same altitude, one before and the other after noon. 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 re- fraction are apt to be very marked. As we have no means of eliminating 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 theoreti- cal considerations assuming a certain law of operation. 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 refraction is so small that the error of the computed value is not likely to exceed other errors which enter into the work. 4. The Observer. — Lastly, we come to the observer himself as the third element in making an observation. Like the ex- ternal conditions, he is a variable factor ; all new observers cer- tainly are. The observer, having put his instrument in adjustment and satisfied himself that the external conditions are favorable, should not begin work if he is seeking the highest degree of accuracy unless he considers that he himself is in his normal condition. If he is not in this condition, he introduces an 6 THE ADJUSTMENT OF OBSERVATIONS unknown disturbing element unnecessarily. He is also more liable to make mistakes in his readings 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-determina- tions, for instance, nothing is gained by observing from dark until daylight. The observer is supposed to have no bias. A good observer, having taken all possible precautions with the adjustments 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 observ^er's experience, the more do his habits of observation become fixed, and the more mechanical does he become in certain parts of his work. But his judgment may be constantly at fault. Thus, with the astronomical transit he may estimate the time of a star crossing a wire in the focus of the telescope invariably too soon or invariably too late, according to the nature of his temperament. If he is doing comparison work involving micrometer bisections, he may consider the graduation mark sighted at to be exactly between the center wires of the microscope when it is constantly on the same side of the center. This fixed peculiarity, which none but experi- enced observers have, is known as their personal error, ox per- sonal equation. In combining one observer's results with those of another observer, we must either find by special experiment the differ- ence of their personal errors and apply it as a correction to the final result, or else eliminate it by the method of observation. 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. It is always safer to eliminate the correction by the method INTRODUCTION 7 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 con- stant, yet some apparently trifling change of conditions, such, for example, as illuminating the wires of the instrument differ- ently, 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 expected to furnish as good a determination of the value of a quantity as might be obtained by varying the form of making the observations or increasing the number of observers. 5. When all known corrections for instrument, for external conditions, and for peculiarities of the observer, have been applied to a direct measure, have we obtained a correct value of the quantity measured ? That we cannot say. If the observa- tion 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 preceding pages. There may have been no change in the conditions of sufficient importance to attract the observer's attention when making the observations, but he may have handled his in- strument differently, turned certain screws with a more or less delicate touch, and the external conditions may have been different. What the real disturbing causes were, he has no means of knowing fully. If he had, he could correct for them, and so bring the measures into accordance. Infinite knowledge alone could do this. With our limited powers, we must e.xpect a residuum of error in our best executed measures, and, instead of certainty in our results, look only for probability. The discrepancies from the true value due to these un- explained disturbing causes we call errors. These errors are accidental, being wholly beyond all our efforts to control. They are as likely to be in excess as in defect. If we can ferret out 8 THE ADJUSTMENT OP OBSERVATIONS the law of their operation, they cease to be classed as errors, and become corrections. A very troublesome source of discrepancies in measured values arises from blunders made by the observer in reading his instrument or in recordmg his readings. Blunders from imper- fect hearing, from transposition of figures and from writing one figure when another is intended, from mistaking one figure on a graduated scale for another, as 7 for 9, 3 for 8, etc., are not un- common, — nor are mistakes of level reading of 5 divisions, of estimation of time of /', and the like. Carelessness may produce the same effect as a blunder in reading or in recording. Thus, handling a striding level roughly, or bringing a heated lamp too near it, may affect a result very seriously. Having, then, taken all possible precautions in making the observations, and applied all possible 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 quantity sought be found from these different observed values ? CHAPTER II THE LAW OF ERROR TJie Arithmetic Mean 6. If a quantity x is to be determined by measurement, and M is a measured value of x, then, if the observation were per- fect, we should have X — M == o. 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 observa- tions are not perfect — that is, that they are affected with cer- tain errors — we should rather write X — i/2 = ^2 X — .!/„ = Ar., (0 where M^, M,„ . . . M„ are the observed values, and A,, A^, . . . A„ are the errors of the observations. We have here ;/ equations and ;/ + i unknowns. What principle shall we call to our aid to solve these equations and so find X, Aj, A„, . . . A„? In answering this question, we shall follow the order of natural development of the subject, which, in the main, is also the order of its historical develop- ment. The value sought must be some function of the olDserved values, and fall between the largest and smallest of them. If the observed values are arranged according to then- magni- tudes, they will be found to cluster around a central value. 9 to THE ADJUSTMENT OP OBSERVATIONS On first thought, the value that would be chosen as the value of X 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 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 objec- tionable, because it gives the preference to a single one of the observed values ; while if these values are supposed to be equally worthy 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 x to be a symmetrical ftmction x^ of the observed values. The simplest symmetrical function of the observed values that can be chosen as the form for x^ is their arithmetic mean — that is, (J/i + J/2 + J/3 • • • M,:) Xrt — n n where 2 is the ordinary algebraic symbol of summation, or n in the system of notation introduced by Gauss. The principle may be stated as follows : If %ve Jiave n observed values of an iinknozvn, all equally good so far as we know, the -most plausible value of tJic unknoivn {best value on the whole) is the arithmetic mean of the observed values. THE LAW OF ERROR 1 1 It may happen that the vahies AI^, M^, . . . M„ are of such nature that some other symmetrical function than the arithmetic mean will satisfy the observation equations better than will the arithmetic mean. That the arithmetic mean is OH the whole the best form of the function may be confirmed by a comparison of results following from this hypothesis with the records of experience. 7. By adding equations (i), Art. 6, and taking the mean, we have, . = m + M = ,v. + M. n n n The last term of this equation will become very small if, n being very large, the sum [A] of the errors remains small. Now, if, after making one observation and before making another, we re- adjust 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 influences 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 blunders in reading the instrument 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 // is a very large number, consider ^ A to be an infinitesimal with re- spect to X. We may, therefore, in this case put that is, xvhen the number of observed values is very great, the aritJimetic mean is the true value. 12 THE ADJUSTMENT OP OBSERVATIONS 8. From the principle of the arithmetic mean, two important inferences may be derived. For, taking the arithmetic mean, ,i-y, of n observed values of an unknown as the most plausible value of that unknown, we may write our observation equations in the form, Xo — Ml = V{ :Vo — M._ = V.^ (0 Xo- M, = 1\. v^ are called the residual errors of obser- where v^, v,^, . . vation, or simply the residuals. (a) By addition, nx, - [M] = [vl and .•. [V] = 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 71 is infinitely great and x is the true value, and a series in which ;/ is finite and the arithmetic mean .i;, is taken as the best value attainable. For in the first case the sum of the ei'j'ors, A, is zero, and in the second the sum of the residuals, V, is zero. {b) Let X be any assumed value of the unknown other than the arithmetic mean, and put X - Ml = V,' ' X- IU = v.: (3) X - M„ = v: From equations (i) and (3), by squaring and adding, [^'^] = nx,x, - 2 X, [M] + [M'], [(vy]= nXX - 2 X [M] + [M^]. Hence by a simple reduction, THE LAW OF ERROR 13 [ivy] = [v'] + " (-V - WJ , Now, [X— ^ — ] , being a complete square, is always positive. .-. [{vj] > [v~] ; that is, t/ie sum of the squares of the residuals v, found by tak- ing the arithmetic mean, is a minimum. Hence the name Method of Least Squares, which was first given by Legendre. The Law of Error of Obsen'ed Quantities. 9. When several independent measures of the same quantity, all equally good, have been made, it must be granted 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 frequently than large ones, and that there is a limit to the magnitude of the error to be expected. If, therefore, a denotes this limit or maximum error, we must con- sider 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 the probability that an error lies between o and A be denoted by /(A), the probability q of an error between A and A + ^A is given by q =/(A + d^) -/(A) =/'(A) d^ = (A) (iA, suppose ; (i) q may be taken to be the probability of the occurrence of the error A, since ^A is small. The function <^ (A) is called the law of distribution of error, or simply the laiu of error. The probability that an error falls between any assigned limits b and a is the sum of the probabilities (/> (A) <'/A extend- 14 THE ADJUSTMENT OF OBSERVATIONS ing from b to a, and is expressed in the ordinary notation of the integral calculus by (A) (/A. (2) «y a Hence it follows that the probability that an error does not ex- ceed the value a is "^^ ' (A.) d_K ° dx ^ (Aj) dx ^A, dx ^ " ' (A„) dx _ 4>' (A.) 4>' (A3) I <^^(A.) ^ ,. A,<^(Aj) 1 ^(^(A^) ^ A„<^(A„) since from Eq. i, Art. 6, d\ _d^ _ _ ^ _ dx dx dx But from the principle of the arithmetic mean, when the number of observed values is very great, A^ + A. + . . . + A,. = o. (6) Also, since equations (5) and (6) must be simultaneously satisfied by the same value of the unknown, we necessarily have, THE LAW OF ERROR 15 Hence for any arbitrary value A, A <^ (A) - '"• Clearing of fractions and integrating, <{> (A) = ce^^^\ where e is the base of the system of natural logarithms and c is a constant. Again, since Q is to be a maximum, -—^ or — ^^ — - must be negative. Now, /C' (A^ + A. + . . . ) A: _ = C"i dQogO) dx (A) it is evident that the probability of an error A will be the larger, the smaller // is, and vice versa. Hence, Ji is a test of the quality of observations of different series. It was named by Gauss tJic measure of precision. In practice, it is more convenient to compare the precision of different series of observations by other methods. II. Proof of the Law of Error on Hagen's (Young's) Hypothesis. — Various proofs of the law of error have been derived. Each is open to some theoretical objection. The following proof of Hagen's hypothesis starts with a clear and definite statement of the assumed nature of an accidental error, namely, that it is the algebraic sum of an infinite number of independent infinitesimal element errors, each of which is as likely to be positive as negative. The law of error being derived directly from this hypothesis, it is clear that if an ob- server wishes to put the errors of observation into this acci- dental class, and, therefore, to make them easy to eliminate, he must use such instruments and methods as will make the errors conform as closely as possible to this definition of an accidental error. It is because Hagen's proof thus indicates clearly to the observer the standard toward which he should struggle, that it is here given in addition to the complete and independent proof in the preceding article. An accidental error of observation does not result from a THE LAW OF ERROR 17 single cause. Thus, in reading an angle with a theodolite, the error in the value found is the result of imperfect adjustment of the instrument, of various atmospheric changes, of want of pre- cision in the observer's method of handling the instrument, etc. Each of these influences may be taken as the result of numer- ous 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 independent 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 con- clude that an error of observation may be assumed to be the algebraic sum of a very great number of independent infinitesi- mal element errors e, all equal in magnitude, but as likely to be positive as negative. Let the number of these element errors be denoted by 2 «, 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 2 ne results, and this can occur in but one way ; if all but one are +, the error {2 n — 2) e results, and this can occur in 2 ;/ ways ; and generally, if ;/ -f ;;/ are +, and Ji — m are — , the error 2 me results, and this can occur m ^^ !^ ways.* Hence 1 2 • • ■ (« — m) the numbers expressing the relative frequency 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 2 ;/th power of any binomial. * Sec Todluinter's or Ntwcomb's Algebra. i8 THE ADJUSTMENT OF OBSERVATIONS 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 2 ne, the maximum error. If, therefore, A denotes the error in which n + m + e's and // — vi — e's occur, and A + ^-/A denotes the consecutive error in which n -\- 111 + i -\- e's and n — m— i — e's occur, we have A = 2 nif. A 4- (/A = (2 m + 2) e, and therefore, A = mc?A. Calling / the relative frequency of the error A, and / + df that of the consecutive error A + ^/A, we have 2 w (2 w — i) • • • (w + m + i) / = n — m\ 2 w (2 w — i) • . • (w + w + 2) n — m — \\ Hence ^by division, f n — m n + m + I ' f\^ df f 2 m + I or w + W + I 2 A + (/A W(/A + A + (/A Now, since ^dL is infinitely small in comparison with A, we may write df f 2A ' W(f A + A Also, since df is infinitely small in comparison with /, 2 A is with respect to udA + A, and we may neglect A in the denom- inator in comparison with ndA. We have, therefore, df^ 2 A THE LAW OF ERROR tg And since A is infinitely small in comparison with ;/c/A, and ^A is infinitely small in comparison with A, it follows that ;/ must be an infinity of the second order. It is, therefore, of a magnitude comparable with ., , and hence, u (^/A)- must be («A)' a finite constant. Calling this constant — , we have Integrating and denoting the value of /, when A = o, by _/" , The errors being separated by the intervals dA, so that o, dA, . . . A, A + rt^A . . . are the errors in order of magnitude, we must, in order to make the system consistent with the defi- nition of probability, and therefore continuous, consider not so much the relative frequency of the detached errors as the rela- tive frequency of the errors within certain limits. Now, by the definition of probability, the probability of an error between the limits A and A + ^A is represented by a fraction whose numerator is the number of errors which fall between A and A + dA, and denominator the total number of errors committed. If we denote this probability by (p (A) we may write where ^ is a constant, 2/ being necessarily a constant for the same series of observations. 12. The Principle of Least Squares. — Let us return to 9, Eq. 4. If the observed values are of the same quality throughout, // is constant and the product becomes ^V~//f-^*l This product is evidently a maximum when [A"] is a minimum ; that is, if we assume that each of a very large number of observed values of a 20 THE ADJUSTMENT OF OBSERVATIONS quantity is of the same quality, the 7nost probable 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 quahty, h is differ- ent for the different observations, and the most probable value of the unknown would be found from the maximum value of ^-[/(iA2]. ^Ys2l^ is, from the minimum value of [//A'-]. Thus, if each of a large number of observed values of a quantity is of dijfcrcnt qjiality, the most probable value of the quantity is found by vinltiplying each error of observation by its h, and making the sum of the squares of the products a mijtimum. The Laiu of Error of a Linear Function of Independently Observed Quantities. 13. We have found the law of error in the 'case of a quantity 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^, + fl„M„ For simplicity in writing, consider two observed quantities, Jfj, M^, only, and let //, , //., be their measures of precision. The probability of the simultaneous occurrence of the errors A in tJ/j and A., in M^ is ^-e-''rV-V^.VA,^A,. (i) TT Now, an error A^ in M^ and an error A^ in J/^ Produce an error in F, according to the relation A = fljAj + a.A,, (2) and this relation can always be satisfied by combining any value . . M, ; that is. when F = a^M^ + 02^2 + here « , (I., . are all constants, TPIE LAW OF ERROR 21 of A„ with all values of A^ ranging from — 00 to -f- 00. The probability, therefore, of an error A in F may be written, <^ (A) r/A = ^' d A., C" e- ^^'^^'- ^^'^^' rfAj. But from (2), and since A^ is independent of A^, JA = a^d^y Hence, which is of the form Ae-'''^VA. Vtt That is, //^^ /^^c of error of the function F is the same as that of the independently measured quantities Af^, M.^ The precision of the function F is found from /,2 = 'hJh ; that is, from i- = ^ 4- ^ = F— 1 This theorem is one of the most important in the method of least squares, and will be often referred to. Ex. — To find the precision of the arithmetic mean of n equally well-observed values of a quantity : We have, F = ' {M, + M.,+ ■ ■ ■ +.U„). n Let //„ = precision of the arithmetic mean ; h = precision of each observed value. 22 THE ADJUSTMENT OP OBSERVATIONS or ho = y/Tih. That is, t/ie precision of the arithmetic mean of n observations is yfti times that of a single observation. 14. Comparison of the Accuracy of Different Series of Observations. — We have seen that the measure of precision h affords a test of the relative accuracy of different series of ob- servations. This test was suggested by the form of the law of error, and is naturally the first that would be chosen for that purpose. The Mean-Square Error. — On account of the inconven- ience of computing h, Gauss suggested the m.ean-sqnare or "to be feared " error as a test of quality. This is defined as a quantity \x whose square is equal to the mean or average of the squares of the individual errors, or when ;/ is very large. ^ n n To find the relation between Ji and /x. The probability of the occurrence of an error A, that is, of an error between A and A + ^A, is ^ (A) di^. The number of errors in the series being ;/, the sum of the squares of the errors in the same interval will be nt^^ (A) ^/A, and the sum of the squares of the errors between the limits of error + a and — a will be for a continuous series. X+rt A2 , n being very large, we have 15. The Probable Error. — The most common m.ethod in use in this country of determining the relative precision of different THE LAW OF ERROR 23 series of observations is by comparing errors which occupy the same relative 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 convenience, the ones chosen. Let the errors in a series, arranged in order of magnitude, be ± 2 a, ■ • • ± r, • • ■ o, 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 oi probable error. If, therefore, n is the total number of errors, the number lying between -f- r and — r is «/2, and the number outside these limits is also «/2. In other words, the probability that the error of a single observation in any system will fall between the limits -f r and — r is 1/2, and the probability that it will fall outside these limits is also 1/2. We have, therefore, from which to find r. If we put //A = t, and the value t = p corresponds to A = r, then -7= f e ' at = -' sJttJo 2 Expanding the integral in a series (see Art, 22), we shall find that approximately the resulting equation is satisfied by p = 0.47694. Now, since Jir = p = 0.47694 and ///x ^2 = i, it follows that r = 0.6745 IX — - P- roughly. 3 e-"=^ 24 THE ADJUSTMENT OF OBSERVATIONS 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 seriesis of considerable 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 commit- ting 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 Vt ' V-n- or I : e-(0-4'694)2^ or I : 0.8. The idea of probable error is due to Bessel {Berlin. Astron. JaJirb., 18 1 8). 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. 16. The Average Error. — It naturally occurs as a third test of the precision 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 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 t], so that [A '' = ^' where [A is the arithmetic sum of the errors. THE LAW OF ERROR 25 An expression for 1] in terms of the mean-square error fx may be found as follows. The number of errors between A and A + ^/A is n (A) (/A, and the sum of the positive errors in the series is Jr»eo Acf> (A) d\. The sum of the negative errors being the same, the sum of all the errors is X+QO A<^ (A) r/A. A^ (A) (/A Af-'-'-^VA = .v 2 h VTr ' IT the relation required. The average error may, as stated above, be directly used as a test of the relative accuracy of different series of observations. 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 [A than [A']. 17. The formulas for fi and r computed in this way are as follows. From the last equation preceding = 1.2533 „-' = 0.8453 and from Art. 14, r = 0.6745 fi n The relations connecting /x, r, and ?; are easily remembered in the following form : THE ADJUSTMENT OP OBSERVATIONS fL 'V2 = - = Vttt;. 9 These relations may also be conveniently arranged in tabular form : M r ■<] n. = r = r) = 1 .0000 0.6745 0.7979 1.4826 1 .0000 I. 1829 1-2533 0.8453 1 .0000 The p. e. which is to be used as a measure of accuracy may be computed from the sum of the squares of the errors and also from the sum of the errors without regard to sign. The question then arises, which of the two methods will give the better re- sult } It may be stated without here setting forth the proof, that the value of the p. e. computed from the squares is some- what more trustworthy than it is when derived from the first powers of the errors. 18. 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 International 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 obtained should always be referred to the probable error computed from Gauss' formula, and not to the mean error." {Prods Verbanxy 1879, p. 'j'].) In the United States, in the Naval Observatory, the Coast Survey, the Engineer Corps, and the principal observatories, the p. e. is used altogether. So, too, in Great Britain, in the Green- wich Observatory, the Ordnance Survey, etc. In the G. T. THE LAW OF ERROR 27 Survey of India the m. s. e. is used, for the reason given by Gauss above. Among German geodeticians and astronomers the m. s. e. is very generally employed. The p. e. has a definite meaning, namely, that the chances are even for and against a given error being greater or less than the corresponding p. e. This frequently furnishes a convenient test as to whether the errors of a given series of observations are dis- tributed according to the assumed law of error. The p. e. will be used in the text of this book nearly always. The Probability Czirve. 19. The principles laid down in the preceding articles may be illustrated geometrically as follows : We have seen that in a series of observations the probability of an error A, that is, that an error will lie between the values A and A + <;/A, is given by the expression (Art. 13) s-f^'^'dX Now, if O is the origin of coordinates, and a series of errors, A, 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 be- tween A and A -f dA, will be repre- sented by the rectangle whose height is k v: Fig. I. ^-a-'a2 and width TT d\ 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 A all values from -f cx) to — 00 and drawing the -^^ 28 THE ADJUSTMENT OF OBSERVATIONS corresponding ordinates, we shall have a continuous curve whose equation may be written y — r This curve is called the probability curve. 20. To Trace the Form of the Curve. — Since A enters to the second power, and j to the first power, the curve is symmet- rical with respect to the axis of j, and the form of the equation shows that it lies altogether on one side of the axis of A. Also, when A = o, — = o ; that is, the tangent at the vertex is par- <:/A allel to the axis of x. K's> A increases from o the values of y continually decrease. When A = ± CO , then y =1 o and dy/di^ — o, showing that the axis of A is an asymptote, /\gain, since dA^ Vtt there is a point of inflection when A = y-^ = /X, " ■V2 and the m. s. e. is therefore the abscissa of the point of inflection. Also, when A = o, d'^y/d^- is negative, showing that the ordinate at the vertex is the maximum ordinate. Hence the curve is of the form indicated in Fig. i, OA representing the maximum ordinate and OM the m. s. e. The values of //, that is, of i//a V2, being different for differ- ent series of observations, the form of the curve will change foi each series, and the curve may be plotted to scale from values of y corresponding to assumed values of A. In plotting the curve, since the maximum ordinate at the ver- tex h Vtt enters as a factor into the values of each of the other ordinates, its value may be arbitrarily assumed. We may there- THE LAW OF ERROR 29 fore 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 principles already laid down in deducing the law of error, and 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 A 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 posi- tive and negative errors are equally probable, is indicated by its symmetrical form with respect to the axis of j. 21. The area of the curve of probability is the sum of the rectangles — - e~'^'-^'dA, for values of A extending from + 00 to Vtt — cc , and may be denoted by unity. If, then, we represent by their area 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 will represent the number of errors to be expected in the series between the values of those limits. Thus, if O is the origm, 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 of positive errors less than OP, the area PRR'P' the number that lie between OP and OR. Fig. 2. 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 ^QQ'-P' would represent the number of errors numerically k-ss than the probable error. 30 THE ADJUSTMENT OF OBSERVATIONS The average error r] is evidently represented by the abscissa of the center of gravity of either the positive or negative half of the probability curve. T/ie Laxv of Error Applied to an Actual Series of Observations. 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. 22. Effect of Extending the Limits of Error to ± go. — The expression —L ^~''''^'<^A gives the value of the probability Vtt of an error between A and A 4- dt^ in an ideal series of observa- tions where the values are continuous between limits infinitely great. In all actual series the possible error is included within certain finite limits, and the probability of the occurrence of an error beyond those limits is zero. Practically, however, the ex- tension of the limits of error to ± co can make no appreciable difference in either case, as the function (A) decreases so rapidly that we can regard it as infinitesimal for large values of A ; in other words, the greater number of errors is in the neigh- borhood 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 de- veloped from another point of view. The probability of the occurrence of an error not greater than <^ in a series of observations is, since the error must lie between -f a and — a, i f"'\-f''^'d^ (see Art. 13). This is the same as the area of the curve of probability be- tween the limits -^a and —a, the total area being unity (Art. 13). Change the variable by placing /lA = t. The expression then becomes, THE LAW OF ERROR 31 \ P ha r, p ha -^ / e-'\U or ~ / e-'V/, and is usually denoted by the symbol (/). If, as in Art. 14, the value / = /o corresponds to A = ;-, we have finally, by eliminating /i, a ®0) = ^ ; r"di. '« = T-/' The value of this integral cannot be expressed exactly in a finite form, but may be found approximately as follows : Expanding e~*' in a series, and integrating each term sepa- rately, we have, Jo Jo V I 1-2 / U 3 1-2 5 This series is convergent for all values of /, but the conver- gence is only rapid enough for small values of t. For large values of t it is better to proceed as follows : Integrating by parts, J J 2t I re-" , 2 J t' — e * = - -.e-"+ T^^-'^+ ll/'-F ^^• 2t Hence T V^V^ = ll! ^ i _ JL + J^ _ 13.:^ + . . .\ Ja 2a\' 2f-^{2fy {2ff^ ^ e-"dt = / e-"dt - / e^^dt Jo J t 2 Jt •. finally, X ' Vtt c '-'J_ I ^ I. ,3 _ I-3-5 j_ . _ ^ 2 2 / ( 2 /- (2 /-)- (2 /-/ ) 32 THE ADJUSTMENT OF OBSERVATIONS 23. Approximate values of the expression / e ^\it may be computed from the above formulas for any numerical value of t. In Table I (Art. 213) will be found the values of the func- tion (/) corresponding to the argument ajr. The reason for arranging the table in this way is that it is more convenient to compute - than o , where r r p — 0.47696. The probability that an error exceeds a certain error a is I _ (M) (/), 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 2 r is 0.177, than 3 r is 0.043, than 4 r is 0.007, than 5 /• is 0.00 1, than 6 r is 0.000 1. Hence, in 10,000 observations we should expect only one error greater than 6 /-, in 1000 only one greater than 5 r, in 100 only one greater than 4;-, and in 25 only one greater than 3 r. 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 observations in any case is usually under 100, we are eminently safe in taking the maximum error at about 5 r or 3 /x. Experience indicates that in general the curve representing the true law of error for a given series of observations departs but little from the Gaussian curve. The actual curve for a given series may quickly be compared with the Gaussian curve by the use of Table I. The degree of departure from the Gaussian curve necessarily indicates the extent to which the facts differ from the assumption of an infinite number of sources of infini- tesimal errors. 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 earl- ier differential comparisons of the compensating base-apparatus THE LAW OF ERROR 33 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 compen- sating. There was instead 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 discovered, which gave a means of applying a further correction. The work done later, after taking account of this new correction, showed nothing unusual. 24. General Conclusion. — On the whole, though we cannot say that the formula -L c'-'''^^ will truly represent 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 the corrections, and have applied them to the measured values, the residuum of error may fairly be sup- posed 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. 12 may therefore be applied to a limited series and be written : If t/ic observed vahws of a quan- tity are of different quality, the most probable valne is found by dividing each residual error by tJie probable error 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 ob- served values. Classification of Observations 25. For purposes of reduction, observations may be divided into two classes — those which arc independent, being subject to no conditions except those fixed by the observations themselves; and those which are subject to certain conditions out.side of the 34 THE ADJUSTMENT OF OBSERVATIONS observations, as well as to the conditions fixed by the observa- tions. In the former class, before 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 conditions. For example, suppose that at a station O the angles AOB, AOC, are measured. If the measures of each angle are inde- pendent of those of the other, the angles are found directly. The angle BOC co\x\6. be determined from the relation AOC=AOB + BOC. The unknown in this case may be said to be observed indirectly, 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, AOC, then these angles are no longer independent, but are sub- ject to the condition that when adjusted AOC = AOB + BOC, and no set of values can be assumed as possible which does not exactly satisfy this condition. 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 (i) Direct observations of one unknown. (2) Indirect observations of several independent unknowns. (3) Conditioned observations. CHAPTER III 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 quan- tity which has been directly observed. We shall consider two cases — first, when all of the observed values are of equal qual- ity, and, next, when they are not all of equal quality. There are in all cases two quantities to be found — first the most probable value of the unknown itself, and next the pre- cision of this value. A. Observed Values of Equal Quality. 26. The Most Probable Value; the ArithmeticMean. — V/e have seen that in a series of directly observed values M^, J/,, . . . J/„, of equal quality, the most probal^le value x^^ of the observed quantity is found by taking the arithmetic mean of these values ; that is, x,= \}l^ln. (i) It has also been shown that the same result will follow by making the sum of the squares of the residual errors a mini- mum. (Art. 12.) As the observed values Tl/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 ,t',, must be. Let X' denote tliis approxi- mate value of .i',j, which may conveniently be taken some round number. Subtract each of the observed values J/,, M.,, . . . Jlf,, 35 36 . THE ADJUSTMENT OP OBSERVATIONS in succession from . respectively. Then, in succession from X' and call the differences l^, /^, . . . 4 X' -M^ = k, X' -M^ = h_, X' - IK = L By addition. But nX' - [M] = [/". Xq — n • • ^0 — ^ "1 n (2) = A"+ x\ suppose. (3) Hence all that we have to do is to take the mean x' of the small quantities, /j, Z^, . . . 4» ^^^cl add the assumed value X' to the result. 27. Control of the Arithmetic Mean. — In least-square com- putations, 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 intervals. 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 or — , or vice versa. Hence, even if the computation is made in duplicate, it is advisable to carry through an independent check which may be referred 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 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 num- ber //, the sums of the positive and negative residuals would not OBSERVATIOXS OF ONE UNKNOWN QUANTITY 37 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 a-mount, the positive residuals would each be too large, and the negative residuals too small, by that amount. Hence the discrepancy to be expected would be // times the amount that the approximate quotient taken as the mean differed from the exact quotient. Ex. — In the telegraphic determination of the difference of longitude be- tween St. Paul and Duluth, Minn., June 15, 1S71, the following were the corrections found for chronometer Bond No. 176 at 15 h. 51 m. sidereal time from the observations of 21 time stars. {Report Chief of Engineers, C/.S A., 1871.) M V v' s s -8.78 + 0.04 0.0016 .76 + .02 4 .85 + .11 121 .78 + .04 s 16 •51 — 0.23 529 .64 — .10 100 .68 - .06 36 •63 — .11 121 .58 - .16 256 .80 + .06 36 •75 + .01 I .78 + .04 16 .96 + .22 4S4 .64 — .10 ICO .65 - .09 81 ■83 + .09 81 .70 - 04 16 .64 — O.IO 100 •79 + .05 25 .90 + .16 256 -8.93 + 0.19 0.0361 Mean, — 8.74 + 1-03 [v = 2.02 -0.99 [7r] = .02756 Taking the observations as of equal precision, we find tlie 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 tiu- most probable value according to the relation, 41fim4 38 THE ADJUSTMENT OF OBSERVATIONS X' - M=v. They are written in two columns for convenience in applying the check, {vl = o. The true mean may be derived by subtracting the mean of the residuals from the approximate mean. 28- It was desired in this case to secure a mean which is correct in the second decimal place, or, in other words, is not in error by more than 5 in the third decimal place. This has ob- viously been secured when the difference between the sums of the + and — residuals is less than (0.005) (21) = 0.105. In general, a mean is correct to a given decimal place when the difference betzvcen the sums of the -f- ajid — residuals is less, expressed in units of that decimal place, than one-Jialf of the number of quantities of ivhicJi the mean is sought. If the difference of the sums of the + and — residuals be found to be too great, an error has been made either in deriv- ing and adding residuals, or in deriving the mean. If it is believed to be the latter, or if the value used in deriving the residuals was only an assumed approximate value, the true mean may be quickly derived by subtracting the mean value of the residuals from the assumed mean used in finding the residuals. 29. Precision of the Arithmetic Mean. — The degree of confidence to be placed in the most probable value of the un- known is shown by its probable error. y/ (a) BesseV s Formula. If we knew the true value x of the unknown, and conse- quently the true errors A^, A^ . . ., we should have, as in Art. 20, for the m. s. e. of an observation, 2 _ [^T ~^* But we have only the most probable value x^ and the residuals i\, v.„ . . . v,^ instead of the true values x, A^, A^ . . . A„. Now, Xq — v^ = Ml = X — \, OBSERVATIONS OF ONE UNKNOWN QUANTITY 39 .Ty — V., = .1/. = .V — \, (l) Xo — Vn = ^L = X — A„. By addition, remembering that [7'] = o, nxo = nx - [A]. (2) Substitute for x^ from equation (2) in equations (i) and nvi = (7? — i) Aj — A, — • . . nv2 = — '^i + (» — i) A, — • . . Squaring, »V = (" - 0' '^i' + A^^ + . . . - 2(w - i) AjA, - . . . n'v^^ = Aj2 + („ _ i)2 A/ + . . . - 2(» - i) A,A, - . . . By addition, assuming that the double products destroy each' other,* positive and negative errors being equally probable, ... [v^] = ^ [A3] = (w — i) /j>?, and /.== = -t^ . (3) » — I which gives the m. s, e. of an observation. Now, from Art. 13, 14, /*0 = Vw which gives the m. s. e. of the arithmetic mean of ;/ observa- tions of equal precision. • If the positive and negative errors are equal in number, «, there will be a preponderance of negative products. This is too slight to affect tlie proof. For the proportion of the exxess of the negative products to the total number of products is as n to 2;/' -;^ or as i to 2« - i, which de- creases as the number of errors 2 u increases. 46 THE ADJUSTMENT OF OBSERVATIONS 30. From the constant relation existing between the m. s. e. and p. e. given in Art. 14, we have for the p. e. of an observa- tion and of the arithmetic mean of n observations respectively, y = /3 '\/2 /A Viv/M,. (, where p V2 = 0.6745 nearly. 31. It is important to note carefully the distinction between residuals and errors. The errors are quantities of which we may never hope to secure the exact values, since they are the differ- ences between the true value and the separate observed values. We cannot secure the true value. We can secure a most prob- able value which is an approximation to the true value. The residuals are the differences between the most probable value and the separate observed values. The residuals are approxima- tions to the corresponding errors just as the most probable value is an approximation to the true value. The residuals are quan- tities which may be used in computations. The errors cannot be so used since they are always unknown. The errors appear in the formulas during the process of deriving them, but they necessarily disappear from the formulas before they are in shape to be used by the computer. 32. 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 convenient formula first given by Dr. Peters.* From the equation, we have approximately, without regard to sign, * AstronoDiische A'achrichten, No. 1034. OBSERVATIONS OF OXE UXKXOWN QUANTITY 41 ''1 = V n ^^' V, = 1 /^ ~ ^ A > n Adding and dividing by ;/, V v/"~ ',. n V „ '^' where [v is the sum of the residuals without regard to sign. But from Art. 16, > 2 » (» — I I 2 1; ^^ ^^^ [v nearly. « Vw (w — i) For the p. e. of an observation and of the arithmetic mean of observations we have respectively yn {n — i) 0-8453 , ro = — I [v. 11 Vn — I 33. Collecting the formulas for finding the p. e. of a single observation and of the arithmetic mean of ;/ observations, we have r = 0.6745 v/-^ . ^ = "-^"^SS , / . ' \ n — I yn(n— I) [V r, = 0.6745 J y^ ^ , r, = 0.8453 —7=- • V n{n — i) n V" — i To save labor in the numerical work, I have c<)mi)uted tables containing the values of the coefficients of V[7''] and [r in llx'se 42 THE ADJUSTMENT OF OBSERVATIONS equations for values of n from 2 to 100. (See Appendix, Tables II, III.)* If Bessel's formula is used, compute first \y^\ then V[z^"] can be taken from a table of squares closely enough. This square- root number multiplied by the number in Table II correspond- ing 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 corresponding to the argument n. 34. Control of [•z'^]. — A control is afforded by the deriva- tion of [7/-] from the observed values and the arithmetic mean directly. We have ^1 = ^0 — ^v Vo^ = x-Q — M^, "Vn = Xo — l/„. Square and add, [v^] = nxo'' - 2 Xo[M] + [AP] = [AP]-[M]x,; (i) since nx^ = [M]- The values of JiP may be found from a table of squares or from Crelle's tables, or, if the numbers 7^/ are large, an arithmo- meter, or machine for multiplying and dividing, may be employed with advantage. 35. Approximate Method of Finding the Precision. — A connection between the p. e. of a single observation and the greatest error committed in the series may be established ap- proximately by the aid of the principle proved in Arts. 22-23. There we saw that in a large series the actual errors may be ex- pected 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 * First published in the Analyst, Des Moines, la., May, 1882. OBSERVATIONS OF ONE UNKNOWN QUANTITY 43 not likely to be more than 5 r. The probability of its being as large as this is only about y qVo^- The same principle will enable us to estimate roughly the p. e, in a series of observations. A glance at the 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 i of the range of tJie errors of observation. The probable error so estimated is, however, rather untrust- worthy, as it depends upon but two of the residuals instead of all of them. Moreover, these are the two residuals which cor- respond to the extreme observations, about which there is fre- quently a reasonable doubt as to whether they should not be rejected. 36. Ex. — We shall now apply the preceding formulas to the example in Art. 27 to find the p. e. of the arithmetic mean and of a single observation. (1) The p. e. of the arithmetic mean. These we may find in two ways : {a) From the sum of the squares of the residuals (Art. 29), Mo Y n (n — I _ / 0.2756 or from Table II at once I X 20 = 0.026, r„ = 0.6745 X 0.026 = 0.017; ro = 0.525 X 0.033 = 0.017. (b) From the sum [7/ of the residuals (Art. 32) : The multiplier in Table III corresponding to the number 21 is 0.009. .•. ru = 2.02 X 0.009 = 0.018. (2) The p. e. of a si?iffte obscyiuition. From Tables 11. and 1 1 1, directly : 44 THE ADJUSTMENT OF OBSERVATIONS r= 0.525 X 0.15 1 — 0.079, r = 2.02 X 0.041 = 0.082. Check (a). Nine residuals out of twenty-one are less than the computed p. e. ^o-o-jGi", whereas, according to theory (Table I), one-half the errors, or io\ out of 21, should be less than -J- 0.079". This is the practical way of using the check. We might have arranged the residuals in order of magnitude when the residual 0.09 will be found to occupy the middle place. Check (/3). See Art. 35. Range = 0.22 + 0.23 = 0.45. 6 The values found by the different methods agree reasonably well. 37. 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 a posteriori proof intimated in Art. 6 as necessary for the demonstration of the law. (i) 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 + residuals is 14 17, 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 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 we enter the table with the arsfument ^^^ = 2.i;o and find o.qi. ^ .08 ^ ^ From this deduct 0.60 and multiply the remainder by 21. This gives 6. The number observed is 5. OBSERVATIONS OF OXE UNKNOWN QUANTITY 45 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 Ekkurs. Theory. Observation , J-. S. 0.00 to O.IO o.io to 0.20 over 0.20 13 6 2 14 5 2 Table I, it will be remembered, is founded on the supposition that the number of observations in a given set is very large. In our example the number is only 21. Perfect accordance between the number of errors given by theory, and the number given by observation is, therefore, not to be expected. 38. Caution as to the Application of the Test of Preci- sion. — 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 test of precision, the probable error. The probable error of an observation, or of the mean of a series of observations, has been defined as a measure of accuracy or in other words of uncertainty. It must be kept clearly in mind that it is a measure only of such uncer- tainties as are due to accidental errors, and has no necessary relations to systematic or constant errors. If all the errors occurring in the observations are of the accidental class, the probable error is a true measure of accuracy. If systematic or constant errors also occur, these give rise to errors in the result, in addition to those arising from accidental errors, and the p. e., therefore, expresses but a part of the uncertainty of the result. The neglect of this principle has led in many cases to erroneous conclusions, and to faulty methods of observing and computing. These in turn have led to wholesale condemnation of the method of least-squares. 46 THE ADJUSTMENT OF OBSERVATIONS 39. The fact that the computed probable error is independent of the constant error in the observations may be shown from the formula from which it is computed. In the derivation of the p. e. from a series of n observed quantities M^, M^, ... we had the observation equations. ^0 — Ml =Vi, CCq "~" IvJ- 2 ~~ ^2 ' Xo — M„ = v„. Also r^=p^/2 ^^^ 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 Jl/^ + c, M,^ + Cy . . . the most probable value, instead of being x^, will be x^ 4- c. Also since V = (X\ + c)- {M + c) = .vo - M, the residuals will be the same as before. Hence r is unchanged, and we see, therefore, that the p. e. makes no allowance for constant errors or corrections to the observed quantity. These are supposed to be eliminated or corrected for before the most probable value and its precision are sought. In leveling, 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 p. e. of observation will consequently be small. If the line is levelled in opposite directions, experience shows that the agreement would not be so good. The p. 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 rising or settling of rods and instruments, the refraction of light, . . . which causes OBSERVATIONS OF ONE UNKNOWN QUANTITY 47 are less likely to be mutually destructive and more likely to be cumulative if 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 p. e. alone would be illusory, for the constant and systematic errors in levelling are in general, especially on long lines, much larger than the accidental errors, and the p. e. is simply a measure of the effect of errors of the latter class. 40. Another common misapprehension is the following : From Art. 13 the relation between the p. e of a single obser- vation r and the p. e. of the mean of n observations r^ is r V« This formula shows that by repeating the measurement a suffi- cient number of times we can make the p. e. of the final result as small as we please. Nothing would, therefore, seem to be in the way of our getting an exact result, and that we could do as good work with a rude or imperfect instrument as with a good one by sufficiently increasing the number of observations. The fallacy lies in the implied supposition that all the errors affecting the observations, are of the accidental class. It is true that the effects of errors of this class will be reduced by increasing the number of observations in the manner indicated above, but experience indicates that in all observations, constant and systematic errors are present as well as accidental errors, being sometimes so small as to be discernible only after the accidental errors have been greatly reduced by many repetitions of the observations, and in other cases so large as to be evident after the first few observations have been taken. The repeti- tion of observations has no effect whatever in eliminating the constant errors, and none in eliminating the systematic errors, unless the conditions under wliich the observations are taken be by accident or design so changed from time to time as to 48 THE ADJUSTMENT OF OBSERVATIONS reverse the signs of the systematic errors. In general the repetition of observations reduces the error of the result very rapidly at first, while the effects of the accidental errors still predominate over those of other classes, and while each change of one unit in ;/ produces a relatively large change in \lii. As the observing is continued, the V// changes more and more slowly, and an elimination of the remaining accidental errors from the mean is correspondingly slow. Much more important than this, however, is the fact that a point is, sooner or later, reached in the repetition of observations at which the unelimi- nated accidental error is smaller than the constant error in the observations and mean. Beyond this point the effect of further observations is simply to reduce the smaller and comparatively unimportant accidental error, and leave the larger serious con- stant error absolutely unchanged. The effect of indefinitely continuing the observations is to make the combined accidental and constant error, the total error of the result, approach very slowly to the constant error as a limit when the number of observations is infinite. 41. It has been erroneously assumed by many persons that the limit of accuracy for a given instrument is the smallest magnitude that can be seen with it, that "what cannot be seen cannot be measured." The limit of accuracy beyond which one cannot go by increasing the number of observations, is fixed by the constant and systematic errors as indicated above and has no necessary relation to the power of the instrument used. Three illustrations may be given of measurements of which the errors of the mean are smaller than the smallest quantity which can be seen with the instruments used. With the best cheodolites now in use in the Coast and Geodetic Survey, the mean of 16 measures of a direction has a probable error in gen- eral from i to i of a second of arc. The checks which are available show these probable errors to be true measures of the accuracy. These observations are made with a telescope with OBSERVATIONS OF ONE UNKNOWN QUANTITY 49 which it is impossible under the actual conditions of observation to see a rod or stripe ^V to ^ of an inch wide placed a mile from the instrument, and therefore, subtending an angle of from J to i of a second. With the zenith telescope, now being used for the work of determining the variation of latitude under the direction of the International Geodetic Association, the probable error of a single observation has frequently been found to be as small as ± o'. 10, which is a small fraction of the width of the line used in bisecting the stars and beyond the power of vision of the observer as aided by the telescope. The probable error of the mean result from a night's work is very much smaller than this, ± o*".04 or less. The checks obtained from combining the work of different observatories show that these probable errors are true measures of the accuracy, or in other words, that the constant and systematic errors are extremely small. In the precise level net of the United States, there are thousands of miles of leveling with the new Coast and Geodetic Survey precise level, for which the largest correction expressed in millimeters per kilometer arising from the necessity of closing all circuits, the most severe test of accuracy which can be applied, is J4 of a millimeter per kilometer. The readings in this kind of leveling are made on a direct reading rod by estimating the position of each of three cross-lines in the telescope as seen projected against a centimeter graduation on the rod. Each reading is taken to the nearest millimeter only. It is abso- lutely impossible to see so small a magnitude as Jj millimeter on the rod through the telescope at the distance at which the rod is ordinarily placed, yet the accidental errors are reduced below this limit for a whole kilometer involving in general 12 to 14 sights. 42. The proposition that one cannot measure smaller mag- nitudes than can be seen is a dangerous error, for the reason that it is liable to lead to pf)or habits and poor methods of ob- servation. The student, for example, while being taught to use 50 THE ADJUSTMENT OF OBSERVATIONS the zenith telescope for latitude observations, on being warned that he must be extremely careful not to apply any longitudinal pressure on the head of the micrometer screw, will sometimes experiment for himself by purposely applying such a light pressure while watching the bisection. He may not be able to see any change, becomes skeptical, and thereafter is slovenly in his handling of the instrument. Similarly, the observer with a precise level, on being told that an extremely small amount of unequal heating of the level vial will cause a bubble to travel, and introduce an error into the results, will experiment and convince himself that the movement of this character ordinarily encoun- tered under actual field conditions is on an average smaller than he can see. He may then use a method for years which is radically defective in not guarding against this source of error. Although the motion of the bubble may be too small to be visible, yet the principal error in his work may be due to this cause. 43. As an example of a case in which the constant errors are so large that little is gained in accuracy after the first few obser- vations, the determination of absolute declinations and right ascen- sions may be cited. With the meridian circle Professor Rogers found the p. e. of a single complete observation in declination to be ± o".^6, and the p. e. of a single complete observation in right ascension for an equatorial star to be ± 0''.026. 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 observations the right ascension within o\02 and the declinations 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 known within these limits." * The explanation is, as intimated, that constant errors are not eliminated by increasing the number of observations. Acci- dental errors are eliminated by so doing. * Ffoc. Amcr. Acad. Scz., 1878, p. 174. OBSERVATIONS OF OXE UXKXOWX QUAXTlTV 51 44. There is a 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. The method of least squares is a method of computation, not of observation, which serves merely to aid the computer to secure in his computed results the highest grade of accuracy possible from a given series of observations, but it can- not increase the accuracy of observations already taken. The observations fix an absolute limit to attainable accuracy. The computer may approach this limit more or less closely according to his skill, but cannot pass it. A third puzzle in con- nection with probable error 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 circumstances. It is ever a principle in least squares to make use of all the knowledge on hand of the po'nt 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 sur- face 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 deter- mination. On the other hand, when we have no a piiori knowl- edge, 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,* we could do nothing but take the value resulting from the observations, unless, indeed, it came out with a negative sign, and then its untrustworthy character would be evident. 45. Systematic Error. — References to systematic error in the preceding articles lead us to nf)lice an example or two of * See, for example, Newcomb, Astronomy, app. vii. £2 THE ADJUSTMENT OF OBSERVATIONS the detection and treatment of this great bugbear of obser- vation. We suspect the presence of systematic errors in a series of observations from finding that the residuals do not bear the rela- tions to each other that they would if the errors were all of the accidental class, or from other departures of the results from the laws which they would follow if the errors were all accidental. Sometimes the sources of error are detected without much trouble. Thus, in measuring an angle with a theodolite, 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 lo" in primary work ; and on reading to a number of signals in order round the horizon, the final read- ing 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, 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 observa- tions might also show that the longer the time taken in going around, the greater the resulting discrepancy. The natural inference 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 day when the sun shone brightly. Measurements were then made at night, using 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 station, * At U. S. Lake Survey station Bruld, Lake Superior, many observations were taken during both day and niglitin July, 1S71, to determine the rate of twist of center-post on which the theodolite used in measuring angles was placed. The conclusion arrived at was that "during a day of uniform sun- shine and clear atmosphere tliis twist seemed to be quite regular, and at the rate of about one second of arc per minute of time, 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." OBSERVATIONS OF ONE UNKNOWN QUANTITY 53 assuming it to act uniformly in the same direction during the time of observation, can be eliminated by the method of obser- vation : 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 systematic error, when possible, by the method of observation, rather than to compute and apply it. 46. 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 of two line meas- ures belonging to the U. S. Engineers, the results found by different observers showed large discrepancies. The micro- meter 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 observer's results being tolerably constant for his own focus. As the value of a revolution of the micro- meter 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 redu- cing the regular observations made with the same focus. Had the microscopes been of high power, it would have been sufficient to determine the value of the screw once for all, since the error arising from change of focus could have been classed as accidental. In trying to avoid or eliminate 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. 54 THE ADJUSTMENT OP OBSERVATIONS B. Observed Values of Different Quality. 47. The Most Probable Value : the Weighted Mean. — It has been shown in Art. 12 that if the directly observed values M^, M^, . . . M„ of a quantity are of different quality, the most probable value is found by multiplying each residual error of observation by the reciprocal of its p. e., and making the sum of the squares of the products a minimum ; that is, with the usual notation, or ^ + ^+ ... +^ = amin., (i) (^^"(%— '"-(■^;=^-- ^^> By differentiation and reduction, We have, therefore, the equivalent rule : If the observed values of a quantity are of different quality, the most probable value is found by multiplying each observed value by the reciproeal of the square of its p. e., and dividing the sum of the products by the sum of the reciprocals. The form of the expression for x\^ suggests another standpoint from which to consider it. Let p^, p.,, • • ■ Pn be the numerical parts of— ^ J — » • • • — 5» such that each is of the type 9 2 r' r T ' \ '2 n (unit of measure)^ P- -^ -^ then equation (3) may be written, oc=^^^ (4) Also, since — ^ > _ > • • • — ^, are similarly involved in the nu- r' r r " 1 2 " merator and denominator of the value of x^, this value will re- main the same if p^, p.^, . . . pn are taken any numbers whatever OBSERVATIONS OF ONE UNKNOWN QUANTITY 55 in the same proportion to^j> -^> • • • -^ J that is, if A, A,, 1 '2 'n . . . pn satisfy the relations where r is an arbitrary vaUie of the probable error corresponding to the arbitrarily assumed unit weight. The numbers p^, p.,, p.^, . . . /„ are called the weigJits, or, better, the combining weights of the observed values, and the mean value x^ is called the lueighted mean. The expression \_pM^/\_p~\ can now be put into words as fol- lows : If the observed values of a quantity are of different zveights, the most probable value is found by multiplying each obsc7\>ed value by its zveight, and dividing the sum of the products by the sum of the weights . In Arts. 13, 14, as indicated in the expressions for the p. e. of a single observation and of the mean, it was shown that for obser- vations of equal precision, or, in other words, of equal weight, the p. e. of a mean of ;/ observations is to the p. e. of a single observation as i is to V;/. By comparison of these with the ex- pressions r .2 4^ A<2 h '2 it may be seen that the meaning of weight p^ assigned to an ob- servation is that it has the same degree of accuracy as the mean of /j observations of unit weight. In combining observations it is treated accordingly. With this understanding, it is evident that to combine observed values J/j, M.„ J/.,, . . . J/,„ of which the weights are /,, p.„ />,,, . . . /„, one should jiroceed as if M^ were the mean of p^ separate observations of unit weight, Af., of />., observations of unit weight, and so on. The arithmetic mean of the hypothetical [/^] observations would evidently be [/i^]/[/], and this is precisely the form used. 56 THE ADJUSTMENT OF OBSERVATIONS If the observed values M are numerically large we may- lighten the numerical work by finding x the method of Art. 26. Proceeding as there indicated, we have = X' -\- x" suppose. 48. Reduction of Observed Values to a Common Stan- ^QxA. — The principle of the weighted mean is evidently an ex- tension 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 the 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 ob- served 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. 47, which may be written, r, = Now, since r,, ;;, . . . r„ are the p. e. of J/,, M^_, . . . M,„ the p. e. of M^ V/,, i/, \f/\,, . . . M„ V^~ would each be the same quantity r. Hence, if a series of observed values M^, M ,,^ . . . M,^ have the weights p^, p.„ . . . p„, tJicy are reduced to the same standard bv multiplying by \^, "v^, . . . V/,^ respectively. For example, given the observation equations, X — Ml = z'l weight p^ , X ~ M2 = v^ weight p^ , X — M^= v„ weight />„, to find the most probable value of x. OBSERVATIONS OP ONE UNKNOWN QUANTITY 57 Reducing to the same standard of weight, we have the equations, and the most probable value of x is found by making (VK^i)' + (^/^2^2)' + • . • + ( V^zO- = a min. ; that is, by making Reducing this equation, we find, as before, [pM] Xq [P] 49, Control of the Weighted Mean. — Eq. 4, Art. 47, may be written, [pi'] = o. The error of any assigned value of x^ is evidently [//]/ [/>], in which the I's are residuals corresponding to the assumed value of x^^. If this error is less than one-half unit in a given decimal place, the assumed value is correct to that place. Ex. — Find the most probable value of the velocity of light from the fol- lowing determinations by Fizeau and others: 298,000 kil. ± 1000 kil 298.500 " i 1000 (( 299,990 " i 200 (( 300,100 " i 1000 (( 299,930 " ± 100 « {Amer./our. Set'., vol. xix.) The weights, being inversely as the squares of the probable errors, are as the numbers i, i, 25, i, 100. (Art. 47.) To avoid handling such large numbers as the AI's, wc may assume a value X' for x^^ by inspection, say 299,900, and llioii proceed as follows : S8 THE ADJUSTMENT OF OBSERVATIONS / P pi + igoo + 1400 - 90 - 200 - 30 I I 25 I 100 + 1900 + 1400 — 2250 — 200 -3000 128 — 2150 The correction os^' to the assumed vakie (- 2150) 128 + 17. and the weighted mean = 299,017. [It is much more important in computations to keep the numbers as small as possible than to avoid minus signs. Such a procedure saves time. It is almost impossible to arrange computations so that the computer will not be obliged to watch the signs. He can watch many minus signs as well as a few. In the long run, rapidity depends rather upon the number of signi- ficant figures used in the quantities handled.] 50. The Precision of the Weighted Mean. — Since the weighted mean x^ is the arithmetic mean of \_p\ observations of the unit of weight, its weight is [/]. Hence the p. e. ;: of x^ is found from 'n — M' where r is the p. e. of an observation of the unit of weight (standard observation). According to Art. 48, the value of r may be found by writing ^{L\, Va^'s' ... for ^',, ^^, ... in the formulas derived for observations of the same weight. Hence, substituting in Bessel's and in Peters' formulas. Arts. 29 and 32, we have .6745V/-E£a ^ n — X or r= 0.8453 '\/pv and therefore Vw(; n OBSERVATIONS OF OXE UXKXOWN QUANTITY 59 ro = -6745 V IP^^'] or ro= 0.8453 V^v M(«-0 " '"" ^[p]n{n-i) These expressions reduce to those for the arithmetic mean where the observed values are of the same weight by putting Ex. — The linear values found for the space 0.00" to 0.05" of inch [a3] on the standard steel foot i F. of the G. T. Survey of India were as fol- lows: 0.050027", 0,049971", 0.050019", 0.050079", 0.050021", 0.05001 1". The numbers 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 deter- minations, required the most probable value of the space and its p. e. The direct solution presents no difficulty. The value of r„ may be found as in Ex. Art. 49, and thence the residuals v. The p. e. follows from the formulas of Art. 50. Assume X' = 0.049971. p / // V v^ Pv^ 6 4- 000056 + .000336 + .000003 .000000000009 .000000000054 6 + + + 59 34S1 20886 15 + 48 4- 720 + II 121 ISI5 15 + 108 4- 1620 — 49 2401 36015 8 + 50 + 400 + 9 81 64S 8 + 40 4- 320 + 19 361 2888 58 • . . . + .003396 • .000000062 ■506 . x" = t^ = +0.000059, r = 0.6745 \/-M^K = 0.000075", d xo=x' -h x"= 0.050030, ro= 0.6745 y (;7:zT)[7]"°"°°°°'°"* an Hence, x^ = 0.050030" ± O.OOOOIO. 51. In the above example an important practical point occurs, and one often overlooked. The p. c. is not computed from the original observations, but from these observations grouped in six sets of means. These means we have treated 6o THE ADJUSTMENT OF OBSERVATIONS as if they were 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, some of the facts having been partially concealed by the process of taking the means of the separate groups. In good work 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, whenever there is a choice, the p. e. should always be deduced from the original observations rather than from any combinations of them. The weighted mean value x^ would evidently be the same whether computed from the partial means or from the original observations. Observed Values Multiples of the Unknown. 52. Let the observed values M^, M^, . . . M„ be multiples of the same unknown X ; that is, be of the form a^X, a.^X, . . . a,^X, where a^, a.,, . . . a^ are constants given by theory for each observation. The values — \ -^, . . . — - of X may be regarded as directly observed values of unequal weight. If r is the p. e. of an observation, that is, of Af^, Af,^, . . ., then, since the p. e. of — ^ is — , of — ? is — , . . . the weights of ^?i rt'j a., a„ these assumed observations are proportional to a^', a. Hence, taking the weighted mean, _[aM] Also, since [^'j is the weight of X, 2 2 OBSERVATIONS OF ONE UNKNOWN QUANTITY 6i rx' = 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 regu- lated by the sliding of a board, was shown to the olxserver. 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 Obser- VATIO.SS. Width. Side of Opening. Mean of Microscgi'k Readings. I 2 3 4 5 6 7 8 9 10 6.0 5.5 5.0 4.5 4.0 3-5 3-0 2-5 2.0 1-5 (left \ right (left \ right (left I right (left 1 right (left 1 right (left \ right (left I right (left ( right (left ( right jleft 1 right ( 28.00 1 37-50 J 28.50 (37-00 ( 29.16 \ 37-i6 ( 30.16 } 36.66 \ 30.50 (37.16 (31.16 J 37.00 ( 32.66 i 36.83 5 33-50 } 36.83 \ 33-83 137.00 1 37-i6 Let X = the most probable value of the angle subtending an opening of I inch. 1 hen we have the observation equations, GX - 9.50 = Vi 3.5 X - 5.84 = T/o 5.5 A' - 8.50 = r/a 3X- 4.17 = 7/7 SX - 8.00 = V3 2.5 X - 3.33 = Va 4.5 X - 6.50 = 7/4 2 .V - 3.17 = Vg 4X - 6.66 = z/s 1.5 .V - 1.66 = 7/,„ From the preceding we have for the individual values of X and their weights, * Account of the Principal Triangulation, pp. 54, 55. 62 THE ADJUSTMENT OF OBSERVATIONS X = 1.58, weighted X = 1.55, weight 5. 5^ weighted mean = a2 , 2 9.5 X 6' + 8.50 X 5.5^ + • . • 6^ + 5-5' + • • • = i-55> or making the sum of the squares of the residuals v a minimum, that is, (6 X - 9.50)2 _^ (•j^ X - 8.50)2 + . . . = a min., we find by differentiation that 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 X in that equation, and add the products. The resulting equation gives the value of X. Precision of a Linear Fiinction of Independently Observed Qiiantities. 53. Suppose that there are ;/ independently observed quanti- ties yl/,, i/^, . . . whose m. s. e. are /a^, /x,^, . . . respectively, to find the p. e. r of F where F = a^M, + a^M^ + . . . + aJh, (i) a^, a^, . . . a^ being constants. If Aj, A,, . . . denote the errors of M^, M^, ... we shall have the true value 7" of /^ by writing M^ + A^, M.^ -f A^, . . . for J/j, M^, ... in the above expression for F ; that is, T = a, (M, + A,) + a, (M, + A,) + . . . + «„ (M^ + A„). Call A the error of F; then, since T = F -\- A, we have A = fliAj + a.A.^ + . . . -}- a„A„, and .-. A^ = a,'\' + (7,2A,2 + . . . + 2 a,a,\A., + ■ . . Let the number of sets of M^, M.,, . , . required to find T be n, and suppose A^ summed for all the sets of values of A^, A^, . . . and the mean taken, then attending to Art. 13, fi2? = a^^fj.^^ + c^V.^ + . . . + 2 afi.^ L_i_2i -f- . . . (2) OBSERVATIONS OF OXE UNKNOWN QUANTITY 63 In forming all possible values of A,A.„ A.,A3, . • ., the num- ber of \alues being very large, there will probably be about as many -f as — products of each form,* and wc therefore assume [\X] = [X\] = . . . = o. Hence f^l? = [(7Vj, (3) and r-r = [a-r]. Ex. I. — The Keweenaw 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 = ^ 0.00034", and of tube 2 = j^ 0.00037", 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 = 0.329" p. e. from tube 2 = 966 x 0.00037 = 0.357" .". p. e. of line = v'0.329^ + 0.357^ = 0.485".] 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 0.03". Show that the p. e. in the length of the line of 1933 tubes arising from the same causes may be estimated at ^ 0.136". [p. e. of I measurement of i tube = -^= V94 p. e. of base of 1933 tubes = "_ V1933 V94 = ±0.136.] Attention is called to these two problems, from the impor- tance 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 indepen- dent of every other, and the errors are as likely to be in excess as in defect, and, therefore, may be expected to destroy one an- other in the final result. * See footnote to Art. 29 64 THE ADJUSTMENT OF OBSERVATIONS 54. If the function F whose p. e. is required is not in the hnear form, we first reduce it to that form. Thus, if F=/{M„AU_, ■ • ■ AL), the true vahie T oi F will result if we write 31^ + dM^, M^ + dM,, ... for M^, M^, ... the differentials representing the errors of these quantities. Then T = /(M + dM, M^ + ^1/2, . . .)• Expanding by Taylor's theorem, and retaining only the first powers of the small quantities dM^, dM,^, . . ., we have, or Error oi F = a^dM^ + a^dM^ -{-... -f aJM^ (i) 8/ 8/ 8/ where ^^ = m,' '''"^ UT,' ' ' ' '^ - UT: This expression is of the same form as (i), Art. 53. Hence, rj? = aC-r,' + a^r^ + • • • + a,,'r:~ = [aV^]. 55. Ex. I. — If r„ r„ are the p. e. of the measured sides AB, BC, of a rectangle A BCD, find the p. e. of the area of the rectangle. [Here F = Af.Af,. .-. by differentiation, dF = M^dM._ + M./IM^, and rp^ = M,^r} + Mir^\^ 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 1° Fahr. are approximately nun. mm. S = 0.0248 J; O.OOOI. Z — 0.0617 rt 0.0003. S ■' I Show that ^ = - zt — nearly. Z 5 400 [For F = ^• .'. dF = y dS — -y-^ dZ, I S^ and (p. e.y = ^ (0.000 1)^ + — (0.0003)^.] OBSERVATIONS OF ONE UNKNOWN QUANTITY 65 Ex. 3. — The base b and the adjacent angles A,C oi "d. triangle ABC are measured. If their p. e. are respectively r^, ;'^, Tq, tind the p. e. of the angle B and of the side a. To find ij^. We have, 5 = iSo + € - .4 - C, where e denotes the spherical excess of the triangle. Hence, A and C being independent of one another, ;y = r^ + r^h , sin A To find r„, ^^ = ^1^^- By differentiation, , siny? ,, , sin (C — el . ,, , ^ , . t> • 1, jr^ da = -. — F. db + b ^-7-71 — sin i" dA -\- a cot B sin i" fl'C, sin B sin- jy and therefore, sin^y? , , ^- sin^ (C — e) sin* i" , , , ^s „ • , „ , r 2 = ■ , „ ;',* ^ ■ , „ r ? -\- a^ cot* B sin* i" r^.'. " sin^ B '> sm*B '^ ^ Ex. 4. — Given the base b and the angles A, j5 of a triangle with p. s. e. r^, r^, r^,, respectively, to find the p. e. r^ of the side a. We have a = b— — =• (i) sin B This might be expanded as in the preceding example, but more conve- niently as follows : Take logarithms of both members. Then log a = log b + log sin A — log sin B. (2) (a) By differentiation, da = "i db + a cot A sin i" dA - a cotB sin i" ^ 15 ft. bar at 32° -f 1 1.314 in. ^ 0.421 in. and 15 ft. bar at 32° = 179.95438 in. ^ 0.00012 in. show that the p. e. of the base is ^ 0.450 in. [p. e. = \/(i325 X 0.000 [2)2 + (0.421)' = :J;0.45oin. We multiply J- 0.00021 inches by 1325 : uncertain which sign it is; but whichever it is, it is, constant all the way through.] 68 THE ADJUSTMENT OF OBSERVATIONS Ex. 7. — If the zenith distance f of a star is observed n^ times at upper cuhiiination, and the zenith distance f of the same star is observed ;/_, times at lower culmination, show that the m. s. e. of the latitude of the place of observation is u / I I 2 y «i «2 M being the m. s. e. of a single observation. [Latitude = 90° - Hf + f')-] Ex. 8. — Given the telegraphic longitude results, h. in. s, s. Cambridge west of Greenwich = 4 44 30.99 -j- 0.23 Omaha, west of Cambridge = i 39 15.04 ^006 Springfield east of Omaha = 25 08.69 J- o. 11 show that Springfield west of Greenwich = 5 58 37.34 :^ 0.26 [p. e. = \/.23- + .06- + .11- = 0.26.] Ex. 9. — Given mass of earth + mass of moon = 1 , 305,879 ± 2271 d mass of moon = prove mass of earth = and mass of moon = mass of earth, 81.44 309,635 rt 2299 [8" 44 "I For (305,879 i 2271) X g^ = 309,635 ± 2299. Ex. 10. — In measuring an angle suppose f'l — p. e. of a pointing at a signal, ;% = 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 ;;/ times, and n readings taken in each position, that p. e. of angle = 4- i / — ^^^ — H ^ '^ Y mil in For one position of the limb. p. e. of angle = zt 1/ — + e-, as the error of graduation remains constant throughout each set of n read- ings. It is important to note that the ii-\ readings in each position after the first reading has been taken reduce the effects of pointing and reading errors but not of errors of graduation. OBSERVATIONS OF ONE UNKNOWN QUANTITY 69 What would the p. e. of an angle have been if each of the »in readings had been taken in a new position ? Ex. II. — 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.)==i{4(p. e.)2ofardg.}] Ex. 12. — 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 reading being a, show that the p. e. of the difference of the results of ?i combined measurements (each being the mean of two measurements) is i < -a. [For p. e. of a reading = a. .-. p. e. of a combined measurement = a, and p. e. of mean of ;/ combined measurements = — - 1 etc.] Vn Ex. 13. — A theodolite is furnished with ;,; reading microscopes, all of the same precision. A graduation-mark on the limb is read on w times with a single microscope, giving the p. e. of a single reading to be Tj. The telescope is then pointed at an object ;// times, and the p. e. of the mean of the micro- scope readings is found to be r,. Show that the p. e. of a pointing is [ r,'- n p. e. of reading (mean of verniers) with ;/ microscopes = — 'rz* •s/n Total error = error of reading -f error of pointing. 'to 1 •^2^ = ^ + (p-e. of ptg.)2 ', etc. Ex. 14. — If ;'i/^,, rjh, are the p. e. of the base measurements, and r^X the p. e. of the ratio X, given by the triangulation, of a base A, to a base /',, show that the p. e. of the discrepancy between the computed and measured values of b., is b.^ \/\r-\. [Discrepancy = b., — b\ = /suppose. .•. dh, — b^d\ — \dbi = dl, and r/l. S- S. 1873, Dec. 24, 19 01.42 -J- 0.044 Dec. 26, 1.37 ± .037 Dec. 30, i-38rt: -036 Dec. 31, 1.45 it -036 1874, Jan. 9» 1.60 J^ .046 Jan. 10, 1-55 it -045 Jan. II, 19 01.57 J3 0.047 show that Weighted mean = 19 01.460^^0.0x6, Weighted mean of first four nights = 19 01.404 ^ 0.019, Weighted mean of last three nights = 19 01.573 J^ 0.027, and from the last two results check the first. Ex. 8. — In the triangulation connecting the Kent Id. Base, Md., and the Craney Id. Base, Va., the length of the line of junction computed from m. HI. Kent Id. Base = 26758.432 :t 0-38, Craney Id. Base = 26758.176^0.43. Show that m. m. (i) Discrepancy of computed values = 0.2564:0.57. (2) Most prob. length of junction line = 26758. 32 4: 0.28. Ex. 9. — In latitude work with the zenith telescope, if ;/ north stars are combined with j- south stars, giving its pairs, to find the weight of the com- bination, that of an ordinary pair, one north and one south, being unity. [Let r = the p. e. of an observation of one north star or of one south star. OBSERVATIONS OF ONE UNKNOWN QUANTITY 73 Then, as though combining the mean of n north stars with the mean of s south stars, the \vt. p of the combination is _ = j . p n s But I = - + - . 1 I 2 71S I 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 weight of 2^^— — ^only. In this and all similar cases I treated the whole 3+4 combination as one pair; that is, I inserted in the blank provided the half- sum of the mean of the declinations of north stars and 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., 1S79, p. 1987.) For a series of examples by Airy on the weights to be given to the sepa- rate results for terrestrial longitude determined by the observations of transits of the moon and fixed stars, see Metn. 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 triangulation of Lake Ontario the angle Walworth-Pal- myra-Sodus was measured as follows: In 1875, with theodolite P. and M. No. i, 74° 25' 05.429", J[:;0.29", mean of 16 results. In 1877, with theodolite T. and S. No. 3, 74° 25' 04.61 1" J; 0.22", mean of 24 results ; required the most probable value of the angle and its probable error . , . (0.22)2 The weights are in the ratio 7 ,, • ^ (0.29)2 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' 10.040", because, no matter how much better one is measured than the other, we can do nothing but take the sum of the two values. Ex. 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. 74 THE ADJUSTMENT OF OBSERVATIONS [r,, rjthe p. e. of a single pointing and of a single reading. With a non-repeating theodolite each measurement of the angle contains (pointing + reading) — (pointing + reading) .-. (p. e.)^ of one measurement = 2 f\^ + 2 r.^, and (p. e.)^ of mean of ;/ measurements = - (2 r? + 2 r.?). n With a repeating theodolite the successive measurements of the angle are (pointing + reading) — pointing pointing — pointing pointing — (pointing + reading). /. (p. e.)^ of 71 times the angle = 2 nr^ + 2 r.}, and (p. e.)^ of the angle = — (2 nr^^ 2 r^). If, then, ^„ p.^ denote the weights of an angle resulting from n reiterations or from n repetitions, - 2 „2 ?■., r. A: A= I + .-7^2 : 1 + ^2 nr{ r^ and hence it would seem that the method of repetition is to be preferred to ;-2 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 propor- tion to the precision of the pointing. This result is contradicted by experience. 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 repeatmg 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 so difficult to eliminate the systematic error in observations with a repeating theodolite, that in spite of its advantages over the direction theodolite in so far as acci- dental errors are concerned, it is still an open question which type of instru- ment will give greater accuracy. In the Coast and Geodetic Survey both types are in use. The direction theodolite is, however, used much more than the repeater on primary triangulation. Of the WeigJiting of Ohseivations. 58. When the sources of error are of such kinds that, so far as we know, they cannot be separated, the p. e. and consequent weight are found as described in the preceding sections. The weight has been defined as a number representing the relative OBSERVATIONS OF ONE UNKNOWN QUANTITY 75 goodness of an observ'ation, or of a result computed from obser- vations, and as a number inversely proportional to the square of the p. e. of an observation or computed result. In assigning or in computing weights it must be kept continually in mind that the intention is to make weights inversely proportional to the squares of the p. e. arising from all sources, and that no avenue of information should be neglected. To assign weights which are simply inversely proportional to the computed p. e.'s which may happen to be available without careful examination as to their trustworthiness is not good practice, nor is it in accordance with the complete theory of the method of least squares. The object of the following paragraphs is to indicate some of the precautions which must be taken in assigning weights. For example, two separate determinations of a millimeter space, made in the same way, gave 1000. 1 ± 0.40, mean of 20 readings, 1000.3 i °'33' rfie^ri of 30 readings. To find the weighted mean of these two sets of measurements, we may proceed in two ways. The number of results 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 20 X 0.1 + 30 X .3 = 1000 H = 1000,22. 20 + 30 Again, since the p. e. of the measurements are 0.40 and 0.33, their weights are as -, to — ,, that is, as 1089 to 1600, and 40- 33" the resulting weighted mean is 1000.22, agreeing with the other computed value to the last decimal place retained. 59, In this example the two methods of computation give exactly the same result. It is not always so. Some " run of luck," or balancing of errors, or constant conditions, might have made the observations of one set fall very closely together, in which case the weight as computed from the p, e, would have 76 THE ADJUSTMENT OF OBSERVATIONS been very large, while in the other set varying conditions might have caused large ranges and the computed weight would have been small. In such a case the two means might differ consider- ably. It would then be desirable to study carefully the question as to which method of weighting is probably the better. In such a study it must be kept in mind that the computed probable errors derived from the observations are subject to uncertainty for that reason, and are peculiarly subject to being largely increased by one or a few large residuals. If two series of ob- servations made under similar conditions by the same observer gave computed probable errors which differed widely, it would be an open question whether the actual p. e. of the two series differed widely or whether the difference was simply apparent and due to each of the computed p. e.'s being based on too small a number of observations to be a close approximation to the truth. The computer must use his judgment in deciding this question and assign the weights accordingly. Thus, in the second set of observations above the first three results were 999.8, 999.8, 999.8. The p. e. computed from these values would be zero and the consequent weight infinite. But no one will doubt that these observations are subject to some error, and that the weights assigned to them should be finite and smaller than the weight assigned to the mean of the thirty observations. 60. An Approximate Method of Weighting. — A long-con- tinued series of observations will show the kind of work an instrument is capable of doing under favorable conditions ; 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 a 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 with telescope reverse, need not be expected to be greater than 0.3". If, therefore, after having OBSERVATIONS OF ONE UNKNOWN QUANTITY 77 measured a series of angles in a triangulation net with these instruments, the p. e. all fell within ± 0.3", 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 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 dis- tant 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 determination] was finally fixed at about 60, it being found that with the 32-in. instrument 60 observations would giv^e a mean result of which the p. e. would be about 4 feet."f This method of weighting is based upon the idea that a closer approximation to the truth is obtained in these cases by assuming that the actual p. e.'s are all equal than by accepting their separate computed values as represent- ing the facts accurately. In other words, such a procedure is equivalent to assuming that the principal cause of variation of the separate computed p. e. from a mean value is the accidental grouping of unusually large or unusually small residuals rather than a real variation in accuracy between the different series. 61. Weighting when Constant Error is Present. — The preceding leads us to the case where the error of observation can be separated into two parts, one of which is due to acciden- tal causes, and the other to causes which are constant through- out the observations. The total error e would, therefore, be of the form e = 1,-^1,. * Professional Papers of the Corps of Engineers U. S. A., No. 24, p. 354. t Report, Survey of the Northern Boundary^ p. 86. 78 THE ADJUSTMENT OF OBSERVATIONS This case has been discussed already in general terms in Art. 40 in explaining the well-known fact that an increase in the number of observations with a given instrument does not lead to a cor- responding increase of accuracy in the result obtained. Let r = the p. e. of the observation arising from the accidental causes, r = the error peculiar to the observation arising from the constant causes. Then r^, r,„ being independent, and being as likely to have opposite signs as the same sign, the total p. e., r of observation may be assumed (Art, 53), ;-2 = r,^ + r.\ If fi observations have been made, we shall have for the p. e. r^ of their mean, since ;; is constant, >V = ^ + r,'. n It is evident that when n is large, r/ becomes the important term, and that in any case the value of ;;, and consequent weight can be but little improved by increasing the number of observa- tions. 62. For the purpose of finding the value of the p. e. arising from the constant sources of error, a special series of observa- tions is, in general, necessary. After this series has been made, the value of ;% found from it can be applied in the determination of the value of r„ in any other series made under like conditions. For illustration let us consider a latitude determination with the zenith telescope. The zenith-distance, ^, of each star being observed, the half-difference of zenith-distances for each pair may be computed, and each of these computed values may be considered an observed value. The values of the declinations 8 are taken from a catalogue of stars. The errors of S are, there- fore, independent of those of K, and are constant for the same pair of stars. The latitude from one pair is given by OBSERVATIONS OF ONE UNKNOWN QUANTITY 79 <^ = i (8' + 8) + Kr - 0- Let r^ = the p. e. of ^ (C — for one observation of one pair, rs = the p. e, of ^ (S' + 8) for this pair, rA, = the p. e. of the resulting latitude <^ from one pair, then for a single observation of this pair, r<^^ = ''5^ + r^, and for u observations of this pair, r^' = rs' + r^ The quantity r^ will be found from repeated observations 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 a considerable period. Collect into groups the latitudes resulting from the observed values of each separate pair. Let ;/,, 7i.,, . . . «,„ 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 p. e. in the usual way. We have : No. OF Night. First Pair. Second Pair. • • • Results. V Results. V . . . I 2 3 0,' 01 7',' 0./ 0/' 02 v.: v.," • Means Wi — I r/ = • ^2 - I If, then, ;/ is the total number of results, and ;;/ the number of groups, by adding the above equations there results r^ = • 11 — m In finding r^ we assume that though errors of declination are constant for each star, still for a latitude found from many pairs in the same catalogue these errors may be regarded as acciden- tal. Let, then, many different pairs of stars be observed on each of n nights at iii' 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, taking the differences between these means considered as single results and their mean for that place. Then, reasoning as above, the p. e. of a latitude resulting from ;/ observations on a single pair of stars is r^ = 0.6745 W^i — =4, V n' — m where w' is the number of different pairs of stars observed, and m' is the number of places occupied. Now, r^ is found from , 2 _ , 2 ^^ >s — >v ' n and is, therefore, known for the star catalogue used. This value may be taken in future work in finding r^ from n and the consequent combining weight of ^ will be as I OBSERVATIONS OF ONE UNKNOWN QUANTITY 8i 63. An example of a similar kind is afforded in finding the weights of the angles measured with a theodolite in a triangula- tion where more rigid values are required than would be found by Art. 60. The actual error of a measured value of an angle arises from two main sources, errors of graduation and errors of observation. The former are constant for each part 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 been eliminated by proper shiftings of the circle. The resultant p. e. r of a single measurement is found from ;- = ;-^- + r.r, and the p. e. r^ of the mean of // measurements made on the same part of the limb from ., , r.r n where r^, r^ are the p. e. of graduation and observation respec- tively. The method of treating this problem is quite similar to that of the preceding : ;-;, is found by reading the same gradua- tion-mark on the limb many times, and r^ by reading the angle between two fixed signals many times, the limb being changed after each reading. Thence i\ is known for the instrument in question, and the combining weights of angles measured with this instrument are at once found. 64. The foregoing leads to another important practical point in the measurement of angles. If the weight of a single obser- vation is unity, then the weight of the mean of n observations made with the limb in one position is r{ + u: ^ r^^ + r^ n For certain instruments, experience has shown that we may safely assume 82 THE ADJUSTMENT OF OBSERVATIONS and therefore it follows that for these instruments 2 11 Hence, in using these instruments, no matter how many obser- vations 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 is evident that to secure the maximum efficiency in the elimination of error, the limb of a direction instrument should be shifted after each reading of an angle. The objection ordinarily urged against such a procedure is that it fails to furnish a sufh- cient guard against mistakes in reading. The present practice of the Coast and Geodetic Survey is to consider that a pair of readings on each signal, one with the telescope in the direct po- sition, and the other with it in the reverse position, together con- stitute one observation, and to shift the position of the limb before the next observation. A comparison of the direct and reverse readings furnishes a rough method of detecting mistakes in reading. 65. Assignment of Weight Arbitrarily. — So far we have deduced the combining weights from the observed values them- selves, or from them in connection with a special series of obser- vations. 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 comparison and reduction, it would not be proper for the computer to deduce a weight for each series from the observations themselves independent of other sources of infor- mation he might have. Some of the most inexperienced obser- vers 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 judgment in classing the observations. He should consider the experience of OBSERVATIONS OF ONE UNKNOWN QUANTITY 83 the obsen^er, his previous record for accurate work, the kind of instrument used, the conditions, and the observer's record of what he saw — whetlier it is clear and precise or hazy in its statements. An arbitrary scale of weights may then be con- structed, and to each set of observations 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 considered 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 elimi- nating this kind of error, we must do so 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 Telescopic Observations of the Transit of Mercury, May 5-6, 1878, Wash- ington, 1879, where, of 109 observations sent in, to only 18 was the highest weight assigned. Professor Eastman, under whose direction they were reduced, says : " . . . Several instances may be found where small weight is given to' observations that appar- ently agree well with those to which the highest weight is assigned, but in most cases the observer's remarks indicate the uncertain character of the observation." 66. 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 may 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 is 84 THE ADJUSTMENT OF OBSERVATIONS available for observing 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. Lamb- ton's geodetic work in India reported that " Col. Lambton's sur- veys, though executed with the greatest care and ability, were carried on under serious difficulties, and at a time when instru- mental 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 bet- ter ascertained than formerly, and all uncertainty on the unit of measure may be removed. The base-measuring apparatus can be improved. The instruments for horizontal angles used by Col. Lambton were inferior to those now in use. . . . The com- mittee express the strong hope that the whole of Col. Lambton's survey may be repeated with the best modern appliances." * 67. 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 sys- tematic 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 com- bining the older star catalogues with the more modern ones. Thus, Safford {Catalogue of Mean Declinations 0/2018 Stars, Washington, 1879) says : " In computing positions I have gen- erally employed Argelander's rule, giving to a modern determi- nation from 1 observation a weight ^, 2 observations a weight f, 3 to 8 observations a weight i, 9 or more observations a weight r^ or 2, * G. T. Survey of India, vol. ii. p. 70. OBSERVATIONS OP ONE UNKNOWN QUANTITY 85 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 observations than they deserve. But this is mostly compensated for by the num- ber of determinations." The weight of a quantity being a function of our knowledge 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 of the Lake Survey, measured in 1872 with the Bache-Wiirdemann compensating apparatus, a portion was measured seven times. The results differed widely, far beyond what was expected with the appar- atus. No reason could be assigned at the time for the discor- dances. At this stage, then, one would have been 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. During 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, says of the Mason and Dixon arc {Ejicyc. Metrop.,\). 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, 1S77, p. 95): " It is. 86 THE ADJUSTMENT OF OBSERVATIONS therefore, only owing to the increased perfection of instrumental means and methods that we now dismiss from further considera- tion 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 trian- gulation net, and the whole system 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, then the m. s. e. of each determination may be computed from the mea- sures of the separate nights, and in the adjustment 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 instruments, 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 Ger- many, France, and Austria reduced by Dr. Albrecht*the obser- vations were made between the years 1863 and 1876. The methods of observation had been much improved in this interval. In assigning the relative weights, a scale of weights was first formed from a consideration of all the knowledge on hand, tak- ing 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 of observers ; usual variety of observations ; non-adjustment of electric current, and so on. Similarly Dr. Bruhns in Verhandhingen der eiiropdischen * Astrononnsche Nachrichfen, 2132. OBSERVATIONS OF ONE UNKNOWN QUANTITY 87 Gradniessung, 1880. See also Coast Smi>ey Report, 1880, Appendix 6 ; 1 897, Appendix 2. 68. General Remarks. — The subject of the weightino- 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 consider- ing whether systematic errors enter or not. As he adds to his ex-perience he will consider outside evidence as well as the evi- dence afforded by the observations themselves. This will be specially true if he has any practical knowledge of how observa- tions are made. Indeed, it is doubtful if a comj^uter can apply the principles of least squares properly unless he is at least an average observer. Of the Rejection of Observations. 69. There is nothing in the whole theory of errors more per- plexing than the question of what shall be done with an obser- vation 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 ; 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 may be given to this question, as we look at it from one point of view or another. When observations are made by one man as an expert ob- server, and reduced by another as an expert computer, the judg- ment of each should be authoritative in his own province. The observer's statements of fact as to the conditions under which the observations were made must be accepted. His statements of opinion as to the effects of such conditions on the accuracy of the observations must be given great weight, and only set 88 THE ADJUSTMENT OF OBSERVATIONS aside when a considerable mass of carefully considered evidence indicates that the opinion is not correct. On the other hand, in judging as to the accuracy of a particu- lar observation or group of observations by the residuals alone, the judgment of the computer is authoritative rather than that of the observer. He has in general a much wider acquaintance than the observer with the law of distribution of error, as he deals during his regular routine with the observations of many different men made under many conditions and on different kinds of work. The observer is apt to form his opinion of a given observation, in so far as he judges it by the corresponding residual, by comparing it with the preceding observations in the same series. The computer bases his judgment on all the ob- servations, preceding and following. 70. There is one respect in which experience shows that the opinion of the observer is frequently erroneous. As he sees the conditions under which observations are made, vary from those extremely favorable to accurate measurement to those extremely unfavorable, he more or less explicitly assigns to the observa- tions varying weights. The range of weights assigned is as a rule much too large. If he believes that the observations made under the most unfavorable conditions should be given i as much weight as those under the best conditions, the chances are that the ratio of the true weights is more nearly A to i. In other words, the observer's best observations are usually poorer than he believes them to be, and his poorest better. He is mis- led by his feelings, and estimates the accuracy of the obser- vations by the difficulty of securing them rather than by a care- ful systematic study of them in the light of all available evidence. The observer is therefore in general too apt to reject obser- vations made under difficult conditions or to decline to observe under such conditions. 71. In the hypothetical case on which the exponential law of error was founded, there were no discontinuous observations taken into account. There we contemplated not only observa- OBSERVATIONS OF ONE UNKNOWN QUANTITY 89 tions made with the best instruments and by the most experi- enced 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 00 to — 00 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 skillful observer. This, to be sure, is the most important, and, as shown in Art. 22, 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 wondered at that no solution yet offered is regarded as entirely satis- factory. When it is proposed to reject a certain observation, it should be kept clearly in mind that the only justification for rejection is that by so doing, (i) the effect of a blunder may be eliminated from the final result, (2) or the effect of some error, from an unusual source, of much greater magnitude than the errors affecting the other observations, may be eliminated. The essen- tial difficulty in deciding what to reject is encountered in decid- ing whether a given large residual is due to either of the causes indicated, or is merely due to the accidental agreement in sign of many small errors from various sources, and is in conformity with the law of error. If the residuals are in strict accordance with the law of error, there will be a few which stand out rather far beyond the general range. (See table i.) If the observations corresponding to these are rejected, the final result is in general reduced in accuracy rather than increased. Various criterions for rejection have been devised and used, some of them being rather complicated as to their theoretical basis and tedious in application. The following criterion commends itself as being simple, quick of application, and upon a sufficiently good theoret- ical basis. It is recommended for general use. 90 THE ADJUSTMENT OF OBSERVATIONS Reject each observation for zvJiich the residual exceeds five times the probable error of a single observation. Examine each observa- tion for %vhich the residual exceeds i\ times the probable error of a single observation, and reject it if any of the conditions nndcr whicJi the observation was made were such as to produce any lack of confidence. The theoretical basis of the rule is evident from an inspection of table I . If the residuals follow the law of error, but one resid- ual in 55 should exceed 3I times the probable error of a single observation, and but i in 1000 should exceed 5 times that value. The presumption is strong, therefore, that rejections made under the rule are justified in most cases. A few observations will be rejected by this rule which should be retained, but only a few, and therefore little damage will be done in any case. The probable error for use in the above rule should, of course, be computed from all the observations not rejected up to the time of the proposed application. 72. Rejected observations should be left in the record and computation and in the publication, being simply marked Re- jected. The facts then appear for the inspection of those who follow, and may serve as a basis for indepeildent conclusions. In examining a large residual, it will sometimes appear that it is so evidently due to a "natural mistake" that it may be cor- rected without 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 revo- lutions out of the way, as shown by the rest of the observations ; and the like. Such corrections should be made with great caution, however, especially if the number of observations is small. In the precise leveling of the Coast and Geodetic Sur- vey, the observer is required to run over each section of the line twice. If a discrepancy between two results is discovered that is greater than is allowable and which is evidently due to a nat- ural mistake, the observer is not allowed, no matter how plain OBSERVATIONS OF ONE UNKNOWN QUANTITY 91 the case may be, to correct the mistake and continue. He must rerun the section until he secures two results which are within the required limit and are not subject to any assumption as to a natural mistake. 73. Again, the computer, instead of trusting to his judgment, may call in the aid of the calculus of probabilities, and seek to establish a test or criterion for the rejection of observations which will serve for all kinds of observations. Of the criterions which have been proposed the earliest is due to Professor Peirce. It is as follows : "Observations 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 Siu'vey 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 observa- tion " is given by Chauvenet in his Astronomy, vol. ii. p. 565. "We have seen that the function (Art. 30) a P: f 'e-f^'^'dA 2 h I ' r 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- pHed by the number of observations //, we shall have the actual number of errors less than a ; and hence the quantity n — )i®(i) = ;?{i - ©(/)} expresses the number of errors to be expected greater than the limit a. Put if this C|uantity is less than } it will follow that an error of the magnitude a will have a greater probability against it than for it, and may, therefore, be rejected. The 92 THE ADJUSTMENT OF OBSERVATIONS limit of rejection of a single doubtful observation is, therefore, obtained from the equation i = ;;{l _©(/)} 2 W — I or (/) = 2 n A third criterion was proposed by Mr. Stone, RadcHffe observer at Oxford, Eng., in Mojith. Not. Roy. Astion. Soc, i868, 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 71 observations of a given class. The probability, therefore, is that any record of his of this class of observations as a mistake is . From the average discordances 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 quantity, as C. Then if the probability in favor of a discordance as large as C is less than that of a mistake, or -> ® n the discordant observation is rejected." CHAPTER IV. ADJUSTMENT OF INDIRECT OBSERVATIONS OF ONE UNKNOWN. Determination of the Most Probable Values. 74. If direct measurements of a quantity have been made under the same circumstances, we have seen that the arithmetic mean of these measures gives the most probable value of the quantity. We now come to the case where the quantity meas- ured 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. Let the equations connecting a series of observed quantities J/j, J/„ . . . J/„, ;/ in number, and the independent unknowns X, V, . . . , «, in number (n > «,), be a^X + b,Y -t ■ • • - A = ^^^1 + ^1 > a,X-\-b,Y + ' ■ ■-L, = M, + v„ (i) CnX + bnY + ■ ' ' - L„ = Mn + V^, where a^, b^, . . . L^ . . . L^ are constants given by theory for each observation, and v^, 'v^, - • • "^n 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 em- ployed (see Art. 26).- Let X', ¥',... be close approximations to the value of X, V, . . . 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,Y-Y' = y,--- where x, f, . . . are the corrections required to reduce the approximate values to the most probable values. 93 94 THE ADJUSTMENT OF OBSERVATIONS Then the observation equations reduce to dn'X + l\y + - • • — ln = 'Vn, where - ^2 = a^X' + h^Y' + ' • • - L3 - M2, /„ = aX -VKY' + ■ ' --K- M, n, and are, therefore, known quantities. 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. 12), we can apply it to the simultaneous solution of the equations. The residual errors must satisfy the relation i^i' + ^2^ + » • • + v^- = a min. ; that is, we must make {a^x -^ h^y -\- • • ' — l^y + {a^x -\- h^y + - • • —Uj + • • • + (<7„.T + 6„)' -(-..._ /J2 _ a min. Now, the variables x, y, . . . being independent, the differential coefficients of the expression for the minimum with respect to each of them in succession must be equal to zero. Hence a^iaiX + h^y -\- ' • ■ - I^ + a^(a^x + b^y + ■ ■ ■ — l^) + ■ . . = o, bi(aiX + b^y + ■ ■ • — /i) + ^2(^2''^" + *2>' + • • ' — ^2) + • • • = o> (3) or, collecting the coefficients of x, y, . . . in each equation, \aa\ X + [a6] y -f \ac\ 2 + • • • = \al\ [a&] X + \hb'\ y + \bc:\ z + - ■ ■ = [W], (4) \ac] X + [be] y + [cc] z + ■ ■ ■ = [d], INDIRECT OBSERVATIONS OF ONE UNKNOWN 95 These equations are called normal equations. The number of these equations is the same as the number of unknowns ; that is, n^. 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 A', F, . . . will result. It is useful to notice that equations (3) may be written \}ro\ = o, [bv] = o, (5) • • • These relations correspond to [^'] = o in the case of the arith- metic mean, and may be used as a check on the computation 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. Let X, V denote the elevations of Ogden and Cheyenne respectively. Then {X - 4301)2 + ( K - 6075)' + (X -V+ 1749)' = a min. Differentiate with respect to X, Kin succession, and 2X - V= 2552 - X+ 2V= 7824. .•. X = 4309 feet. V = 6067 feet. 75. If the observation equations are of different weights /,, A' • • • A' ^^^T^> reducing each equation to the same unit of weight by multiplying it by the square root of its weight, we have (Art. 48), V^iQ^.r -I- V^^T + • ■ • - V^^ = ^/A^l' with [fv^-] = a min, 96 THE ADJUSTMENT OF OBSERVATIONS Substituting the values of \^j v^, '^p.[o^, ... in the minimum equation, and differentiating with respect \.o x, y, , . . as inde- pendent variables, we have the normal equations \^aLi\x + Ipal?^ y -{-■.. = Ipall, \pah'\ X + Xphh'l V + • • • = ipWl, (2) from which x, y, . . . may be found. The relations for weighted equations corresponding to those of Eq. 3, Art. 74, are evidently \_pav\ = o, [pbi'] = o,. . . (3) Formation of tJie Normal Eqiiations. 76. Instead of forming the minimum equation and differenti- ating 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 themselves. The first, irom. equations 3, Art. 74, may be stated as follows : To form the normal equation in x, multiply each observation equation by the coefficient of x in that equation, and add the re- sults. To form the normal equation in y, multiply each observa- tion equation by the coefficient oi y in that equation, and add the results. Similarly for the remaining unknowns. The second is suggested by the complete form of the normal equations as given in equations 4, Art. 74. According to this plan we compute the quantities [aa'], [ab^, ■ ■ • [<^/J, etc., sep- arately, 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 quantities. Thus with three unknowns, x, y, z, all the unlike coefficients are contained in + \aa\ X + [al)] y + [ac] 2 = [a/], + [bh'\'y+[hc]z = \hl-], + [cc]z={cl]. INDIRECT OBSERVATIONS OF ONE UNKNOWN 97 Instead, therefore, of computing 1 2 quantities, only 9 are neces- sary, as the remaining 3 can be at once written down. With ji unknowns the saving of computation amounts to I + 2 + 3 -}- • • ■ + (ji— i) = \n{n— i) quantities. If the observation equations are of different weights, the for- mation of the normal equations may be carried out precisely 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 coefficients [^paii], \_tab'], ... to multiply the observation equations by the square roots of their weights, and form the auxiliary equations i, Art. 75, 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. 77. Ex. 1. — Given the observation equations, all of equal weight, X = I X +y =3 X — y + z = 2 — X — y + z = I show that the normal equations are 4X + y =5 x+T,y — 2S = — 2y + 22 = 2 Ex. 2. — The expansions .v„ x.,, x^,, x^ for 1° Fahr. of four standards of length were found by special experiment to be connected by the following relations at a temperature of 62' Fahr. (m = one micron.). 1^ + Xi = 39-945 weight i + x^ = 5932 " '6 + r^ = 5-371 " 4 + jiTj — 1.0937:1-3 = 0.006 " 8 + AX^ - x,= - J .335 " 3 + X, -x,= + 14-833 find their most probable values. 6 98 THE ADJUSTMENT OF OBSERVATIONS [The normal equations are + 7 ^'i - 6x^= + 1 28. 943 + 72.000,1-2— 8.7500.-3— i2X^= -f 78.940 - 8.750.^+13.569.1-3 =+ 21.432 - 6.r, - i2.ooo.j-2 + ()Xi= - 84.993 and X, = 39.913, -r, = 5.932, x, = 5.405, x, = 25.075.] An example will now be given to illustrate the method of forming a series of observation equations : Ex. 3. —At Washington the meridian transits of the following stars were observed to determine the correction and rate of sidereal clock Kessels No. 1324, April 12, 1870, at 11 hours clock time. Star. Observed Clock Time of Transit, T. Right Ascension OF Star, a. T Leonis. V Leonis. /3 Leonis. Virginis. V Virginis. Virginis. /t. in. s. II 21 17.98 II 30 20.41 II 42 2S.57 11 58 38.15 .12 13 18.37 13 3 16-36 J. 16.00 18.51 26.57 36.20 16.37 14-39 Let x = corr. of clock at 1 1 hours clock time, y = rate per hour of clock. Now, from theoretical considerations * it is known that the equation X +y{T-ii) =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 ■*" + 0-35J' =- 1-98 X + 0.50^ = — 1.90 X + o.ji y =— 2.00 x+ 0.98 J/ =-1.95 X + 1. 22 J/ =— 2.00 X + 2.05J = - 1.97 For the remainder of the solution, see Art. 114. Ex. 4, — In the triangulation of Lake Superior executed by the U. S. Engineers there were measured at station Sawteeth East the angles * See Chauvenet, Asirono^ny, vol. ii. chap. v. INDIRECT OBSERVATIONS OF ONE UNKNO\YN 99 62' 59' 40.33" weight 5 64° II' 3492" " 7 100° 20' 29.12" " 4 37° 20' 49.55" " 7 36- oS' 55.86" " 4 Farquhar- Porcupine Farquhar-O liter Farquhar-Bayfield Porcupine-Bayfield Outer-Bayfield required the adjusted values of the angles. AH of the angles may evidently be expressed in terms of FSP, FSO, FSB. Let X, V, Z denote the most probable values of these angles, and let X', Y\ Z' be assumed approximate values of these most probable values, and .r, y. z their most probable corrections. Denoting the measured angles in order by M^, M^, . . . J/5, and their most probable corrections by v^, v^, . . . v^, we have X' 4- X = X = M, + v, 1-'+ j = V = /J/, + v^ Z' ^ z = Z = J/3 + 7/3 X' -x + Z' + z^ - X + Z = Mi+ v^ Y' -y -\- Z' + Z = - V+ Z = M,+ V, Fig. 3- 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 = V, + y + z = V2 weight 5 " 7 " 4 " 7 " 4 — X + z — 0.76 = 7/4 — y + z — 1.66 = z/j Hence the normal equations 12 j^- - 7^=_ 5.32 + I I J' - 4 z =— 6.64 — yx— 4J' + 15 5r = + 11.96 Solving these equations, we find x = - 0.05", J = - 0.36", z =+ 0.68". Hence, t/, = - o 05", v., = — 0.36", 7/3 = + 0.68", v^ = - 0.03", v^ = - 0.62", and the adjusted values of the angles are 62° 59' 40.28" 64° II' 34.56" 100° 20' 29.80" 37^ 20' 49.52" 36° 08' 55.24" lOO THE ADJUSTMENT OF OBSERVATIONS We might have used 7'i, 7/0, . . . 7/5, for the corrections without introducing the symbols X, J/, z at all. Ex. 5. If the unknown x occurs in each of the n observation ecjuations - X + b,y ■\- c,z -V . . ■ = h weight 1, — A- + b^_y + c.,s + . . . = 4 " I, these equations are equivalent to the reduced observation equations, b^y ^ c^z ^ . . . = /i weight i, bny + 6-0^ + . . . = 4 " 1, [^]j' + U->+. ■ •=[/] " -)• [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 + . . . = / weight /, we may write gax + qby + . . . = ql weight —^_- 78. 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 equation the sum of the coeffi- cients of X, V, . . . and of the absolute term / in that equation, and treat these added terms just as we do the absolute terms. Thus let s^, J-.,, . . . s,^^ denote the sums, so that a^ + l\ + Ci + ■ ■ •+/!== si , a., + b.. + r, + • • • + /., = ,v, , (Tn + ^>n + (',1 + ■ ■ • + hi = -^n Multiply each of these expressions by its a and add the prod- ucts, each by its d and add, and so on ; then [aa] + [ab^ -!-••• + [al] = [as] , [ob] + [bb] + ■ . . + [W] = [bs] , [.//] + [/>/] + • • ■+[//] = [/.v]. If these equations are satisfied, the normal equations are correct. Thus each normal equation is tested as soon as it is formed. PROPERTY OF MELVIN D. CASLFR, PORT PT.ATN N. Y. INDIRECT OBSERVATIONS OF ONE UNKNOWN loi Since \_aa'], \^ad'\, . . . [^?/] have been computed in forming the normal equations, the only new terms to be computed in applying the check are \_as^, [/^j-], . . . [/jt], [//j. Various modifications may readily be applied to suit individual tastes. Thus the absolute term may be placed 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 horizontal row equal to zero. 79. Forms of Computing the Normal Equations. — When the number of unknowns in the observation equations is large, or when their coefficients contain several figures, it is convenient to have a fixed form for the computation of the terms of the nor- mal equations. It lightens the labor much either in forming, solving, or in finding the precision of the unknowns from these equations, if the computation 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 [Crelle's], a table of reciprocals, a table of log- arithms, and an arithmometer, or machine for performing multi- plications 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. In the Coast and Geodetic Survey there are in use (1904) the Thomas or Burckhardt machine, the Brunsviga machine, and the Thatcher slide-rule. With the Crelle multiplication tables as good speed can be made as with the machine if the number of significant figures required in the products is so small that little or no interpolation is required in the tables. Form (a). With Crelle's tables, or with a machine, the prod- ucts aa, ah, ... are found directly, and all that is then to be done is to write them in columns and take their sums \ii(f\, \ab\ . . . With a Thomas machine, however, each product may be added to all that precede, so that the final values result at once. 102 THE ADJUSTMENT OF OBSERVATIONS Let us, for example, take the observation equations — 1.2 X + 0.2 y + o.g = v^ , + 3.0 :x; — 2.1 y + i.i ^ v^ , + 0.7 X -\- 1.6 y — 4.0 = Vs . Arrange as follows, the headings indicating the nature of the numbers underneath : a /; / s — 1.2 + 0.2 + 0.9 — 0.1 + 3-0 — 2.1 + I.I + 2.0 + 0.7 + 1.6 - 4.0 - 1-7 aa 1.44 9.00 0.49 + IO-93 ab - 0.24 - 6.30 + 1. 12 -5.42 al - 1.08 + 3-30 - 2.80 - 0.58 as + 0.12 + 6.00 - 1.19 + 4.93 - 542 bb 0.04 4.41 2.56 + 7.01 b/ + 0.18 -2.31 — 6.40 -8.53 bs — 0.02 — 4.20 -2.72 -6.94 • • • • • • • • - 0.58 • • • • • • • • • • • • -8.53 // 0.81 1. 21 16.00 /s — 0.09 + 2.20 + 6.80 + 8.91 + 18.02 Hence the normal equations, with the check all ready for solu- tion, are =+ 10.93 X— 5. 42 J— 0.58 +4-93» o=— 5.42 :x; + 7.01 ;y— 8.53 —6.94. 80. Fo7'm {b). If logarithms alone are used, form a table of the log coefficients of the observation equations as follows : INDIRECT OBSERVATIONS OF ONE UNKNOWN 103 log J,, log/;,, . • -log/,, logij, log a,, log b,,--- log/,, log 5,, log (7„, log /;„, ■ ■ • log /„, log .s-„. Write log a^ on a slip of paper and carry it along the top row, forming the products, log ajCi, log aibi, • • • log aj^, log 0,5, . Similarly with log a.^ form the products from the second row, log 0,^2, log aj)^, • • • log a.,!.,, log Cj^j, and so on till log a^ is reached. The numbers corresponding to these logarithms are next found, so that we have a^dj^, ajbi, ■ ■ ■ ajy, a^s^, By addition we find [iia], [ah], ■ ■ ■ [ill], [as], the coefficients of the unknowns in the first normal equation. Proceed in a precisely similar way with log />^, log /;.„ . . . <^„, omitting the term [al^'j already found ; with log r,, log r.„ . . . log c„, omitting the terms \^ac'\, [^r] already found ; and so on till the last quantity is reached. 81. Form (c). If we wish to use a table of squares alto- gether, then, since ab = is{(a+by - a^ - b-} and therefore [./.] = H[(« + /0'-']-M-M} (0 we form the square sums to4 THE ADJUSTMENT OF OBSERVATIONS [aa],[ia-{-bf],[(a-^cyi [bb],[(b + cy], ■[(a+iy],[{a + sy], [iillQ + sy], and perform the necessary subtractions. In doing this, first take from the table of squares the squares aa, bb, . . . II, ss, and sum them ; next write the coefficients a of .r on a slip of paper and carry them over the coefficients oi y, z, . . . , forming the sums a^ -\- b^, a^ -\- c^, . . . ; a.^ -\- b.„ a., + <:.„ . . . Take out the squares of these numbers and sum them. Proceed similarly with the coefficients oi jy, s, . . . Finish as indicated in (i). Thus in the preceding example, aa 1.44 9.00 0.49 10.93 bb 0.04 4.41 2.56 7.01 // 0.81 16.00 18.02 ss O.OI 4.00 2.89 6.90 a^ b {a + bf a + I {a + If a + s {a + sf i.o 1. 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 1.0 1. 00 7.10 27.79 27.69 [aa] -\-[bb] = 17.94 [aa] + [//] = 28.95 [aa] + [ss] = 17-83 - 10.84 - 1. 16 9.86 - 5-42 -0.58 4-93 = [ab] = [al] = [as] giving the same results as before. 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 equa- tions, each involving two unknowns, will show that to form the normal equations, using a log table only, 24 entries in the table INDIRECT OBSERVATIOXS OF ONE UNKNOWN 105 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 advantage that, as we deal with squares, all thought with regard to sign is done away with. Be- sides, 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. 82. It is worth noticing that whichever method of formation of the normal equations is adopted, labor will be saved by chang- ing the units in which the unknowns are .expressed if the coeffi- cients of the different unknowns are very different. Thus, suppose we had the observation equations, Check Sums. 1000 .V + O.OOOI }' = 4.11 1004. 1 lOI 999 .V -(- 0.0C02 y = 3-93 1002.9302 from which to find x andj' By placing x' = loo.T, y' = o.oi y the equations reduce to Check Sums. lo.v' -t- O.OI y' = 4.11 14.12 9.99 x' + 0.02 y' = 3.93 13.94 which are in more manageable shape for solution. 83. Before beginning the solution of a series of normal equa- tions we should consider whether the object is to find: (i) the unknowns only, or (2) the unknowns and their weights ; io6 THE ADJUSTMENT OF OBSERVATIONS 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 numerous, the method of substitution introduced by Gauss and the Doolittle method are more suitable. Each has its advantages. Both are quite mechanical in operation and are well suited for use with an arithmometer, which is as great a help in solving as it is in forming the normal equations. 84. The Method of Substitution. — For convenience in writ- ing, take three unknowns, x, j, ^, the process being the same whatever the number. The normal equations are [ad] X + [ah] y + [ac] z = [al], [ah]x+[hb]y+ [bc]z = [bl], (i) [ac]x+[bc]y+[cc-]z = [r/]. From the first equation """ [aay MM* Substitute this value in the remaining equations, and, in the convenient notation of Gauss, there result [/;^;.i] v+ [bc.i]z = [bl.i], [bc.i]y+[cc.i]z = [cl.i], (3) [bb.i] = [bb]-^^^[ab], [W.i] = [W]-b^[a/], (4) [cci] = [cc] — pM, [ac], where INDIRECT OBSERVATIONS OF ONE UNKNOWN 107 Again, from the first of equations 3, which value substituted in the second equation gives (5) where [CC.2] (6) (7) Having thus found c, we have j' at once by substituting in (5), and thence x by substituting for j/ and ;:- their values in (2). The first equations of the successive groups in the elimination collected are [aa] X + [ab] y + [ac] z = [al] , [bb.i]y+[bc.i]z = [bl.i], (8) [fc.2] z = [cl.2] . These are called the derived normal equations. Divide each of these equations by the coefficient of its first unknown, and \bc.\\ [W.i] ( . ^ \bb.i\ \bb.\\ [r/.2] [rr.2] 85. 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 io8 THE ADJUSTMENT OF OBSERVATIONS 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 [^/I's], l^^s'\, . . . [/.y], and operating on them in the same way as on the absolute terms [(i'/], [^^/]» • • • 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 coefhcients [^aa'], [<5^], ... of the normal equations, and [aa^, [Z-"/?. i], [^v.2], ... of the derived normal equations, are always positive. For [^^f-?], [/^^. i], . . . being the sums of squares, are posi- tive. Also [aa],[65.i] [aa] , [ah] [ab] , [bb] Co, b^ + (7j , &i 03 . b^ 2 + a positive quantity. (c) By equations 5, Art. 74, the residuals found by substitut- ing for X, J/, s their values in the observation equations must satisfy the relations [av] = [bz'] = • • • = o. (d) A very complete check is afforded by the different methods of computing [vv'] the sum of the squares of the resi- duals, (See Art. 106.) 86. Forms of Solution. — In applying the method of substi- tution to any special example it is important that the arrange- INDIRECT OBSERVATIONS OF ONE UNKNOWN 109 ment of the computation be convenient and that every step be written down. Experience teaches that simpUcity and uniform- ity of operation are great safeguards against mistakes. Form (a). Solution without logarithms. The following form has been found by experience to be con- venient. 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, a. The computation is divided into sections, each section being formed in a precisely similar way, and in each section one un- known is eliminated. Given the normal equations, No. X y 3 Check. Remarks. I. \Ifci ^^"■'\bb:r] IX. X [bc.i] VIII. \ee.x] [e/.,] [es.i] XI, \.cc.i\ [e/.2] [es.2] VIII.— X. [C/.2] [CS.2] |.-'-.2l r,v.ri no THE ADJUSTMENT OF OBSERVATIONS ■ ^ [cc.2] From Eq. IX. ^ _ [bc.i] [M.i] ^~ ^[bb.i]^ Ibb.i]' From Eq. IV, ^^_ W_,M + \^. [aa] [aa] [aa] To eliminate the first unknown, x. In the first line write the quotients -p — i, -t — J- , , . . that is, the coefficients of the first normal equation divided by [aa^, the coefficient of x in that equation. The first line is now multiplied in order by [<^^], \_ac'\, form- ing the second and fifth lines. In the third and sixth lines write equations II. and III. The fourth hue is the sum of the second and third, and the seventh the sum of the fifth and sixth. This concludes the elimination of x, and the results in the fourth and seventh lines involve _y and -c only. Take now these results and proceed in a precisely similar way to eliminate _y. The value of the last unknown, ;:;, next results. Now proceed to find j and x. Thus, substitute for s its value in the eighth line, and we have j; and for j/ and s their values in the first line, and we have x. 87. 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 dividing by [aa], [d^.i], \cc.2'], it is better to multiply by the reciprocals of these quantities, and, in order to avoid subtractions, to first change the signs of the reciprocals. We shall then have to perform only two simple operations — multiplication and addition. By trans- ferring the terms [jil^, [^/], [^/] to the left-hand side of the equations before beginning the solution, the values of the un- knowns will come out with their proper signs. INDIRECT OBSERVATIONS OF ONE UNKNOWN iii (2) Equations VI. and VIII. are the normal equations with x eliminated. An inspection of them shows that the coefficients of the unknowns follow the same law as the coefficients of the unknowns in the original normal equations with respect to sym- metry of vertical and horizontal columns. Hence in the elimi- nation it is unnecessary to compute these common terms more than once. Thus \bc.\\ 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. C3) In a numerical example it is evident that since \ad\^ \bb.\\, \cc.2\ do not in general divide exactly into the other co- efficients of their respective equations, and that only approxi- mate values of the unknowns can at best be obtained, it will give a closer result to divide by the larger coefficients and multiply by the smaller than vice versa. Attention to this by a proper arrangement of the coefficients before beginning the solution re- sults in a considerable saving of labor, as the successive coeffi- cients in the course of the elimination need not be carried to as many places of decimals to insure the same accuracy that a dif- ferent arrangement would require. Ex.— To make the preceding perfectly plain we shall solve in full the normal equations formed in Art. 79. (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. Re.marks. I. II. + '093 — 5-42 -S-42 -I-7.01 4-0.58 + 8.53 + 6.09 -|- 10.12 III. + I — 0.496 + 0.053 + O.S57 I. -^ 10.93 IV. II. .... -f 2.688 -i-7.01 — 0.288 + 8.53 — 3-o'9 4- 10.120 III. X — 5-42 II. V. . . . . + 4-322 + 8.818 + >3'39 II. - IV. VI. . . . • + ' + 2.040=;;/ + 3-040 v. -h4-322 VII. III. + "i" — 0.496 — 0.496 — I. 01 2 + 0.053 - ,.508 + 0.557 VI. X —0 4';" in. VIII. + I -|-I.c6s = .r + 2.0(.S III. \ii. Hence X = i.oO ;' .04 112 THE ADJUSTMENT OF 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 y / Check. Remarks. I. II. 0.0915 + 10-93 — 5-42 — 5.42 + 7.01 — 0.58 -8.53 — 4-93 + 6.94 .... III. . — I + 0.496 + 0.053 + 0.451 I. X —0.0915 IV. II. V. 0.2314 . . . — 2.688 -j- 7.010 + 4-322 — 0.288 — 8.530 — 2-445 + 6.940 + 4-493 III.X-5-42 II. II. + IV. — S.818 VI. . . . . — I + 2.040 = y — 1.040 V. X — 0.2314 VII. III. VIII. .... — I — 0.496 + 0.496 + 1. 012 + 0-053 — 0.516 + 0.451 — 0.065 VI. X 0.496 III. III. + VII. — I + 1 .065 r= .r In order to find the values of the unknowns to two places of decimals, the computation should be carried through to three places, and the third place dropped in the final result. 88. Form {I?). The logarithmic solution. As an example of the logarithmic method let us take the gen- eral form of the preceding example, when R and 5 are substi- tuted for the absolute terms 0.58 and 8.53 respectively. 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 oper- ation the slip would have on it 8.96138 ;/, 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 consequently 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 x, /, . . . and the other for the coefficients of R, INDIRECT OBSERVATIOxNS OF ONE UNKNOWN 113 S, . . . Though unnecessary in our example, it is inserted for illustration. It will be noticed that the coefficients of R, S in the values of X, y follow the same law of symmetry as the normal equations. A little consideration will show that this is always the case. Hence, attending to this, we may shorten the computation by leaving out the common terms. We have, therefore, one term less to compute for each unknown, proceeding from the last to the first. The case is precisely analogous to that of Art. j6. x y Check. K 5- Check. Remarks. I. II. III. IV. V. VI. II. VII. VIII. IX. X. XI. 10.93 -5.42 1.03862 (8.96138 «) — 5.42 + 7.01 0.73400 « 9.69538 0.42938 « — 2.688 +7.010 + 4.322 0.63568 (9.36432 «) — I — 5-51 — 1-59 0.74115 n 9-70253 0.43653 n — 2.732 — 1-590 — I 8.96138 + 0.091 9.69538 « — 0.496 — I — I — I 0.0 9.36432 + 0.231 9.05970 + 0.115 + I + I 8.96138 n — 0.091 9.69538 + 0.496 + I + 1-40 0.17493 953925 « — 0.346 9.23463 n — 0.172 — 0.091 log I. III. — log 10.93 Nos. IV. -log 5.42 Nos. II. VI. + 11. log. VII. VIII. -log 4.322 Nos. IX. +log5-^' Nos '°-''J — 4-322 0.6356S « 0.00000 + • — 0.496 0.69548 » 9.05980 + 0.115 8.75518 + 0.057 + 0.091 + 0.148 XII. XIII. — I . . . + 1 + 0115 — 0.263 .•.X = 0.148 7? + 0.1 15 S, _K = o. 1 1 5 /? + 0.23 1 S. Substituting for A' and J>' their values, we have, as before, X = 1 .06, y = 2.04. 8q- Ex. I. — In tlie elimination of ;/ normal equations by the method of substitution, show tliat the total number of independent coefficients in the ..,,,., , ,. . fiin + i)(« + s) origmal and derived normal equations is ^ [The sum is ^ {1.4 + 2.5 + . . . n {n+ 3)}.] Ex. 2. — If the elimination of the unknowns in the normal equations is carried out by the method of substitution, the product [aa], [M-i], [cc.z] . . . has the same value whatever order has been followed. 114 THE ADJUSTMENT OF OBSERVATIONS 90. A method of indirect elimination by successive trials and approximations has been suggested by Gauss. It will be found in Coast Survey Report, 1855, Appendix 44; von Freeden, Die Praxis der Methode der kleinsten Quadrate y p. 96 ; Vogler, Aus- gleicJiungsrecJinung, p. 129. It is not given, because our experience has been that it is in general intolerably tedious. 91. The Doolittle Method of Solution. — This method of solution is due to Mr. M. H. Doolittle of the Computing Division of the Coast and Geodetic Survey.* In it there is a combination of improvements on the Gaussian method of substitution. Its advantage lies mainly in the arrangement of the work in the most convenient form for the computer. This makes the solu- tion more rapid than by the other method, the gain in speed be- ing the more marked the greater the number of equations. In order to make the process employed readily followed, the solution of the three normal equations, [a./] A- + [(//)] V + \ac'\z = \ar\, lah-\x^\hh'\y^\hc-\z=\bl\, \ac\x + \hc'\y\\cc'\z= [r/] , is given in general terms according to this form, i The coefficients and absolute term of the first equation arc written in line i, Table A. The reciprocal of the diagonal coefficient \aa'\ is taken from a table of reciprocals and entered in the front column with the minus sign prefixed. The remain- ing terms of line i are multiplied by this reciprocal, and the products written in line 2. This gives x as an explicit function of J/ and z. The coefficients and absolute term of Eq. 2 (omitting the co- efficient of x) are written in line i, Table B. The terms in line 2, Table A, beginning with that under j, 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 written in line 3, Table A. * See Appendix 8, C. and G. Survey Report for 1878, pp. 115-118. INDIRECT OBSERVATIONS OF ONE UNKNOWN 115 Line 4, Table A, is found from line 3 in the same way as line 2 was found from line i. This givesj/as an explicit function of ^r. The coefficients and absolute term of Eq. 3 (omitting the co- efficients of X and j') are written in line 3, Table B. The terms in lines 2, 4, Table A, beginning with those under c, are multi- plied by [«r x'\ r , rr y = y + y , z = / + z'\ B Recip. X y z I [aa] ^[aa] X =: + [ab] [ab] Vaa] ^[ac] [ac] [aa] -[al] .Ul] '^[aa] 1 [bb.A y — + [5f..] [bc.^] -[bU] AblA '^[bb.^] [M.i] [bb.{] I + [c-..2] z = -VI -A [Cl.7] ^ \CC.2] [a-.2] y z + [M] +M -[bl] • a • ^'U-^' \.cc] -[>■/] • • • -sm ^'^^^"^ ■ • • -ISifjt- +'£]=-> D y z [ab] [aa] • • • [ac] [aa] [bc.^.] [bb.i] X y z A.an '^[aa] [aa]y [bl.x] ^[bb..] [bc.z] [ib.i] [C/.2] z' y' x' INDIRECT OBSERVATIONS OF ONE UNKNOWN 117 X y ~ -[al\ -[bc.i\ -[ci-A F X > z + [aa] ["-^J y. 1 [^/-i]. z" y x" 93. Addition of New Equations. — It often happens that after the adjustment of a long series of observations, additional observations are made leading to additional equations. To make a solution de novo 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 [adX X + [bd] y + [cd] z + [dd] w = [d/] , ( 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 [aa] X + [ab] y + [] y + U>^] ^ + [M] w = [hi] , + [cc]z-{- [cd]w = [cl], + [dd]w = [dl]. 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 approximations to the values of X, y, z resulting from the above four normal equations. Sub- stitute in (i), and \ad] X + [bd] y' + [ r/] ^ 4- [dd] 7c = [d/]', (2) where x', y, ,c^ are corrections to the approximate values of X, y, z. The solution is now finished as follows : Form Table C {a) by adding the extra column zv to Table C. II 8 THE ADJUSTMENT OF OBSERVATIONS The term — i — i is found by multiplying \_ad] by the first re- [aa ] ciprocal. The coefficients of the new equation, (2), are written in the first line of Table G. Since corrections 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 za, that is, \_dd.T,'], in 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 1 • \(^n' ■ absolute term — [d/]' of the new equation, that is, r ,, y is an approximate value of zv. This value of zv is multiplied by the terms in the last column of Table C (a), and the products are written in the first line of Table H. Column s gives the cor- rection to 2. Table H is now completed in the same way as Tables D and F. C (a) G y z IV I [aa] I [ab] [aa] • • • • • > [aa] [bc.^] [ad] [aa] [bd.i] [bb.i] I [bb.y] [bb.'i] [cd.2] [cc.z] [CC.2] I [dd.3] [ad] [bd] •r ^[ad] \aa\ [bd.i] [cd^ [tc] [aa] [bc.i] [ad] [3-759 — -29 z + 2.354 — -433 — 2.157 + -356 -3-658 + -673 > + 4-323 — •7'33 3 4 7 8 9 12 13 14 15 + X •9 .8 !4 ?6 y -- I --12 — 3 759 82 54 5' z — 1. 213 — -944 + 3-470 + 1.900 + .626 + 7- '24 — 1. 019 — .768 — 4.101 + 2.856 + i-4('7 + 8-742 — 2.950 — '-254 + 0.685 + 1.584 + 1-539 — 3.104 5 6 10 1 1 —.114 + 8-77 + 5 -996 — .6S4 + 4.538 — -5'73 7 8 16 •7 —.809 .... 2 =: — -57 The first column in each of the above tables gives the number of the line, and the second the order of procedure. It is to be observed that the numbers in Table B have but a single use, while those of Table A are used over and over ; and when the number of equations is large, it is of great advantage that they should be thus tabulated by themselves in a form com- pact and easy of reference. c D Reciprocal. X y z IV X y z 1 2 3 4 -..84 -.165 — .114 — .809 — .401 + .807 — .29 — -433 + •356 — .6S4 2 3 4 + -673 ■ + .2468 — .1024 + -3524 --7'33 — .2029 + .0368 -.879 — -5'73 + -3899 — -"7 --57 1 =yi = «'i ="-' "''1 + "-"7 I20 THE ADJUSTMENT OF OBSERVATIONS E F I 2 3 4 ■w JK y Z I 2 3 4 — .0165 • • • + .0.69 + .0066 + -0047 — -0133 — .0068 + -0039 + .007 1 + .0084 -j- .0102 + .0296 I 2 3 4 + .0030 + .0103 + .0146 + .0071 - .0039 .0085 .0052 .0176 + .0018 + -0163 — 0239 ) + -0235 + .0181 — -0154 + -035 => As the multiplication is performed by Crelle's Tables, no mul- tiplier is allowed to extend beyond three significant figures. Other numbers may be extended to four ; but it would be a waste of time to extend any number farther, except in the pro- cess of substitution for the determination of residuals. If an arithmometer is used, four significant figures may be re- tained throughout. It will seldom be advisable to retain more significant figures than this. For the sake of perspicuity in explanation and convenience in printing, we have here made some slight departures from actual practice. For instance, in the solution of a large number of equations, it would be inconvenient to pass the eye and hand out to a vertical column of reciprocals ; and they are better written in an ol^lique line near the quantities from which they are de- rived and with which they are to be employed. By this process, Mr. Doolittle solved in five and one-half days, or 36 working hours, with far greater than requisite accuracy, 41 equations containing 174 side coefficients counting each but once, or 430 terms in all. 95. Suppose that after the solution of the foregoing equations and the consequent adjustment a new condition is established, resulting in the following normal equation : o = — 2.0475 «' -\- 0.8362 X -f 1.85677 — 1.3 149 - + 8.2527 ;/ — 1.8372 ; with the addition of the term — 2.0475 ^^ to the first of the previous equations, +0.8362 ii to the second, etc. The absolute term 1.8372 is supposed not to be an original discrepancy, but an out- INDIRECT OBSERVATIONS OF ONE UNKNOWN 121 Standing residual, after the foregoing solution has fully entered into the adjustment, as is generally the case with azimuth and length equations, in a triangulation adjustment. C u; 1 (1 Recip- rocal. .r y 2 u w jr y z u -..84 -..6s — .114 -.809 — -.45 — .401 + .807 — .29 — •433 + •356 — .684 + •377 — .274 + .0307 — .282 I 2 3 4 5 — 2.05 + .836 + .822 + .166 + 1.851 — 1.654 - .481 — '•SIS + .888 + -59' + -184 + 8.253 — ^773 — -455 — .008 — .098 - .269 .... + .348 + 6.gig H 7V X y z u I 2 3 4 S + .1007 + .0326 + .0482 + .0469 + .228 — -0732 — .0268 — •0173 + .0082 + -OS'S + -0597 — •0753 ( + -267 1 = «1 ' = J'3 = -»-3 = «'3 — .117 The Precision of the Most Probable {Adjusted) Values. 96. The problem now before us is to find the p. e. of the un- knowns, .r, J/, ... as determined from a series of normal equa- tions. If the observation equations are reduced to the same unit of weight, which we shall take to be unity for convenience, the general form of the normal equations is \aa\ X -f [a6] v + • ■ • = []v + • • • = [W]. (0 Let r = the p. e. of a single ob.servation. ^x. ''v. • • • = the p. e. of .Y, )', . . . Pxj py ,• • • = the weights of .r, y, . . . From Art. 47 we have 122 THE ADJUSTMENT OF OBSERVATIONS PxrJ' = Pyry^ = . . . = /-2. (2) In order, therefore, to determine r^., r^, ... we must make two computations, one of the weights p^, py, . . . and the other of ;', the p. e. of a single observation. It is evident from an inspection of the normal equations that X, y, . . . are linear functions of /^, l^, . . . Let, then, X = aj^ + a/o + • • • + a„/« = [a/] , y = fi,k + N-, + • • ■ + PJn = W , (3) in which a^, a,, ...; /3^, ^,,, ...;... are functions of «,, d^, ... \ a^, b.„ ...;.. . their values being as yet undetermined. Now, r being the p. e. of each of the observed quantities M^, M2, ■ ■ ■ M.,„ must be also the p. e. of /,, 4 . . . 4» which differ from M^, M,,, . . . M„ by known amounts (see Art. 74). Hence, since /,, /.,,... /„ are independent of each other (Art. 13), r.2 = ,-2[aa],;-/ = r[/?;8],. . . (4) and therefore ^^=[^' ^""^wy " ' ^^^ We shall first of all determine the weights /,^, py, . . . 97. The demonstration may be carried out simply by the ap- plication of the principles of undetermined coefficients. Thus, substitute [a/], [/3/], . . . for ,r, j, ... in the normal equa- tions (i), and [aa] [a/] + [ab-] [(31] + • • • = [al], [ab] [a/] + [bb] [Pl] + . . • = [bl], (6) or, arranging according to Z^, /^, . . . {[aa]a, + [ab](3,+ a,}l, + {[aa~\ a^ + [ab] /S, + • • --ajl^^ {[ab]a, + [bb]^, + - ■ --M/i + {[] (3^ + [ab]a, + [bb]ft,+ - bi = o, — b., = o, (7) are simultaneously satisfied by the same values of Uj, U-2 J ' • • ' Hi' r'2' •••!••• Multiply the equations of each set by a^, a.^, . . . in order, and add ; then necessarily [aa]=i,[a(3] = o,[ay] = o, ... (8) In a similar way, multiplying by d^, b^, . . . \ c^, c^y . . . , etc., and adding, there result [6a] =0, [6/3]= i,[6y] = 0, . . . [ca] = o, [c(3] = o, [cy] =1, . . . Again, multiply the first set by a,, a., • . • , the second by yS,, y8.„ . . . , and so on, and add, and we have the sets of equa- tions, I [aa] [aa] + [ab] [a/3] + ■ < [ab] [aa] + [bb] [a(3] + ■ = [h.\\ [abf aa_ • X being found first, Px = [««" - [abf [bb] 5 X Pv = X r n ' (7(7 A = [aa] [bh — ab] [ab With three unknowns, x, y, .■:, performing the elimination of the normal equations in the order -c, y, x, we have Pz = [f"^-2] , [bb.i-\[cc.2-\ ^'- [cc.q ' _ [«'/] [bb.-i] {cc-2] ^' ~ m[^^]^M [be] ' which expressions arc easily transformed into \ ^' ~ [aa] [bb] - [abf ' _ X ^^ ~ [aa] [cc] - [acf ' [26 THE ADJUSTMENT OF OBSERVATIONS where X = [aa] [bbl [cc] + 2 \ab'\ \bc\ [ac] - [aa] [bcY - [bb] [acf - [cc] [ab]\ 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 cumber- some. Ex. — To find the weights of the adjusted angles in Ex. 4, Art. 77. Here X = 12 X II X 15 — I2X 16 — II X 49 = 1249 and J, ^-49 149 = 8.4, , 1249 A =-^ = 9-5, 1249 A = -^ = 95. If Ux, Tiy, Ui denote the reciprocals of ^;c,^y,/2 respectively, then Ux = 0.1193, Uy = 0.1049, Ui = 0.1057. loi. Modification of General MetJiod. — To carry out the method of Art. 98 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, z, and to the ordinary form of the normal equations as arranged for solution add the columns 100 o I o 001 the check being carried throughout. INDIRECT OBSERVATIONS OF ONE UNKNOWN 1^7 Perform the elimination exactly as stated in Art. 86, and find the values of the unknowns in the usual way. We have then .r :• A' -s- /• Check. \aa [^^1 [tc] [an r ' o o as] + I ab M l6c W] o 1 o 6s + I \ac 6c [cc \.cn -< o o I .-.■ +, ai] [ac] [al] I ... 1 ' "" n aa] aa\ [»'A f ^M.l] 6c.i 6l.x'\ r ^1 o 6c: i] cc.i] ./..] [tc] ■> . . . [6c.,] [^/.,] I . . . [66.i] CC.2] [66. i] cl.z [66.,] [66.1] ^ [cC.2] Q\ . • . [ccA L [cC.2] • ■ * where o = o = [aa] + ''^ ' M , [bc.i] [aa] [bb.i] Rl + ^2 [bc^ [bb.l] + S. Now, taking the first column in the table under the heading R, and attending to Art. 98, we have M= ""' M = [re. 2] ' [bb.l] [bb.l] M [bb.l] [rc.2]' I [«^]r..«i ['"■] ^'- = '^'"^ = M-Mf"^^ + A'r' + r:- M [aa] [bb.l] [cc.2] Similarly for the column under S, 128 THE ADJUSTMENT OF OBSERVATIONS [M = [CC.2] ' ^ ^ -' [M.I J ICC.2\ and for the column under T. "z = [yy] = [cC.2] Also it is evident that [d.2] [cC.2] The forms of the expressions for [aal, [/3/Q], [77], . . . 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 ;/,,, ?/y, ?/, 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 re- quired. 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 re- quired, with a trifling increase of work. 102. Ex. I. — Given the normal equations ' 12 ;i- — 7 z = J? + 11 y — A 2 = S - T X - 4y + IS 2 = T to find the weights oiy and z. INDIRECT OBSERVATIONS OF ONE UNKNOWN 129 X y 2 S T + 12 - 7 I + II - 4 + II - 4 I - 7 - 4 + 15 - 0-5833 - 4 + 10.9169 - 0.3636 + 9-4625 I +-;■■) + 0.0909 ) + 0.3636 j + 0.0384 i+ ■ ■ ■ ="A- (4) 132 THE ADJUSTMENT OF OBSERVATIONS 105. To Find the p. e. r of a Single Observation. — If the errors A were known — that is, if the n observation equations were <^i^o + ^o'o + • • • - A = ^1. «2^o + b.,y^ + ... — /, = A,, (i) where ,1-,, j^ . . . are the true values of the unknowns — we should have at once /^" n But we have only the residuals v with the observation equations a^x 4- l\y + • . ■ — /j = 7\, a^x + b.j +...—/, = z;^, (2) where x, y, . . . are the most probable values of the unknowns. We must, therefore, express [A"] in terms of the residuals v in order to find \i. From the two sets of equations, by subtracting in pairs, Aj = v^ + «j (.To — x) + b^ (j'o - y) + ■ ■ ' A3 = 7'3 + a.,_ (a-o - x) 4- ^2 (J'o - J') + • • • (3) Now, taking the m. s. e. yi,., /u-^, ... to be the errors of x, y, . . , , that is, to be equal to x^ — x, f^ — y, . . . , we have from Eq. 4, Art. 96, and therefore Aj = v, +,i(a, VH + ^ VC^ +•••). A. = V, + IX (a, V[H + ^3 V[i8y8j +...)• Squaring, adding, and attending to equations (9), Art. 97, we have approximately, n^ being the number of unknowns, [A^] = [z'^] + n,f.\ (4) Putting [A-] = ;//*", there results INDIRECT OBSERVATIONS OF ONE UNKNOWN 133 ^== ^M__ and ,.= . 6745 \/J!l;. (s) the expression required. Reasoning as in Peters' formula, Art, 32, we easily deduce from (4) r= 0.8653 -=J^=, (6) yn {// — fi-) which is known as Liiroth's formula {Astron. NacJir., 1740). When ;/,. = i, equations (5) and (6) reduce to Bessel's and Peters' formulas respectively (Arts. 29, 32). 106. Methods of Computing [^r]. — {a) The ordinary method is to substitute the values of the unknowns found from the solu- tion of the normal equations in the observation equations, and thence find z\, t'.„ . . . The sum of the squares of these resid- uals will give [zr] . The residuals having to be found, for the purpose of testing the quality of the work this method of computing [zr] is on the whole as short as any. As checks on the values of [t'"] found in this way the follow- ing are of value : {b) If we multiply each observation equation by its v and take the sum of the products, then, remembering that [(rv'\ = o, [dv'j = o, . . . , we find (c) If we multiply each of the observation equations by its / and take the sum of the products, [a/] X + [l>l] y+ ... -[//] = [vl] = - [v"]- {d) We have for two unknowns, x and j, [7/^] = [(ax + by - 1Y\ = [aa] x'' + 2 \a/>]xy + [/V'J v- - 2 [a/] x - 2 \/>/] v + [//J 134 THE ADJUSTMExNT OF OBSERVATIONS + [//..l-[|^[M.] and generally for tn unknowns, \f\ = [(ax -\.by+ ■ ■ ■ - /)-] V HI' M/ Now, from (9), Art. 85, the coefificients of [aa'], [l>d.i^, . . . are each equal to zero. Hence [vv] = [//.;;/] [alf [bl.if [r/.2]2 = [//] [aa] [bb.i^ [cc.2] This expression was first given by Gauss (De Elenientis Ellipticis Palladis, Art. 13). Its form suggests that if we add an extra column to the normal equations, as shown in the follow- ing scheme, we shall find [z/^] at the same time as the first un- known. This is analytically very elegant, and, as the check (see Art. 85) can be carried with this column through the solu- tion of the normal equations, it may be used for finding [z'^J, if one is computing alone. Only one extra term [//] has to be computed while forming the normal equations. The scheme is as follows : INDIRECT OBSERVATIONS OP ONE UNKNOWN 135 X y z [aa\ • • + [bb] • • • + M ^\bc] + Vcc] al] bl] cl] ill] I • • _^{ab] \aa] + \bb.\] • • • ^\ac] [aa] + [^^.i] + icc.x] [al] [aa] \bl.\] \_cl.\] [11.] = [//] - [f^] [./] • • • ■ I ,\bc.x] ^\bb.^] \cc.z] \bl.A [bb.\] [c/.2] 107- Ex. I. — To find the m. s. e. of the adjusted values of the un- knowns found in Ex. 4, Art. 77. The first step is to find [/<:/] 2 = -\- 11.96 X + 0,68 = 8.13 i^iAA = o 10.79 pjj^ = o /3AA = AAA = 7 X (0.76)^ = 4-04 PJJ^= 4 X (1.66)' =11.02 15.06 4.27 = Li^^'] (^) We find lpll\ = 15.06. The solution of the normal equations, with the extra column for \_pll\ added, would be, according to the foregoing scheme, X y z 12 • • + II • • • • • • - 1 - 4 + 15 - 5-32 ' - 6.64 4- 11.96 + 15.06 > 2.36 I • • • • + II -' 0-583 - 4 + 10.917 - 0.443 J - 6.64 + 8.857 + 12.70 (= 15.06 - 2.36) I - 0.364 + 9.462 • • • — 0.604 + 6.422 + 8.69 (= 12.70 - 4.01) I \pvv\ = + 0.68 + 4-3° (= 8.69 - 4.39) Mean value of [pv-] = 4.29. Hence (Art. 105) =v/^=-' and (see Ex., Art. 103) M. = 1.47" Vo.i 193, ixy = 1.47" \/o. 1049, M. = 1.47" V0.1057 = 0.51" = 0.48" = 0.48' IXDIRECT OBSERVATIONS OF ONE UNKNOWN' 137 Ex. 2. — Show that [\v] = [v^. [Multiply equations i, Art. 105, by.7'1, 7'.,, . . . and add. Then, since, [az'] = o, [i7'] = o, . . . .-. [At-] = - [/T-] ]. Ex. 3. — Show that L^-" ^ ^ [aa] [M.I] [Form the normal equations from equations (3), Art. 105, and [aa] U-o - -r) + [ad] ( jo - j) + ■ • ■ = [« ^], [ad] (a-„ - X) + [dd] ( Jo -y) +...=[/, 4 since [a7/] = [dv] = • • • = o. Hence, from Art. to6, -" [«aj [od.i] •" Ex. 4. — From the equation . [a/]x+[d/]j+ [//] = -[7'=], and deduce [aa] [dd.i] [cT.2] ^~ [M.I] "^[rf.2] -^ [aa] [dd.i] Ex. 5. — Prove that [av] = [fizz] = • • • = o. Ex. 6. — From ^''i-\-^i [aa] [dd.i] deduce the formula n — Hi 108. To Find the Precision of any Function of the Ad- justed Values X, Y, ... — This is the more general ease ot the problem just discussed. The method of solution is : Fit's t, to find r, the p. e. of an observation of weifj^ht unity, and next /y,. the weight of the function, whence the p. e. of the function is given by r "^up, where Uj.^ is the reciprocal of Pp. 138 THE ADJUSTMENT OP OBSERVATIONS The value of r is computed from (5) or (6), Art, 105. Next, to find Up. Let the function be F=f{X, F, . . . ) (i) in which X, F, . . . are functions of the independently observed quantities Mj, M^, . . . M„. Reducing the function to the linear form, we have, adopting the notation of Art. 74, J^=/(X' + x, Y'+y,. . .) SF BF =/(X\ F', . . . ) + ^,a- + ^y,y + . . . (2) or, as it may be written, ^F= G,x + G,y + . . . (3) Now, since x, y, . . . are not independent, but are connected by the equations \aa']x + {iit>\y + • • • = [a I'], [ab]x + [bb]y+ . . . =[/'/], we must get rid of this entanglement by expressing these quan- tities X, y, . . . in terms of l^, 4, . . . , which are independent of each other. From Arts. ()6-<^'j we may write X = [a/], y = [^/], . . . where a^, a^, . . . ; yS^, yS^, . . . ; . • . are functions of a^, b^, . . . ; a.^, b„, ...;.. . Hence, substituting in (3), dF= {G,a, + G.fi, + . . .) /j + (G^^a, + G.fi.^ + . . .) /^ + . . . and, therefore (Arts. 13, 47), since the observation equations have been reduced to weight unity, Up = {G,a, + G,p, + ■■■)' + (G,a^^ + G,I3, + . . .)2 + . . = G,'[aa] + 2 G,G^[aft]+ • • • + G,' im + • • • (4) + • • • where [^aa] , [a/S] . . . may be found in the manner indicated in Arts. 98, loi, or 103. Hence ?/^ is known. IVDIRECT OBSERVATIONS OF ONE UNKNOWN 139 109. {b) Eq. 4 may be written Up =G,Q,+ G,Q, + • . . + (9„(2„, (s) where (2i = [ott] G, + [a^] G. + - ■ - Q^^^[ati]G, + {mG. + - ■ • (6) that is, where (see Eq. i, Art. 103) G, = [aa] Q, + [../.] <2, 4- • • • G, = [ab] Q, + [M>] Q,+ - ■ • (7) Hence the weight of a function G,x + G.,y+ ■ ■ - of several independent unknowns x, y, . . . is found from Up = [GQ-\, where Q^, Q.^, . . . satisfy the equations M<2i + MG. + • • • = <^i. [ab] Q, + [bb] Q, + - ■ ^G,, Therefore, we conckide that, if in a scries of obsei'vation equations the values of the unknowns x, y, . . . are required, as tvell as their weights or the zveight of any function of them, these results can be found at one time by making a solution of the normal equations for finding x, y, . . . in general terms, and then sub- stituting for [rt:/], [^/], . . . their numerical values on the one hand and the values of G^, G.,, . . . on the other. 1 10. {c) This result may be stated in other forms. Thus, from Eq. 4, by substituting for [^aa], [^a^'], . . . their values from Art. loi or by substituting for Q^, Q.,, . . . their values in (6) as expressed in Art. 10 1 we have, after a simple reduction, _ G,^ ^ {G,R, 4- G.^' {G,R, + G.S.^^G^l Comparing this expression with (d), Art. 106, it is evident that 140 THE ADJUSTMENT OF OBSERVATIONS the several terms are such as would result from the following elimination (Ex. three unknowns) by finding the products of the quantities bracketed : X y z [aa] [ab] [ac] [ab] [bb] [be] [ac] [be] [ee] - G2 G, ^i = o, -^2 = - 0.0440 iiF= (0.0833)2 X 12 + (0.0440)2 X 9.4625 = 0.1016 Also m/= 1.47" ^o- 1 02 = 0.47". Ex. 2. — Given the observation equations a^x + b^y = /i, an-v + bny = In, to find the weight of fx + gy. [The normal equations are [ad\x + [ab]y = [«/], 1 13. 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^ is M^ + V . From Art. 74 it follows that the weight of M^ + v^ is the same as the weight of l^ + Vy Now, /j + z/^ = a^x + b^y + ■ ' • (i) Hence if P^ is the weight of the adjusted value M^ + t\, that is, is the weight of the function a^x + b^y + . . ., and/j, /g . . . are the weights of /^, Z^, . . . , we have ^ = a,Q,-^b,Q,+ - • . (2) -^ 1 where (see Eq. 4, Arts. 104, 109) Q^ = [uaa] a^ + \ilo.^\ /^i + • • • = U^o.^, Q^ = [uafi] a, + [.//8/3] /;^ + . . . = u,li,. (3) Therefore by substitution of Q^ Q., - • ■ in (2), ^ = a,a, + b,l3, + -^ 1 INDIRECT OBSERVATIONS OF ONE UNKNOAVN 143 Similarly P ^ = a^a, + K_^^_^ + • • • Hence by addition r|l = [aa] + [/'/8] + . . . to //i terms = ;/j, the number of independent unknowns. Hence the average value of [///^] is njn, or in words, //;^ average value of the ratio of the weight of the observed value of the quantity to that of its adjusted value is the ratio of the number of independent iniknowns to the number of obsef-i'ed quantities. This result may be very readily derived directly as follows : In (i) put [a/] ior X, [/S/] for j, . . . , and we have /, + V, = {a,a, + b,p, + ■■ ■)h + {'h<^2 + I' A + ■ ■ ■)k + - ■■ Hence, since l^, I.,, . . . are independent, ^ = (.^,a, + b,p, + . . -r-j^ + {a,a, + i>A + ---yj + --- + i,^\[ua(3]a,-\-[um^-^■■ ■} = u, (a^a, + b,^, + • • • ) (4) as before. Ex. — To check the weights of the adjusted values of tlic angles found in Ex. 4, Art. 77. The weights of the measured values of the angles are 5. 7, 4, 7, 4 The weights of the adjusted values are (Ex., Art. 100 and Art. 112), 8.4, 9-5. 9-5> 9-8, 7-5 Ai 5,7,4,7,4 ^^^° 8T4 + r5''9r5""9':8 + r5 = 3 = the number of independent unknown.s, as it should. 144 THE ADJUSTMENT OF OBSERVATIONS Two Special Artifices. 114. The labor of solving and finding the values of the un- knowns may be often shortened by taking advantage of some principle inherent in the form of the observation equations them- selves. 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 x^-b^y = /i, weight /i, X + b.2.y = 4, weight /2- Forming the normal equations in the usual way, we have [/] x + [pb] }' = [/>/ 1 [pb]x + [/>P]y = [pb/], whence eliminating x, Now, if the first normal equation is divided by [/], so that and from this equation each of the observation equations in suc- cession is subtracted, there result the equations, The normal equation for findings from these equations is, \[p] [#'] - [#']| y = [/] [/^'^] - [^^] t^^l' the same as results from the elimination o^ x in the normal equations. INDIRECT OBSERVATIONS OF ONE UNKNOWN 145 This process is specially convenient if the original observation equations are numerous, and the coefificients /;, , b,,, . . . and the terms /,,/!,, . . . are large and not widely different. Ex. — To solve the equations in Ex. 3, Art. 77. The mean of the equations is X - o.gyy = - 1.97. Subtract each equation from this mean equation, and + 0.62 y = + o.oi, + 0.47 J =- 0.07, 4- 0.26 y = + 0.03, — 0.01 J = — 0.02, — 0.25 jK =+ 0.03, — i.oS y = 0.00. The normal equation formed from these equations is + I 91 J = - 0.27, and .'. y = — 0.014'. Substitute for j its value in the mean equation, and x=- 1.95'. 115. Again, we may take advantage of the presence in the problem of some arbitrary quantity to which a convenient value may be assigned. Thus, to find the difference of the coeffi- cients of expansion of two standards A and B from observed differences of length at certain fixed temperatures. Let X = the excess in length of A over B at an arbitrary tem- perature /y, y = the excess of the coefficient of expansion of A c)\'er that of ^, A> 4. • • • = the observed differences in length at temperatures /^, /,, . . , and whose weights are/p/._,. • • • We have then the observation equations ^ + (/j - /„)7 - /, = t/j, weight /i, ^ + (4 — Qy - 4 = ^'2' weight /.,, (i) 146 THE ADJUSTMENT OF OBSERVATIONS and the normal equations [/>]x+ {[/>f]-[/>]Qy = [J>n {[//] - [/] ^,\ X + L/ (^ - 4)^'] J^ = [(^ - O /^]. (2) As the vakie of /„ is arbitrary, the normal equations will be simplified by taking it equal to the weighted mean of the tem- peratures ; that is, / - ^ • (3) and they will then become [/].r = [//] from which x and j 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/ if the mean value of x found from [/] ~r = [//] is substituted in the observation equations before the normal equation in j/ is formed. We should then have from which to find y. It is evident that the value of j found in this way is the same as before. For [/ {f - Q (^ - ^^)] = [/ (^ - o n - \[pn - [/] Q ^ = [p{t- 4) n^ since the coefficient of x is equal to zero from Eq. 3. The quantity / — x comes in very conveniently in computing 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. /i of a single observation, INDIRECT OBSERVATIONS OP ONE UNKNOWN 147 llpvv] and (*■ [/J h, The length at any temperature /' is x+{t' - Q y, and its m. s. e. \x^ is found from V-/ = /^.^ + it' - /o)2 /^2 /^^ , it' -t.y . ~\pV\pit-hyf' The weight is greatest when ^x,^ is least, that is, when ° [/] 116. Ex.— The following were among the observations made for the determination of the difference of length between the Lake Survey Standard Bar and Yard ; and also for the difference between their coefficients of ex- pansion. The unit is jooW 'i^ch. Required the difference of length at 62° Fahr. and at any other tempera- ture /. Date. 1S72, March 5 " 14 26 April 4 " 12 20 Observed Temp. if). 37 61 49 66 71 Bar -Yard (/)• w EUiH r (/» 791 5 811 1 83^ 6 820 () 847 8 s.,<, S 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 A- + (/ - 62) J - / = 7'. 148 THE ADJUSTMENT OF OBSERVATIONS The computation is arranged in tabular form as follows : p i>t pi /-/o /-x 5 123-5 3955 - 32.2 -40 I 37-1 811 - 19.8 — 20 6 370.2 4998 + 4-8 + 2 6 295.8 4920 - 7.6 — II 8 534-4 6776 + 9-9 + 16 8 34 572-0 6792 28,252 + 14-6 + 18 19330 • • 4 = 56.9° X = S31 .... P{^- io? P(t - /J (/ - .r) y^t- /„) V pvv 5184.20 6440.0 — 40.6 -0.6 1.8 392.04 396.0 - 24-9 -4-9 24.0 138.24 57.6 + 6.0 + 4-0 96.0 346-56 501.6 - 9.6 + 1.4 11.8 784.08 1267.2 + 12.5 -3-5 98.0 1705.28 2102.4 + 18.4 + 0.4 1-3 8550.40 10,764.8 .... .... 232.9 y = 1.26 " = \/i^ = '■«. M* = M» = 7-^ V34 1-3, 7.6 0.1. Value of ■\/855o.40 .^ = 831 at 56.9° 6-4 5-1° 837-4 62.0° Me/ =(1.3)' +(5- 1 )' X(o.l)== 1.9. These values may be checked by computing by the ordinary process. CHAPTER V ADJUSTMENT OF CONDITIONED OBSERVATIONS 117. We now take up the third division of the subject as laid down in Art. 25. So far the quantities we have dealt with, whether directly observed or functions of the quantities ob- served, 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 observation — they are said to be conditioned by these relations, or the relations are spoken of as conditions. All problems relating to conditioned observations may be solved by the rules laid down in the preceding chapters. Let, with the usual notation, X^, X^, . . . X,^ denote the most probable values of n directly observed quantities liT^, M'.,, . . . Mj^ whose weights are /j, p.,, . . . /„ respectively. Let the n^ conditions to be satisfied exactly by the most probable values, when expressed by equations reduced to the linear form, be a'X^ + a"X., + • ■ ■ - r = o, b'X^ + b"X., + • • .- L" = 0, (i) where a', a'\ . . .\ b', b" , ...;...;//, L" , . . . are known constants. If v^, V.,, . . . v^ denote the most probable corrections to the observed values, so that X.,-M., = v.„ (2) jr„ - M„ = z'„, 149 150 THE ADJUSTMENT OP OBSERVATIONS we have the reduced condition equations a'v^ + a"v^ -f- ...—/'= o, b'v^ + b'^v-" +...-/" = o, or {av\ — /' =0, [bv] - I" = o, (3) where V = L' — \aM\ l" = L" - \bM\ . . . , and are, there- fore, known quantities. The most probable system of corrections is that which makes \pv'^\ = a minimum, w suppose. The problem is to solve this minimum function when the corrections v are subject to the above 7i^ conditions. Direct Solution — Method of bidependent Unkfiowns. 118. It is plain that n^ of the corrections can, by means of the condition equations, be expressed in terms of the remain- ing n — n^ corrections, and that by substituting these n^ values in the minimum function, we should have a reduced minimum function containing u — n^ independent unknowns. This func- tion can be found in the usual way by equating to zero its differential coefficients with respect to each unknown in suc- cession. The 71 — n^ resulting equations, taken in connection with the 7tg condition equations, determine the tt corrections v^, v^, . . . v^, Thence {pvv'\ is found. The solution of the 7i — 71^ equations can be carried through by any of the methods of Chapter IV. The precision of the adjusted values, or of any function of them, can also be found as in Chapter IV. Ex. I. — Take that already solved in Ex. 4, Art. 77. Let 7/1, 7/3, 7/3, 7/4, 7/5 be the most probable corrections to the measured angfles, then the conditions to be satisfied are PSB + 7/4 = FSB + 7/3 - FSP - V, , OSB -h 7/5 = FSB + 7/3 - FSO - 7/2. ADJUSTMENT OF CONDITIONED OBSERVATIONS 151 Substituting for FSB, FSB, etc., their measured values, the condition equa- tions may be written Vi- Vs + 7/4 = - 0.76, V2 — V3+ v^ = — 1 .66, with 5 z'i' + 7 •z'o" + 4 '^3" + 7 '^4^ + 41'^ = ^ rnin. Substitute for v^, v-^ in the minimum equation, and 5 z/f + 7 -vi -^ \v^ ^ 1 {%\ - z/g + 0.76)2 + 4 (7/0 - 7/3 + 1 .66)' = a min. Hence, differentiating with respect to t/j, v.,, v^, as independent variables, we have the normal equations 12^1 -77/, = - 5.32, 1 1 z/, — 47/3 = — 6.64, - 7 7/1 - 4 '^'2 + 15 ■^^s = " -96, whence Vi= — 0.05", '^'2 = ~ o-36". ^3 = + 0.68"; and from the condition equations, V^=- 0.03", 7'5 = - 0.62". These results are the same as those already found in Art. 77. Ex. 2. — The angles A, B, 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 = iSo + €, (i) where e is the spherical excess of the triangle. Putting Ml + 7/1, Mo + Vo, M^ + v-^ for A, B, C, the condition equation becomes 7/, + 1I2 + 7/3 = 180 + e — [M'\ = /, suppose. (2) Also 7/,' + v.? + v^ = 3. min. Substitute for 7/3 from (2) in the minimum function, and v^ + 7a/ + (7/1 +Vn,- ly = a min. Differentiating with respect to the independent variables 7/,, 7/0, and (3) / which give Also, from Eq. 2, v^ = Hence t/ie correction to each anffle is onc-thinl of the difference of the theo- retical and nicastired sutns ol' the three unifies. ^1 + Vo -- = /, ^1 + 2 V2 = = /, ^1 = 7/2 = / — • 3 152 THE ADJUSTMENT OP OBSERVATIONS To find the weight of the adjusted value of an angle, as A. The function is dJ^ = 7^1 . Hence, following the method of Art. 101 (i), where G^ = i, and Q^, Q., are found from 2 Q, + Q, = I, that is, weight oi A = | if weight of measured value is unity. Check. Weight of direct measure oi A =1, Wt. of indirect meas. (= 180 + e — ^ — C) of ^ = J, Weight of mean = li. 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. rr ,sin^ Here F = a = b -. — ^ , sm B .'. dF = a sin i" (cot Av^ — cot Bv^. The weight is found from Up = a sin \" cot AQ^ — a sin \" cot BQ^, where 2n ^2 satisfy the equations (Art. loi), - 2i + 0,2= ^ sin \" cot A, Qi + -Qi=- ^ sin i" cot B. Hence, Up = § a} sin^ i" (cot^ A + cot^ B + co\. A cot B). Indirect Solution — Ulethod of Correlates. 119. If the unknowns in the condition equations are much entangled the direct sohition would be very laborious. It is in general, therefore, advisable, instead of eliminating the n^ un- knowns directly, to do so indirectly by means of undetermined multipliers, or correlates, as they are called. If we multiply the condition equations (3), Art. 117, in order by the correlates C^, C,, . . . , we may write 0, = C; ([«7'] - /') + a ([H - n + •••-!- [/H = a min. (i) and determine C^, C, , . . , accordingly. By differentiation, ADJUSTMENT OF CONDITIONED OBSERVATIONS 153 du, = (a'C, + ^'C + ■ • • + 2 Ar,) dc', + (a"C, + b"C, + • . . + 2/,zg dv.^ + . . . (2) If we place equal to zero the coefficients of ji^. of the differen- tials (h\, (/::,, . . ., we shall have «^ equations from which to find C^, C,, . . , Substitute these Ji^. values in the expression for dco, and there will remain ;/ — u^ differentials which are independent of one another. In order that the function may satisfy the condition of a minimum, the coefficients of each of these differentials must be equal to zero. This gives ;/ — ;/,. equations, which equations, taken in connection with the u^ con- dition equations, give the n unknowns v^^, t'.,, . . . v„. The practical solution would, therefore, be : Form u equations by placing equal to zero the differential coefficients of the mini- mum function with respect to each of the quantities v^, v.,, . . . v„. From these n equations and the n^ condition equations determine the ;/ -|- u^ unknowns C^, C,, . . . , t',, ?',,, . . ., and thence the function \_pvv]. 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 C; {[at'] - I') + C, {[bi'] - I") + \ [pvz'] = a min. (3) Differentiating, we have the n correlate equations a'C, + l/C\_ +. . . =/^v„ a"c,+/rc, + . . •=//'3, (4) Substituting for v^, v.,, ... in the condition equations their values derived from these equations, and the normal c(|uati()ns result. They are 154 THE ADJUSTMENT OF OBSERVATIONS Solving, we obtain C^, Co, • • • , and thence v^, v^, and X^,X^,. . . from (2), Art. 117. The normal equations may be written [uaa] Cj + [nad'] C^-\- • • ■ = I' , [jiab] C^ + [iM] C, + . . . = I", from (4), (6) where ii^, n^^, . . . denote the reciprocals of the weights p^, po, . . . The form of these equations shows that the coefficients \7iaa\ \iiab\ . . . may be computed as in Art. 79, the corre- sponding scheme being, c. a Q • • • ■^'l »l a' b' c' -^2 "2 a" b" c" . . . * • ' ' • ■ • • If the elimination of the normal equations is performed by the method of substitution (Art. 84), we have, by collecting the first equations of the successive groups, \iiaa\ Cj + \iiab\ C + \iiac\ Q + • • • = /' + \tibb.\\ 4 + lubc.\\ C; -j- . • • = l".x + \ucc.2\ C3 + . . . = I'". 2 (7) where /^ l".\, I'". 2, correspond to [al], [bl.\\ [c/.2], . . . respectively. These equations being precisely similar in form to Eq. 8, Art. 84, the elimination gives (see Art. loi), 7' 7" /'" ^ Q = [uaa] [ubb.i] r.i where o = [ubb.i] [7/ab] + [iia:2'\ l"'.2 [ucc.2] S" + (8) [/^rtrrt-J + ^', ADJUSTMENT OF CONDITIONED OBSERVATIONS 155 o = [^ + [i^l:ii^' + ^", (9) [u(7a] [uoo.i] r.i = I'R' + / ', l"'.2=l'R" ^l"S" \1"\ (10) 120. Ex. I. — Take that solved in Ex. i, Art. 118. The condition equations are t^i - 7/3 + 7^4 = - 0.76, 7/2 — 7/3 + ■z's = — 1.66. (i) The correlate equations consequently are Cj = 5 ^1 . - Ci- €2 = 4'Z'3. (2) Co = 4V^. To form the normal equations, we may substitute for 7/,, t/,) • • • from (2) in (i), or proceed by means of the tabular form in Art. 1 19. We find 0.59 C, + 0.25 C = — 0.76, 0.25 C, + 0.64 C = — 1.66. The solution of these equations gives Ci = - 0.23, Q = - 2.491 whence, from the correlate equations, v,=- 0.05", z'2 = - 0.36", 7/3 - + 0.68", 7/4 = - 0.03", 7/5 = - 0.62". Check. — The results satisfy the condition equations. Ex. 2. — The angles A, B, C, of a spherical triangle are measured with their weights,^,,/,,,/;,; required their adjusted values. The condition equation may be written (see Ex. 2, Art. 118) with [pv^ = a min. The correlate equations are C = ^,7/, , C = /2'?^2 » C=A,7/3, and the normal equation [u] C — I. Hence the adjusted values are known. When the weights are equal, thi n 156 THE ADJUSTMENT OF OBSERVATIONS the same results as in Ex. 2, Art. 118. Note. — If a condition equation is of the form po, . ■ .] ?i. go, • ■ • respectively Hence the correlate equations, + cos 6^Cl + sin ^iQ + [a cos 0] = o, + [a sin 6] = o, (U = Pi, ^i, «i sin 0^Cl + a, cos ^,6*2 = ^1, j'l, and the normal equations. (2) Fig. 4. e cos 0' I [cos- 01 f a- sin' g "] I f [ sin cos ^ "1 _ f a- sin gcosg ]] iL'^J^L ^ Jj ' iL ^ J L ^ Jl sin cos i> 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 C, [a] =-[a cos 61], C2 [«] = - i'^ si" ^l from which C„ Cj are known. The weights indicated correspond to the assumption that the errors of measurement of the lengths are directly, and of the same bearings inversely, proportional to the square root of the lengths of the corresponding courses. The corrections x„ x., . . . ; y„ ^2 • • -^ are known from (2). The errors in latitude (see Eq. i) now reduce to cos ^, J', - fl, sin 0,yi =- a^ \a cos 0\ 158 THE ADJUSTMENT OF OBSERVATIONS • n [^ cos 6] cos 62 X2 — «2 sin ^2^2 = ~" ^2 — rzi — ' and the errors in departure to [a sin e] sin ^1 Xi + «! cos ^1 Ji = — «i — r—. — » • ^ [^ sin 0] sin 6.2 X2 + ao sin ^2^2 = — <^2 — f-i — ' 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. To Find the Precision of the Adjusted Values, or of any Func- tion of them. 121. The method of proceeding is the same as in Art. 108. The first step is to find r, the p. e. of a single observation, and next the weight, pp, of the fimction, whence the p. e. of the function is given by r'slup, 7ip being the reciprocal of the weight. {a) To find r. In Art. 105 it was shown that in a system of observation equations the p. e. r of an observation of the unit of weight is found from r = 0.674c; ' n- Hi V ;, ADJUSTMENT OF CONDITIONED OBSERVATIONS 159 where [Pt/] is the sum of the weighted squares of the residuals, V, n is the number of observation equations and ity the number of independent unknowns. Hence, in a system of condition equations, ;/ being the num- ber of observed quantities and 11^ the number of conditions, the number of independent unknowns is n — n^, and .= 0.6745 v 'f'^ , = 0.6745 y/^ ■ (.) Lijroth's formula (Art. 105) may be used as a check on the value of r. Checks of \pv'\ — When the number of residuals is large, in order to guard against mistakes [/•?'"] should be computed in at least two different ways. The following check methods will be found useful : (a) The correlate equations 4, Art. 1 19, may be written, V/i?'^ = V/v^'Cj + \fu^b'C, + . . • Square and add, and [/zr] — [uaa] CjC, + 2 [ual'] C,C, + 2 [itac] CjCj + • • • + [ubi>] c.,c^_ + 2 [ni>c] cq, + - . . + [iicc] C3C3 + • • • (2) + • • • = [Cl] from (6), Art. 119. = I'c^ + r'c, -f ■ . . \\tcad\ \ubb.\\ \ucc.2\ } I rl'"-' I '"'■' s" I \ \[«/Vv.lJ [?/r^.2] / + \uaa] \ 11 bb. i ] \jicf. 2 ] by addition attending to ICq, lo, Art. i F9. i6o THE ADJUSTMENT OF OBSERVATIONS This expression is very readily computed from the solution of the correlate normal equations, as shown in Ex. 2 following. Compare the computation of [vv] from the scheme in Art. io6. The sum [pir] can in general be computed more rapidly by these methods than by the direct process of summing the weighted squares of the residuals. 122. Ex. I. — The three angles of a triangle are measured with the weights;^,, p.^, /,; required the mean-square error of a single observation. Using the values of v^, v^, v^, found in Ex. 2, Art. 120, we have L/'^'J - [,,p + [uj ^ [uj ence / M = — ; — ' V[«] since «, = i. Check (i). [pv'] = [CI] as before. Check (2). [pv'^} = F-+i directly from Eq. 3, since [tiaa] = i. Ex. 2. — To find the p. e. of a single observation in Ex. i, Art. 120. The first step is to find the value of [Pv-]. Three methods are given : (0 (2) p V pv'' 5 7 4 7 4 — 0.05 — 0.36 + 0.68 -0.03 — 0.62 .01 .91 1.85 .01 1-54 4.32 = [^7/2] c 0.230 2-493 0.76 1.66 a 0.17 4.14 4.31 = [pv'] ADJUSTMENT OF CONDITIONED OBSERVATIONS 161 (3) From the solution of the correlate normal equations : c, a + 0.5929 + 0.2500 + I + 0.2500 + 0.6429 + 0.4217 + 0.537s + I -0.76 = /' - 1.66 = /" - i.28i8=p-^' . - 1-3395 = /"-• ^ /".I •'• [/>'^'^] = 0.76 X 1.2819 + 1-3395 X 2.492 = 4-3i36- Hence, the number of conditions being two, r = 0.6745 v/¥ = °-<"'' 123. {^) To find ?(p. Let the function whose weight is to be found be F=f{V^. K,. .. K), and let it be conditioned by the ;/^. equations (4) (5) Expressing Fin terms of the observed values, 3f^, M„, . . . Jl/„, which are independent of one another, and reducing to the linear form, we have ,_ 8F ^ SF ^ Hence, as in Art. 108, 8/''V /8/'-V , "^ WfJ (6) (7) where u^, u^, . . . are the reciprocals of the weights of the observed values. Ex. 3. — To find the m. .s. e. of a side, a, in a triangle whose angles have been measured with the weights p^, p., />■„ the base, l>, being free from error. i62 THE ADJUSTMENT OF OBSERVATIONS The function equation is „ ,s'm A F = a = o -. — jT, sin B and the condition equation ^ + 5 + C = i8o + e. Hence from Ex. 2, Art. 120, expressing A, B in terms of the observed values, A = M^^- .^, {180 + e - {M^ + M.,_ + M^^, B = M.^ + p- {180 + € - {Ml + M._ + 3/3)}. Now, ■^ " [dA 5M, + dB dM,r' ^ [SASM^ SB UlJ'- ^ \5A UI^ "^ 5^ SmJ''' Therefore, ..^=«^sin^i"(j[i -^)cot^ +gcot5|^. and iJ-p = M '^«iP'> where « is the m. s. e. of a single observation. \i the weights ^„^2» A ^i"^ ^^.ch equal to unity, this reduces to Uj^^ = I ] d-h . . . = \ubf\ (5) from which T,, C . . . arc found. These equations being precisely of the form of ordinary nor- mal equations, it follows, as in {c) and (d). Art 106, that ,,,„ = [;,^] - \_uaf\ C, - [ubf] C, (6) n,^ = Wf] - (-^^^^^^-j - ^^^^ (7) The form of the last expression for ///,. shows that it may be found by means of the following scheme, in which [naf], [iibf\ ... are added as an extra column in the solution of the corre- 164 THE ADJUSTMENT OF OBSERVATIONS late normal equations (5), in the manner shown in /.rt. 106. For three correlates the scheme would be c. Ca C3 [tiaa] [ nab] [ nbb] [ Jiac ] \ubf\ • • • ■ [ubb.i] [ tibc. I ] [ llCCl . . . . [ub/A] [ iicf . I ] .... .... [ ncc.2 ] \licf .2\ [n//.2] .... • • ■ • ■ • • • .... 125. Ex. 4. — To find the weight of the angle FSB in Ex. i, Art. iif Here dF = — Vi + v^, From the condition equations «' = + I, b-' =+ I, a'" = - I, ^"' =-i, fl""= + I, b'""=+ 1, .-. [ naf] = ^x-i + ix-i=- 0.45, [nb/]=-h [nc/] = +o.4S- The correlate normal equations with the extra column for finding n^ : c, c / + 0.5929 + 0.2500 + 0.2500 + 0.6429 — 0.7600 — 1.6600 — 0.4500 = + 0.2500 =- [nbf] -r 0.4500 = [!///] + 1 + 0.4217 + 0.5375 - I. 2818 - 1-3395 - 0.7590 — 0.0602 = - [Ub/A] + 0.3416 + 0.1084 = [t n ^ the result required. If the triangles are equilateral, this reduces to « an = 2 nP sin' i' (3) (4) ADJUSTMENT OF COXDITIOXED OBSERVATIOXS 167 Hence in a chain of equilateral triangles the weights of the sides decrease as we proceed from the base, b, through the successive triangles, inversely as the number of triangles passed over ; that is, are as the fractions 1 i 1 1 i' ■!■> 3» }> • • • (b) Taking logs of both members of Eq. i, and differentiating, d log a„ = j^ log sin A^ (A,) - -^ log sin B^ (B,) + • • • = [S^U)-Sj,(B)l ' (5) or expanding the first member, where 5 is the tabular difference for one unit for the number W\ /\ 7 of equilateral triangles, all / \ / \ / Vi/ \ / \ /*■ / \ / \ / \/A B\ / \/ of the angles being equally ^7 ^ ^ ^^-^ '-^ ^ well measured, and the sides Pip g^ BC, CD, . . . being in the same straight line. Find the m. s. e. of the line BJ\^, which is n times the base. Take first the simple case of u = 2. _ „,,. . sin (T. , , sin y^, sin ^, sin C, J^ = BJV = b -. — 5^ + b -. — „' . „- . — Y~ ■ sm Bj sm B, sm B.^ sm B^ .-. dB" = (cot A, (Ai) - 2 cot B, (Bt) + cot C, (C,) + cot A., (A.^) - cot B._ (B,) - cot B3 (B3) + cot C3 (C,)}b sin 1". Also, we have the condition equations (A,) + (B,) + (C.) = /', (A,) + (B,) + (Q = /", {A,) + {B,) + (Q = /'". i6S Hence THE ADJUSTMENT OF OBSERVATIONS [aa] = 3, [af] = o, [M.i] = 3, [^/i] = o, [CC.2] = 3, [Cf.2] = O. |j7J= (cot' ^1 + 4 cot' 5, + cot^ C, + cotM 2 + cot^ ^2 + cot' ^3+ cot^ C3) ^^ sin^ I' = ^ P sin' i", since cot' 60° = ^. Substituting in Eq. 7, ^bn^ ~t'^^ ^'^^ ^"> and therefore, ^^ y = fj.^/ -^ b sin i", where m is the m. s. e. of an observed angle. Generally, dF = {ti- \) cot A^{A^) - n cot B^ {B^) + cot Ci(Ci) + (« - i) cot ^2(^4,) - (« - i) cot ^0(^2) - (« - I) cot ^3 (^3) + cot C3 (C3) + ^ — 3 «' + 5 ;/ ^ and Uji2f= ^'sin' I' ,/4«! ') If the chain proceeds in the opposite direction until AN'= BN, then, since H-an'^^ I^bj^/^ ^^^ NN' = 2 bn approximately, we have + 5 ^^^,'= MA^A^' sin x-yi^^^^ If NN' is « times the base (putting « = — J ^,,, = .^^-si„."y/IZZ|ZZI". Hence it follows that in a chain of equilateral triangles where one base only is measured, it is better to place the base at the center of the chain rather than at either end. Ex. II. — If two similar isosceles triangles on opposite sides of the base ACzre measured independently, thus forming a rhombus (vertices B, B'), then, taking the weight of each angle unity, ^f^b sin i" ,B f^BB' = ^ 7=~ ^°^^^" " V 24 2 and if BB' is n times the base b, then, since cot — = n, fiBB' sin i' f^BB' iV^ n + -]■ nl Caution. — If we solved for the rhombus directly, it would not do to take BB'= ^cot -, Fig. 7. and then form \^bb'- '^^^ result would be \/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 ADJUSTMENT OF CONDITIONED OBSERVATIONS 169 BB' = - ( cot — + cot IB B'\ I cot — + cot - 1 , with the condition equations (^') + (5')+(CC') = /., corresponding to the angles of the two triangles. Solution in Two Groups. 127. 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 equa- tions are thus divided into two groups for solution, and the com- plete 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 ex- pended in the solution of the first group in determining the additional corrections due to the second group. The solution is rigorous, and, being broken into two parts, is more easily managed than if all the equations had been solved simultaneously. Let the first group of equations be the observation equations, n in number and containing ;/„ unknowns (;/ > ;/„), a^x -I- ^,jK -h • • • - /, = ^'i , weight/,, a^x -f Ky + • • ■ - /j = ?', , " Pv (0 and the second group the condition equations, «<. in numbi-r, involving the same unknowns {n^ < ;/„), a'x + a"y +...-/'= o, b'x -f b"y +...-/" = o, (2) 170 THE ADJUSTMENT OP OBSERVATIONS The most probable values of the unknowns x, f, . . . are those which are given by the relation [pir] = 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 sec- ond, (i), (2), . . . arising from the condition equations, thus: X = (x) -{- (i), J = (^)+(2), (4) • ••••• Now, overlooking for the present the condition equations, and taking the observation equations only, (x), (j), . . . would be found by solving these equations in the usual way. We have, therefore, reducing all to weight unity for convenience in writing, the normal equations [aa] (x) 4- [a/>] (y) -\- . . ■ = [a/], [ai,](x)-i-[^^](y) + ... =[^/], (5) The solution of these equations gives (see Art. 103) (x) = [au] [a/] + [a(3] [^/] + • • • (y) = m M + [m m + •'• (6) Hence (x), (j), . . . are known. To find the condition corrections (i), (2), . . . , eliminate 7>^, v^, . . . t'n by substituting in the minimum equation, which then becomes, [aa] XX + 2 [ab] xy + • ■ ■ — 2 [«/] x, + myy+ ■■■ - 2[bl]x, (7) + [//] = a min. This equation is conditioned by equations 2. Thus, the solution is reduced to that already carried out in Art. 1 19. Calling /, //,... the correlates of equations 2, we have the correlate equations ADJUSTMENT OF COxNDITIONED OBSERVATIONS 171 [aa]x + [aF\y + • • • - [a/] = a / + l>' JJ + ■ ■ ■ [ad\x + [M\j + • • • - [^/J = ^"^ + ^"^^ + • • • These equations, taken with (4) and (5), give the relations [aa\ (i) + [al?\ {2) + ■ • ■ =a' I + b' JI + ■ • ■ =[T] suppose [ab\ {i) + m ^2)+ . . ■ = a"/+ 1>"/I+ • ■ • =0^ " (8) which being of the same form as (5), their sokition gives (2) = [a/3] g + [m + • • • (9) or substituting for [T| , [2] , . . . their values from (8), (i) = a' /+ p/ //+ c' ///+ • • • (2) = a"/+ p/'//+ c"///+ • . . (10) where A' = [aa] a' 4- [al3] a" + • ■ ■ B' = [aa]y + [aiSjr + • • • (u) and are 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, j, . . . their values from (4) in the con- dition equations, and a' (i) + «" (2) + • ■ • = /;, ^'(l) + r(2)+ ••• =C, O2) where /; = /' -a' (x) - a" (y) /;' = /" - b' (x) - b" 0') - ■ ■■ (^3) and are, therefore, known quantities, since (.r), (;'), . . . arc known. 172 THE ADJUSTMENT OF OBSERVATIONS Substitute the values of (i), (2), . . . from (10) in (12), and we have the correlate normal equations, [^] 7+ [^] //+ . . . =/;, [ob] /+ [^] 77+ . . . - 4", (14) where [oa] = [aa]d',/ + [a/S ]^ = a^x + b^y + • • • — l^ = a, (x) 4- b, ( j) + . . . - /, 4- ^1 (i) + ^ (2) + • . . = v,' + a,{i) + b,{2)+ .. . Similarly ^'2 = ^'2° + ^2 (0 + ^'2 (2) + • • • where v,'> = a,(x)-{-b,(y) +...-/„ v^^ = a^{x) + b.{}')+ • • • - A, 174 THE ADJUSTMENT OF OBSERVATIONS that is, v^, v^, . . . are the residuals arising from taking the observation equations only. Attending to Eq. 5, Art. 1 27, it follows evidently that [av''\ = o [/>v"] = o, . . . Square the residuals 2\, v.^, . . . and add, then [7-2] = [tP^fi] -^.[la (i) + /. (2) +•••!-] _ [2;02;0j _j_ [2e/a/] suppose. The total sum [v^] may therefore be found in two parts, one from squaring the residuals of the observation equations, and the other from the corrections (i), (2), . . . We proceed to put [zaw] in a more convenient shape for computation. [WW] = [la(i) + /;(2) + ■ . • |2] = (i)l[H(i) + [«'^](2)+ • •• } + (2)\[a/^](i) + [M]{2)+ ... I + = (i) + (2) [7] + . . . from Eq. 8, Art. 127. Substitute for (i), [T|, (2), . . . their values from equations 8 and 10, Art. 127, and expand; then [wTo] = { [oa] I -\- [^] II -\r ■ ■ ■] I + [[^]/+ [^] //+ •■•}// + which may be transformed, by means of Eq. 14, into the form lwuJ\ =1^1^ i^' 11+ ... or, as in Art. 121, into the form \_aA\ L^B. ij |_] (v) + • • • = [a/], [ad] (x) + [M] 0') + ■■■ = [/V], the problem is reduced to that already solved in Art. 108. If, therefore, //y,, is the reciprocal of the required weight, uf = [GQ] (22) where <2i = [aa]G,-h[a/3]G,+ ■ ■ • Q, = [a/3] G, + [m a + • . . (23) the quantities [aa], [a^], . . . being as in the weight equa- tions 9. Putting for G^, G„, . . . their values from (21) in these equations, and attending to (u), we find Qi = ^1 — a' ^i — b' /', — . . . Q., = q.,- A"k^ - b"A, - • • . (24) where 9i =K]<^i + ["■P]g-z + • • •' ^.=^[-(i]gy + [ms.+ ■ • ■ (25) Substituting in (22) for G^, G,^, . . . Q^, Q„, . . , their values from (21) and (24), [GQ] = [gq] - U-A] ^, _ [ .b] 4 - . . . — [ rt^ ] Z'l — [ /^ ] /'2 — • • • + !H ^1 + M ^2+ • • • }^i + {[^] ^1 + [7i^]^'2+ • • • }^2 But from (11) and (25) [aq] = [gA], [dq] = [^-b], . . . Hence, attending to (19), the above expression reduces to ADJUSTMENT OF CONDITIONED OBSERVATIONS 177 [c;<2] = M - U-A] I', - [^^] k,- ... or to L<^<2] = L giy • • • » ^^"^^ '^^'^' ^'^^ [iV] = [""] .^'i.^'i + - [«/3] A^A'. + • • • + ■ • • where \aa\, [a/3], . . . may be taken from the weight equations. The remaining terms of the second form of [GQ] may be found from the solution of the normal equations, as shown in Art. 121. Solution by Successive Approximation. 131. This method of solution (due to Gauss) is of the great- est importance in adjustments involving many conditions. It may be stated as follows : The condition equations may be divided into groups, and the groups solved in any order we please. Each successive group will give corrections to the values furnished by the pre- ceding groups, and the corrected values will be closer and closer approximations to the most probable values which would be found from the simultaneous solution of all the groups. For suppose we have the condition equations az\ + ^"^'a + • • • = 4. h'v^ + h"T., -f . . . = 4, k'v^ -\- k"iu + . . • = 4, with [pv"^] = a mm. Let t'/, <, . . . be the values of %\, v„, . . . obtained from solving the first group alone ; that is, from 178 THE ADJUSTMENT OF OBSERVATIONS a z\ + a 7'/ + • • • = 4» b'v; + b"v.l + . . . = 4, [/z/'2] = a min. If now (^'/), (t'Z), ... are the corrections to these values re- suhing from the remaining equations, then since the condition equations are reduced to b' (v,') + b" {v^) + ... = /;, h' M + /i" {v.!) + ■ ■ • = //, with [/ {^')'] = ^ min., and the values of {v') found from the simultaneous solution of these equations, added to the values of v' found from the solu- tion of the first set, would be equal to the value of v found directly. Similarly, if v^', v,y, ... be the values of {v') obtained by solving the second set alone, and {v^'), {v/'), ... be the cor- rections to these values resulting from the remaining equa- tions, then since the condition equations are reduced to a' {v^') + a" (7-/0 -h . . . = /;^ y (7;/') + //' (7'/') -F . . . = 4", h' (v,") + h" (7^;') + • • • = 4", with [^ (z/')2] = a min. ADJUSTMENT OF CONDITIONED OBSERVATIONS 179 The quantities [/>v'% [pv"% . . . being positive, the minimum 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 con- tinuing 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. METHOD OF ANGLES 132. The adjustment of the measured angles of a triangula- tion 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. 133. For clearness we will explain in some detail the pre- liminary work necessary for the formation of the condition 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, BC by the ordinary rules of trigonometry. In plotting the figure, the point (Tcan be located in but one way, as only the measurements necessary for this purpose have been made. Fig. 8. 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, accordins; 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. 180 ADJUSTMENT OF A TRIANGULATION — ANGLES i8i If the exterior 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 we have considered so far that in a triangle, only two of the angles are measured. 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 CAB + ABC + BCA = 180° + 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 : (i) 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 necessary 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 ^^«^r.«/ conditions. 134. The number of conditions to be satisfied will depend on the measurements made. Each condition can be stated in the form of an equation in which the most probable values of the measured quantities are the unknowns. The number of equations being less than the number of unknowns, an infinite i82 THE ADJUSTMENT OF OBSERVATIONS 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 correc- tions a minimum. The form of the reduction depends on the methods employed 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 ^'2- 9- 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. The first method of measurement is known as the method of independently measured angles, or, if each angle is mechanically multipUed before being read off from the circle, it is called the method of repetition ; and the second method is known as the method of directions. The Method of Independent Angles. 135. 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 inde- pendently observed and angles which are independent in the sense that no condition exists between them. Thus at the ADJUSTMENT OF A TRIANGULATION — ANGLES 183 station O, above, the angles yi 6>i>, BOC, AOC might be 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 tlie measured angles can be expressed by means of the conditions connecting them. In the present case any two of the three angles AOB, BOC, AOC may betaken as independent, and the third angle would be dependent. Angles may be measured independently either with a repeat- ing 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 follows : 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 com- bined result, and is free from the error arising from uniform twisting of the post or tripod on which the instrument is placed, or from "twist of station," as it is called, provided the rate of observing is constant. The telescope is next turned 180° in azimuth and then 180° in altitude, leaving the same pivots in the same wyes, and another combined result is obtained. The mean of the two combined results is free from errors of the instrument arising from imper- fect adjustments for collimation, from iiicciuality 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." i84 THE ADJUSTMENT OF OBSERVATIONS 136. Besides those mentioned, there are two kinds of system- atic error in measuring angles that deserve special attention. 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 ob- servation, 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 consecu- tive microscopes in case of two-microscope and three-microscope instruments respectively. Thus, if 71 is the number of pairs of combined results desired, the changes would be and re- n u spectively with the instruments mentioned. The operation of — reversal in case of a three-microscope instru- /\ /^^ ment causes each microscope to fall at the / \ / \ middle of the opposite 120° space, the limb L V j remaining unchanged. Thus, if the full lines y / \ / in Fig. 10 represent the positions of the \ / \ / microscopes with telescope direct, the dotted ^ -^ lines show their positions with telescope re- Fig. 10. verse. In this lies the greatest advantage of three microscopes over two, since with the latter, in revers- ing, 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 ordi- nary care, it will usually be less than o. i''. In case, however, of a signal having a high altitude above the horizon, the error from this source may be greater, and then special care should be taken in leveling. For an expression for its influence in any case, see Chauvenet's Astronomy, Vol. II, Art. 211. It is desirable to make the observations under various condi- tions so as to avoid constant errors. See Appendix No. 4, U. S. C. & G. Survey Report for 1903, pp. 843-844, 869. ADJUS-niENT OF A TRIANGULATION — ANGLES 185 137. 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 quadri- lateral N. Base, S. Base, Lester, Oneota. LNO =124° 09' 40.69" weight 2 SNL = 113° 39' 05-07" " 2 ONS = 122° 11' 15.61" " 14 NSO = 23° 08' 05.26" " 23 ZSJV = 47° 31' 20.41" " 6 ZSO = 70° 39' 24.60" " 7 SOJV= 34° 40' 39-66" " 31 JVOL = 43° 46' 26.40" " I OZS = 30° 53' 30.81" - 8 These angles we shall denote by M^, 3f.,, . . . , M^ respectively. The length of the line N. Base — S. Base (Minnesota Point) is 6056.6 m., and the latitudes of the four stations are approxi- mately N. Base, 46° 45' Lester, 46° 52' S. Base, 46° 43' Oneota, 46° 45' 138. The Local Adjustment. — When in a system of triang- ulation 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. 133, it is readily seen that at a station only two kinds of conditions are possible : (rt) that an angle can be formed from two or more others, and C/;) that the sum of the angles round the horizon should be equal to 360°. The second of these is included in the fir.st, and the method of adjustment may be stated in general terms as follows: [86 THE ADJUSTMENT OP OBSERVATIONS An inspection of the figure representing the angles at the station will show how all of the measured angles can be ex- pressed in terms of a certain number of them which are inde- pendent of one another. These relations will give rise to condition equations, or local eqtmtiojis, as they are called, which may be solved as in Chapters IV or V. Thus, if AT^, M^, . . . 31 „ denote the single measured angles, and v^, Vo, . . . v„ their most probable corrections, then if any of the angles M/„ Mj. can be formed from others, we have, by equating the measured and computed values, the local condition equations, Mj, + 7', = M^ + z'l •+ M. + z'2 + • ■ • Mk + Vk = M^ + 7-1 + M. + 7'2 + • • . or i\ + e'o + ■ ■ ■ - v„ = /;, suppose, t\ + 2'2 + • • • - ?'i- = 4 suppose, with where /^,/2' • • • + A^'a' + /i^'j' + • • • + Pn-^'n = a minimum /„ denote the weights of the angles. The solution may be in general best carried out by the method of correlates, as in Chapter V. 139. The following special cases are of frequent occurrence : (i) At a station O the n — i single angles A OB, BOC, . . . are measured, and also the sum angle AOL, to find the ad- justed values of the separate angles, all of the measured values being of the same weight. The condition equation is Af, + z\ + J/2 + ^'2 + ■ + J/„_i + 7'„_i = M„-\- Z'„ or = / suppose, with [7'^] = a minimum. + ^„-i) ADJUSTMENT OF A TRIANGULATION —ANGLES 187 The solution gives (Art. 118 or 119), _ _ _ _ / that is, t/^c correction to each angle is - of the excess of the siivi n angle over the sum of the single angles, and the sign of the co7- rection to the sum angle is opposite to that of the single angles. (2) At a station O the ;/ single angles ^(9^, BOC, . . . LOA are measured, thus closing the horizon, to find the adjusted values of the angles. The condition equation is z'x + '2 + • • • + ^'« = 360° -{M^ + M^+ ■ .. + M„) = I suppose, with [/^'^] = 3. minimum. The solution gives I ^1 = ''1 ri' 7'5 = ?/2 f-^ » It where u. = — A , U„ = — , . . . and [//l = A If the weights are equal, then V, = z,= ...=z, = ^; that is, the correctioji to each angle is - of the excess of 360° over n the sum of the measured angles. Ex. I. — The angles at Station N. Base close the horizon; required to adjust tliem. We have (Art. 137), ■^1 + 'J'l = '24° 09' 40.69" + 7/, weight 2 M^\v^= 113° 39' 05.07" + V. " 2 M:, + 7/3 = 122° [l' 15.61' + 7/, " 14 Sum = 360° 00' 01.37" + 7/, + v.. + 7'3 Theoretical sum . . = 360° 00 ' 00.00" .-. Local equation is . = 0° = 1.37 " + 7^, + v^ + v^ i88 THE ADJUSTMENT OF OBSERVATIONS Hence (Ex. 2, Art. 120), 7\ =- -t~, — "1 . 1 X 1.37 = -0.64", V2=— 0.64", ^3 = - 009". and the adjusted angles are, 124° 09' 40.05" 113° 39' 04.43" 122° 11' 15.52" Check-sum = 360° 00' 00.00" Ex. 2. — Precisely as in the preceding we may deduce at Station S. Base the local equation, o = 1.07" + Vi + v^- v^, and the adjusted angles 23° 08' 05.13", 47° 31' 19.91", 70° 39' 25.04". 140. 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 i- — i. 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 contradiction in these two values would give rise to a local (condition) equation. If, there- fore, 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. 141. The General Adjustment. — With a single measured base, the number of conditions arising from the geometrical re- lations existing among the different parts of a triangulation net can be readily estimated. F'or 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 determine a fourth ADJUSTMENT OF A TRIANGULATION — ANGLES 1S9 point, and so on. Hence, to determine the s — 2 points, 2 (s — 2) angles are necessary. If, therefore, n is the total num- ber of locally independent angles, 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 j- is the number of stations, show that the number of conditions to be satisfied is j- — 2 ; and in a chain of quadrilat- erals, with both diagonals drawn, the number of conditions is 2 .y — 4. The equations arising from these conditions are divided into two classes, angle equations and side equations. 142. The Angle Equations. — The sum of the angles of a triangle drawn on a plane surface is equal to 1 80°. The sum of the angles of a spherical triangle exceeds 180° by the spheri- cal excess (e) of the triangle, which latter is found from the relation area of triangle e = r^ sin I 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 dimen- sions 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 V7v\\'', where i?, N, are the radii of curvature of the meridian and of the normal section to the meridian respectively at the point corresponding to the mean of the latitudes (f> of the triangle vertices. . Hence we may wrap our triangulation on the spheroid in question by conforming it to the spherical excess computed from the formula, «/, sin C, c (in seconds) = — ^ ,, . 77 . ^ ' z i?A^sm I where a^, b^, are two sides and T, is the included angle of the triangle. 190 THE ADJUSTMENT OF OBSERVATIONS For convenience of computation we may write, e := inaj\ sin Cj, when log m may be tabulated for the argument ^. The follow- ing table is computed with Clarke's values of the elements of the terrestrial spheroid of 1866 corresponding to latitudes from 10° to yoP. The meter is the unit of length to be used. Table of log m. Lati- tude. Log m. 18 00 18 30 19 00 19 30 20 00 20 21 21 22 22 30 00 00 23 00 23 30 24 00 24 30 25 00 25 30 26 00 26 30 27 00 27 30 28 00 28 30 29 00 29 30 30 00 31 31 J^ 32 00 32 30 00 1.40639 636 632 629 626 623 619 616 612 608 605 601 597 594 590 586 5S2 57S 573 569 565 560 556 552 548 544 539 534 530 1.40525 Lati- tude. / 00 33 30 34 00 34 30 35 00 35 30 3f^ 00 36 30 37 00 37 30 3S 00 3S 30 39 00 39 30 40 00 40 30 41 00 41 30 42 00 42 30 43 00 43 30 44 GO 44 30 45 GO 45 30 46 00 4^' 30 47 00 47 30 Log »t. 1.40520 5" 506 501 496 491 486 482 477 472 467 462 457 452 446 441 436 431 426 421 416 411 406 400 395 390 385 380 ^•40375 Lati- tude. 48 00 48 30 49 00 49 30 50 00 50 51 51 52 52 00 30 00 30 53 00 53 30 54 00 54 30 55 00 55 30 56 00 56 30 57 00 57 30 58 00 58 30 59 00 59 30 60 00 60 30 61 00 61 30 62 GO Log »i. 1.40369 364 359 354 349 344 339 334 329 324 319 314 309 304 299 295 290 285 280 276 271 266 262 257 253 249 244 240 ■235 1.40231 Lati- tude. Log 63 GO 63 30 64 00 64 30 65 GO 65 30 66 00 66 30 67 00 67 30 68 GO 68 30 69 GO 69 30 70 00 70 30 71 GO 71 30 72 GO 1.40227 223 219 215 210 207 203 199 195 192 188 185 181 178 174 171 168 164 T.4G161 ADJUSTMENT OF A TRIANGULATION — ANGLES 19k To find a^^, b^, cf), a preliminary geodetic computation must first be made of the tri-angulation 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. An error of less than 3' in the latitude will not under any circum- stances produce an error of overo.ooi" in the computed spheri- cal excess, and in general therefore the latitudes may be taken from a map or sketch of the triangulation. 143, 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 OA^S, LSO will be found to be 0.05" and 0.37" 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 1 80°, together with the spherical excess, gives a condition equation.* This is called an ajigle equation^ or by some a triangle equation. Ex. — In the triangle N. Base, S. Base, Oneota, if 7^3, 7/4, 7'^, denote the corrections to the three angles, we have for the most probable values, 0NS=\2z° II' 15.61" + 7/3 NSO = 23° 08' 05.26" + v^ SON ^ 34° 40' 39.66" + v^ Sum . . . . = 180° 00' 00.53" + 7/3 + 7/4 + 7/7 Theoretical sum = 180° 00' 00.05" = '8°° + * and the angle equation is formed by equating these sums. The result is, 7/3 + 7/4 + 7/7 + 0.48" = o. Similarly, from the triangle Lester, S. Base, Oneota, the angle equation is, T/g + 7/7 + 7/g + 7/0 + 1. 10" =- O. 144. Number of Angle Equations in a Net. — It is to be expected that in a triangulation net some of the lines will be * We confine ourselves throughout to triangles to wliicli I.cgendre's tlioo- rem is applicable. For very large triangles other formulas for spherical excess must be used if great accuracy is rccjuired. Sec The Transcoittitwiital Triangulaiion of C. ^r" G. Survey, pp. 51-5.). 192 THE ADJUSTMENT OF OBSERVATIONS sighted over in both directions, and some only in one direction. If these latter lines are omitted, the number of angle equations will remain unaltered. Thus in our Lake Superior quadrilateral (Fig. II) the line NL has been sighted over from N, but not from Z, so that we have only two angle equations : namely, those re- sulting from the triangles ONS, OLS, just as if the figure had been of the form of Fig. 1 3, in which the line NL is omitted. Let s be the total number of stations in a figure or series of figures regardless of whether the stations are occupied or not, j-„ the number of unoccupied stations, / the total number of lines in the figure, and /^ the number of lines which are observed over in one direction only. The number of angle equations will be / — /j — i- + ^„ + /. This may be proved by plotting the figure in detail, adding at each step of the process one new point and all observed lines connect- ing that point with points already shown on the figure. The formula is true in any case, because, as the figure is drawn point by point and line by line as indi- cated, it holds for the simplest pos- sible figure the triangle ; for the first two lines to any new occupied point, I — s is, increased by one and the value of no other symbol is changed in the formula and one new angle equation appears ; for each new complete line to a new occupied point after the first two, / is increased by one, and one new angle, equation appears ; for each new line observed in one direction only to a new point after the first two lines to it are drawn, I — l^ remains unchanged, and no new angle equation appears ; the addition of an unoccupied station with any number of lines to it which are necessarily observed in one direction only does not change the value oi I — l^ — s -\- s^ -{- I, and no new angle equation is introduced. ADJUSTMENT OF A TRIANGULATION — ANGLES 193 It may seem that there are more angle equations than have been indicated, but it will be found in every such case that the supposed additional equation may be derived algebraically from those already used, and is therefore not a new independent equation. Ex. — In the quadrilateral A BCD, in which all of the S angles are meas- ured, show that there are three independent angle equations, and that these may be found from the fol- a - B lowing 8 sets of figures : ABD, ABC, A CD; ABD, ABC, ABCD; ABD, A CD, ABCD; BDA, BCD, BCA; BCA. BCD, BCDA; CDB, CAB, CD A; CDB, CD A, CDBA; DAB, DBC, DAC. Fig. 14. 145. 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. 13 any side can be computed from A^S in but one way. The only condition equations apart from the local ,L2 equations would be the two angle equations. But in Fig. 11, in which the line iVL is sighted over from iV, we have the further condition that the lines OL, NL, SL intersect in the same point, L. The figure plotted from the measured values would be of the form of Fig. 15. To express in the form of an equa- tion the condition that the three points /.,, L^, L^ must coincide, we proceed as follows: Starting from the base NS, we may com- pute SL^ directly from the triangle SNL^, and SL^ from the triangles SON, SOL.,. This gives Fig. IS. side SN sin .S'ZjTV side SLx sin SNL■^^ ' 194 THE ADJUSTMENT OF OBSERVATIONS side SJV _ sin SOJV sin SZ^O ^ side SZs sin SNO sin SOL^ but 5Zj must be equal to SL^. Hence the condition equation is sin SLN sin SNO sin SOL _ sin SNL sin SON sin ^Z6> ~ ^' which is called a side equation or j^/;/^ eqnatioti. The side equation, sin ^ZiV^ sin SOL sin ^iVC> = I, = I, sin SNL sin SLO sin 56>iV^ gives the identical relation, side SN side SL side ,5(9 side SL side .S't) side »SA^ Hence, in forming a side equation we may proceed mechani- cally in this way. Write down the scheme SN SL_ S0_ SL SO SN~ ^' the numerator and denominator each being formed by the lines radiating from the point 6" 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 J^o/e of the quadrilateral for this equation. It should be noted that side equations formed from a pole as starting-point are the most convenient to be used of the many that are possible. Not necessarily do all of the lines radiate from a point or pole that enter a side equation. The side equations thus formed by the use of lines radiating from a selected pole are not the only ones possible. They are the convenient ones. As an example of a side equation not formed with a pole, take the following from Fig. 23, supposing Zj, L^, Z3, to form one point L. The equation is ADJUSTMENT OP A TRIANGULATION — ANGLES 195 sin ONS sin NLS sin NOL sin LSO ^ sin SON sin Z^iV sin ZA^6> sin OLS ~ '' This equation expressed in the form of ratios of sides is SO SNNL OL SN NL OL SO = I. It is evident that the sides involved do not il radiate from one point. Such side equations as these need not be used, as a sufficient number of the more conveniently formed side equa- tions of the kind which involve a pole may always be secured. Ex. — In the figure ABCDr^D^A, the three angle equations, D,AB + ABD, + BD,A = 180° + e^, ABC + BCA + CAB = 180° + e,, BCD^ + CD^B + D^BC = 180° + 63, given by the triangles Z),^5, ABC, BCD^, may- be satisfied, and yet the figure not be a perfect quadrilateral. Show by equating the values of BD^ and BD^ that the further condition necessary is sin ABD sin BCA sin CDB ^ sin BBA sin CAB sin BCD ~ ' Fig. 16. 146. 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 OJVS, OLS, the side equation with pole at 5 means that the points L^ and T.., must coincide. The side equation with pole at N means that Zj, Z2 coincide, and with pole at O that Z^, L.^ coincide. If any one of these conditions is satisfied, the others are also satis- fied, as each amounts to the same condition that Z is not three points, but one point. Similar reasoning will show that by plotting the figure from L.ONS, ONS, the side equations formed by taking the poles at N, Z, 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 llie figure definitely and removes all contradictions from it. 196 THE ADJUSTMENT OF OBSERVATIONS 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 A BCD, in which all of the 8 angles are measured, show that of the 15 side equations that may be formed, not all of which are of the polar kind, 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 3 angle and i side equations necessary to determine the quadrilateral. Ex. 2. — Examine the truth of the following statement. In a quadri- lateral an angle equation may be replaced by a side equation, so that the quadrilateral may be determined by 3 angle equations and i 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. If the triangulation net, instead of involving quadrilaterals only, involves central polygons, such that, in computing the lengths of the sides, we can pass from one side to any other through a chain of triangles, the same pro- cess is followed in forming the side equations as in a quadri- lateral. Thus, in the figure which repre- sents part of the triangulation of Lake Erie west of Buffalo Base, there are side equations from The quadrilaterals CDHG, GHFA, The pentagons GABCH, HGDEF. The scheme for the pentagonal side equation GABCH, for ex- ample, would be just as in the case of a quadrilateral, taking G as pole, GA GB GC GH ^ GB GC GH GA ^' and the side equation. ADJUSTMENT OF A TRIANGULATION — ANGLES 197 sin GBA sin GCB sin GffC sin GAIT = I. sin GAB sin GBC sin GCH sin G-/^^ 147. Reduction to the Linear Form. — Thus far we have considered the side equations in their rigorous form. But in order to carry through the sokition by combining them wth the other condition equations, they must be reduced to the hnear form. We proceed to show how this may be done. Let the side equation be sin V]^ sin V^ (0 sin V^ sin V^ where V^, V.,, . . . denote the most probable vahies of the angles. Let J/j, J/g, . . . denote the measured values, and Vy, z'2» • • • the most probable corrections to these values ; then the equation may be written, sin (J/^ + v^ sin {M^ + 7-3) ^ ^ ,. sin (J/, + ^'2) sin {M^ + i\) ' ' ' ^- \-) 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^ -f j^ (log sin M^ v^ - j log sin M., + j^ (log sin J/,) z;, ( + ...= o, (3) which may be written in two forms for computation : First, if the corrections to the angles are expressed in sec- onds, we may put ^ (log sin J/j) = S'. dM^ where 8' is the tabular difference for \" for the angle J/, in a o 1 table of log sines. Then we have, S'z/j — 8"7'2 + • • • + log sin J/j — log sin M.^-\- • ■ ■ = o\ that is, [8^] = ^. (4) where / is a known quantity. 198 THE ADJUSTMENT OF OBSERVATIONS Secondly, we may replace — — r (log sin il/j) by mod sin i" cot Af^, where mod denotes the modulus of the common system of logarithms. Eq. 3 may then be arranged, cot M^7\ — cot M^v^ + • • • = —. ^. n (log sin M, - log sin M, -\- ■ ■ •), (5) 10^ mod sm i if the seventh place of decimals is chosen as the unit. The first of these two forms is preferred in the computing work by the computers of the Coast and Geodetic Survey. 148. Check Computation. — The side equation deduced from spherical triangles must also follow from the correspond- ing plane triangles, the angles of each spherical triangle being transformed according to Legendre's theorem ; that is, for ex- ample, we should obtain the same constant term / by reducing to the linear form the equation, sin SZJV sin SOL sin SJVO sin SJVZ sin SLO sin SOJV or the equation. = I sm (SZIV- ^A sin (sol - -) sin IsNO - 'A sm (SJVL - ^A sin (SLO - -) sin IsOJV- A where e,, €.,, e,,, denote the spherical excesses of the triangles SNL, SOL, and SON, respectively. It is, in general, simpler to use the spherical angles than the plane angles. It affords a check on the accuracy of the numeri- cal work to compute the side equation with both the spherical and the plane angles. It is hardly worth while, however, to spend the time required for this check, as it will take as long to apply the check as to have a duplicate computation made. Ex. — The quadrilateral N. Base, S. Base, Oneota, Lester (Fig. 19). Take the pole at Lester. ADJUSTMENT OF A TRIANGULATION — ANGLES 199 ,.r u ,1 LS LN LO We have the scheme y~^t fD Ys ^ ^' from which we write down the side equation, sin LNS sin LON sin LSO _ sin LSN sin LNO sin LOS ^ ' sin (M^ + 7^2) sin {M^ + v^ sin {M^ + v^ ^ ^ that is, ^.^ ^^j^^ _l_ ^^^ gjj^ ^^j^^ ^ ^^^ gjj^ ^^^ + j/^ + ^^ + ^^) = Fi'rsf Form of Reduction. log sin (113° 39' 05.07" + v^ = 9.9618970 - 9.2 v^ log sin ( 43° 46' 26.40" + T/g) = 9-8399903 + 2.0 7/g log sin ( 70° 39' 24.60" + ■z/g) = 9.9747657 + 74 'y« 530 log sin ( 47° 31' 20.41" + 7/5) = 9.8677860 + 19-3 "^5 log sin (124° 09' 40.69" + v^ = 9.9177470 - 14-3 '"x log sin ( 78° 27' 06.06" + 7/7 + T/g) = 9.991 i^So + 4-3 '^"i + ^i) 510 Hence the side equation in the linear form is 14.3 7/i - 9.2 v^ - 19.3 7/5 + 7.4 v^ - 4.3 7/7 + 17.7 7/g + 20 = o, the unit being the seventh place of decimals. Check off 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. Log Sin. Angle. Log Sin. 113° 39' 05.00" 9.9618970 47° 31' 20.34" 9.8677858 43° 46' 26.36" 9.8399903 124° 09' 40.65" 9.91 7747 1 70° 39' 24.48" 99747656 78° 27' 05.02" 9.991 1 180 529 509 4- 20 agreeing closely with the value found from the spherical angles. Second Form of Reduction. Log Sin. Log Sin. 9.9618970 — 0.438 V2 9.8677860 + 0.916 Vi 9.8399903 + 1.044 7/3 9.9177470 - 0.679 Vi 9.9747659 + 0.351 V 9.991 1 180 + 0.204 (^7 + ^3) 53S 5£0 20 log 1. 30103 — i-^ — r, log 8.67664 10' mod sm i ° 9.97767 0.955 and the side equation is 0.679 7/1 - 0.438 v., - 0.916 7/5 4- 0.351 7/0 - 0.204 7/7 + 0.840 7/s + 0.95 = 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 ioo THE ADJUSTMENT OF OBSERVATIONS absolute 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 logarithmic sines have been carried to seven decimal places only. This is sufficient for the most accurate primary triangulation. An error of one in the seventh place of decimals corresponds to an error of less than i part in 4,000,000. In the most accurate primary tri- angulation, discrepancies of more than ten times the amount occur in at least half of the figures. The uniform present practice of the Coast and Geodetic Survey is to use /-place logarithms. In the past, eight or more places have been frequently used, 149. We have seen that the coefficients of the corrections 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. There will be less liability to mistakes on account of misplaced decimal points, and less difficulty arising from omitted decimal places in the solution, and especially in connection with a check column, if the coefficients throughout the condition equations are of the same order of magnitude. Since the coefficients of the corrections in the angle equations are + i or — i, it fol- lows 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 coefficients 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 ^1 — 0-92 z'2 — 1-93 2^5 + o-74"'6 — o-43 ^7 + i-77 ^'s + 2.00 = o. It would have been equally correct to multiply each of the angle equations by 10, and so have put them on the same ADJUSTMENT OF A TRIANGULATION — ANGLES 201 footing as the side equations. Dividing the side equations is, however, to be preferred, as the coefificients are made smaller throughout, and the formation and solution of the normal equations are consequently easier. A striking difference between condition equations and obser- vation equations is here brought out. As a condition equation expresses a rigorous relation among the observed quantities al- together 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. 48.) 150. Position of Pole. — In a quadrilateral, taking any of the vertices as pole, the conclusion was reached in Art. 145 that 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 further notice. Two points are to be considered — precision of results and ease of computation. As regards the first, since the differences in a table of log sines are more sharply defined for small angles, and these differences are the coefificients 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 choosing 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 equation, for then the normal equations would be more easily formed. If the approximate form of solution in Art. 1 3 1 is employed, it is advantageous to choose the pole at the intersection, O, of the diagonals, as will be seen in the sequel. 202 THE ADJUSTMENT OF OBSERVATIONS 151. 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 {s — 2) lines besides the base ; that is, 2 j- — 3 lines in all. Starting from the base, each line in this figure can be com- puted 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 number of side equations, as indicated by the number of superfluous lines, is /— 2 i- + 3. 152. Check of the Total Number of Conditions. — The no- tation already given may be summarized as follows : In any figure or series of figures, / is the total number of lines, /^ the number of lines observed over in one direction only, s the total number of stations, s,^ the number of unoccupied stations, and ;/ the number of angles measured which are independent in so far as local conditions are concerned. At each station occupied, the number of locally independent angles is one less than the number of lines observed from that station, hence « = 2 / — /j — J + ^„. (i) From Art. 144 the number of angle equations in the figure is / - /, - ^ + ^„ + /. (2) From Art. 151 the number of side equations in the net is / - 2 J + 3. (3) More accurately, this is the number of side equations which are independent of each other and of the angle equations. Adding (2) and (3), the total number of angle and side equa- tions in the net is 2 / - /j - 3 J + J„ + 4- ADJUSTMENT OF A TRIANGULATION — ANGLES 203 Combining these with (i), it becomes n — 2 s + 4, as was proved in Art. 141 to be the total number of conditions in a figure developed from a single base. The condition equations referring to lengths, azimuth, latitude and longitude, which arise when there is more than one line in a figure fixed by previous adjustment, will be treated later. 153. Manner of Selecting the Angle and Side Condition Equations. — In the selection of side and angle equations in a triangulation net, four dangers must be guarded against. First, that some necessary condition equation may be omitted ; second, that some unnecessary condition equation may be introduced ; third, that a condition equation which is chosen may not be in- dependent of those already selected ; and fourth, that the con- dition equations used may be so selected from the many available that the solution of the normal equations will be an unstable one ; that is, a solution in which the effect of omitted decimal places on the derived values of the required unknowns is large, and in which it is therefore necessary to carry a large number of decimal places in the solution to secure the unknowns with certainty to a small number of decimal places. A good method of avoiding the first, second, and third of these dangers is to start from some line as base and plot the figure point by point. As each point is added, draw all observed lines connecting it with points previously located, and express the conditions arising from the new lines. For each new point, the number of new angle equations is one less than the num- ber of lines observed in both directions connecting it with pre- vious points, and the number of new side equations is two less than the number of lines connecting it with previous points, regardless of whether these lines are observed in both directions or only in one direction. For example, let Fig. 18 represent a triangulation net, plotted in detail as follows: First draw the line connecting the points Tobacco Row and Spear. Add the new point Long, and con- 204 THE ADJUSTMENT OF OBSERVATIONS TOBACCO ROW 5 FLAT TOP SPEAR SMITH Fig. i8. nect with Tobacco Row and Spear. This furnishes one angle equation from the triangle Long-Tobacco Row-Spear. Next add the new point Smith, and draw lines from it to Tobacco Row, Spear, and Long. This furnishes two new angle equations corresponding to the two new triangles, and one side equation. Complete the figure by adding the new point Flat Top, and draw four lines to Tobacco Row, Spear, Long, and Smith. This introduces three new angle equations and two new side equations, making the total number of angle equations six, and of side equations three. These numbers may be checked by the formulas in sec. i 54. If in this triangulation net the line Spear-Smith had been observed in but one direction, Spear to Smith, and Flat Top had been an unoccupied station, the drawing of the figure in the same three steps would have indicated respectively one angle equation, one angle equation and one side equation, and two side equations. The total number of side equations would have been three as before, and of angle equations two. These numbers may be checked by the formulas in sec. 154. 154. If the above process of selecting the condition equations is followed strictly, there will be little danger of choosing mutu- ally dependent condition equations. If, however, such mutually dependent equations have been chosen, it will become evident in the course of the solution by the appearance there of two equations which are identical. In this case one of the corre- lates becomes indeterminate. The danger of selecting such condition equations that the solution will be somewhat unstable ADJUSTMENT OF A TRIANGULATION — ANGLES 205 is a much more difficult one to avoid. In such cases the skill of the expert computer gives him a decided advantage. The computer may be guided by the following suggestions and conditions based on experience. These suggestions will tend to make the solutions of the equations less laborious as well as more stable. 1. In selecting angle equations, preference should be given to the triangles which have one or two sides on the exterior of the figure. This tends to avoid entanglements with other conditions. It is expedient to exclude triangles with small angles in order to avoid entanglement with side equations having large coeffi- cients. It is often desirable, though of less importance, to exclude triangles that adjoin small angles, and so have a side in common with them. 2. In selecting the side equations, it is desirable, as already indicated in Art. i 50, to secure large coef- ficients, and therefore small angles should be used. If, however, such side equations are selected that the small angles of the figure are used more than once, it may be found that the solution is unstable be- cause in some of the normal equations there are side coefficients of about the same magnitude as the diagonal coefficient. The rule in select- ing side equations should therefore be to use the small angles of the figure once and only once. 3. While it is true in general that time will be saved by using side equations having a small number of terms, there are exceptions to the rule. For example, in dealing with the fol- lowing figure, experience shows that it is advisable to use a side equation having its pole at P and involving all four of the lines which radiate from it, and containing eight terms, although a sufficient number of side equations could be written, each of which would contain but six terms. Experience .shows that in such a case as this the solution may be somewhat unstable, Fig. ig. 2o6 THE ADJUSTMENT OF OBSERVATIONS unless this side equation of large scope is used, apparently because the points B and D, which are not connected by a line of sight, are not otherwise sufficiently bound together. 4. In the process of elimination, it is advisable to avoid the introduction into an explicit function of an unknown quantity from which the corresponding original equation is free. In line 4, Table A, the value of x contains the unknown quantity J/ which was absent from the second equation. Had the third equation changed places with the first, the terms of the original second equation might have been introduced directly into Table A, and the number of lines in Table B would have been two less, and the number of columns one less. The order of solution in a figure adjustment can be best decided by inspection of the figure. The work should com- mence with an angle equation from a triangle having a side (or better, two sides) on the exterior of the figure ; and no angle equation from a triangle with a new interior side should ever be introduced till after the entrance of every angle equation not thus exposed to entanglement with conditions yet untouched. A side equation should usually be postponed till after the introduc- tion of all the angle equations that relate to the same points and no others, but should immediately follow them, so as to precede all equations that extend beyond its domain into new territory. The suggestions in the preceding paragraphs will be illustrated in Art. 180, in connection with the method of directions. 155. Adjustment of the Quadrilateral NSOL (Fig. 10). The method of forming the condition equations having now been ex- plained, we are ready to adjust the quadrilateral NSOL, as promised in Art. 137. The condition equations have all been formed in the preceding sections. Collecting them, we have : Local equations (Ex. i, 2, Art. 139), 7', + %>n + 7'3 = - 1-37, V^+ 7/5 - T/g =- 1.07. Angle equations (Ex. Art. 143), 7/3 -I- 7/4 + 7/7 = — 0.48, '4 + V^ + Vg + 7'9 = — 1. 10. ADJUSTMENT OF A TRIANGULATION — ANGLES 207 Side equation, the unit being the sixth place of decimals (Ex. Art. 14S), 1.43 7'i - 0.92 7'. - 1.93 7/5 + 0.74 T/fi - 0.43 V. + 1.77 7's = - 2.00. The methods of solution have been explained in Chapter V, and we shall proceed in the order there given for the three forms. 156. First Solution — MctJiod of Independent Unknowns There being 9 unknowns and 5 condition equations connecting them, there must be 4 independent unknowns. We shall choose t/,, 7/3, 7/4, 7/5. Expressing all of the unknowns in terms of these four, we write the equa- tions in the form of observation equations, as follows (see Art. 118) : ^'1 = + 7/2 = ^3 = - ^4 = -Vf, = ^6 = ^7 = + weight 2 + 7/, - ^, + 7/2 ^2 + + Vn — + - 1-37 7^5 + 1.07 + 0.89 7/g = — 0.565 7/1 + 0.763 V2 — 0.661 7/4 + 0.672 7/5 — 1.36 1 7/9 = — 0.435 '^1 "~ 1-763 "^2 + 0.661 7/4 — 1.672 7/5 — 1.699 Hence the normal equations 14 23 6 7 31 I 8 (I) ^l V, ^'4 ^5 Const. + 48.83 + 50.70 + 72.45 - 32-93 - 40.83 + 64-93 + 5-44 + 24.09 — 2.29 + 35-82 - 53-45 - 69.70 + 28.18 - 29.30 + 83.79 '=[^'ni Solving these equations (Art. 157), we have the values of the corrections, 7/, = — 0.»2' 7/4 = — 0.22" 7/5 = - 0.47' ^8=- 1-33 7/,, = — 0.08' -'^2=- 0.36", and thence from the condition equations, 7/3 = - 0.19", 7/6 = + 0.38", 7/7 = - 0.07", These corrections applied to the measured values of the angles give the most probable values as follows : AT, = 124° 09' 39.87", ^0 = 70° 39' ^2=113° 39' 04.71", iJ/7 = 34° 40' M^= 122° II 15.42 i^4= 23° c8' 05.04", ^6= 47° 31' 19-94", M, = 43° M, = 30° 46' 53' 24.98", 39-59", 25-07", 30.73" 2o8 THE ADJUSTMENT OF OBSERVATIONS 157. The Precision of the Adjusted Values. (a) To find the m. s. e. of an observation of the unit of weight (Arts. 105, 108). From the above values of the residuals v, ipv-\ = 7.53- Check of {pv^X Carrying through the solution of the normal equations the extra column required by the sum {pll\ we find (p. 209), li>xr-\ = 7.54. Hence, '^=v/, = . / 7.54 9-4 = ±1.23". {b) To find the weight and m. s. e. of the adjusted value of an angle. Take the angle NLS. Proceeding as in Art. 108, we have, F = NLS = 180 + e - {M^ + V2+ M^ + 7/5). ■'. dF = — 7'2 — Vy Hence, from the extra column, the sixth, carried through the solution of the normal equations (p. 209), Up = 0.053, and therefore, h-p = 1-23^/0.053 = 0.28". (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 ,sin OA''S s'm LSO = JVS = NS sin SOJV sin OLS sin (Ms + V3) sin (Mg + v^) sm I sin {M^ + Vt) sin (J/g + v^ For check we shall proceed in two ways. (i) Expand /^directly ; then, ,^ I^F dF 8F SF \ = - 0.0505 Vs + 0.0282 Z/g — O.I 160 7 - 0.1342 Vg = - 0.007 Vi + O.17I V2 + 0-056 4 + 0.253 Vs, by substituting for v^, Vf,, v^, Vg, their values from equations (i). Carry through the solution of the normal equations the extra column re- quired by these coefficients, and np = 0.0019. Hence, Mir- = 1-23 v^o.0019 = 0.05 m. ADJUSTMENT OF A TRIANGULATION — ANGLES 209 (2) Take logs of both members of the equation ; then, log F = log NS + log sin (J/j + v^) + log sin {M^ + v^ - log sin {Mj + Vj) — log sin {Mg + f,). But since NS is constant, we have, in units of the sixth place of decimals, d\og ^ = - 1.33 t^a + 0.74 Vg - 3.04 vj - 3.52 V,, = — 0.18 ?;, + 4.50 Z', + 1.45 Vi + 6.63 7/5, from equations (i). Hence, from the last column added to the solution of the normal equations, "loeF ^ '5° '" units of the sixth place of decimals. Also, Miog ir = 1-23 ^^^^ = 1.5 in units of the sixth place of decimals. -^r mod. F= 16556 w, .'. ixF = 0.06 m. The solution of the normal equations, with the extra columns required by the weight determinations, is as follows : Now, since d\ogF and Z'l V2 '^i Z'5 / /(angle). /(side). / (side). + 48.83 --50.70 -- 72.45 — 32-93 — 40.83 + 64.93 + 5-44 + 24.09 — 2.29 + 35-82 [///J = — 53-45 — 69.70 -1- 28.18 — 29.30 + 83-79 — I — I — 0.007 + o-'7i + 0.056 + 0.253 — 0.18 + 4-50 + 1-45 + 6.63 + 1 + 1.038 -4- 19.808 — 0.674 - 6.643 + 42-725 - - 0.1114 - 18.4420 - 1-3784 1-35.2140 = - 1.0946 - 14-2038 - 7-8652 - 23.3454 - 25.2S30 — I — I — 0.000 -j- 0.176 + 0.053 + 0.253 — 0.004 + 4-687 + 1-328 + 6.650 -|- O.OOI + ' — 0.335 + 40.497 + 0931 + 7-563 + 18.044 -f 0.7176 4- 12.6322 -- 10. 1 1 14 + 15.0910 -f- 0.050 — 0.335 — 0.069 -f- 0.050 -(-0.009 -|- 0.112 ■4- 0080 + 0.002 + 0.237 -j- 2.900 --2.287 + ' . >o9 + ' + 0.187 + 16.63a + 0.3112 + 7-753 + "-'S' — 0.008 — 0.006 + 0.003 + O.OOJ + 0.069 -l- 0.000 + 0.072 + '745 -f 0.208 + > [prvv.] = -|- 0.466 + 7-538 — 0.000 0.000 0.050 o.o<>3 .000 -f- 0.000 0.000 0.002 0.000 0.000 + 0.105 + 0.183 0.000 1.109 0.208 0.183 1.500 O.OS33 0.002 210 THE ADJUSTMENT OF OBSERVATIONS The solution has been carried to four places of decimals in certain parts, on account of loss of accuracy arising from dropping figures in multiplications. The resulting values of the corrections have been cut down to two places of decimals. The work was done with a machine, as explained on p. io6, the reciprocals of the diagonal terms being used so as to avoid divisions. Thus the first reciprocal is 0.02048. 1 58. Second Solution — Method of Correlates. Arranging the condition equations in tabular form, we have J'l I'l J'3 ^'4 5^5 ^'o v~ ^'8 Z'o weights 2 2 •4 23 6 7 31 I 8 + ••43 — 0.92 + • + I -)- I + I — ••93 + 0.74 + > — 0.43 + 1 ' ■ — I + ••77 + 1 ' + 1 — 2.00 — '-37 — 0.48 — 1.07 I.IO T/ic Corretatc Equations. 2 ^'1 2 7'2 14 7^3 23 6 7 v^ 31 7'7 8 7's I. II. III. IV. V. + 1-43 + I — 0.92 + I . . . . . . • • • • + I + I .... + I + I - 1-93 . . . + I + 0.74 . . . — I + I -0-43 + I . . . + I + 1-77 ::: + I + I The N'onnal Equations. I. II. III. IV. v. /. + 5- 284 + 0.255 + 1. 07 1 — 0.014 + 0.071 + 0.147 - 0.427 + 0.043 + 0.353 + 1.862 + 0.032 - 0.143 + 1-300 — 2.GG - 1-37 -0.48 - I.G7 — I.IO The solution of these equations gives (see page 212) ADJUSTMENT OF A TRlAXGULATION — ANGLES 211 I. = - 0.3973, II. =- 1.0749, III. = — i.6oc6, IV. =-3.5721, V. = — 0.6301. Substituting these values in the correlate equations, the same values of the corrections result as before. Also, [pv"-] = 7.53. 159. The Precision. {a) To find the m. s. e. ^ of an observation of weight unity. From the values of 7/ we find directly, [pv'] = 7.53- Checks of [pv-\ These are worked out in the solution of the normal equations on p. 178, according to the formulas of Art. 121, and give 7.54 and 7.55 respectively. Hence, taking the mean, [pv'-] = 7.54, and the number of conditions being 5, = 1.23", as before. Compare Ex. 2, Art. 122. {b) To find the weight and m. s. e. of the adjusted value of an angle. Take the angle NLS. . ■ . dF = — V2 — v^. From the values of u, a, b, . . . \n the condition equations in connection with the values of /"given by this function, we have \_uaf] =+0.782, [udf] = — 0.167, [ ubf ] = -0.500, [ uef] = o. [ zee/ ] = o, [ u//"] = + 0.667. Hence, from the seventh column in the solution of the normal equations below, u^ = 0.053 and M^= 1.23 Vo. 053 = 0.28". Compare Ex. 4, Art. 125. (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), Art. 157, we have, dF = — 0.0505 Vz 4- 0.0282 i^u — o.ii6oz^7 — 0.13425 v^. Also from the condition equations, 212 THE ADJUSTMENT OP OBSERVATIONS [ 7^«/] = + 0.0046, [ iidf\ = - 0.0040, [ ubf\ = - 0.0036, [ uef'\ = - 0.0165, [uc/] =- 0.0073, 1^'//]=+ 0.0030. Hence, from the eighth column in the solution of the normal equations, Uj,^ = 0.0023, and finally, Mjr = 1-23 '^•0023 = 0.06 /«. Solution of the Normal Equations. I. II. 111. IV. V. / /(Angle) /(Side). + 5.284 + 0.2S5 + I-071 — 0.014 -- 0.071 + 0.147 — 0.427 + 0.043 + 0.353 + 1.862 + 0.032 — 0.143 + 1-300 — 2.00 — 1-37 — 0.48 — 1.07 — 1. 10 + 0.782 — 0.500 — 0.167 + 0.667 + . 0046 — 0.0036 — 0.0073 — 0.0040 — 0.0165 + 0.0030 + I + 0.0483 + 1-0587 — 0.0026 + 0.0717 + 0.1470 — 0.0S08 + 0.0206 4 0.0419 + 0.3185 + 0-3524 — 0.0902 -- 0.0369 --0.007S -- 0.6438 — 0.3785 — I. 2731 — 0.4853 — I. 2316 — 0.3952 + 0.7570 + 0. 1480 -70.5377 + O.O02I — 0. 1035 — 0.2756 + 0.5510 + . 0009 — 0.0038 — 0.0073 — 0.0036 — 0.0182 + 1 + 0.0677 + 0. 1421 + 0.0204 +0.0404 +0.3181 - - 0.0851 -0.0434 -0.0093 - 0.6361 — 1.2025 — 0.3991 — 1 . 2068 — 0.5037 + 1.5309 0. 5078 + 0.0385 — 0.0930 — 0.3214 + 0.2780 — 0.0036 — 0.0070 — 0.0035 — 0.0185 + 0.0630 + 1 +0.2843 +0.3066 + 0.3054 — 0.0030 + 0.6228 — 2 . 8086 — 1-0933 — 0.3818 + 1. 1209 + 0.2709 — 0.1039 — 0.3332 + 0.2676 — 0.0493 — 0.0015 — 0.0164 + 0.'0027 Values of the Unk I. =1 — 0.39 II. = - T.07 Ill.rr- 1.60 naivns : 73. 49. 06, + 1 — 0.0098 + 0.6228 — 3.5659 — 0.3924 + 3-8986 — 0.3389 — 0.3342 + 0.2324 — 0.0049 — 0.0164 + 0.0027 IV. = -3. 5721, V. rr — 0. 6301. + 1 — 0.6301 = V + 0.2472 — 0.5366 + 0.0531 — 0.0264 + 0.0023 I. X /' = — 0.3973 X — 2.0 11. X /" = — 1.0749 X — 1.3 III. X /"' = — 1.6006 X —0.4 IV. X /"" = — 3-5721 X — 1.0 V. X /'"" — — 0.6301 X — I.I = 0.7Q 7 = 1-47 8 = 0.77 7 = 3 82 =r . 69 7-S4 = [/ivv] 0.7570 I - 5309 1. 1 209 3.8986 0.2472 7 - 5546 = [pzn'] The values of [pv-] are found from Equations 2, 3, Art. 121. ADJUSTMENT OP A TRIANGULATION — ANGLES 213 160. Third Solntiojt — Solution in Tzvo Groups. The form given in Art. 128 is followed. The Local Adjustment. {a) At North Base. The Observation Equations. p U-.) ix,) / 2 + I 0. 2 . . . + I 0. 14 — I — I - 1-37 The Normal Equations. 16 + 14 = — 19.18 = [pal} suppose, 14 + 16 = — 19.18 = [pl^l] suppose. Solving in general terms, (jr,) =+ 0.267 [pal] -0.233 [pbll (^0) = - 0.233 [P<^i] +0.267 [pbl]. Hence, and (:r,) = - 0.64", {x^) = - 0.64", {X,) =+0.64" + 0.64" = 1.37", = - 0.09", Local Angles. 124'- 113' 122^ 09' 40.05", 39' 04.43", 11' 15.52". To find the m. s. e. of a single observation. The value of [pv''] = [Pxx]^ 1.75. Hence, for this station, the number of condit ions being 3-2, "-v/j^ = I-3"- {b) At South Base. The OiisERVAT ON Kouations. p (^4) (^5) / 23 + I + I 0. 0. 7 + I + I -'•37 214 THE ADJUSTMENT OP OBSERVATIONS Thb Normal Equations. 30 + 7 = - 7-49. 7 + 13 = - 7.49- Hence Also, (X,) = - 0.13", (x,) = - 0.50", (x,) = - 0.13" - 0.50" + 1.07", = + 0.44"- Local Angles. 23^ 08' 05.13", 47° 31' 19-91", 70° 39' 25.04". [/z/2] = Ipxx] = 3.24. = 1.8". The General Adjustment. Mosf Probable Angles. At N. Base, 124° 09' 40.05" + (i), 113° 39' 04.43" + (2), 122° 11' 15.52"- (i)- (2). At S. Base, 23° c8' 05.13" + (4), 47° 31' 19-91" + (5), 70° 39' 25.04" + (4) + (5). At Oneota, 34° 40' 39-66" + (7), 43° 46' 26.40" + (8). At Lester, 30° 53' 30.81" + (9). The Angle and Side Equations, (a) Triangle, N. Base, S. Base, Oneota. Angle SJVO 122° 11' 15.52" - (i) - (2) " ATSO 23° 08' 05.13" + (4) " JVOS 34° 40' 39-66" + (7) Sum = 180° 00' 00.31' 180 + e = 180° 00' 00.05' o = 0.26" - (I) - (2) + (4) 4- (7) ADJUSTMENT OF A TRIANGULATION — ANGLES 215 {d) Triangle Lester, Oneota, S. Base. Angle jVSO 70° 39' 25.04" + (4) + (5) SOL 78° 27' 06.06" + (7) + (8) " OLS 30° 53' 30.81" + (9) iSo° 00' 01.91' 180° 00' 00.37' 0= 1.54" + (4) + (5) + (7) + (8) + (9) (c) Quadrilateral N. Base, S. Base, Oneota, Lester, sin LA'S sin LSO sin LON sin LNO sin NSL sin LOS = 1. LNS = 113° 39' 04.43" + (2), LNO = 124° 09' 40.05" + (i), LSO = 70" 39' 25.04" + (4) + (5), NSL = 47° 3^' >9-9i" + (S), LON= 43° 46' 26.40" + (8), LOS = 78° 27' 06.06" + (7) + (8). 9.9618975,6 - 9,22 (2) 9.9747660,1+ 7>39K4) + (5)| 9.8399903,4 + 21,98 (8) 539.1 509.4 29.7 9-9177479.3 - 14,29 (0 9.8677849,8 + 19,28 (5) 9.9911180,3 + 4,30 1(7) + (8) \ 509,4 Check by deducting \ of the spherical excesses of the triangles from the angles. "3° 39' 04.36", 124° 09' 40.01", 47" 31' 19.84", 70" 39 24.92 , 43"^ 46' 26.36", 9.9618976,2 9.9747659,3 9.8399902,5 38,0 _8^ 29.7 78° 27' 05.93' 9-9177479.9 9.8677848,6 9.9911179,8 "83 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. 1.43 (i) - 0.92 (2) + 0,74 (4) - 1. 19 (5) - 0.43 (7) + 1.77 (8) + 2.97 •= o. 2l6 THE ADJUSTMENT OF OBSERVATIONS The Weight Equations. (i) = - 0.233 |T) + 0.267 [U (2) = + 0.267 [T] - 0.233 S (4)= + 0.038 (4] - 0.02 1 ui (5)= — 0.021 [4] + 0.088 [Jj (7) = + 0.032 [7] (8) = + 1.000(8] (9) = + 0.125 SJ The Correlate Eqtiations. I. II. III. Check. m=-i + 1-43 = 0.43 0=-! — 0.92 + 1.92 [4l=+i + I + 0.74 - 2.74 = + I - 1. 19 + 0.19 0=+' + I - 0.43 - 1-57 [8;i = + I + 1-77 - 2.77 ® = + I — 1. 00 The check is formed by adding each horizontal row (Art. 78). Expression of the Corrections in Terms of the Correlates. I. II. III. Check. + 0.233 - 0.333 + o.ioo — 0.267 — 0.246 + 0.513 + 0.613 — O.I 15 - 0.447 (I) = — 0.034 — 0.267 + 0-233 (2) = — 0.034 + 0.038 (4) = + 0.038 — 0.021 + 0.017 — 0.021 + 0.088 — 0.579 + 0.382 + 0.214 + 0.596 + 0.028 + 0.024 + 0.052 — 0.016 — 0.105 - 0.562 + 0.038 +0.028 —0.104 — 0.021 + 0.024 — 0.004 — 0.108 + 0.058 + 0.017 (5) = - 0.021 + 0.067 — 0.121 + 0.075 (7) = + 0.032 + 0.032 — 0.014 — 0.050 (8) = + I. + 1.770 - 2.770 (9) = + 0.125 — 0.125 ADJUSTMENT OF A TRIANGULATION — ANGLES 217 The Corrections in Terms of the Correlates {Collected). I. II. III. (I) = — 0.034 - 0.579 (2) = — 0.034 + 0.596 (4) = + 0.038 + 0.017 + 0.052 (5) = — 0.021 -4- 0.067 — O.I2I (7) = + 0.032 + 0.032 — 0.014 (8) = + 1. 000 + 1.770 (9) = + 0.125 Formation of the A'of/nal Equations, I. II. III. Check. - (i) = + 0.034 + 0.579 — 0.613 - (2) = + 0.034 - 0.596 + 0.562 + (4) = + 0.038 + 0.017 + 0.052 — 0.108 + (7) = + 0-032 + 0.032 — 0.014 — 0.050 + 0.138 + 0.049 + 0.021 + 0.209 (4) = + 0.017 + 0.052 — 0. loS (5) = + 0.067 — O.I2I + 0.075 (7) = + 0.032 — 0.014 — 0.050 (8) = + I. + 1.770 - 2.770 (9) = + 0.125 + I. 241 — 0.125 + 0.049 + 1.687 - 2.978 I. II. III. Check. + 1-43(0 + 0.852 — 0.804 - 0.92(2) + 0.533 - 0.564 + 0.74(4) + 0.038 — 0.080 - 1-19(5) + 0.144 — 0.089 - 0.43(7) + 0.006 + 0.022 + 1-77(8) ■i- 3-133 - 4903 + 0.021 + 1.687 + 4.706 — 6.418 The Normal Equations {Collected). I. II. III. + 0.138 +0.049 +0021 =—0.260 + 1. 241 + 1.687 =— 1.540 + 4.706 = - 2.970 The solution of these equations gives (page 220) I. =-1.597 II. = — 0.642 III. =- 0.394 Substitute for I., II., III., their values in (4), and we have the general corrections. 2l8 THE ADJUSTMENT OF OBSERVATIONS Adding the local corrections and general corrections together, the total corrections to the measured angles result and are as follows : Local. General. Total. / /:'2 Final A NGLES. n n ti / n A-i = — 0.64 - 0.18 = -0.82 2 1-34 124 09 3987 A-2 = — 0.64 + 0.28 = — 0.36 2 .26 "3 39 4.71 ^3 = — 0.09 — O.IO = — 0.19 14 .50 122 II 15.42 X, = -0.13 — 0.09 = — 0.22 23 1. 10 23 08 5.04 -1'5 = — 0.50 + 0.04 = — 0.46 6 1.27 47 31 19-95 ^'6 = + 0.44 — 0.05 = +0.39 7 1.06 70 39 24.99 ^7 = .... — 0.07 = — 0.07 31 •15 34 40 39-59 A-g = .... - 1-33 = - 1-33 I 1.77 43 46 25.07 ^9 = .... -0.08 = - 0.08 8 .05 30 53 3073 ypzfi\ = 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. 161. The Precision of the Adjusted Values. {a) To find the m. s. e. of an observation of weight unity. Computation of \_pv'^\ (i) From the preceding table \_pv'^'\ has been found directly; thus, [^Z;2] = 7.50. (2) Check (Art. 129). From the station adjustments find \zPv°\ N. Base gives (p. 213) 1.75 S. Base gives (p. 214) 3.24 4-99 = b°'z^°]- From the general adjustment find \avw\ (a) i: n (^) 4' C-i X I. X II. X III. 4' X p^ X [feT]' = — 0.26 X — 1.597 = -f 0.42 = — 1.54 X — 0.642 = + 0.99 = — 2.97 X — 0.394 = + 1. 16 + 2^ = — 0.26 X — I.S85 = + 0.49 1.45 X - 1. 183 =+ 1.72 ADJUSTMENT OF A TRIANGULATION — ANGLES 219 l'".z X " [tc.2] = - 0.94 X - 0.394 = + 0.37 and Hence, taking the mean of the values of \pv'^\ + 2.58 .'. [ww] = 2.58 {pv-'\ = 4.99 + 2.58 = 7-57. » / 7-54 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. ... dF = -{2) - (5) = +0.055 I- ~ 0.067 II' + 0.700 III. from the weight equations. From equations (25), Art. 130, qx = ['^°-]^i + [«^]i"2 + • • • (72 = [«/3]^i + im^r^ + • • • The values of [aa], [a/3] . . . are given in the weight equations. Hence, gi=+ 0.267 X o - 0.233 X - I = + 0.233, §'2 = — 0.233 X o + 0.267 X — I = — 0.267, ^3 = + 0.038 X o — 0.021 X — I = + 0.021, q^ = — 0.021 X 0-+ 0.088 X — I = — 0.088. g' q g^q — I — I + 0.233 — 0.267 + 0.021 - 0.088 0.267 0.088 [.gH.i]' [£'C.2f [C C.2] 0.022. = 0.006 = 0.274 0.302 (.See the solution [.i,^i/] = 0.355 of the normal — 0.302 equations.) «^. = + 0.053 fip= i.23"Vo.o53 = 0.28". 220 THE ADJUSTMENT OF OBSERVATIONS {c) To find the m. s. e. of a side in the adjusted figure. Side = Oneota-Lester. Therefore, sin^OTV^ sin OLS' dF= 1.33 (i) + 1.33 (2) + 0.74 (4) + 0.74 (5) - 3.04 (7) - 3.52 (9) in units of the sixth place of decimals, = —0.174 I- — 0-475 ^I- + 0.015 III. from the weight equations. The solution is carried through exactly as in the preceding case. We \_gq\ = 2.01 1 and Up = 1.49. ix.\fup^ i.23\/i.49 = 1.5 in units of the sixth place of decimals, log OL = 4.2189699 OL = 16556 m. find, Hence, Now, r -J 16556 " m. s. e. of side = — -^^^ x 0.0000015 0.434 = 0.06 7n. Solution of the Normal Equations. I. 11. III. / y(ANGLE). + 0.138 + 0.049 + I. 241 + 0.021 + 1.687 + 4706 - 0.260 - 1.540 - 2.970 + 0.055 — 0.067 + 0.700 + I + 0-355 + 1.224 + 0.152 + 1.680 + 4-703 - 1.885 - 1.448 - 2.930 + 0.399 — 0.087 + 0.692 + 0.022 • + I + 1-373 + 2.396 - 1-183 - 0.945 — 0.071 + 0.811 + 0.006 . . . . + I - 0.394 + 0.338 + 0.274 ADJUSTMENT OF A TRIANGULATION — ANGLES 221 162. Ex. I. — Adjust the observed differences of longitude * given in the following table : HEARTS CONTENT FOILHOMMEBUM Fig. 20. 'WASHINGTON Dates. Observed Differences. Correc- tions. 1851 1857 1866 1866 1866 1872 1872 1872 1872 1872 I 869-1 870 1870 1867 ( 1872) 1872 Cambridge-Bangor Bangor-Calais Calais-Heart's Content . , . Heart's Content-Foilhommerum Foilhommerum-Greenwich , . Brest-Greenwich Brest-Paris Greenwich-Paris St. Pierre-Brest Cambridge-St. Pierre .... Cambridge-Duxbury .... Duxbury-Brest Washington-Cambridge . . . Washington-St. Pierre . . . h. }/i. s. s. 9 23.080 ^ 0.043 6 00.316^^0.015 55 37-973 ±0.066 2 51 56.356:^^0.029 41 33-336 ±0.049 17 57.598 J:; 0.022 27 18.512 j- 0.027 9 21.000 J- 0.038 3 26 44. Sio -t; 0.027 59 48.608^^0.021 I 50.191 J:; 0.022 4 24 43.276^0.047 23 41.041 J:; 0.018 23 29.553 ±0.027 [Number of condition.s = /i — s + i, where n is number of observed dif- ferences of longitude, and s is number of longitude stations. The condition equations are s. — v^ + V-, — Vg = + 0.086 - V, V, - V3 The weights are taken inversely as the squares of the p e. Solution by method of correlates, as in Art. 119.] Ex. 2. — The system of triangulation shown in the figure was exe cuted by Koppe in the determination of the axis (Airolo-Gos- chenen) of the St. Vt- Vs + V^ + Vg + 7/,o = + 0.045 + '^u + ^12 = - 0.049 + ^13 - ■Z'm = - 0096 - V. a ^10 + ^10 vni xn Fir. 21. * Coast and Geodetic Survey Report ^ 1880, app. No. 6. 222 THE ADJUSTMENT OF OBSERVATIONS Gothard tunnel* In the following table the adjusted values are given side by side with the measured values. It is proposed as a problem of adjustment. At Goschenen II. O III. 44 IV. 69 V. 124 At II. III. IV. 37 V. 60 VI. 77 Goschenen 93 VII. 124 At III! VIII. IX. 53 VI. 99 IV. 102 Goschenen 138 VII. 144 II. 180 At VIII. XI. XII. 18 X. 43 IX. 50 VI. 106 V. 112 III. 130 At IX. VI. V. 8 III. 18 VIII. 63 X. 76 XI. 79 Airolo 109 XII. 123 Measured. Adjusted. / // // 00 00.00 00.00 ^T, 10.88 10.03 30 12.51 11.62 58 4.23 5.13 00 00.00 00.00 53 54-33 52-97 29 33-'^2 33-82 4 5-67 8.17 II 41.69 40.57 16 33-98 33.27 00 00.00 00.00 58 14.48 15.49 47 50-21 50.86 32 51-36 51-90 44 28.81 29.70 28 12.47 11.40 59 38.94 39-11 00 00.00 00.00 56 17.43 17.54 50 24.03 24.70 18 22.52 20.27 30 15.04 15.37 28.72 29.24 II 30.81 41.54 00 00.00 00.00 28 17.13 15.06 33 3-27 5-00 41 28.63 28.55 59 50.89 51-48 10 36-33 36-34 45 39-23 39-33 16 23.76 24.23 * Zeitschr. fiir At IV. V. VI. VII. Goschenen II. III. At V. IV. VIII. IX. VI. VII. Goschenen II. At VII. II. III. IV. V. VI. At XI. XII. Airolo IX. VIII. At XII. IX. Airolo X. VIII. XI. Measured. Adjusted. o / // /' o 00 00.00 00.00 15 41 3-57 6.29 74 12 20.55 19.86 So 32 48.99 50.12 135 44 49.77 50.91 199 24 11.56 10.73 o 00 00.00 00.00 78 40 5.91 6.72 140 44 43-51 44-45 215 32 45-41 43-45 286 19 25.30 27.21 316 00 44.92 43.61 338 20 33.53 31.74 o 00 00.00 00.00 19 II 58.44 59.03 32 4 49.32 48.68 64 II 54.08 56.05 90 05 39.47 37.00 o 00 00.00 00.00 16 55 55-o6 54-38 37 13 59-79 58-43 152 26 30.24 30.44 o 00 00.00 00.00 30 31 2.30 3.39 42 13 20.53 21.33 90 3 2.22 1.74 98 40 14.95 13-72 At Airolo. XI. c Vermess.^ vol. iv. 00 00.00 00.00 ADJUSTMENT OF A TRIAXGULATION — ANGLES 223 AtX. XII. 94 54 56.06 55.26 XII. 00 00.00 00.00 IX. 230 53 7.51 6.98 Airolo 9 49 30.02 37-92 X. 296 26 49-43 51-" IX. 91 30 5.16 5-96 VIII. 252 43 46.75 47-49 XI. 275 12 8.44 9-74 The distance X-XII is 4416.8 in. There are 19 angle equations and 15 side equations in the adjustment. Solution by Groups. 163. The rigorous forms of solution which have been given are suitable for a primary triangulation where the greatest accuracy is required. In secondary or tertiary work it is fre- quently not advisable to spend so much labor in the reduction. For work of this kind the group method of solution is to be preferred. The solution by groups may be made by either of two general methods. First, each condition or set of conditions may be 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 and second groups of condition equations closely enough, the process must be repeated until the required accuracy is attained. This is the method outlined in general terms in Art. 131. Second, each group may be adjusted in turn while the results of the adjustment of the preceding groups are preserved by insuring that the conditions which have been satisfied remain so. This is an exact method. 164. To make the operation as simple as possible, let us take but a single condition at a time. (i) Local equation at N. Base, ^1 + ^-2 + ^'3 + 1-37 = o- The soluti'jn is given in Ex. i, Art. 139, i), = — 0.64", v.^ = — 0.64", 7'3 = — 0.09. 224 THE ADJUSTMENT OF OBSERVATIONS (2) Local equation at S. Base, 7\ + 7'5 —t'e + 1.07 = o. The solution is given in Ex. 2, Art. 139, Vi = — 0.13", 7-5 = — 0.50", 7'6 = + 0.44". (3) Angle equation, 7^3 + '^'i + ^'7 + 0.48 = o. Using the values of v^, v^, already found as first approxima- tions, the equation reduces to ^3 + ^'i+ ^'7 + 0-26 = o. The method of solution is given in Ex. 2, Art. 120, 7'3 = — 0.13", 7'i = — 0.08", V = — 0.05". The successive approximations found so far, when added, give n= - o-S5"» z'e = + 0.44", 2/7 = — 0.05". Proceed similarly with the remaining two condition equations. The resulting values will agree closely with the rigorous values already found. 165. In order to bring out still more clearly the advantages 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. ?'l = — .64", Z'2 = — .64", 7's = — 0, .22", ^'i = — 0, .21", ADJUSTMENT OF A TRIANGULATION — ANGLES 225 VANDERLIP CT Fig. 22. 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. .Station Occupied. Sir John . Carlton . Kingston . Wolfe . . Angle as Measured Between Carlton and Kingston Wolfe and Kingston. Wolfe and Sir John . Kingston and Sir John Sir John and Wolfe . Carlton and Wolfe Wolfe and Amherst . Duck and Carlton . . Amherst and Carlton Kingston and Carlton Sir John and Carlton 90 17 24 1 20 4S 62 03 64 ?,! 88 1 88 140 84 25 40 02 19 07 1 2 18 44.91 09.77 06.54 27.56 50.91 04 • 43 14.70 18.54 34-44 14 •.S4 16.80 LoC.\I.LY Cork. Angles. 226 THE ADJUSTMENT OF OBSERVATIONS Locally Station Angle as Measured Between CORR. Occupied. Angles. o f n // Amherst . . Kingston and Wolfe. . . 35 41 23.02 22.69 Kingston and Duck . . . III 45 28.46 28.68 Wolfe and Duck ... . 76 04 06.32 05 --99 Grenadier and Duck. . . 54 38 00.34 Duck and Vanderlip. . . 71 IS 25-43 25-32 Vanderlip and Kingston . 176 59 06.11 06.00 Duck . . . Oswego and Vanderlip. . 104 08 58.93 59.10 Vanderlip and Amherst . 70 26 31-99 32.16 Amherst and Wolfe . . . 56 01 12.47 12.64 Wolfe and Grenadier . . 18 45 43-36 43-53 Grenadier and Stony Pt. . 49 53 12.77 12.94 Stony Pt. and Oswego . . 60 44 19.46 19.63 Grenadier . Stony Pt. and Duck . . . 78 13 33-64 33-84 Duck and Amherst . . . 50 35 04.28 04.19 Duck and Stony Pt. . . . 281 46 25.89 26.16 Amherst and Stony Pt. . 231 II 22 .04 21.97 Stony Point. Oswego and Duck .... 88 22 00.86 . . Duck and Grenadier. . . 51 53 12.60 12 70 Grenadier and Duck. . . 308 06 47.21 47-3° Oswego . . Sodus and Vanderlip . . 80 29 46 10 46-59 Sodus and Duck .... 107 19 03.28 03.96 Sodus and Stony Pt. . . . 138 12 49-44 48.28 Vanderlip and Duck. . . 26 49 16.61 17-37 Vanderlip and Stony Pt.. 57 43 01 .g6 01 .69 Duck and Stony Pt. . . . 30 53 42.88 44-32 Vanderlip . Amherst and Duck . . . 38 18 07 .12 07.30 Amherst and Oswego . . 87 19 53-47 53-16 Duck and Oswego. . . . 49 01 45-54 45-86 Duck and Sodus .... 87 59 12.55 12.42 Oswego and Sodus . . . 38 57 26.55 26.56 Sodus and Amherst . . . 233 42 40.41 40.28 Sodus . . . Vanderlip and Oswego. . 60 32 57-55 . . . The local and general equations are formed as usual (see Arts. (133-155). 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 con- ditions 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. ADJUSTMENT OF A TRIANGULATION — ANGLES 227 166. The Local Adjustment at Each Station. (a) Adjust for each sum angle separately. Rule and example in Arts. 138-139. (d) Adjust for closure of the horizon. Rule and example in Arts. 138-139. 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 adjusted, as in Art. 138. The resulting values are given in the table. If we break the adjustment into two parts, as in (a) and (d), we have : (a) Sum Angle. Measured Values. Adjusted. / // // // Kingston-Wolfe, 35 41 23.02 — 0.29 22.73 Wolfe-Duck, 76 04 06.32 — 0.29 06.03 III 45 29-34 28.76 Kingston-Duck, III 45 28.46 + 0.29 28.75 check, 3)0.88 0.29 Closure of Horizon. / // // // Kingston-Wolfe, 35 41 22.73 ~ °-°^ 22.65 Wolfe-Duck, 76 04 06.03 ~ °-°7 05.96 Duck-Vanderlip, 71 15 25.43 — o.oS 25-35 Vanderlip-Kingston, 176 59 06.11 — 0.07 4 ) 00.30 06.04 00.00 check 00.075 The adjusted values agree closely with those from the simultaneous solution, as given in the table. 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. 139.) 167. The General Adjustment. — The local adjustment being finished, we shall consider the adjusted angles to be inde- 22S THE ADJUSTMENT OF OBSERVATIONS pendent of one another and to be of the same weight. We are therefore at hberty to break up the net into its simplest parts. We have in our figure, first a quadrilateral SCWK, next two single triangles KWA, A WD, next a central polygon DAGS and, lastly, a single triangle VOS. These three figures include most cases that arise in any triangulation net, 1 68. {a i) Adjustment of a Quadrilateral: Approximate Method. — In the quadrilateral SCK W all of the eight angles 1,2,. . . 8 are supposed to be equally well measured. (i) Tke Angle Equations. The angle equations from the triangles SCW, CWK, WKS, may be written in general terms I'l +2^2+ '3 + ^4 = A' 7'3 + ^'4 + Z'5 + ^'6 = /z' ^'5 + ^'6 + ^7 + ^'8 = 4- As these equations are entangled, if we adjusted for each in succession a great many repetitions of the adjustment would be neces- ^^''^' sary 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 C^, Q, C„, C^, the correlates of the equations in order ; then the correlate equations are C^ = v^, C,+ Q = v„ c; = v,_. c; + c; = v„ <^l + C = Z'3, Cg = V^, c; + c; = Vi, Cz = v^, and the normal equations. 4 C\ + 2 C, = /j, 2 Cj + 4 C, + 2 Cg = /,, 2 C + 4 Cg = /g. ADJUSTMENT OP A TRIANGULATION — ANGLES 229 Solving these equations, there result, Ci = n+ 3 A - 2 /, + Q, C,= \{- 2/j + 4/, - 2/,), Q=H+ /i-2/, + 3/3). Substitute these values in the correlate equations, and ^'i = 2'2 = i (+3A - 2/2+ 4), ^'8 = 7'4 =i (- /i + 2/, - /,), 2'5 = 7'6 = i(- A + 2/3+ Z^), which may be written, ^i = ^'. = Ki-H^.-^4- Wi), z's = ?'6 = 14 + H4- ^4-^/1), whence follows at once the convenient rule for adjusting the quadrilateral, so far as the angle equations are concerned: (a) Write the measured angles in order of azimuth in tzvo sets of four each, the first set being the angles of SCW, and the seco7id those of WKS. (/3) Adjust the angles of each set by one fourth of the dif- ference of this sum from 180° + excess of triangle, arranging the adjusted angles in two columns, so that the first column zvill show the angles of SCK, and the second those of CWK. (7) Adjust t lie first column by o?ie-fourth of the difference of its sum from 180° + excess of triangle, and apply the same cor- rectioji, with the sign changed, to the second column. This rule was first published by me in the Journal of the Frankli^i Institute, June, 1880. The spherical excesses of the triangles ^CH>', CJVA'. U'k'S, being 0.16", 0.35", and 0.47", respectively, the adjustment of the quadrilateral may be arranged as follows : 230 THE ADJUSTMENT OF OBSERVATIONS Measured Angles. Adjusted Angles. o / /f ff // 33 53 35-14 35-56 35-40 62 03 27.56 27.98 27.82 58 44 38-98 39-40 39-56 25 18 16.80 17.22 17-38 179 59 58-48 00.16 check 180 + e = 180 00 00.16 4) 1.68 0.42 58 54 57-54 58.11 58-27 37 02 04.43 04.99 05-15 27 38 46 • 48 47.04 46.88 56 24 9.77 10 -33 10.17 179 59 58-22 00.91 00.47 check 180 + € = 180 00 00 . 47 4)2.25 00.28 4)0.63 0.56 0.16 169. (2) T/ic 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 SCO sin CWO sin WKO sin KSO = I ; or writing it in general terms, when reduced to the linear form, where ■z'/, vj, . . . are the corrections resulting from the side equation. Solving as in Ex. 2, Art. 120, we have the corrections These corrections may be found still more rapidly as follows : Since the side equation may be so transformed that the coeffi- cients «j, a.„ . . . are approximately equal to unity numerically ADJUSTMENT OF A TRIANGULATION — ANGLES 23 1 (see Art. 149), wc may take each of them to be unity, and then t'x = ^3 = ^'5' = ^'7' = + i 4» V,' = 7'/ = z'e' = v/ = - i 4 ; that is, t/ie corrections to the angles are nmnerically equals but are alternately + ^^id — . This plan has the additional advantage of not disturbing the angle equations. The rule gives approximate results which are the more nearly correct, the more nearly the coefficients a^, a.„ . . . are equal to each other and the smaller is the absolute term of the side equation. In many cases it will give results which depart widely from those found by the exact process. Returning to our numerical example, we first reduce the side equation to the linear form o OSC = 33 53 35.40 + v„ SCO = 62 03 27.82 + -y., OCIV = 58 44 39-56 + 7/„ CIVO =25 18 17.38 + V, OWK = 58 54 58 . 27 + v„ WKO = 37 02 05 . 15 + 7/,„ OKS = 27 38 46.88 + 7/„ KSO = 56 24 10.17 + z/g. 9-7463587 + 31-3^1 9-9461673 + II. 2 7/2 9.9318952 + 12.87/3 9.6308691 + 44.sVi 9.9326832 + 12. 7-^5 9.7797125 + 27.97/9 9.6665301 + 40.27/7 9.9206181 + 14.07/g 72 70 70 2 Dividing by 20, which will reduce the coefficients to unity approximately, and 1 . 56 7/,' — o . 56 7/0' + o . 64 7// —2.22 7/4' + o . 64 7'/ — 1 . 40 7// + 2.01 7/7' — o . 70 7// +0.10=0. Hence, [aa] =15, and 7// =— 0.01", 7/2' = 0.00", 7/..,' = 0.00", 7// =+ o.oi", etc. By the second rule the corrections would be ^ ^-— , that is, ^ 0.0 1", alter- 8 nately, 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. 170. (a 2) Adjustment of a Quadrilateral : Rigorous Method. — By the following artifice the quadrilateral may be rigorously ^32 THE ADJUSTMENT OF OBSERVATIONS adjusted for the side equation without disturbing the angle equation adjustment, 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 al- ready explained in (a i). If v/, v./, . . . v^', denote the cor- rections arising from the side equation, the condition equations may be written •Vi + v^ + v^ + vl = o, vl + vl + %% + vl = o, v/ + Ve' + v^' + Vs = o, a^v,' + a.v^' + a^v^ + i^Vi + ihv^ + a^v^ + aiv^ + a^v^' = //• By writing the corrections in the form v^ = + v - v', v^ = + v — v", ^3 vl = — v - v'\ v^ =-v — V , the first* three condition equations become = identically, and we have therefore to deal only with the single condition equation («! -\- a^ + ch + ae - «3 - ^h - c^ - ^s) ^' + (36 THE ADJUSTMENT OF OBSERVATIONS Given Angles. Log Sines. DiF. l". Squares. Prod- ucts. Sums. f If // 54 38 00.34 50 35 04.19 74 46 55-83 180 00 00.36 180 00 01.21 = 00.62 04.47 56.12 = i8o + e 9.91 14060 9.8879338 + 14.9 + 17-3 222.0 299-3 257.8 - 2.4 3)0.85 0.28 78 13 33.84 51 53 12.70 49 53 12.77 34-49 13-35 13-42 9.9907654 9.8958618 + 4-4 + 16.5 19-4 272.2 72.6 — 12. 1 59-31 1.26 1-95 0.65 88 22 00.86 30 53 44.32 60 44 19.46 04.64 3-47 00.47 43-03 19.07 9.9998235 9.7105188 + 0.6 + 35-2 .4 1239.0 21. 1 -34-6 1. 17 0.39 26 49 17.37 49 01 45-86 104 08 58.93 2.16 18.15 46.64 59-70 9.6543842 9-8779750 + 41-6 + 18.3 1730.6 334.9 761.3 + 23.3 4.49 2.33 0.78 38 18 07.3a 71 15 25-32 70 26 31.99 04.61 1. 71 2.90 0.97 06.33 24.36 31.02 9-7922537 9-9763353 + 26.7 + 7-1 712.9 50.4 189.6 + 19.6 6328 6247 6247 81 3 -"4n 4881. 1 1302.4 6183.5 2 1302.4 - 6.2 12,367.0 The Normal Equations. Local Equation at Station Duck. 12,367 C, 6.2 C, + 6 . 2 C, = - + 10 Cj = C, = — o .020 ■243 2 .01 c. + 0.213 74 46 56.12 49 53 13.42 60 44 19.07 104 08 59.70 70 26 31.02 359 59 59-33 360 00 00.00 00.67 3 2.01 ADJUSTMENT OF A TRIANGULATIOX — ANGLES 237 Corrections. (i: = -0.39 (2: = + 0.26 (3; = + 0.13 (4: = — . 24 (5: = + 0.18 (6: = + 0.06 (7. = -0.31 (s: = + . 40 (9) = — 0.09 (10: = -o-75 (III = + 0.45 (12J = + 0.30 (13) = - 0.47 (14) = + . 20 (15) = + 0.27 Adjusted Anc .LES / // 54 38 00.23 50 35 04 73 74 46 56 25 78 13 34 25 51 53 13 53 49 53 13 48 88 22 00 16 30 53 44 33 60 44 18 98 26 49 17 40 49 01 47 09 104 08 60 00 38 18 05 86 71 15 24 56 70 26 31 29 Approximate ]\Ictliod of Finding t/ie Precision. 174. 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 m. 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 ^l' of the m. s. e. of an angle in a triangu- lation net after adjustment is easily seen from Art. 1 13 to be , In - n^ where n = number of angles observed, n^ = number of local and general conditions, fx,m= iTi- s. e. of a measured angle of average weight ; or, if all the angles arc of equal weight, it is the m. s. c. of a measured angle. The value of /x, the m. s. e. of an angle of unit weight, is, by the usual formula, 238 THE ADJUSTMENT OF OBSERVATIONS in which p^ is the average weight of a measured angle. To find the m. s. e. of a side of a triangle, a single chain of the best-shaped triangles between the base and the side is selected, all tie lines being rejected. Then, assuming the base to be exact and the m. s. e. of each adjusted angle to be ix', we have from Ex. 9, Art. 126, where S^, S^j are the log differences corresponding to i" for the angles A, B m 2i table of log sines. 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. 157) [pv'^'\ = 7.54. -N/f = 1-23", pm = 10.4 and .-. /A„ "• = v^^"- = §/*» = 0.26". CHAPTER VII APPLICATION TO THE ADJUSTMENT OF A TRIANGULATION. METHOD OF DIRECTIONS 175. This method is clue to Bessel. Various modifications of Bessel's plan of making the observations are in use on differ- ent surveys. The following is that used in the Coast and Geo- detic Survey on primary triangulation at the present time. Each series of observations consists of successive pointings on the various stations in order, from left to right, with corre- sponding readings of the horizontal circle with three micrometer microscopes, followed immediately by pointings on the same stations in the reverse order, after reversing the position of the horizontal axis of the telescope in the wyes, and turning the alidade 180° in azimuth, each pivot remaining in contact with the same wye as before. Each observation of an angle consists therefore of two pointings on each station involved, one in each position of the telescope, together with the corresponding microm- eter readings, twenty-four in all, both a forward and a back- ward reading of each micrometer being made in each of its positions. Sixteen such series of observations are taken upon each station, one in each -of sixteen positions of the horizontal circle. As implied in this statement of the method, the instrument used carries an accurately divided horizontal circle which is read by micrometer microscopes. The circle may be shifted to different positions in azimuth, but is not provided with such a clamp and slow motion tangent screw controlling the position of the graduated circle a»s is needed on any instrument used with the method of repetitions. In the repetition method of angles, one angle, between two 239 240 THE ADJUSTMENT OF OBSERVATIONS Stations, is measured at a time. In the direction method, as practiced in the Coast and Geodetic Survey, a single series of observations serves to measure all the angles between stations, or to determine the relative directions to all stations, observed upon in that series. In the repetition method, the unknowns which are being measured, are the angles ; and, in the direction method described, the unknowns are relative directions, and the difference of any two such unknowns is an angle. As this dif- ference exists in the method of observation, it also exists in the method of adjustment. In the direction method of adjustment, the unknowns are directions, one for each line observed over from each station, and angles appear only as differences of directions. 176. In the direction method of adjustment, it is assumed that there is an error inherent in each direction observed which affects every angle involving that direction. For errors which are due to the instrument, and also those due to the observer (with one exception, noted below), there is no sufficient reason for assuming that errors are inherent in the directions, rather than in the angles. All errors due to external conditions, that is, to conditions outside the instrument and the observer, on the other hand, must be assumed to be inherent in the separate directions observed, rather than in the angles. The external errors may be due to various causes, including phase and asym- metry of the object pointed upon, eccentricity of the signals pointed upon, or of the instrument, and lateral refraction. Such errors tend to recur with one sign for each observation over a given line from a station, and therefore to affect that direction in a constant manner which is independent of the other directions which happen to enter that series of observa- tions. In this class of external, constant errors must also be placed, whatever tendency the observer may have to misjudge the position of the center of the image pointed upon. If an observer makes every pointing on every object too far to one side, say to the left, by a constant amount, this error will not ADJUSTMENT OF A TRIANGULATION — DIRECTIONS 241 affect the final results. But any such tendency to a one-sided pointing is reasonably certain to be modified by the brightness and size, as well as by any asymi" Hry, of the image pointed upon. It is therefore probable that an error of bisection exists which is peculiar to each direction observed, as long as the image of the object pointed upon in that direction remains con- stant in appearance. It matters little in choosing the method of adjustment, whether the above outline of the manner in which errors are inherent in particular directions is accurate in detail or not. If the great mass of evidence available indicates strongly that the principal errors in angle measurements are of the external class, and that they are inherent in separate directions rather than in angles, the direction method of adjustment should be used in preference to the angle method. That the direction method should be used, and for this reason, is the conviction to which years of critical observation have led the computers of the Coast and Geodetic Surv^ey. The direction method of adjustment, as set forth in this chapter, is now almost exclusively used in that Survey, even when the observations have been taken by the method of repetitions. 177. In the first part of the preceding section the direction method of observation in use in the Coast and Geodetic Survey is described. The U. S. Lake Survey is at present using a method of observation with a direction instrument which differs widely in several important respects from the Coast and Geo- detic Survey method. The most important difference is that the different angles are measured independently, that is, there are but two signals pointed upon in each series of observations. The advocates of the method of independent angles with a direction instrument urge (i) that if the twist, due to the action of the sun's rays, of the tower upon which the instrument is mounted, be considered; (2) and if the influence on distiiutness of vision of tlie use of the same focus for lines of different lengths; (3j the interruptions that may occur iu the course of 2 42 THE ADJUSTMENT OF OBSERVATIONS a long series ; and (4) the more uniform line that may always be had when the number of signals in use at any one time be small, — be also considered, the conclusion must be reached that this method will give a greater accuracy than the other. The advocates of the method of including all the signals which are then visible in each series, believe that the second and fourth considerations are of minor importance under actual average conditions ; that the third consideration is of little importance if one does not wait (and he should not) for a signal which is not showing when a pointing is desired; and that on the towers, as now built, in the Coast and Geodetic Survey, the twist is so small as to be difficult to discover, even by special observations for that purpose, extending over long inter- vals of time, and therefore that the first consideration is of little importance. As against the considerations set forth above bearing upon the accuracy to be obtained, these advocates call attention to the fact that in their method the pointings upon each signal are scattered over the whole of the observing period during which that signal is visible ; whereas, in the method of independent angles, the observations on each signal are confined within a few short periods ; and that, therefore, a greater variety of conditions are encountered, and a tendency to greater accu- racy secured in the former method. The great disadvantage of the method of independent angles lies in the greater time and cost required to secure a given number of observations. As compared with the method of independent angles, the method used in the Coast and Geodetic Survey requires but three- fourths as many pointings for a given number of observations of each angle if three signals are observed in each series on an average, and but five-eighths as many if there are five observed in each series. It has been urged in favor of the method of independent angles that it simplifies the local adjustment. This argument had considerable force as against the method formerly used in the Coast and Geodetic Survey, but not against the present ADJUSTMENT OF A TRIANGULATION — DIRECTIONS 243 practice in that Survey, with which no local adjustment what- ever is found to be necessary. 178. The Local Adjustment. — If all signals are observed in every series, the local adjustment becomes simply a process of taking means and differences. If a broken series is observed, that is, a series in which one or more of the signals are missing, because they were not visible at the particular time when needed, the Coast and Geodetic Survey observers are at present directed that, "the missing signals are to be observed later in connection with the chosen initial, or some other one, and only one, of the stations already observed in that series." The observations on the signal which is common to the two frag- ments of a series are used to connect these fragments, and when so connected the series is used as if it had all been observed at one time, and no local adjustment is necessary. In this process the series which are made up of joined fragments, are given a slightly greater weight than they should be. But this is of very little importance, especially as the principal errors in angle measurements are of the systematic class and do not appear until a figure adjustment is made. If the observations are made by the method of independent angles with a direction instrument, as in the Lake Survey, the local adjust- ment may be made by the methods stated in articles 121, 122. 179. The Figure Adjustment.* — The following directions were ob- served, among others, at the stations named. It is required to adjust the quadrilateral shown in Fig. 25 by the method of directions, the line Two- Fig. 25. Rock having been completely fixed by the adjustment of the preceding figures. * Throughout this chapter all formulae and examples have been given upon the assumption that all observed directions arc lo be assigned equal POINT 244 THE ADJUSTMENT OP OBSERVATIONS Obsen'ed Directions. At Two. At Rock. o 1 ff / // Hill . . . 145 33 38.1 Two ... 53 30 44 . 3** Point . 212 09 30.8 Hill ... 78 36 08.7 Rock . 269 41 25 .2* Point . . . 127 43 40.0 * Corrected direction, 26.3 // ** Corrected direction, 42.7". At Hill. At Point. / n / // Rock . . • 00 00.0 Rock ... 165 04 04.8 Two . • • 30 46 43-1 Two . . . 213 19 10.7 Point . • • 315 44 36.8 Hill ... 251 41 04.6 The signal chosen as the initial happens to appear in these lists in only one case, Rock being the initial at Hill. There is no difference in the treatment during adjustment of the initial station and any other station. The ten directions for which it is proposed to derive correc- tions are identified by numbers on Fig. 25. The convenient notation used below indicates the corrections to these figures by the same numbers inclosed in a parenthesis, thus, ( i ) stands for the correction to the direction numbered i, namely, Point to Rock. No corrections are to be derived for the directions Two to Rock and Rock to Two, as these directions have already been fixed by previous adjustment. The four condition equations are as follows, the three angle equations being given first : Condition Eqtiations. o=-i.3-(r) + (2)-(8)+(io) o=- 2.7 -(5) + (6) -(7)+ (9) o =- 7.1 - (2) + (3) - (4) + (6) - (7) + (8) . o == - 12.0 + 1.9 (i) - 4.6 (2) + 2.7 (3) + 0.6 (4) - 3-5 (5) + 2.9 (6) - 4-S (9) + 0-6 (10) In the angle equations each required correction to an angle is expressed as a difference of two corrections to directions, except when one of the directions is already fixed, in which case the weight. This is the case which most frequently arises. If it is desired to assign unequal weights. Chapter V shows how the weights are to be intro- duced. ADJUSTMENT OF A TRIANGULATION — DIRECTIONS 245 correction to the other direction involved in the angle becomes the same (with the proper sign) as the correction to the angle.* The triangles are so small in this case that it is not necessary to take the spherical excess into account. The side equation has its pole at Two. The numerical work of forming this side equation is shown below in a convenient form. (For an explanation of the meaning of the side equation, see Art. 126.) Plus Terms. Directions. Uncorrected Angles. Logarithmic Sines. Log Sine, Dif. for + 10 -2 + 3 -5 + 6 / // 74 12 57.3 38 21 53.9 30 46 43 . 1 9.983308 9.792860 9 ■ 709035 i" + 0.6 + 2.7 + 3-5 Sum = 9.485203 MlNi:s Te RMS. Directions. Uncorrected Angles. Logarithmic Sines. Log Sine, Dif. for -1 + 2 -4 + 6 + 9 / n 48 15 05.9 75 02 06.3 25 05 26.0 9.872783 9.985015 9.627417 + 1.9 + 0.6 + 4-5 Sum . . Difference . = 9.485215 . = — 12 For such directions as occur twice in the formation, as, for example, direction 2, the corresponding coefficient is the algebraic * "The first two angle equations were purposely selected so that they refer to triangles of which the fi.xed line Two-Rock forms one side. Each such angle equation contains but four terms, whereas otherwise it would contain six, and the normal equations have two side coefficients which are zero. The work of forming and solving the normal equations is therefore reduced by this selection." 246 THE ADJUSTMENT OP OBSERVATIONS sum of two log sine differences for i", each taken with the sign fixed as indicated in the first and fourth or fifth and eighth cokimns of the above form, together with the headings, "plus terms" and "minus terms." The log sine differences for i" are given in units of sixth decimal place. This triangulation is of tertiary character, and hence the sides are computed to six decimal places only in the logarithms. The correlate equations are shown below in the form indicated in equations (4) of Art. 1 10. Correlate Equations. -C, + 1.9 Q = (i) + Cj — Q - 4-6 C, = (2) + 6; + 2.7C4 = (3) - C3 + 0.6 q = (4) -c, - 3.5 Q = (5) + C + q, + 2.9 c, = (6) - c; - C3 (7) -c, + C3 (8) -f c; - 4.5 Q = (9) + c; + 0.6 Q = (10) Note that the coefficients in the first column of these correlate equations are the same as the coefficients in the first line of the condition equations, and that the second column and the second line correspond, and so on. The normal equations are shown below, formed from the correlate equations and condition equations as indicated in equations (5) of Art. 119. Normal Equations. o = — 1.3" + 4.0 Cj — 2.0 C3 — 5.9 C4 o=— 2.7" +4 C + 2.0 Cg -f 1.9 Q o = — 7.1" — 2.0 Cj +2.0 6*2 H- 6.0 Q + 9.6 C4 o = — i2.o" — 5.9 Cj +1.9 C2 + 9.6 Q +73.7 Q ADJUSTMENT OF A TRIANGULATION — DIRECTIONS 247 The solution of these equations gives : C, =+1.173" Cj = — O.I 10 Q = + 1.511" C4 = + 0.063 These values substituted in the correlate equations give the required corrections: (i) =- 1.05" (6) =+1.58" . (2) =-0.63" (7) =-1.40" (3) =+1.68" (8) =+0.34" (4) =- 1.47" (9) =-0.39" (5) =-o.ii" (10) =+I.2l" After applying these corrections to the directions, the com- putation of triangle sides shows every triangle with the sum of its corrected angles equal to 180° 00' 00.0", there being no spherical excess in these small triangles, and shows the lengths of the lines to be the same when computed in two possible ways through the triangles. The correctness of the adjustment is thus checked. The procedure in adjusting an}' figure, however complicated, in which the only thing fixed by previous adjustment is one line, is not essentially different in any respect from that here illus- trated by the simple case of a quadrilateral with one fixed line. The number of angle and side condition equations in any particular figure is to be determined as indicated in Arts. 144, 151, 152. For examples of condition equations for complicated figures, see Appendix 4 of the Coast and Geodetic Survey Report for 1903. 180. The Best Side Equations. — In Arts. 153, 154, certain suggestions are given as to the manner in which side equations should be selected to avoid the danger that the solution of the normal equations may be an unstable one, that is, a solution in which the effect of omitted decimal places on the derived values of the required unknowns is large, and in which it is, therefore, 248 THE ADJUSTMENT OF OBSERVATIONS necessary to carry a large number of decimal places in the solution to secure the unknowns with certainty to a small number of decimal places. The suggestions are difficult, but important, to follow. This article will serve to illustrate these suggestions by a concrete case,* namely, that shown in Fig. 25, The observed directions are as follows: At Spear. At Tobacco Row. Oft/ o ' tt Long . . . o 00 00.000 Spear ... o 00 00.000 Smith . . . 6 04 57.749 Long ... 72 37 08.593 Flat Top . . 37 CO 48.900 Smith . . . 118 11 11. 341 Tobacco Row, 47 03 16.925 Flat Top . . 159 40 31.200 At Long. o Smith o 00 00.000 Flat Top 57 52 28.128 Tobacco Row . . . 108 47 15.636 Spear 169 06 53.169 At Smith. At Flat Top. 01 o / // Flat Top . . o 00 00.000 Tobacco Row, o 00 00.000 Tobacco Row, 30 12 41.103 Spear ... 10 17 00.258 Spear . . . 51 03 16. 151 Long . . . 42 01 51.794 Long . . • 55 51 27.879 Smith . . . 108 18 02.385 The figure requires three side equations. Let each side equation be represented symbohcally by the abbreviation inclosed in a parenthesis for the station used as a pole, followed in order by the abbreviations for each of the other stations at the ends of the sides involved. As in Art. 179, the required correction to a direction will be indicated by the number of that direction inclosed in a parenthesis. The numbers assigned to the direc- tions are indicated on the figure. The side equations repre- sented by the symbols (L.)-Sm.-T. R.-Sp., (L.)-Sm.-F. T.-Sp., and (L.)-Sm.-F. T.-T. R. are * This article, as well as the suggestions in Art. 154, is based on pp. 118- 120 of the C. and G. S. Rep. for 1878 (App. No. 8) written by Mr. M. H. Doolittle. ADJUSTMENT OF A TRIANGULATION — DIRECTIONS 249 o = + 1.2" - 17.80 (i) + 19.76 (2) - 1.96 (4) - 0.66 (5) + 2.72 (6)- 2.06 (7) - 4.38 (14) + 25.05 (15) — 20.67 (i6). (0 o = - 1.9" - 16.97 (i) + 19.76 (2) - 2.79 (3) - 1.43 (13) + 25-05 (15) - -362 (16) - 3.40 (18) + 4.33 (19) — 0-93 (20). (2) o =- 1.9" - 1.95(6) 4- 2.06(7) -0.1 1 (8) - 1.43(13) + 4.38 (14) - 2.95 (16) - 2.33 (17) + 3.26 (19) — 0.93 (20). (3) The directions i, 2, 15, and 16 are the sides of the small angles at Spear and Smith. In the first two equations the coefficients of the corrections (i), (2), (15), and (16) so largely predominate over everything else, and for corresponding terms are so nearly equal, that the two equations may be considered approximately identical. The effect of this would be to make certain side coefficients in the normal equations about as large as the corresponding diagonal coefficients, and the solution would be unstable. An attempt at solution with factors extending to but three significant figures would not be likely to furnish even an approximation to the values of the unknown quantities. Adding the third equation to the first, and subtracting the second from the sum, the following equation results: = 4- 1.2" - 0.83 (i) 4- 2.79 (3) - 1.96 (4) - 0.66 (5) 4- 0.77 (6) - 0.1 1 (8) - 2.33 (17) + 3.40 (18) )7 (19)- C-^) I.O' It is evident that if the first, third, and fourth equations are satisfied, the second must also be satisfied. Hence, the fourth equation may be safely substituted for the second, or, as will easily appear, for the first, if it be preferred to retain the second. The fourth equation corresponds to the symbol (T-)-^^ T.- T. R.-Sp., and might liavc been obtained directly in the u.sual way. The second suggestion in Art. i 54, to use small angles once, and only once, would have led to the selection of the fourth equation in the ])lace of the second. 250 THE ADJUSTMENT OF OBSERVATIONS The second suggestion of Art. 154 may be carried out still more fully by so selecting one of the side equations as to involve the small angles of the triangle Sp.-T. R.-F. T. Let the side equation, represented by the symbol (T. R.)-Sp.-L.- F. T., be used. It is : o = - 4.0" + 1.96(1)- 11.89(3) + 9-93 (4) + 1.71(10) -2.91 (11) + 1.20(12) + 9.27 (17) - 11.60(18) + 2.33 (19). (5) To carry out the third suggestion of Art. 154, that it is sometimes desirable to use a side equation of large scope, let (L.)-Sm.-F. T.-Sp.-T. R. be used. It is : o = + 3.1" - 0.83 (i) + 2.79 (3) - 1.96 (4) - 0.66 (5) + 2.72 (6) - 2.06 (7) + 1.43 (13) - 4.38 (14) + 2.95 (16) + 3.40 (18) - 4.33 (19) + 0-93 (20). (6) The three side equations recommended as best are, then, the sixth, third, and fifth. It is interesting to note that the sixth equation is the same as the first minus the second. 181. Length, Azimuth, Latitude and Longitude Condition Equations. — In the adjustment of a triangulation, it has been shown thus far how in a net joining several stations the condi- tions arising from the closure of triangles and from the equality of lengths or sides as computed by different routes can be satisfied. Cases frequently arise in which other condition equa- tions must be satisfied, in addition to the angle and side equa- tions. These will now be treated briefly. The principal cases which arise are the following three, and various combinations of them : I. A section of triangulation which starts from a line which is fixed in length, either by direct measurement or by previous triangulation, may end on a line which is similarly fixed in length. In this case a length condition equation must be used to make the length of one of these fixed lines as computed ADJUSTMENT OF A TRIANGULATION — DIRECTIONS 251 through the adjusted triangulation from the other fixed line agree with its fixed value. 2. A section of triangulation may include two lines which are each fixed in azimuth, either by previous triangulation, or by astronomic observations, which in the particular case furnish a determination of the azimuth stronger than that given by the triangulation. It is then necessary to use an azimuth condition equation to insure that the azimuth as carried by computation through the adjusted triangulation from one fixed line shall agree at the other fixed line with the azimuth as already fixed there. 3. The latitude and longitude of each of the two stations in the section of triangulation may have been fixed by previous triangulation, and it may be desired to retain these positions unchanged. This may be done by writing a latitude condition equation, and a longitude condition equation, such as to insure that the latitude and longitude as computed from the first fixed station through the adjusted triangulation shall agree at the second fixed station with the fixed latitude and longitude there. As a belt of triangulation is gradually extended, it frequently happens that it returns upon itself in such a manner as to form a complete circuit. In such a case it usually happens that before the circuit is closed by the field operations a considerable portion of the triangulation in the circuit has been adjusted, the results used in various ways, and perhaps published. It is then desired to adjust the last portion of the circuit, that is, the por- tion still unadjusted when the last of the field work is done, so that no discrepancies of any kind remain at the closing line. In this case it is necessary to use, in this last portion, in addition to the angle and side equations, one condition equation of each of the other kinds. In any case, when tlie necessary condition equations referring to length, azimuth, latitude and longitude have been formed, they may be placed with the angle and side condition equations, and the formation of correlates and of normal equations and the 252 THE ADJUSTMENT OF OBSERVATIONS solution of the normal equations may be made as indicated in Art. 179. 182. Length Condition Equations. — A length condition equation is written in the same form as a side equation, and is essentially of the same character. It serves to insure that the length of the second of the two lines which are fixed in length by direct measurement, or by other triangulation, shall, as com- puted through the adjusted triangulation from the first fixed line, agree with its fixed value. The form of the equation is, when expressed in terms of angles : o = (log a) + h, {A 1) ^B, (^1) + ^A, {A.) hn, {B,) + 8,,^ (^3) K (^3), • • • in which, in a selected chain of triangles from the first to the second fixed line, A^, A.,, A^ . . . is in each case the angle opposite the required side, and B^, B.,, B^ . . . is the angle opposite the known side.* {A^), {A,), (A^) . . ., (B^), (B,), (B.) . . . , are the required corrections to these angles. Sj^, Sj, Sj . . ., 8jj. ^/j, 8j. . . ., are the differences for i" in the logarithmic sines of the angles A^, A.,, A^ . . . , B^, B^, B . . . . (log a) is the necessary correction to the logarithm of the second fixed side, a, as computed through the selected chain of triangles from the first fixed side, b, by the law of the propor- tion of sines, using the uncorrected angles A^, A,, A^ . . . , i?,, B.^, B^ . . . , to make it agree with the fixed value of log d. The proof that the formula given above is a proper expression of the length condition, is similar to that given in Art. 147, for the form of the side condition equation there derived. To express this equation in terms of corrections to directions, all that is necessary is to substitute in each case for (A^), (A.), (A.^ . . ., (B^), (B,), (B^) . . ., which are corrections to angles, the corresponding differences of two corrections to directions, if neither side of the angle is fixed, or one correction to a direc- * These angles .4,, A.,, A^ . . ., B,, Bj, B^ . . ., may appropriately be called distance angles. Their locations in a chain of triangles are illus- trated in Fig. 25. ADJUSTMENT OF A TRIANGULATION —DIRECTIONS 253 tion with the proper sign, if one side of the angle is a line fixed in direction. Any chain of triangles between the fixed lines may be selected. It is well to select a chain containing a minimum number of triangles, thereby making the number of terms in the condi- tion equations, as small as possible, and reducing the labor of solution. It is well, also, in order to avoid instability in the solution of the normal equations, to select the strongest possible short chain, that is, a chain in which small angles are avoided as far as possible. The number of independent length condition equations in any figure is one less than the number of lines which are fixed in length. As, for example, if there are three fixed lengths in a figure, and length condition equations are written fixing the ratios of the first and second, and first and third, of the fixed lines as computed through the adjusted triangulatlon, the ratio of the computed lengths of the second and third lines is thereby fixed. Any third length condition equations will therefore be derivable from the condition equations already written, not independent of them. The rigorous adjustment is made by adding the length con- dition equation, or equations, to the other condition equations, proceeding with the formation of correlates and normal equations as indicated in Art. 179. Various approximate methods of adjustment have been proposed and used in the place of the rigo^us adjustment. The experience of the Coast and Geodetic Survey indicates that it is seldom advisable to use any of these approximate solutions, other than those of the nature indicated in Art. 1 86. 183. Azimuth Condition Equations. — An azimuth condi- tion equation serves to insure that the azimuth of the second of the two lines which are fixed in azimuth, by observations external to the section of triangulatlon being adjusted, shall, as computed through the adjusted triangulatlon from the first fixed line, agree with its fixed \'alue. 254 THE ADJUSTMENT OF OBSERVATIONS Let the selected chain of triangles, for a section of triangula- tion used in writing the length condition equation as indicated in the preceding article, be illustrated by Fig. 26. Let it be supposed that the order of computation is that indicated by the numbering of the triangles i to 6. The distance angles used in forming the length condition equation are marked by the letters A and B. Let it be supposed that the lines DE and NO are fixed in azimuth. For the same reason that in forming the length condition equations the number of terms was kept as small as possible, namely, to save work in the computation, so here the azimuth condition equation should directly involve as few angles as possible. It is possible to carry the azimuth by computation through a chain of triangles by using only one angle in each triangle (the lengths being supposed known), namely, the angle marked C in Fig. 26. This third angle in each triangle will, for convenience, be called the azimuth angle to distinguish it from the distance angles. The azimuth condition equation should involve the azimuth angles only in the selected chain of triangles through the figure used in writing the length equation. As a given correction to any of the azimuth angles in Fig. 26 affects the azimuth of the line NO as computed through the adjusted triangulation by precisely the amount of correction to that angle, it is evident that the general form of the azimuth condition equation is 0=-{a)+^{Cp)-^{C£), in which {a) is the correction to the azimuth of the second fixed line, as computed through the unadjusted azimuth angles from the first fixed line, necessary to make it agree with the fixed azimuth there, and 2 Cp^ is the sum of the corrections to the azimuth angles which are on the right-hand side of the chain of triangles when proceeding in the direction of computation, and 2 {Cj) is the sum of corrections to the left-hand azimuth angles. ADJUSTMENT OF A TRIANGULATION — DIRECTIONS 255 For example, in Fig. 26 the azimuth condition equation is o = - (a) + (CJ - (Q - (Q + (Q) + (Q) + (Q, in which (a) is the fixed azimuth of A'O minus its azimuth as computed through the unadjusted angles C^, C,, C.^, . . . Cg from the fixed azimuth of DE. To express this condition equation in terms of directions, it is necessary simply to substitute for each correction to an angle the difference of the two corrections to directions, one of which is necessarily zero if that particular direction is already fixed. As the number of length condition equations is one less than the number of fixed lengths in a figure, so the number of azimuth condition equations is one less than the number of fixed azimuths. 184. Latitude and Longitude Condition Equations. — A latitude condition equation serves to insure that the latitude of the second of the two points which are fixed in latitude, by observations external to the triangulation being adjusted, shall, as computed through the adjusted triangulation from the first of two points fixed in lati- tude, agree with its fixed value. A longitude condition equa- tion serves the same purpose with respect to longitudes. In Fig. 26 the latitude and longitude of each of the points E and N are supposed to be fixed. It is supposed that the position <^„X„ of j^oint iV as computed through the unadjusted triangulation differs from its fixed position )i a arc i" and B^ is the value at the vertex of any C angle of ^_ (I -.2 sin^c^)! « (i — /^) arc i" In these expressions for A and B, a is the equilateral radius of the spheroid on which the triangulation is computed, and e is its eccentricity. A and B are factors used in the computation geodetic positions. For their values on the Clark Spheroid of 1866, see Appendix 9 of the Coast and Geodetic Survey Report for 1894. In each triangle, reckoning all the angles in seconds, a cor- rection {A) in a distance angle A will produce a correction in the computed latitude at N of " ~ M ' ^"^ ^ correc- tion in a computed longitude of — ""- jr^j^ • A correc- tion to a distance angle B produces similar corrections to the latitude and longitude at N, but with the reverse algebraic sign. <^,.X, is the position of the vertex of the C angle in the triangle containing the angle which is supposed to be corrected. Let any azimuth angle on the right side of the chain of triangles proceeding in the direction of progress of computation ADJUSTMENT OF A TRIANGULATION — DIRECTIONS 257 be designated by 6*^, and any such angle on the left side by C^,. Then any correction to a right azimuth angle produces a correc- tion in latitude at yV of + i\ {\ — \.) (C/,.) and in longitude of — ;-, {,.) (Cji). Similarly, any correction to a left azimuth angle produces the corrections — 7\ (\„ — \.) (C/,) and + '2 (^« ~" ^c) {Cl) to the latitude and longitude respectively at N* Hence, °-^"-^" ^\ M M + r, (A„ - A,) (Cn) - r, (A„ - A,) (cA (i) (A„ - A,) 8a (A) (K - K) ^n (B) o = A„ - A„' + 2 [ M M - r, („ - ,) (Cn) + r, (c^„ - <^,) (CjA (2) With sufificient accuracy and greatest convenience (^„ — (f)^ and \ — \ may usually be reckoned in minutes and tenths. Then, multiplying the equations by — = ( ) in order to re- 60 V 60 / move a constant factor from the coefficients of (A) and (B), o = .00724 (<^„ - <^,/) + 2 [(<^„ - <^,) 8a {A) - (<^„ - <^,) 8ji {B) + -4343 ^i {K - K) (Cn) - -4343 (K - K) (O,)]- (3) o = .00724 (A„ - A„0 + 2 [(A„ - A,) 8a (A) - (A„ - A,) 8n (B) - -4343 ''2 (^n - <^c) (Q) + .4343 ^2 (<^« - "^c) (Q)]- (4) 185. If these condition equations are used in this form, and if there is a length condition equation and an azimuth condition equation covering the same chain of triangulation, it will be found that the normal equations have large side coefficients, and their solution will be unstable. The following transformation facilitates the solution by avoiding this difficulty. * To sliow the florivalion of tlic formuLx for y, and /-_, would re<|uirc considerable space. This derivation has no bearing upon tlie method of least squares. It suffices for the present purpose to note tliat r, and /_, are functions oi the latitude, anrl tliat the formuhi- tjiven show tlie relation be- tween (C) and the corre.sponding corrections in latitude and longitude at A'. 258 THE ADJUSTMENT OF OBSERVATIONS Assume a point and (f>,,'X./, approximately in the mean latitude and longitude of the points <^,X, which are at the vertices of the C angles. Multiply the length condition equation given in Art. 152, which is in the form o = (log a) + 2 Sj (A) 2 8/.. (B), by c^, - ,„ and the azimuth condition eciuation given in Art. 153, namely, o = - (a) + 2 (Cb) - 2 (C/,), by .4343 '\ (K - ^«)< and add the products to equation (3). Also multiply the length equation by (\ — X„) and the azimuth equation by — .4343 r^ ((^^^ _ <^J and add the products to equation (4). Then the latitude and longitude condition equations are as follows, in which all that portion within the first square bracket in each case constitutes the absolute term : o = [.00724 (cj>„ - ct>„') (^h - >) (log ^) + -4343 '\ (^h - K) (a)] + 2 [(<^, - <^.) 8.J (A) - (h - c) {Cu) +,4343 r, {u - 4>c) (Cl)]. (6) This transformation is allowable since the adjusted angles which satisfy the length and azimuth equations and equations (5) and (6) must evidently satisfy {3) and (4) also, and no new condition has been put into the solution, as (5) and (6) are mere combinations of the original conditions. The transformation has the effect of transferring the point at which the discrepancy in latitude and longitude is supposed to have developed from the end of the triangulation (p,^\ to a mean point ^/,\/,. Equations (3) and (4) correspond to the supposition that the computation of latitudes and longitudes has progressed continuously from the beginning of the section of triangulation being adjusted to (p„\„. Equations (5) and (6) correspond to the supposition that the computation of latitudes and longitudes has been made in two sections, one from the ADJUSTMENT OF A TRIANGULATION — DIRECTIONS 259 beginning of the' section being adjusted to (j)/,'\./,: and the other from the end of the section (at /,\, this portion of the position computation being supposed to be made by starting with the fixed length and azimuth found at a hne of which (t>n\ is one end; and that the discrepancy in latitude and longitude is developed at ^/^\ by the junction of the two position computations there. To express the latitude and longitude condition equations as written above in terms of corrections to directions, it is necessary simply to substitute for each correction to an angle the differ- ence between two corrections to directions, one of which is necessarily zero if that particular direction is fixed. The number of latitude condition equations in a figure is one less than the number of groups of points fixed in latitude, each group being composed of points which are tied together by lines already fixed in length and azimuth, and being separated from other groups by lines not so fixed. The removal of the latitude discrepancy for one point of such a group removes it for all. Similar statements are true for the longitude condition equations. On tJie Breaking of a Net of Triangnlation into Sections for Convenience of Solution. 186. In a long chain of triangulation, or in a complicated net, the simultaneous solution of the condition equations, which are required by theory to secure the ideal most probable results, would be very troublesome, not from any principle involved, but from its very unwieldiness. For example, such a simultaneous solution for all the primary triangulation now forming a con- tinuous net in the United States would involve more than two thousand condition equations. Accordingly it is necessary to break the triangulation into sections and adjust each section by itself. As this breaking into sections causes more or less dis- turbance of the ideal solution at or near the lines of breaking, the exercise of judgment is required in the selection of these 26o THE ADJUSTMENT OF OBSERVATIONS lines. The larger the sections are made, the nearer the approxi- mation to the ideal solution, and on the other hand the greater will be the labor of computation. The present practice in the Coast and Geodetic Survey is to divide the triangulation into sections, adjust each section by the rigorous method set forth in this chapter, and, in commencing the adjustment of each new section, to hold as absolutely fixed in all respects all lines which have entered into a previous adjustment. This method has the great advantage of giving the final results by a single adjustment. It is believed that with a proper selection of lines of separation between adjustments the results obtained are so close an approximation to the ideal best results that no further expenditure of energy in computation is warranted. Two principles guide in selecting the lines of separation. First, such a line should be one which is strongly determined. Second, it should be a line which enters in but few conditions in one of the two sections. For example, a line at the edge of a base net and forming the first line of the chain of triangulation connecting that base net with the next is frequently selected as a line of separation. It is strongly deter- mined in length by the base and base net. It is usually involved in but few conditions on the side toward the chain of triangulation. In the separation into sections in primary tri- angulation the most frequent cases are: 1. A section which begins with a line at the margin of one base net and extends to a line at the far side of the next base net, thus including the second base net. 2. A section which includes two base nets and the triangula- tion between them. 3. A section which is limited at each end by a line fixed by previous adjustment in azimuth, latitude, and longitude as well as in length. In secondary and tertiary triangulation a great variety of cases arise, and, in general, the adjustment is made in smaller sections than the primary triangulation. CHAPTER VIII APPLICATION TO BASE-LINE MEASUREMENT AND TO LEVELING 187. Precision of a Base-Line Measurement. — For clear- ness it will be necessary to outline the principles on which the measurement is made. First, we must find the length of the measuring bar or tape in terms of some standard of length; and as the measurements of the line itself are made at various temperatures, the coeffi- cients of expansion of the metals in the measuring apparatus must also be known. Comparisons must, therefore, be made with the standard during wide ranges of temperature; and as these comparisons are fallible, the results found for length and expansion will be more or less erroneous. 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 precision not looked for in chaining, such as for aligning the measuring bar, for finding the inclination of each position of the bar, and for estab- lishing fixed points for stopping at and starting from in measure- ment. But these make no change in the essential principle. The errors in the value of a base line may, therefore, be con- sidered to arise from two principal sources, — comparisons and measurement. Experience has shown that a considerable portion of the error in the length of a base arises from the errors in the comparisons which serve to determine the length of the measuring bar or tape. These errors differ essentially in character. An error arising from the comparisons, being the same for each bar measure- ment, is cumulative for the whole base, while errors arising in the measurement of the base itself, were t]ic mcasuienicnts 261 262 THE ADJUSTMENT OF OBSERVATIONS repeated often enough and the conditions sufficiently varied, would tend to mutually balance, and could, therefore, be treated by the strict principles of least 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 com- paratively 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 instead of being able to compare results at the end points only, we 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 pre- cision of measurement. It may be stated as follows: A base is measured in n sections with a bar of a certain length, each section being measured n^ times. By the first measurement the first section contains M^ bars, the second M^' bars, . . . ; by the second measurement the first section contains M' bars, the second M,y bars, . . . ; and so on. The weights of the meas- urements in opder being //, //, . . . ; /'/^ //^ ...;... respectively, required the m. s. e. of the most probable value of the base. Let x^ = most prob. value of first section, x^ = most prob. value of second section. then we have the observation equations: First section, x^ — J// = i\ wt. //, x^ - m;' = vl' wt. //', Second section, x^ — M^^ = v^^ wt. p^, x,-M," = v," wt.//', and so on. Now, either of two assumptions may be made. APPLICATION TO BASE-LIXE MEASUREMENT 263 188. {a) In the first place, that the precision of the measure- ment of each bar length is the same throughout the different sections. We have, then, nn^ equations containing n unknowns, and the normal equations are + [A] ^2 = [A^^2]. whence x^'^ x.„ . . . are known, and therefore the whole line X = x^ -\- X., -\- . . . -\- x^'is known. The probable error r of an observation of weight unity — that is, of a single measurement of a bar length — is given by (see Art. 105) . = o.6„5 \J ^'-''^ No. of obs. — No. of indep. unknowns y n (n^ — i) 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 bar lengths measured. For since r is the p. e. of a measurement of a single bar length, the p. e. of the measurement of a length of AI bars is r ^IM. Hence — is the weight of a measurement of length J/ when the weight of a measurement of the unit of length is unity. Writing, therefore, for the weights / their values in terms of M, =°'^?«v^K^^)[i]' In the case usually occurring in practice, where the line is measured twice, we may put this formula in a form more con- venient for computation. For if llic first measurement of the 264 THE ADJUSTMENT OF OBSERVATIONS «j sections gave lengths M^, M,^, ■ . . , and the second meas- urements gave lengths M^ + ^i, M^ 4- ^i^, ... for the same sections in order, then, since b'] 'U(^f [.p] we have for the m. s. e. of one measurement of a bar length and for the mean of two measurements respectively, ^h d'' M Hence the p. e, of a single measurement and of the mean of the two values of the whole base are the number of bar lengths in the line being [i^/]. 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 Bar Lengths. Northern Part Sec. I Sec. 2 Sec. 3 Southern Part Sec. I Sec. 2 Sec. 3 L -0.183 + 0.094 — 0.013 — 0.007 + 0.095 + 0.757 116 87 61 264 92 60 ^ 283 Hence the m. s. e. of the northern part, arising from errors of measurement only, is I 7264 l.\%f .0942 .oii^\ . ^ * Das rhehiische Dreiecksnetz. Berlin, 1876. APPLICATION TO BASE-LINE MEASUREMENT 265 and the m. s. e. of the southern part is 2 V 3 \, 92 ^ 60 ^ 131 y =^ ^-^• 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 rt 0-386, J^ 0.313 for the northern part, zt 0.391, rt 0.335 for the southern part. Remembering that these latter errors are systematic, we have, finally, p. e. of base = .6745 V.093^ +-327^ +(-386 + .391)' + (-313 + -SSS)'' L — 0.72. 189. (/') 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., X [A] ' ^,,2 = ^ \^/'^^ ^ = -7^^ since A' = A" =••• = /. ^ _[A^ '~ [A] ' ^^J = [A^'2'] = y^"^ since// =/./' = . . . = -i, If X denotes the whole line, so that X = .Tj + X, + • • • + .T„, then, since the measurements are independent, 2 ''I 2 I /*« = M«r + /*«/ + • • • 266 THE ADJUSTMENT OP OBSERVATIONS and the (m. s. e.)'" of a single measurement of the line «j — I The number of bar lengths in the line being [^/], we have for the average value of the (m. s. e.)^ of a single measurement of a bar length I [^\'] + [^'2'] + «j — I [M] If, for example, the line has been measured twice, and d^, d,, . . . ^4 ^re the differences of the measurements of the several sections, then — > 2 [^r] and therefore ".' = 1? and the (m. s. e.)^ of a single measurement of a bar length Ex. — The Chicago Base, measured in 1877 with the Repsold metaHic- thermoineter apparatus belonging to the United States Engineers. The base was divided into 8 sections, and was measured twice. Section. No. OF Bar Lengths. DiF. OF Measures. vun. I. 227.25 - 1-3 II. 230-25 + 2.5 III. 234-50 + 2.3 IV, 232 + 0.7 V. 231 + 1-5 VI. 225 + 1.1 VII. 300.50 + 1-3 VIII. iq6.8o — 0.2 APPLICATION TO BASE-LIXE MEASUREMENT 267 Taking the errors of the different sections as independent, the p. e. of the mean of the two measures of the base is 0-6745 y-j^= I-46- The p. e. arising from the other sources of error were (i) Measuring bar zt 6-38, (2) Metallic thermometer . . . . rt:^-82, (3) Elevation above mean tide, N.Y. rt 0-36. Assuming these to be independent, the p. e. of the Chicago Base at sea level is V1.462 + 6.38- + 2.82- + 0.36- = 7.14. APPLICATION TO LEVELING. 190. Lines of levels are usually run in duplicate, each por- tion being leveled over twice, sometimes both levelings being in the same direction, but preferably in opposite directions. The probable error of the mean result for a single kilometer, or for the whole line, may then be computed from the discrepancy between the two levelings over each section by the formulae given in the preceding article, in connection with base line measurements. Whenever a series of such lines becomes so connected as to form a network, an additional determination of the accuracy of the work is afforded by the closing errors of the various circuits forming the net. Experience shows that the prolmble errors as thus computed from the adjustment are usually considerably larger than as computed from the discrepancy on short sections between the two runnings of each line, thus indicating that there are systematic errors in the leveling which are not eliminated by duplicating each line. The rigorous adjustment of a level net may be made in cither of two ways, namely, by the method set forth in Chai:)ter I\' in dealing with indirect observations, or by the method of Chapter V which is applicable to conditioned ob.scrvation. 268 THE ADJUSTMENT OF OBSERVATIONS 191. Method of Indirect Observations. — Let it be supposed that s is the total number of junction bench-marks in a net of levehng, each of these junction bench-marks being common to three hnes of the net. Let / be the number of lines in the net, each connecting two junction bench-marks. Now if it be assumed that the unknowns desired are the elevations of each of s — I oi these junction bench-marks, referred to the remaining bench-mark as a zero point, the case \n hand is one of indirect observations with the no conditions. There are s — i unknowns, and / observation equations each of the form X — y = / or X = /^ see equations (2) of Art. yy, and the solution is carried out in accordance with Chapter IV. 192. Method of Conditioned Observations. — As before, let it be supposed that s is the total number of junction bench-marks in the net / and the number of lines each connecting two junc- tion bench-marks. If it be assumed that the required unknowns are the / differences of elevations between the junction bench- marks at the ends of each of the / lines, the case in hand is one of conditioned observations. There will be / observation equa- tions each of the form .r = I expressing the direct observation in each case of one of the required unknowns. The number of differences necessary to fix the relative elevations of s bench- marks is J- — I. The number of observed differences in excess of this required number is / — (j- — i), and therefore the num- ber of condition equations \s I — s -\- i. These conditions exist as the requirement that each circuit in the net must close, that is, the sum of the differences of elevation in order around each circuit must be zero. The solution may be carried out as indi- cated in Chapter V by the Method of Correlates (see Art. 119). 193. One question which arises at the outset with either of these two general methods of solution is, what relative weights shall be assigned to the different lines. If all the leveling is done with one type of instrument by a uniform method and APPLICATION TO LEVELING 269 under similar conditions, the problem of assigning weights is simply that of determining the relation between the errors in the lines and their lengths. If all of the errors are of the acciden- tal class, the accumulated errors in different lines tend to be as the square roots of their lengths, and therefore the proper weights are inversely as the lengths. If, on the other hand, the principal errors are of the systematic class, the accumulated errors are proportional to the length, and the weights to be assigned are inversely as the squares of the length. The de- cision as to the assignment of weights must be based on the computer's judgment as to whether the lines should be assigned to one or the other of these classes, his judgment being based upon special investigation, if possible. 194. If the leveling combined in a net is done with various instruments and methods and under strongly contrasting condi- tions, the difficulty of assigning proper weights is largely in- creased. In such a case the basis for judgment as to the relative weights which should be assigned will be frequently found to be rather insecure. If the net of levels is extensive, and if there are several lines in each of the classes of leveling, the following principle may be used to determine whether the weights assigned are approximately correct and what modifications, if any, are de- sirable. As the correct weights are inversely proportional to the squares of the probable errors, it follows that in general, un- less there is an extraordinary difference as to the manner in which the different classes of lines are involved in the net, that the average value of pii^ for each class of lines should be the same, if the assigned weights are correct. If a certain class of lines be assigned too great weight in a given adjustment, the average pv^ for that class will be larger than for other classes, and vice versa. New relative weights may be assigned as indi- cated by the average values oi pir for the different classes, and a new adjustment made which will, in turn, furnish a new test of the assigned weights. In making the changes in weights, it should be kept in mind that as the weights,/, lor a given class 270 THE ADJUSTMENT OF OBSERVATIONS are decreased, there will be a tendency for the ^^s in that class to increase. It is not important to make a very close approxi- mation to the ideal weights, for the reason that it will be found in general that a considerable change in weights is accompanied by only a small change in the numerical results of the adjust- ment. For an example of the application of this method of testing assigned weights in connection with the adjustment of the precise level net of the United States, see Appendix 8 of the Coast and Geodetic Survey Report for i899,pp. 437, 438,447,448. The same principle may be applied to testing the assigned re- lation between the lengths of the lines and their weights. If the values oi p'-J' resulting from an adjustment for the various lines of a single class of leveling are tabulated in order of the lengths of the lines, the values should show no progressive change if the assigned relation between length and weight is correct. If, for example, on the other hand, weights inversely proportional to the length have been assigned to a given class of leveling, where- as they should have been inversely proportional to the square of the length, it will be found that the average pir for the long lines is much greater than for the short lines. For an example of the application of the principle, see pp. 445-447 of the Appendix referred to in the preceding section. This method of determining the proper weights becomes more reliable the more extensive the net and the greater the number of lines in each class. 195. It sometimes happens, as, for example, in the adjustment of the precise level net covering the eastern half of the United States, that the elevations are all referred to mean sea level by tidal observations connected directly with the net at various points. In this case the method of adjustment is the same as outlined above, sea level at any point being treated as if it were the one bench-mark in the above statement to which all levels are referred.* * For examples of complicated level net adjustments, see Appendix 8, Coast and Geodetic Survey Report for 1899, and Appendix 3 of 1903. APPLICATION TO LEVELING 271 196. The adjustment of a net of trigonometric levels does not differ essentially from that outlined above. In this case, from measurements of vertical angles at various triangulation stations, combined with the horizontal distances between these stations as determined by the triangulation, the differences of elevations between the inter-visible stations taken in pairs, are computed. These computed differences of elevation are treated as the direct results of the observation, and the relative elevations of the various stations are derived from an adjustment of the net. CHAPTER IX APPLICATION TO THE SELECTION OF METHODS OF OBSERVATION 197. The preceding chapters have dealt with the principles of least squares as furnishing a method of computing the most probable results from given observations. To confine the attention entirely to this application, as is frequently done, even by persons thoroughly familiar with the principles, is to over- look a still more important though less extensive use to which these principles may be applied, namely, to the selection of methods of observation. When, in any class of precise measurement, the method of observation is selected, the maximum accuracy attainable is thereby fixed. The computer cannot improve upon what the observer has done, he can but bring out all its excellence. The observer may, by improvements in his method of observation, not only raise the standard of maximum accuracy attainable in his own observations, but by example raise it for all later ob- servers. Or he may for himself and later observers greatly reduce the amount of observing necessary to attain a given standard of accuracy, and thereby greatly reduce the cost of the work. Improvements in methods of observation are far- reaching in their effects, and the application of least squares to this end correspondingly important. The expression, "selection of methods of observation," is here used in a general sense such as to include the selection of the instrument and its mounting and protection, and the selection of conditions under which to observe, as well as the mere selection of a method of manipulation and a program of observation. 198. A clear understanding of the relative influence upon 272 SELECTION OF METHODS OF OBSERVATION 273 the results of errors arising from various sources is a prime requisite on the part of one who proposes to improve methods of observation. It is also of great importance to the observer attempting to secure a maximum of accuracy with a minimum of expenditure. A working knowledge of the principles of least squares is essential to this clear understanding. In general, methods of observation are to be improved : 1. By reducing at their source" the errors which have pre- dominating influence. 2. By transferring the errors from any given source from the systematic or constant class into the accidental class, by a change in instrument or method. 3. By introducing such simplifications in instruments and methods as will increase the rapidity of observing, possibly at the expense of making slight increases in such errors as have little influence on the final results. In general, the third sug- gestion is important not simply for economic reasons, but also because any increase in the rapidity of observing is likely to lead indirectly to an increase in accuracy by reducing instru- mental errors. A treatment of the application of the principles of least squares to the selection of methods of observation within the limits of a single chapter must be suggestive rather than com- plete. In the examples given to illustrate this application, con- siderable knowledge of instrument and method of observation will be assumed to be possessed on the part of the reader. Attention is invited to the principles sketched rather than to the particular numerical estimates of magnitude of each class of errors in the examples. 199. Distinction Between Accidental, Systematic, and Con- stant Errors. — In discussing errors, and especially when dis- cussing them with reference to their ultimate effects, it is quite important to keep clearly in mind the distinctions between accidental errors, constant errors, and systematic errors. A constiiJit error is one which has the same effect upon all the 2 74 THE ADJUSTMENT OF OBSERVATIONS observations of the series or portion of a series under consider- ation. A systematic error is one of which the algebraic sign, and, to a certain extent, the magnitude, bears a fixed relation to some condition or set of conditions. Accidental errors are not constant from observation to observation, they are as apt to be minus as plus, and they presumably follow the law of error which is the basis of the theory of least squares. Thus, for example, the phase error in observations of horizontal direc- tions is systematic with respect to the azimuth of the sun and of the line of sight. The personal equation of an observer introduces a constant error into the observations of the sepa- rate stars in a time set. The expression " constant error " is often used loosely in contradistinction to " accidental error," in such a way as to include both strictly constant errors and systematic errors. The effect of accidental errors upon the final result may be diminished by continued repetition of the observations and by the least-square method of computation. The effects of con- stant errors and of systematic errors must be eliminated by other processes, for example, by changing the method or pro- gram of observations, by special investigations or by special observations designed to evaluate a constant error or to deter- mine the exact law of a systematic error. 200. More Accurate Definition of Probable Error. — It cannot be emphasized too much or too frequently that the theory of least squares applies to accidental errors only, that the least-square method of computation is designed to secure efficient elimination of the effects of accidental errors, but may or may not be efficient in reducing the effects of errors of other kinds, and that a probable error derived directly from residuals is an adequate measure of accidental errors only. The most frequent, and perhaps the most serious, mistakes made in applying the theory of least squares arise from disregarding the points here emphasized. The definition of probable error as ordinarily given is equiva- SELECTION OF METHODS OF OBSERVATION 275 lent to saying that the chances are even for or against the proposition that a certain stated value having given a probable error does not differ from the truth by more than that probable error. That is, if the probable error of a single observation of an angle is stated to be ± i.oo" it is understood that the chances are even for and against the proposition that the result of any one observation is within i.oo'^ of the truth. If the probable error in question is one which is based directly upon residuals, this definition should be modified to limit it so as to refer to the accidental errors only. This may be done by stating that in the above case the chances are even for or against the proposition that the result of any one obser- vation is within i.oo" of the mean value which would result from an infinite number of such observations made under the same average conditions as the observation in question. This form of definition is non-committal as to possible constant errors affecting all the observations, and as to systematic errors which might change in magnitude and algebraic sign, if there were changes in the conditions. Such a definition differs from the ordinary definition in stating that the probable error is a meas- ure of the departure to be expected from a mean of an infinite number of such results as are being considered, rather than from the truth. The supposed mean of an infinite number of results would be free from the effects of accidental error, would be as much affected by constant error as any one observation, and would be less subject to systematic error only to the degree determined by the frequency and extent to which the conditions were allowed to vary. There is no sharply defined line of separation between acci- dental and systematic errors. The perfect type of accidental error, the type upon which the theory of least squares is based, is an error* which is the algebraic sum of an infinite number of independent infinitesimal elemental errors, all equal in magni- tude, and each as likely in any given case to be positive as * See p. i6x. 276 THE ADJUSTMENT OF OBSERVATIONS negative. In any actual case the number of elemental errors is finite, each of them is finite in magnitude, and the elemental errors due to different causes may be of very different average magnitude. Each elemental error actually depends both in magnitude and sign upon certain conditions, though those con- ditions may be unknown, and is, therefore, a systematic error. The conditions upon which different elemental errors depend may not be independent. Indirect evidence shows nevertheless that in many cases the errors of observation were sensibly of the perfectly accidental type, that is, the relation between the numbers of errors of various magnitudes and signs is sensibly that which must hold for the perfect type of accidental error (see p. 31 and Table I), and therefore all deductions based upon this law of error are sensibly true. In other contrasting cases, certain of the elemental errors may be easily proved to be in the systematic class, and may be segregated from the re- maining elemental errors which then, as combined, belong sensibly to the accidental class. In the intermediate and most frequent cases, there are slight observable departures from the law of error corresponding to the perfect type of accidental error, and the deductions based upon that law are slightly, but observably, at variance with the facts. In such a case some systematic error exists which is almost, but not quite, detected, and one is forced to treat all the errors as being of the acci- dental class. 201. Detecting Systematic or Consatnt Errors. — It is im- portant that the computer should detect systematic or constant errors in order that he may modify his method of computation to correspond. It is still more important that the investigator of methods of observation should detect systematic and constant errors in order that they may thereafter be avoided. It is much more important to avoid, or to provide special means of eliminating, systematic or constant errors than accidental errors, for the reason that accidental errors are rapidly eliminated by the mere process of increasing the number of observations. SELECTION OF METHODS OF OBSERVATION 277 A considerable variety of methods may be used for the detection of systematic and constant errors. Each method will be found as a rule to correspond to one or another of the fol- lowing five cases: Case i. Systematic errors may sometimes be detected by noting a tendency of the residuals to have a certain algebraic sign when certain conditions exist, and the opposite sign when these conditions are absent or the opposite conditions occur. Case 2. Errors which are constant in each of a number of groups of observations may sometimes be detected by noting that the disagreements between the mean results for each group are greater than can be accounted for by the probable errors of those means as computed from the residuals within each group. Or, what is equivalent, errors which are constant for each of a number of gi'oups of observations may sometimes be detected by noting that the probable error of the mean of all the obser- vations is apparently much larger if computed from the residuals of the mean of each group from the mean of all, than if computed from the residuals of each observation from the mean of all. When it is recognized that there are errors which are constant for each group, these errors may in some cases be proved to belong to the systematic class by detecting a relation between the errors peculiar to each group and some condition peculiar to that group. Case 3. Systematic errors may sometimes be detected by comparing the relative frequency of residuals of different mag- nitude and sign with the theoretical relation, according to the law of error, between the magnitude and the frequency of er- rors. This comparison may be made by using Table I, or by plotting the actual curve of the residuals with the theoretical curve of error superposed upon it. The comparison may show that there are relatively many more very large residuals than would be the case if the errors were all accidental. The e.xami- nation of the conditions corresponding to these large residuals may then lead to the detection of the law and cause of these large systematic errors. 278 THE ADJUSTMENT OE OBSERVATIONS Case 4. That systematic or constant errors may arise from a given source may sometimes be proved by special observations for that purpose. Case 5. Either systematic or constant errors may sometimes be detected by comparing the results of the observations in question with results obtained independently from observations of an entirely different kind. The detection of systematic or constant errors necessarily involves least squares as a basis, but this must be supplemented by something else, as the method of least squares deals with accidental errors only. Examples of each of these cases will be found in the text which follows: 202. Zenith Telescope Latitude Observations. — Observa- tions with a zenith telescope for latitude are especially interest- ing as illustrating errors which are a close approximation to the perfect type of accidental error. The zenith telescope and in- strument is an example of proper selection. It is difficult to im- prove upon it for the reason that the errors from every important source are of the accidental class, and the effects of the errors arising from various sources upon the final results are so nearly of the same magnitude that little gain in accuracy may be se- cured except by reducing the errors from several sources. As indicating how the error in a result from an observation of a single pair of stars with a zenith telescope approaches the ideal accidental error, which is supposed to be the algebraic sum of an infinite number of independent infinitesimal errors, all equal in magnitude, and each as likely in any case to be positive as negative, sixteen independent elemental errors in this result may be named, each capable of introducing an accidental error of from ±0.01" to ±0.16'' into the result, of which the probable error due to all causes is ± 0.20" to zt 0.30''. Several of the sixteen errors named could if desired be separated by more minute examination into other elemental errors as suggested in the text, so that the number of elemental errors is SELECTION OF METHODS OF OBSERVATION 279 really several times sixteen. The list of elemental errors as given is suggestive rather than complete. 203. Elemental Errors in Zenith Telescope Observations. — I. The error of bisection of the star image by the microm- eter line, depending among other things upon the observer's perception and his control of the muscles of his fingers, the shape of the image, the defects of the observer's eyes, the irregular motion of the image due to momentary changes in refraction, irregularities in the line used in making the bisection, the magnitude of the star and the lighting of the field of view and the line. 2. The error of reading the position of the bubble in the level, depending upon the lighting and upon parallax as well as upon the observer's estimate of tenths of a division and his perception. 3. The error caused by unequal heating of the level vial and consequent movements of the bubble. 4. The error caused by the error in the assumed value of one division of the level. The error in the value of a division is due in part to errors in determining it and in part to variation of the actual value from time to time. 5. The error of reading the micrometer head. 6. The error due to the error in the assumed value of one turn of the screw. The remark made in connection with the level value applies here also. 7. The error due to non-uniformity of the screw throughout its length, 8 . The error due to periodical errors, having a period of one turn in the screw and its nut. 9. The error due to the inclination of the bisecting line to the horizon and to the difficulty of making all bisections on exactly the same part of the line. 10. The error due to inclination of the horizontal axis of the instrument. 1 1. The error due to the azimuth error of the instrument. 28o THE ADJUSTMENT OF OBSERVATIONS 1 2. The error clue to the colHmation error of the instrument. 13. The error due to variations in the angle between the tangent to the level vial at its middle point and the hne of col- Hmation, this variation being due to changes in temperature in different parts of the instrument as well as to stresses. 14. The error due to variation of the differential refraction from its assumed mean value. This variation is dependent upon the conditions as to temperature and pressure at all points along many miles of each of the two lines of sight. 15. The error due to the error in the declination of each star as used in the computation. The error of declination is due in general' to the separate errors in dozens or perhaps even hun- dreds of observations made at various times at many different observatories, each observation being in general affected by as many elemental errors as are suggested in the preceding fifteen numbers of this list. In estimating the number of elemental errors affecting a result which are represented by this list, it should be noted that each result depends upon the observations on two stars. 204. In determining the latitude of a station by zenith tele- scope method, a program frequently followed is to observe sev- eral pairs of stars, say twenty, on each of several nights, five, for example. The probable error of a single observation is com- puted from the residuals of each observation from the mean of the five observations on that pair. When the mean results for the different pairs are compared, they are found to show dis- agreements which are greater than can be accounted for by the residuals within each group. This test, which is an illustration of Case 2, shows that there is some error in the results which is constant for each group. In this case it seems obvious that this constant error in each group is, in the main, simply the error in the mean of the two declinations of the stars of the pair. This declination error, which is constant for each pair, belongs mainly in the accidental class when the results from various pairs are considered. SELECTION OP iMETHODS OP OBSERVATION 281 There may be other errors which are constant for each pair and which combine with these dechnation errors; for example, errors No. 6 and 7, and some parts of No. i of the above list. For the present purpose it is not sufficient to stop with the obvious conclusion that all or nearly all of the error which is constant for each pair is due to declinations. It is desirable, if possible, to prove it. Two lines of proof are available and have been used. Let Cj, be the probable error of the mean result from a single pair as derived from the residuals of these various mean results from the mean of all for the station. Let e be the average probable error of a mean result for a pair as derived from the residuals of the separate observations on that pair from the mean result for that pair. Let e*j, be the average probable error of the mean of two star declinations for the particular stars observed. The apparently obvious assumption suggested in the preceding two paragraphs is expressed by the formula : * ^i = £ + e-**. e*, is the only unknown, and its value may therefore be computed from the latitude observations. Its value may also be derived from the computations made by the astronomer in combining the observations at various observatories and at various times. These two computations give usually nearly the same values for r*t the value from the latitude observations being, upon the average, very slightly larger than that computed by the astronomer. Hence, the errors which are constant for each pair are nearly, but not quite, all due to errors of declinations. If several latitude stations along the same parallel have been occupied and the same list of pairs used at all stations, it becomes possible to apply in a slightly different way the principles involved in the preceding paragraph. Tliis was done for twelve * See Appendix 7 of the Coast and (ieodetic Survey Report for iS^cS, Longitude, Latitude, Azimutli, ]). 358. 282 THE ADJUSTMENT OF OBSERVATIONS Stations along the Mexican boundary. It confirmed the con- clusion of the preceding paragraph.* 205. Let it be supposed that the probable error of the mean of the two declinations, e**, is rto.i6", and that the probable error of a single observation of a pair is zto.30". The first of these probable errors represents the accuracy to be expected in the declinations which are now available. The second is easily attained with a good portable zenith telescope. Under these conditions, which is the better program, to observe 20 pairs on each of five nights, 100 observations in all, five on each pair, or to observe 100 pairs each once only, the observations being scattered over as many nights as may be necessary to secure them.? If 20 pairs are observed on each of five nights, the probable error of the final result will be, V^' e^** e 1 + = V(.o36)^ + (.030)^ = ± .047. 20 100 If 100 pairs are each observed once, the probable error of the final result will be N/'^ -\ — V(.oi6)- + (.030)- = ± .034, 100 100 showing a very decided advantage in favor of this program of observation. The advantage becomes still more evident when it is noted that if 54 pairs be each observed once, the probable error of the final result will be V/: ^* + — = V(.022V + (.041)2 = ± .047, 54 54 V . / V ^ the same as would be obtained from 100 observations on 20 pairs, but secured with little more than one-half as much observ- ing. It would evidently be a great improvement to change * Report of the Boundary Commission upon the Survey and Re-marking of the Boundary between the United States and Mexico, West of the Rio Grande, 1891-1896, p. 105. SELECTION OF METHODS OF OBSERVATION 283 from the plan usually followed in the place of observing each pair four or more times to the plan of observing each pair but once. This improvement is one which almost inevitably suggests itself to a person looking at the subject from a least-square point of view, whereas the writer's experience indicates that those who do not take this view-point fail to appreciate the desirability of this impro^•ement. For an example of the com- parative results by the two plans, see the Mexican Boundary Report referred to on the preceding page, pp. 106, 107. At the typical station represented by that series of observations, 108 observations on 72 pairs gave a much more accurate result than could have been obtained even from an infinite number of observations on 18 pairs. 206. Telegraphic Longitude Observations. — Determinations of differences of longitude by tlie telegraphic method furnish illustrations of the detection of systematic or constant errors by the methods of Case i and Case 4 and the following three illustrations of Case 2. 1. For the transits ordinarily used in telegraphic longitude determinations in the Coast and Geodetic Survey, the probable error of an observing transit of a star over a single line is usually less than ±0.10", as computed from the residuals of the separate lines from the mean of the 11 lines on which obser- vations were taken.* On this basis, the probable error of the transit across the mean of the 1 1 lines would be less than ± 0.03". This same probable error, as computed from the residuals of the separate stars from the mean for the time set, is usually considerably larger, say it 0.04" upon an average. This indicates that the errors of the observations would not be much reduced by increasing the number of lines, say from II to 21. 2. From fifteen Coast and Geodetic Survey longitude determi- * The numerical estimates of errors used in tiiis chapter are taken, as a rule, from Appendix 7 of the Coast and Geodetic Survey Report for iSy8, Time, Longitude, Latitude, and Azimuth. 284 THE ADJUSTMENT OF OBSERVATIONS nations involved in the primary longitude net, it was found that whereas the probable error of a difference of longitude from one night's observation as computed from the residuals of different stars from the mean for the night was ± 0.013", if this prob- able error were computed from the residuals of different nights from the mean of the ten nights concerned in the determination (the correction for relative personal equation being already applied), it was enough larger to indicate that the constant error peculiar to each night was i 0.022. Hence there would be little appreciable gain in accuracy if the number of observa- tions per night were greatly increased. 3. From many longitude determinations involved in the pri- mary longitude net, and each consisting as a rule of ten nights of observation, it was found that whereas the probable error of the mean of the ten nights as computed from the residuals of the separate nights from the mean was ±0.01 1", the probable error as computed from the adjustment of the net was so much larger as to show that there was a constant error peculiar to each mean of ten nights of i 0.022". It follows that a reduc- tion in the number of nights to six or even four would result in but a slight increase in accuracy, — say 10 per cent. In the longitude determinations referred to above, the usual procedure was for the two observers to exchange places after the first five of the ten nights of observation. In all other respects except this, the last five nights of observation were made under conditions as nearly as possible identical with those during the first five nights. If the mean of the ten nights is taken, and the corresponding residuals written out, it is evident, as a rule, that there is a tendency for the residuals of one group of five to have one sign, and of the other group to have the opposite sign. This is an illustration of Case i, and indicates that there is a systematic error in the result which bears a fixed relation to the relative position of the observers, and is there- fore due to their relative personal equation. Accordingly it is eliminated by taking one-half the difference of the two groups SELECTION OF METHODS OF OBSERVATION 285 of five as the relative personal equation, correcting each night's result by this amount, and deriving the final difference of longi- tude from these corrected values. A confirmation of the sup- position that this systematic error is the relative personal equa- tion, is furnished by comparing successive values of the relative personal equation as thus derived for the same pair of observers in successive longitude determinations. In connection with some longitude determinations, the rela- tive personal equation of the two observers has been determined by a personal equation machine, or by observations by the half-transit method. Either of these is an illustration of Case 4. Similarly the method of Case 4 has been applied to prove that the systematic error in a longitude determination arising from the action of the single relays which connect the mean tele- graphic line at each longitude station with the chronograph circuit must be in the thousandths of seconds only, not in hun- dredths. This was done by making special observations of changes which occur in the time of operation of these relays when extreme changes are made in their adjustment and in the strength of the currents operating them. A study of the sources of error in telegraphic longitude determinations by the methods suggested in this chapter, but necessarily in much greater detail than it is possible to give here, inevitably leads to the conclusion that a considerable portion of errors which are constant for a night are due to variation of the relative personal equation of the two observers. This suggests that the line along which some improvements in the methods should be sought is that of securing some- means of making the relative personal equation and its variation zero. For this purpose a new attachment to the astronomical transit, known as a transit micrometer, was put into use several years ago by the Prussian Geodetic Institute with great success, and is now in use in this country.* * See Appendix 8 of the Coast and Geodetic Survey Report for 1904, The Transit Micrometer. 286 THE ADJUSTMENT OF OBSERVATIONS 207. Other Illustrations. — As another illustration of Case 2, it may be noted that in the measurement of angles in primary triangulation the general experience is that the probable error of an angle as computed from the residuals of the various meas- ures of that angle from their mean, or even computed from the residuals which occur in a local adjustment involving all the angles at a station, is very much smaller than the probable error of an angle computed from the figure adjustment, which involves a much larger group of observations. This indicates that there are errors which are constant for a station and are not affected by either increasing or decreasing the number of measurements of an angle. Acting upon this reasoning, the Coast and Geodetic Survey has recently reduced the number of observations of each angle in primary triangulation from 22 to 34, to 16. This very important saving in time and money has not been accompanied by any appreciable decrease in accuracy. 208. An examination of observations of astronomical azimuth at many stations has shown that frequently the residuals for each night tend to stand in a group by themselves, all having one sign. This examination is one form of application of the method of Case 2. It indicates that there are constant errors, in some instances, at least, peculiar to each night in azimuth observations, and that therefore, if the highest degree of accu- racy is desired, the observations must be extended over several nights. The same test applied to latitude observations made with a zenith telescope, indicates in general that there are no constant errors peculiar to each night, though to this statement some exceptions have been noted. 209. Two interesting illustrations of Case 2 are furnished by the use of a five-meter iced bar in the standardization of base apparatus in the Coast and Geodetic Survey. In determining the length of the bar, it was found that the residuals apparently indicated that the bar was 2 to 4 microns longer when its length was determined while its A-end lay to the left of the observer SELECTION OF METHODS OF OBSERVATION 287 than when it lay m the reverse position.* Though this is a very small quantity, it was persistently shown by the observa- tions. It was recognized, when it had been carefully studied, as a systematic error due to the personal equation of the observers in making bisections of the graduations on the bar. To secure the highest possible degi'ee of accuracy in determining the length of the bar and in using it as a standard, it is therefore necessary to make one-half the observations with the bar in each position and therefore eliminate this systematic error. Again, in using this iced bar in the open air to measure a standard length of 100 meters, it was found that the residuals from the mean of all the measures was of one si^rn if the meas- uremient had progressed toward the sun, and of the opposite sign if the measurement had progressed away from the sun.f This systematic error was eliminated, in part at least, by making the measurements in pairs with the progress in opposite direc- tions in the two measurements of each pair, and with as short an intervening time interval as possible. This systematic error is believed to be due to slight motions of the microscopes due to changes of temperature. 210. A good illustration of Case 3 is furnished by a com- parison of observed and predicted tides at Sandy Hook, N.J4 The difference between the observed and predicted heights of high and low water was due to many separate elemental errors, the errors in the six years of tidal observations from which the tidal constants used in the prediction were derived and in the one year of observations with which the comparison was made, the errors in the theory involved in the computation of the tidal constants, and the errors in the operation of the predicting machine itself. It was desired to determine as fully as pos- * Appendix 8, Coast and Geodetic Survey Report for 1S92, the Hohon Base, pp. 382-391. t Appendix 3, Coast and Geodetic Survey Report, 1901, the Measurement of Nine Hases, p. 245. t Appendix 15, Coast and Geodetic Survey Report for ICS90, and espe- cially illustrations Nos. 66 and 67 of that Report. 288 THE ADJUSTMENT OF OBSERVATIONS sible what systematic errors existed, tlie magnitude of the acci- dental errors, and especially how large were the errors due to the machine. There were four groups of f oo each to be considered, arising respectively from the prediction of high-water heights, low-water heights, high-water times and low-water times. In addition to other tests applied, the probable error of a single prediction was computed for each of these groups (after the constant error had been removed), the curve of distribution of errors for each group drawn on a large scale (the magnitudes of the error being the abscissae, and the number of such errors the ordinates of the curve), and the theoretical law of error (Art. 27) drawn to the same scale was superposed on it. For the two curves corresponding to predicted heights it was at once apparent that there was a systematic difference in char- acter between the actual and theoretical curves. The actual curve in each case showed about 13 times as many errors as the theoretical curve greater than 4^ times the computed prob- able error, and about 79 times as many greater than 5f times the computed error. It also showed about J as many errors as the theoretical curve less than the computed probable error, and about f as many between one and three times the computed probable error. A difference of this kind between the actual and theoretical curves indicates that there is, in addition to the actual errors, some large systematic error which occurs occa- sionally, in this case about one in twenty times upon an aver- age. The unusual number of large residuals would be caused directly by such a systematic error. An indirect effect of these large residuals, due mainly to the large systematic error superposed on the accidental errors, would be to make the com- puted probable error much too large. This in turn would cause the theoretical curve to depart from the actual between errors of zero and three times the probable error, in a manner similar to that noted above. Following up the conclusion that the particularly large residuals indicated by the outer portions of SELECTION OF METHODS OF OBSERVATION 289 the actual curve were due to a systematic error, considerable evidence was found that they were due to effects of storms upon mean sea level. As the principal purpose of the test was to determine how great were the errors due to the action of the tide predicting machine, this conclusion that the large errors were not chargeable to the machine was an important one. The comparison between the actual and theoretical curves for predicted times of high and low water showed a very close agreement. Storms are known to have but little effect upon the time of high and low water, hence the systematic errors due to this cause should be expected to be small, as the curves indicated them to be. 211. Trigonometrical leveling, that is, leveling by observations of vertical angles taken in connection with triangulation, frequently connects points that are also connected by precise leveling, and thus furnish an illustration of Case 5. The test applied by the precise level usually, but not always, indicates that the systematic and constant errors in theoretical leveling are so small as to be almost or quite concealed by the accidental errors. 212. The following three illustrations of Case 5 are all taken from The Solar Parallax and Its Related Constants, by William Harkness. 1. The aberration constant has been determined many times and by many methods, among which are : a, by observations of right ascensions of stars with an instrument in the meridian; /;, by observations of the declinations of stars with an instrument in the meridian ; c, by observations with an instrument in the prime vertical; d, by zenith telescope observations. A com- parison of the results by the various methods indicates clearly that they are, as a rule, subject to constant or systematic errors much larger than the uneliminatcd effects of accidental errors. 2. The flattening of the earth has l^cen derived: a, from geodetic arcs; b, from pendulum observations; r, from the observed precession and nutation ; d, from perturbations of the 290 THE ADJUSTMENT OF OBSERVATIONS moon. The comparison of the various results indicates that the systematic or constant errors in some if not all of them are much larger than the uneliminated effects of accidental errors. 3. The mean density of the earth has been determined : a, by observations of the attraction of mountains as measured by the deviations of the plumb-line in the immediate vicinity; b, by observing the attraction of known masses of matter either with a torsion balance or a pendulum; c, by pendulum observations at different distances from the center of the earth, near mean sea level and on mountain tops, or at the surface of the earth and in mines ; d, by observations with balances of the ordinary form either of the attraction of a known mass or of the change in the attraction of the earth upon a known mass when it is moved to a higher or a lower position. The comparison shows systematic or constant errors in the results which are large in comparison with the uneliminated accidental errors, as a rule, when methods a and c are used, and in some cases even when other methods are used. APPENDIX 213. Values of 2 f'p VirJo ^-« dt. See Art. 22. (/) is the probability that the error will be less numerically than the limit which is expressed in the first column in terms of the p. e. Thus, the sixth line of the table means that out of every thousand errors the chances are that 264 will be less than one-half as great as the p. e. The probability that an error a is greater than r is 0.5, than 2 r is 0.177, than 3 r is 0.043, than 4 r is 0.007, ^^'^^^^ 5 ^' is 0.00 1, than 6 r is 0.000 1. TABLE I. See Art 27,. a &{i) Difference. a ©W Difference- r r 0.0 0.000 .. 2-5 0.908 O.I 0.054 54 2 6 0.921 13 0.2 0.107 53 2 7 0.931 10 0.3 0. 160 53 2 8 0.941 10 0.4 0.213 53 2 9 0.950 9 0.5 0.264 51 3 0-957 7 0.6 0.314 50 3 I 0.963 6 0.7 0-363 49 3 2 0.969 6 0.8 O.4II 48 3 3 0.974 5 0.9 0.456 45 3 4 0.978 4 I.O 0.500 44 3 5 0.9S2 4 I.I 0.542 42 3 6 0.985 3 I .2 0.582 40 3 7 0.987 2 1-3 0.619 37 3 8 0.990 3 1.4 0.655 36 3 9 0.991 I 1-5 0.688 33 4 0-993 2 1.6 0.719 31 4 I 0.994 I 1-7 0.748 29 4 2 0.995 I 1.8 0-775 27 4 3 . 996 J 1.9 0.800 25 4 4 0.997 I 2.0 0.823 23 4 5 0.998 1 2.1 0.843 20 4 6 0.998 2.2 0.862 19 4 7 . 998 2-3 0.879 17 4 8 0.999 I 2.4 0.89s 16 4 9 0.999 2 . ^ 0.908 13 5.0 0.999 291 292 THE ADJUSTMENT OF OBSERVATIONS TABLE II. 214. Factors for BcsscVs Probable Error Formulas. See Art. 33. « .6745 \/« — I .6745 « .6745 ■6745 ^n(n— I) \/n (« — I) 40 olioSo 0.0171 . • • . 41 .1066 0.0167 2 0.6745 0.4769 42 •1053 0.0163 3 .4769 •2754 43 .1041 0.0159 4 •3894 .1947 44 . 1029 0-0155 . 5 0-3372 0.1508 45 0.1017 0.0152 6 .3016 -I231 46 • 1005 .0148 7 •2754 .1041 47 .0994 .0145 8 •2549 .0901 48 .0984 .0142 9 •2385 -0795 49 .0974 •0139 10 0. 2248 .0711 50 0.0964 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 IS 0. 1803 0.0465 55 0.0918 0.0124 16 .1742 •0435 56 .0909 .0122 17 .1686 .0409 57 .0901 .0119 18 .1636 .0386 58 .0893 .0117 19 .1590 -0365 59 .0886 -OI15 20 0.1547 0.0346 60 0.0878 0.0113 21 .1508 .0329 61 .0871 .0111 22 .1472 .0314 62 .0864 .0110 23 .1438 .0300 63 .0857 .0108 24 . 1406 .0287 64 .0S50 .0106 25 0-1377 0.0275 65 0.0843 0.0105 26 •r349 .0265 66 -0837 .0103 27 •1323 •0255 67 .0830 .0101 28 .1298 .0245 68 .0824 .0100 29 •1275 .0237 69 .0818 .0098 30 0.1252 0.0229 70 0.0812 C.OO97 31 .1231 .0221 71 .0806 .0096 32 .1211 .0214 72 .0800 .0094 33 .1192 .0208 73 •0705 .0093 34 .1174 .0201 74 .0789 .0092 35 0.1157 0.0196 75 0.0784 0.0091 36 .1140 .oiqo 80 •0759 .0085 37 .1124 .0185 85 .0736 .0080 38 .1109 .0180 90 •0713 .0075 39 .1094 •0175 TOO .0678 .0068 APPENDIX 293 TABLE III. 215. Factors for Peters Probable Error Formulas. See Art. n. ft .845J .8453 n •8453 •8453 V" (« — ') «\/« — I \/« (« — •) « \/« — I • • 40 0.0214 0.0034 • • • . . 41 .0209 •0033 2 0.5978 0.4227 42 .0204 .0031 3 ■ 3451 •1993 43 .0199 .0030 4 .2440 .1220 44 .0194 .0029 5 0. i8go 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 51 .0167 •0023 12 .0736 .0212 52 .0164 •0023 13 .0677 .0188 53 .0161 .0022 14 .0627 .0167 54 •0158 .0022 IS 0.0583 0.0151 55 0-0155 0.0021 16 .0546 .0136 56 .0152 .0020 17 •0513 .0124 57 .0150 .0020 18 .0483 .0114 58 .0147 .0019 19 •0457 .0105 59 .0145 .0019 20 0.0434 0.0097 60 0.0142 0.0018 21 .0412 .0090 61 .0140 .0018 22 •0393 .0084 62 •0137 .0017 23 .0376 .0078 63 •0135 .0017 24 .0360 .0073 64 •0133 .0017 25 0-0345 . 0069 ^'5 . 1 3 1 0.0016 26 •0332 .0065 66 .0129 .0016 27 •0319 .0061 67 .0127 .0016 28 .0307 .0058 68 .0125 .0015 29 .0297 •0055 69 .0123 •0015 ?,° 0.0287 0.0052 70 0.0122 . 00 I 5 31 .0277 .0050 71 -0120 .0014 32 .0268 .0047 72 .0118 .0014 33 .0260 .0045 73 .0117 . 00 I 4 34 .0252 .0043 74 .0115 • 00 1 3 35 0.0245 0.0041 75 . IT 3 . 00 1 3 36 .0238 .0040 80 .0106 .0012 37 •0232 .0038 85 .0100 .0011 38 .0225 •0037 90 .0095 .0010 39 .0220 •0035 100 .0085 .0008 INDEX The figure refers to the page Accidental error, nature of, 90, 273. Accuracy not limited to what can be seen, 48. Adjustment, figure, 263. Adjustment, general, 188. Adjustment, local, 185. Adjustmentof a central polygon, 234. Adjustment of a level net, 208. Adjustment of a quadrilateral, 206, 228, 231. Adjustment of triangulation, method of angles, 180. Adjustment of triangulation, method of directions, 230. Angle equations, 189, 191. Angle equations, selection of, 191. Approximate method of finding pre- cision, 237. Arithmetic mean, 9, 35. Artifices, two special, 144. Assignment of weight arbitrary, 82. Assignment of weights in a level net, 270. Average error, 24. Average ratio. of weights, 142. Azimuth condition equations, 253. Base-line measurement, precision of, 261. Bessel's formula for probable error, 38. Best side equations, 247. Blunders, 8. Bowditch's rule for balancing a sur- vey, 158. Breaking a net into sections, 259. Caution about tests of precision, 45. Classification of observations, 34. Comparison of average, mean square and probable errors, 26. Comparison of observatioii and theory, 45. Computation of normal equations, 132. Computation of [t'-], 32. Computing machines, loi. Condition equations for length, azi- muth, latitude, and longitude, 250. Conditions, general, number of, 202, Conditions, local, number of, 202. Conditions, number of, 202. Conditioned observations, 169. Constant error, detection of, 276. Constant error present, weighting, 77- Constant errors, 51, 273. Control of [v^], 42. Control of arithmetic mean, 36. Control of formation of normal equations, loc. Control of solution of normal equa- tions, 107. Control of weighted mean, 54. Control of weighted mean, 57. Correlates, method of, 152, 210. Correlate equations, method of di- rections, 246. Curve of probability, 27. Detection of systematic or constant error, 276. Direct observations, one unknown, 35- Direction method of adjustment of triangulation, iSo. Direction method of observing an- gles, 289. 295 296 INDEX Distinction between accidental, con- stant, and systematic errors, 273. Doolittle method of solving normal equations, 114. Elemental errors in latitude observa- tions, 279. Error, accidental, nature of, 9, 273. Error, average, 24. Error, effect of extending limits of, 3°- Error, law of, 13. Error, mean square, 22. Error of a given magnitude, proba- bility of, 15. Error, probable, 274. Error, reduction of, by repetition of observations, 47. Error, systematic, 274. Errors, distinction between acci- dental, constant, and systematic, 273- Errors distinguished from residuals, 12. Errors, instrumental, 274. Errors, observer's, 5. Experience, law of error tested by, 44. External conditions, 4. Factors for Bessel's probable error formula, 292. Factors for Peters' probable error formulas, 293. Figure adjustment, 263. Formation of normal equations, 96. Function of adjusted values, preci- sion of, 137. General adjustment, 188. Groups, solution by, 69, 213, 223. Hagen's hypothesis, 16. Iced-bar measurements, 286. Improvement in methods of observa- tion, 273. Independent angles, methods of, 180. Independent angles, method of ob- servation, 183. Indirect observations, 93. Instruments, i. Latitude and longitude condition equations, 255. Latitude observations, zenith tele- scope, 278. Law of error tested by experience, 44. Law of error, 13. Least squares, principle of, 19. Length condition equations, 52. Level net, adjustment of, 268. Level net, assignment of weights, 270. Limit of accuracy not the limit of vision, 48. Linear function, law of, error of, 20. Linear function, precision of, 137. Local adjustment, 185. Local conditions, number of, 188. Logarithmic solution of normal equations, 112. Longitude condition equations, 250. Mean, arithmetic, 9, 35. Mean, weighted, 54. Mean square error, 22. Mean square error vs. probable error, 20. Methods of computing [7/^], 183. Methods of observation, selection of. Multiples of the unknown, observed, 60. Normal equations, 98. Normal equations, forms of compu- ting, lOI. Normal equations, method of corre- lates, 152, 210. Normal equations, solution of, 105, 177. Number of angle equations, 189, 191. iNDfiX 297 Number of general conditions, 202. Number of local conditions. 202. Number of side equations, 193. Observations, classification of, 34. Observations, conditioned, 149. Observations, indirect, 93. Observations, weighting of, 94. Observed values, multiples of the unknown. 60. Observer's errors. 5. Observing angles by direction method, 239. Observing by method of independent angles, 1S3. One unknown, direct observations, 35- Personal equation, 6. Peters' formula for probable error, 40, 293. Pole, position of, 195, 201. Precision, approximate method of finding, 237. Precisionof adjusted values, 121, 158, 162, 208, 211, 218. Precision of arithmetic mean, 38. Precision of base-line measurements, 261. Precision of function of adjusted values, 137. Precision of a linear function, 62. Precision, measure of, 16. Precision of weighted mean, 58. Predicted tides, 2S9. Principle of least squares, 19. Probability curve, 27. Probability of error of a given mag- nitude, 32. Probable error, 22. Probable error formula, factors for, 292, 293. Probable error, independent of con- stant error, 46. Probable error of single observation, 132. Probable error vs. mean square error, 20. Probable error, more accurate defi- nition of, 274. Quadrilateral, adjustment of, 206, 22S, 231. Ratio of weight of observed to ad- justed value, 143. Rejection of observations, 87. Relation of, probable error to aver- age of errors, 43. Residuals distinguished from errors, 12. Residuals, squares of, a minimum, 13- Residuals, sum of = zero, 12. Repetition of observations to reduce error, 22. Sections, breaking a net into, 259. Selection of methods of observation, 272. Selection of side and angle condition equations, 189, 191. Side equation, reduction to linear form, 197. Side equations, 193, 243. Side equations, best, 247. Side equations, number of, 202. Single observation, probable error of, 132- Solution by groups, 169, 213, 223. Solution of normal equations, 105. Solution by successive approxima- tion, 177. Squares of residuals a minimum, 19. St. Gothard tunnel, 221. Substitution, method of, 106. Summation, symbol of, 10. Telegraphic longitude observations, 283. Tests of precision, caution, 45. Time of solving a set of normal equations, 120. 298 lNDE:?t Triangulation, adjustment of, method of angles, 180. Triangulation, direction method of adjustment, 239. Trigonometric leveling, 289. Weights, 55. Weights arbitrarily assigned, 82. Weights in a level net, assignment of, 270. Weights of unknowns, 124, 129. 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