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ON THE 
 
 ALGEBRAICAL AND NUMERICAL 
 
 THEORY OF ERRORS OF OBSERVATIONS 
 
 AND THE 
 
 COMBINATION OF OBSERVATIONS. 
 
ON THE 
 
 ALGEBRAICAL AND NUMERICAL 
 THEORY 
 
 OF 
 
 ERRORS OF OBSERVATIONS 
 
 AND THE 
 
 COMBINATION OF OBSERVATIONS. 
 
 By SIR GEORGE BIDDELL AIRY, K.C.B. 
 
 ASTRONOMER ROYAL. ' 
 
 SECOND EDITION, REVISED. 
 
 Hon&on : 
 aIACMILLAN AND CO. 
 
 1875. 
 
 [All Eights reserved.] 
 
QA 
 
 Camfcrt'trge : 
 
 PRINTED BY C. J. CLAY. MA 
 AT THE UNIVERSITY PRESS 
 
PEEFACE TO THE FIEST EDITION. 
 
 The Theory of Probabilities is naturally and strongly 
 divided into two parts. One of these relates to those 
 chances which can be altered only by the changes of 
 entire units or integral multiples of units in the funda- 
 mental conditions of the problem ; as in the instances 
 of the number of dots exhibited by the upper surface 
 of a die, or the numbers of black and white balls to 
 be extracted from a bag. The other relates to those 
 chances which have respect to insensible gradations in 
 the value of the element measured ; as in the duration 
 of life, or in the amount of error incident to an astro- 
 nomical observation. 
 
 Tt may be difficult to commence the investigations 
 proper for the second division of the theory without 
 referring to principles derived from the first. Never- 
 theless, it is certain that, when the elements of the 
 second division of the theory are established, all refer- 
 ence to the first division is laid aside ; and the original 
 connexion is, by the great majority of persons who use 
 the second division, entirely forgotten. The two divi- 
 sions branch off into totally unconnected subjects ; those 
 persons who habitually use one part never have occasion 
 for the other ; and practically they become two different 
 sciences. 
 
 In order to spare astronomers and observers in 
 natural philosophy the confusion and loss of time which 
 are produced by referring to the ordinary treatises em- 
 bracing both branches of Probabilities, I have thought 
 
VI PREFACE. 
 
 it desirable to draw up this tract, relating only to Errors 
 of Observation, and to the rules, derivable from the 
 consideration of these Errors, for the Combination of 
 the Results of Observations. I have thus also the 
 advantage of entering somewhat more fully into several 
 points, of interest to the observer, than can possibly be 
 done in a General Theory of Probabilities. 
 
 Xo novelty, I believe, of fundamental character, will 
 be found in these pages. At the same time I may state 
 that the work has been written without reference to 
 or distinct recollection of any other treatise (excepting 
 only Laplace's Theorie des Probabilites) ; and the me- 
 thods of treating the different problems may therefore 
 differ in some small degrees from those commonly em- 
 ployed. 
 
 G. B. AIRY. 
 
 Royal Obseevatoey, Greenwich, 
 January 12, 1861. 
 
 PREFACE TO THE SECOND EDITION. 
 
 The work has been thoroughly revised, but no im- 
 portant alteration has been made : except in the intro- 
 duction of the new Section 15, and the consequent 
 alteration in the numeration of articles of Sections 1(5 
 and 17 (formerly 15 and 10) : and in the addition of the 
 Appendix, giving the result of a comparison of the 
 theoretical law of Frequency of Errors with the Fre- 
 quency actually observed in an extensive series. 
 
 G. B. AIRY. 
 
 February 20, 1875. 
 
INDEX. 
 
 PART I. 
 
 FALLIBLE MEASURES, AND SIMPLE ERRORS OF OBSERVATION. 
 
 Section 1. Nature of the Errors here considered. 
 
 PAGE 
 
 Article 2. Instance of Errors of Integers 1 
 
 3. Instance of Graduated Errors : these are the sub- 
 
 ject of this Treatise 2 
 
 4. Errors of an intermediate class .... ib. 
 
 5. Instances of Mistakes ib. 
 
 C. Characteristics of the Errors considered in this 
 
 Treatise 3 
 
 8. The word Error really means Uncertainty . . 4 
 
 Section 2. Laic of Probability of Errors of any given 
 amount. 
 
 i). Reference to ordinary theory of Chances . . ib. 
 
 10. Illustrations of the nature of the law ... 5 
 
 11. Illusfration of the algebraic form to be expected 
 
 for the law 6 
 
 12. Laplace's investigation introduced ... 7 
 
 13. Algebraical combination of many independent 
 
 causes of error assumed ib. 
 
Vlll INDEX. 
 
 PAGE 
 
 Article 15. This leads to a definite integral S 
 
 16. Simplification of the integral 10 
 
 17. Investigation of J dt.e~& 11 
 
 Jo 
 
 f" 
 IS. Investigation of I dt. cos rt. e~ 1 ' 2 . . . .12 
 Jo 
 
 20. Probability that an error will fall between x and 
 
 I a- 2 
 
 x + 8x is found to be — t- .e~c 2 .8x . . .14 
 cJtt 
 
 21. Other suppositions lead to the same result . .15 
 
 22. Plausibility of this law ; table of values of e c 2 • ib> 
 
 23. Curve representing the law of Frequency of Error . 16 
 
 Section 3. Consequences of the Law of Probability or 
 Frequency of Errors, as applied to One System of 
 Measures of One Element. 
 
 25. It is assumed that the law of Probability applies 
 
 equally to positive and to negative errors . .IS 
 
 26. Investigation of "Mean Error" . . . .19 
 
 27. Investigation of "Error of Mean Square" . . 20 
 
 28. Definition of "Probable Error" . . . .21 
 
 1 /""' 
 
 29. Tableof--/ dic.€~ K2 , and investigation of Pro- 
 
 's/ 7 '' Jo 
 bable Error 22 
 
 30. Remark on the small number of errors of large 
 
 value 23 
 
 31. Table exhibiting the relations of the Modulus and 
 
 the several Errors ib. 
 
 32. Introduction of the term "Actual Error" . . 24 
 
INDEX. i X 
 
 Section 4. Remarks on the application of these processes 
 in particular cases. 
 
 PAGE 
 
 Article 33. "With a limited number of errors, the laws will be 
 
 imperfectly followed 24 
 
 34. Case of a single discordant observation . . .25 
 
 PART II. 
 
 ERRORS IX TIIE C0MBIXATI0X OP FALLIBLE MEASURES. 
 
 Section 5. Law of Frequency of Error, and values of 
 Mean Error and Probable Error, of a symbolical or 
 numerical Multiple of One Fallible Measure. 
 
 35. The Law of Frequency has the same form as for 
 
 the original: the Modulus and the Mean and 
 Probable Errors are increased in the proportion 
 expressed by the Multiple 26 
 
 36. The multiple of measure here considered is not 
 
 itself a simple measure 27 
 
 37. Nor the sum of numerous independent measures . ih. 
 
 Sectiox 6. Law of Frequency of Error, and values of 
 Mean Error and Probable Error, of a quantity formed 
 by the algebraical sum, or difference of two independent 
 Fallible Measures. 
 
 39. The problem is reduced to the form of sums of 
 groups of Errors, the magnitudes of the errors 
 through each group being equal. . . .29 
 
 43. Results : that, for the sum of two independent 
 Fallible Measures, the Law of Frequency has the 
 same form as for the originals, but the square of 
 the new modulus is equal to the sum of the 
 squares of the two original moduli . . .33 
 
X INDEX. 
 
 PAGE 
 
 Article 44. The same theorem of magnitudes applies to Mean 
 Error, Error of Mean Square, and Probable 
 Error 33 
 
 45. But the combined Fallible Measures must be ab- 
 solutely independent 34 
 
 47. The same formulae apply for the difference of two 
 
 independent Fallible Measures . . . .30 
 
 49. In all cases here to be treated, the Law of Fre- 
 quency has the same form as for original obser- 
 vations 37 
 
 Section* 7. Values of Mean Error and Probable Error 
 in combinations which occur most frequently. 
 
 50. Probable Error of kX+l If 3s 
 
 51. Probable Error of R + S+T+U+&C . . . ib. 
 
 52. Probable Error of rIZ + sS+tT+uU+&.c. . . 33 
 
 53. Probable Error of X 1 + A r 2 + ...4-X„, where the 
 
 quantities are independent but have equal proba- 
 ble errors (b. 
 
 54. Difference between this result and that for the pro- 
 
 bable error of nXi 40 
 
 55. Probable Error of the Mean of X 1 , X i} ...X„ . 41 
 
 Section 8. Instances of the application of these Theo- 
 rems. 
 
 5G. Determination of geographical eolatitude by obser- 
 vations of zenith distances of a star above ami 
 below the pole 42 
 
 57. Determination of geographical longitude by trans- 
 its of the Moon 43 
 
INDEX. XI 
 
 Section 9. Methods of determining Mean Error and 
 Probable Error in a gicen series of observations. 
 
 PAGE 
 
 Article 58. The peculiarity of the case is, that the real value of 
 
 the quantity measured is not certainly known . 44 
 
 59, For the Mean Error, the rule is the same as 
 
 before ib. 
 
 60, For Error of Mean Square, and Probable Error, 
 
 the divisor of sum of squares will be n— 1 instead 
 
 of n 45 
 
 61, Convenient methods of forming the requisite num- 
 
 bers 47 
 
 FART III. 
 
 PRINCIPLES OF FORMING THE MOST ADVANTAGEOUS COMBINA- 
 TION OF FALLIBLE MEASURES. 
 
 Section 10. Method of combining measures ; meaning of 
 " combination-weight ;" principle of most advantageous 
 combination ; caution in its application to "entangled 
 measures." 
 
 62. First class of measures ; direct measures of a 
 
 quantity which is invariable, or whose variations 
 are known 49 
 
 63. Combination by means of combination-weights . 50 
 
 64. The combination to be sought is that which will 
 
 give a result whose probable error is the smallest 
 possible ib. 
 
 65. To be found by the algebraical theory of complex 
 
 maxima and minima 51 
 
 6G. Sometimes, even for a simple result, there will 
 occur " entangled measures." Caution fur the 
 reduction of these ib. 
 
XU INDEX. 
 
 PAGE 
 
 Article 67. Second class of measures ; when the corrections to 
 several physical elements are to he determined 
 simultaneously ; this is also a case of algebraical 
 complex maxima and minima .... 52 
 
 Section 11. Combination of simple measures; meaning 
 of " theoretical weight;" simplicity of results for theo- 
 retical weight; allowable departure from the strict 
 rules. 
 
 68. Independent measures or results are supposed 
 
 equally good ; the investigation shews that they 
 must be combined with equal weights . . 53 
 
 69. Independent measures or results are not equally 
 
 good ; their combination-weights must be in- 
 versely proportional to the square of the probable 
 error of each 54 
 
 70. If the reciprocal of (probable error) 2 be called 
 
 "theoretical weight," their combination-weight 
 oucrht to be proportional to their theoretical 
 weight ; and the theoretical Aveight of result 
 = sum of theoretical weights of original mea- 
 sures ">.") 
 
 72. Instance : eolatitude by different stars . . . •"><; 
 
 73. We may depart somewhat from the strict rules for 
 
 formation of combination-weights without intro- 
 ducing material error of result . , . 58 
 
 Section 12. Treatment of entangled measures. 
 
 74. Instance (1). Longitude is determined by lunar 
 
 transits compared with those at two known 
 stations 60 
 
 75. Reference must be made to actual errors ; result 
 
 for combination-weights, and for theoretical weight 
 of result (jl 
 
INDEX. Xlll 
 
 PAGE 
 
 Article 76. Partition of theoretical weight of result . . .62 
 
 78. Partition is applicable in other cases . . .04 
 
 79. Instance (2). Theodolite observations of the meri- 
 
 dian and of distant signals ; theoretical weight for 
 each azimuth found by partition .... ib. 
 
 SO. Instance (3). Zenith distances of stars are ob- 
 served at three stations of a meridional arc; to 
 find the amplitude of the first section . . . ib. 
 
 81. All valid combinations must be considered, and, 
 
 being entangled observations, must be treated by 
 actual errors .65 
 
 82. Equations formed and solved 06 
 
 84. The result for the first section and that for the 
 second section are entangled, and cannot be com- 
 bined to form the result for the whole ; difference 
 between actual error and probable error . . 6S 
 
 86. General caution for treatment of entangled obser- 
 vations 70 
 
 Section 13. Treatment of numerous equations applying 
 to several unknown quantities; introduction of the term 
 " minimum squares.'" 
 
 87. General form of such equations . . . .71 
 
 88. Obvious method of combining them in order to 
 
 form the proper number of determining equa- 
 tions 72 
 
 89. Symbolical equations for x, one of the unknown 
 
 quantities 73 
 
 90. Symbolical equations for making the probable 
 
 error of x minimum 74 
 
 91. Synthetical solution of the equations . . .75 
 
XIV INDEX. 
 
 PAGE 
 
 Article 93. Complete exhibition of the form of solution . .77 
 
 94. This form is the same as the form of solution of 
 the problem, " to reduce to minimum the sum of 
 squares of residual errors, when the errors are 
 properly multiplied." Introduction of the term 
 " minimum squares." Danger of using this term 78 
 
 96. Expression for probable error of x . . . .79 
 
 97. Approximate values of the factors will suffice in 
 
 practice SI 
 
 Section 14. Instances of the formation of equations ap- 
 plying to several unknown quantities. 
 
 99. Instance 1. Determination of the personal equa- 
 tions among several transit-observers . . .82 
 
 100. Instance 2. Consideration of a net of geodetic 
 
 triangles So 
 
 101. The probable error of each measure must first be 
 
 ascertained ; different for angles between sta- 
 tions, for absolute azimuths, for linear measures ib. 
 
 102. Approximate numerical co-ordinates of stations are 
 
 to be assumed, with symbols for corrections . S6 
 
 103. Corresponding equations for measures mentioned 
 
 above ih. 
 
 104. These equations will suffice SS 
 
 105. Generality and beauty of the theory ; it admits of 
 
 application to any supposed measures ; instance S9 
 
 10G. No objection, that the measures are heteroge- 
 neous ih 4 
 
 107. Solution of equations is troublesome . . .90 
 
INDEX. XV 
 
 Section 15. Treatment of Observations in which it is 
 required that the Errors of Observations rigorously 
 satisfy some assigned conditions. 
 
 PAGE 
 
 Article 110. Instance 1. In a geodetic triangle, of which the 
 three angles are observed, and their sum proves 
 erroneous : to find the corrections for the several 
 angles 91 
 
 111. Equations for probable errors ib. 
 
 112. Assigned condition introduced .... 92 
 
 113. Result ib. 
 
 114. Instance 2. In a series of successive azimuthal 
 
 angles, whose sum ought to be 360°, the sum 
 proves erroneous : to find the corrections for 
 the several angles 94 
 
 115. Result ib. 
 
 116. Instance 3. In a geodetic hexagon, with a central 
 
 station, all the angles are subject to error . . 95 
 
 117. Assigned conditions introduced .... 96 
 118,119,120. Eliminations, and equations . 97,99,100 
 122. Practical process, which may be preferable . .101 
 
 PART IV. 
 
 OX MIXED ERRORS OF DIFFERENT CLASSES, AND CONSTANT 
 ERRORS. 
 
 Section 16. Consideration of the circumstances under 
 which the existence of Mixed Errors of Different 
 Classes may be recognized ; and investigation of their 
 separate values. 
 
 124. The existence of Error of a Different Class is not 
 
 to be assumed without good evidence . . .103 
 
 125. Especially without evidence of possibility of such ■ 
 
 Error ib. 
 
XVI INDEX. 
 
 PAGE 
 
 Articlel26. Formation of result of each group .... li>4 
 
 127. Discordance of results of different groups . . ib. 
 
 12S. Investigation of Mean Discordance, supposed to 
 
 be a mutter of chance, and its Probable Error . 105 
 
 129. Decision on the reality of a Mean Discordance . ib. 
 
 130. Much must depend on the judgment of the Com- 
 
 puter 106 
 
 131. Simpler treatment when Discordance appears to 
 
 be connected with an assignable cause . . . ib. 
 
 Section 17. Treatment of observations when the values 
 of Probable Constant Error for different groups, and 
 probable error of observation of individual measures 
 within each group, are assumed as known. 
 
 132. "We must not in general assume a value for Con- 
 
 stant Error for each group, but must treat it 
 as a chance-error 107 
 
 133. Symbolical formation of actual errors . . . 10S 
 
 134. Symbolical formation of probable error of result ; 
 
 equations of minimum 109 
 
 135. Resulting combination-weights . . . .110 
 
 136. Simpler treatment when the existence of a definite 
 
 Constant Error for one group is assumed . .Ill 
 
 CONCLUSION. 
 
 137. Indication of the principal sources of error and 
 
 inconvenience, in the applications which have 
 been made of the Theory of Errors of Observa- 
 tions and of the Combination of Observations . 112 
 
 APPENDIX. 
 
 Practical Verification of the Theoretical Law for the Fre- 
 quency of Errors • . .114 
 
COEKIGENDA. 
 
 Page 47, dele line 1, and substitute the following: — 
 
 Mean Square of Sum of Errors a + b + c + d + &e. 
 
 Page 61, between lines 6 and 7, insert "final apparent results, as 
 affected by the " 
 ,, line 12, for ' actual error of ' read ' apparent '. 
 
 ,, line 14, for ' actual errors of the ' read ' apparent '. 
 
 ,, line 19, for ' actual error ' read ' result '. 
 
- . 
 
 OF THE 
 
 UNIVERSn 
 
 ON THE 
 
 ALGEBRAICAL AND NUMERICAL THEORY 
 
 OP 
 
 ERRORS OF OBSERVATIONS 
 
 AND THE 
 
 COMBINATION OF OBSERVATIONS. 
 
 PART I. 
 
 FALLIBLE MEASURES, AND SIMPLE ERRORS OF 
 OBSERVATION. 
 
 § 1. Nature of the Errors here considered. 
 
 1. The nature of the Errors of Observation which 
 form the subject of the following Treatise, will perhaps 
 be understood from a comparison of the different kinds 
 of Errors to which different Estimations or Measures are 
 liable. 
 
 2. Suppose that a quantity of common nuts are put 
 into a cup, and a person makes an estimate of the num- 
 ber. His estimate may be correct ; more probably it 
 will be incorrect. But if incorrect, the error has this 
 
 A. A 
 
2 SIMPLE ERRORS OF OBSERVATION. 
 
 peculiarity, that it is an error of whole nuts. There can- 
 not be an error of a fraction of a nut. This class of errors 
 may be called Errors of Integers. These are not the errors 
 to which this treatise applies. 
 
 3. Instead of nuts, suppose water to be put into the 
 cup, and suppose an estimate of the quantity of water to 
 be formed, expressed either by its cubical content, or by 
 its weight. Either of those estimates may be in error by 
 any amount (practically not exceeding a certain limit), 
 proceeding by any gradations of magnitude, however mi- 
 nute. This class of errors may be called Graduated 
 Errors. It is to the consideration of these errors that 
 this treatise is directed. 
 
 4. If, instead of nuts or water, the cup be charged 
 with particles of very small dimensions, as grains of fine 
 sand, the state of things will be intermediate between the 
 two considered above. Theoretically, the errors of esti- 
 mation, however expressed, must be Errors of Integers of 
 Sand-Grains ; but practically, these sand-grains may be 
 so small that it is a matter of indifference whether the 
 gradations of error proceed by whole sand-grains or by 
 fractions of a sand-grain. In this case, the errors are 
 practically Graduated Errors. 
 
 5. In all these cases, the estimation is of a simple 
 kind ; but there are other cases in which the process may 
 be either simple or complex ; and, if it is complex, a dif- 
 ferent class of errors may be introduced. Suppose, for 
 instance, it is desired to know the length of a given road. 
 
NATURE OF THE ERRORS. 3 
 
 A person accustomed to road-measures may estimate its 
 length ; this estimation will be subject simply to Graduated 
 Errors. Another person may measure its length by a yard- 
 measure ; and this method of measuring, from uncertainties 
 in the adjustments of the successive yards, &c. will also be 
 subject to Graduated Errors. But besides this, it will be 
 subject to the possibility of the omission of registry of 
 entire yards, or the record of too many entire yards ; not 
 as a fault of estimate, but as a result of mental confusion. 
 / In like manner, when a measure is made with a micro- 
 meter ; there may be inaccuracy in the observation as 
 represented by the fractional part of the reading ; but there 
 may also be error of the number of whole revolutions, or 
 of the whole number of decades of subdivisions, similar to 
 the erroneous records of yards mentioned above, arising 
 from causes totally distinct from those which produce in- 
 accuracy of mere observation. This class of Errors may 
 be called Mistakes. Their distinguishing peculiarity is, 
 that they admit of Conjectural Correction. These Mistakes 
 are not further considered in the present treatise. 
 
 G. The errors therefore, to which the subsequent in- 
 vestigations apply, may be considered as characterized by 
 the following conditions : — 
 
 They are infinitesimally graduated, 
 
 They do not admit of conjectural correction. 
 
 7. Observations or measures subject to these errors 
 will be called in this treatise "fallible observations," or 
 " fallible measures." 
 
 A2 
 
4 SIMPLE ERRORS OF OBSERVATION. 
 
 8. Strictly speaking, we ought, in the expression of 
 our general idea, to use the word "uncertainty" instead of 
 "error." For we cannot at any time assert positively 
 that our estimate or measure, though fallible, is not per- 
 fectly correct ; and therefore it may happen that there is 
 no " error," in the ordinary sense of the word. And, in 
 like manner, when from the general or abstract idea we 
 proceed to concrete numerical evaluations, we ought, instead 
 of "error," to say "uncertain error;" including, among 
 the uncertainties of value, the possible case that the un- 
 certain error may = 0. With this caution, however, in the 
 interpretation of our word, the term " error " may still be 
 used without danger of incorrectness. When the term is 
 qualified, as "Actual Error" or "Probable Error," there is 
 no fear of misinterpretation. 
 
 § 2. Law of Probability of Errors of any given amount. 
 
 9. In estimating numerically the "probability" that 
 the magnitude of an error will be included between two 
 given limits, we shall adopt the same principle as in the 
 ordinary Theory of Chances. When the numerical value 
 of the "probability" is to be determined a priori, we 
 shall consider all the possible combinations which pro- 
 duce error; and the fraction, whose numerator is the num- 
 ber of combinations producing an error which is included 
 between the given limits, and whose denominator is the 
 total number of possible combinations, will be the "pro- 
 bability " that the error will be included between those 
 limits. But when the numerical value is to be deter- 
 
LAW OF PROBABILITY OF ERRORS. 5 
 
 mined from observations, then if the numerator be the 
 number of observations, whose errors fall within the given 
 limits, and if the denominator be the total number of 
 observations, the fraction so formed, when the number of 
 observations is indefinitely great, is the " probability." 
 
 10. A very slight contemplation of the nature of 
 errors will lead us to two conclusions : — ■ 
 
 First, that, though there is, in any given case, a pos- 
 sibility of errors of a large magnitude, and therefore a 
 possibility that the magnitude of an error may fall be- 
 tween the two values E and E + he, where E is large ; 
 still it is more probable that the magnitude of an error 
 may fall between the two values e and e + he, where e is 
 small ; he being supposed to be the same in both. Thus, 
 in estimating the length of a road, it is less probable that 
 the estimator's error will fall between 100 yards and 
 101 yards than that it will fall between 10 yards and 
 11 yards. Or, if the distance is measured with a yard- 
 measure, and mistakes are put out of consideration, it is 
 less likely that the error will fall between 100 inches and 
 101 inches than that it will fall between 10 inches and 
 11 inches. 
 
 Second, that, according to the accuracy of the methods 
 used and the care bestowed upon them, different values 
 must be assumed for the errors in order to present com- 
 parable degrees of probability. Thus, in estimating the 
 road-lengths by eye, an error amounting to 10 yards is 
 sufficiently probable ; and the chance that the real error 
 may fall between 10 yards and 11 yards is not contemptibly 
 
b SIMPLE ERRORS OF OBSERVATION. 
 
 small. But in measuring by a yard-measure, the proba- 
 bility that the error can amount to 10 yards is so insigni- 
 ficant that no man will think it worth consideration ; and 
 the probability that the error may fall between 10 yards 
 and 11 yards will never enter into our thoughts. It may, 
 however, perhaps be judged that an error amounting to 10 
 inches is about as probable with this kind of measure as 
 an error of 10 yards with eye-estimation ; and the probabi- 
 lity that the error may fall between 10 inches and 11 inches, 
 with this mode of measuring, may be comparable with the 
 probability of the error, in the rougher estimation, falling 
 between 10 yards and 11 yards. 
 
 11. Here then we are led to the idea that the alge- 
 braical formula which is to express the probability that an 
 error will fall between the limits e and e + Be (where Be is 
 extremely small) will possess the following properties : — 
 
 (A) Inasmuch as, by multiptying our very narrow 
 interval of limits, we multiply our probability in the same 
 proportion, the formula must be of the form cf> (e) x Be. 
 
 (B) The term <p (e) must diminish as e increases, and 
 must be indefinitely small when e is indefinitely large. 
 
 (C) The term <£ (e) must contain a constant symbol 
 or parameter c, which is constant in the expression of the 
 probabilities under the same system of estimation or mea- 
 sure, and is different for different systems of estimation or 
 measure. If (as seems likely), upon taking a proper pro- 
 portion of magnitudes of error, the law of declension of 
 the probability of errors is the same for delicate measures 
 
laplace's investigation of their law. 7 
 
 and for coarse measures, then the formula will be of the 
 
 form yjr (-) x 8 (-) , or i//- (-J x — ; where c is small for 
 
 a delicate system of measures, and large for a coarse 
 system of measures. 
 
 [The reader is recommended, in the first instance, to 
 pass over the articles 12 to 21.] 
 
 12. Laplace has investigated, by an a priori process, 
 well worthy of that great mathematician, the form of the 
 function expressing the law of probability. Without enter- 
 ing into all details, for which we must refer to the Theorie 
 Analytique des Probabilites, we may give an idea here of 
 the principal steps of the process. 
 
 13. The fundamental principle in this investigation is, 
 that an error, as actually occurring in observation, is not of 
 simple origin, but is produced by the algebraical combina- 
 tion of a great many independent causes of error 1 , each of 
 which, according to the chance which affects it inde- 
 pendently, may produce an error, of either sign and of 
 different magnitude. These errors are supposed to be of 
 the class of Errors of Integers, which admit of being 
 treated by the usual Theory of Chances ; then, supposing 
 the integers to be indefinitely small, and the range of their 
 number to be indefinitely great, the conditions ultimately 
 approach to the state of Graduated Errors. 
 
 1 This is not the language of Laplace, but it appears to bo the iinder- 
 standing on which his investigation is most distinctly applicable to single 
 t-rrors of observation. 
 
8 SIMPLE ERRORS OF OBSERVATION. 
 
 14. Suppose then that, for one source of error, the 
 errors may be, with equal probability, 
 
 -«, -7i + l, -?i + 2,...-l, 0, +1, 2, ... Ti-2, n-1, », 
 
 the probability of each will be ^ — : — - . 
 
 Suppose that, for another source of error, the errors 
 may also be, with equal probability, 
 
 -n, -n + 1, -ra+2, ...-1, 0, + 1, 2, ...n-2, n — 1, n, 
 
 and so on for s sources of error. And suppose that we 
 wish to ascertain what is the probability that, upon com- 
 bining algebraically one error taken from the first series, 
 with one error taken from the second series, and with one 
 error taken from the third series, and so on, we can pro- 
 duce an error I. The first step is, to ascertain how many 
 are the different combinations which will each produce I. 
 
 15. Now, if we watch the process of combination, we 
 shall see that the numbers are added by exactly the same 
 law as the addition of indices in the successive multipli- 
 cations of the polynomial 
 
 -n0 v'-l i g-(n-l) e V-l i g-(n-2) fl V"l i £ ('i-2) V~l i £ (n-l) \M . e »0 >H 
 
 by itself, supposing the operation repeated s — 1 times. 
 And therefore the number of combinations required will 
 be, the coefficient of e wvM (which is also the same as the 
 coefficient of e~ WvM ), in the expansion of 
 
 f € ~n0 V-l i € -[n-l)0 V-l _i_ g-1*-8)0\'-l i e (»-2!0 V"l i ^"-1)0^-1 , ^0 v'-l U 
 
laplace's investigation of their law. 
 
 This coefficient will be exhibited as a number uncom- 
 bined with any power of e 9 ^" 1 , if we multiply the expansion 
 
 either by 6 mV_1 , or by e~ Wyht , or by \ (e 16 ^ 1 + e' 16 ^ 1 ). 
 
 The number of combinations required is therefore the same 
 as the term independent of in the expansion of 
 
 - ( 6 wV-i_|_ e -z0V-i\ r e -»£» V-i _|_ e -(«-D e V-i _[_ & c< + e !"- 1 »9v / -i^. e «0V-ii.s 
 
 or the same as the term independent of in the ex- 
 pansion of 
 
 cos 10 x {1 + 2 cos + 2 cos 20 + + 2 cos n0} s . 
 
 And, remarking that if we integrate this quantity with 
 respect to 0, from = to = ir, the terms depending on 
 will entirely disappear, and the term independent of 
 will be multiplied by ir, it follows that the number of 
 combinations required is the definite integral 
 
 - . \"d6 . cos 10 x (1 + 2 cos 6 + 2 cos 20 ... + 2 cos n0} s , 
 
 7T Jo 
 1 [* 
 
 or — . | c70 . cos 10 x 
 
 7T jo 
 
 And the total number of possible combinations which 
 are, a priori, equally probable, is (2n + l) s . 
 
10 SIMPLE ERRORS OF OBSERVATION. 
 
 Consequently, the probability that the algebraical com- 
 bination of errors, one taken from each series, will produce 
 the error 1, is 
 
 z . 2n+l 
 f » /sin — s — 
 
 1 1 f* / 2 
 
 dd . cos 19 x 
 
 [2n + l)*'7r J 
 
 In subsequent steps, n and s are supposed to be very 
 large. 
 
 16. To integrate this, with the kind of approximation 
 which is proper for the circumstances of the case, Laplace 
 assumes 
 
 2>i+l n 
 
 sin 
 
 1 
 
 (2?i + 1) . sin - 
 
 — e s 
 
 (as the exponential is essentially positive, this does not in 
 
 2n + 1 
 strictness apply further than — - — 8 = it; but as succeed- 
 ing values of the fraction are small, and are raised to the 
 high power s, they may be safely neglected in comparison 
 with the first part of the integral) ; expanding the sines in 
 
 powers of 6, and the exponential in powers of , it will 
 be found that 
 
 AJ{n{n+ l)s}\ s J 
 
 where B is a function of n which approaches, as n becomes 
 
laplace's investigation of their law. 11 
 
 very large, to the definite numerical value T ^. The expres- 
 sion to be integrated then becomes, 
 
 1 VG 
 
 7T V{H-('tt + 1) S\ 
 
 ' m i, r ^v<3 f, b„ „ 
 
 a£.cos -77-, _,. -, H — fr+<xc. 
 l_Vi«0H-l)sj V « 
 
 .€ ' 2 .(l+— £ 2 +&C 
 
 s 
 
 To simplify this integral, it is to be remarked that e~ c ~ 
 multiplies the whole, and that this factor decreases with 
 extreme rapidity as t increases. While t is small, the 
 
 terms — f in the argument of the cosine are unimportant; 
 
 and when t is large, it matters not whether they are retained 
 or not, because their rejection merely produces a different 
 length of period for the periodical term which is multi- 
 plied by an excessively small coefficient. Also it appears 
 (as will be shewn in Article 19) that the integration of 
 such a term as cos mt .e f \ oBt" introduces no infinite term, 
 and therefore when it is divided by the very large number 
 s, this may be rejected. The integral is therefore reduced 
 to this, 
 
 \/G f a 1± It \/G 
 
 at . cos 
 
 7r ' \j\n (n + 1) s] J ' \/{n (n + 1) s} ' 
 
 17. As the first step to this, let us find the value "of 
 
 dt . e~ l \ There is no process for this purpose so con- 
 
 . 
 
 venient as the indirect one of ascertaining the solid content 
 of the solid of revolution in which t is the radius of any 
 
12 SIMPLE ERRORS OF OBSERVATION. 
 
 section, and z the corresponding ordinate = e" f \ Let # and 
 y be the other rectangular co-ordinates, so that f = x* + y~. 
 Then the solid content may be expressed in either of the 
 following Yfays: 
 
 By polar co-ordinates, solid content 
 
 = 2tT. dt.t.€~ P =7T. 
 
 Jo 
 By rectangular co-ordinates, solid content 
 
 = ( dx . J dy . e~^ + ^ = ( ^ . e- l \ | cfy . r^ 
 
 - — oo - — 00 * — =o " — ac 
 
 =( 4 //- e i x (/."'^- e i =i CC' ? '' £ i' 
 
 since, for a definite integral, it is indifferent what symbol 
 be used for the independent variable. 
 
 Hence, 4 ( dt . e~^ ) = ir, 
 
 and I dt . e ' 2 = -— . 
 
 18. Next, to find the value of \ dt . cos ri . e~ l \ Call 
 
 Jo 
 
 this definite integral y. As this is a function of r, it can bo 
 differentiated with respect to r; and as the process of inte- 
 gration expressed in the symbol does not apply to r, y 
 can be differentiated by differentiating under the integral 
 sign. Thus 
 
 ^=-f° dt.t :sin rt.€~-\ 
 dr J o 
 
laplace's investigation of their law. 13 
 
 Integrating by parts, the general integral for ~ 
 
 = - sin rt . e~ r ' — - jdt . cos rt . e~ fi , 
 
 in which, taking the integral from t = to t = oo , the first 
 
 term vanishes, and the second becomes — ^ y. Thus we have 
 
 dy _ r 
 di- = ~ v l y ' 
 
 Integrating this differential equation in the ordinary 
 way, 
 
 y= C.e'i. 
 
 Now when r = 0, we have found by the last article that 
 
 val 
 tin ally 
 
 kiTT 
 
 the value of y for that case is --- . Hence we obtain 
 
 at . cos rt .e l = ~ . e i . 
 
 19. If we differentiate this expression twice with 
 respect to r, we find, 
 
 I dt.f. cos rt . e~ t2 = ^1 - 1 e - 'I ; 
 
 and expressions of similar character if we differentiate four 
 times, six times, &c. The right-hand expressions are 
 never infinite. This is the theorem to which we referred 
 in Article 16, as justifying the rejection of certain terms in 
 the integral. 
 
14 SIMPLE ERRORS OF OBSERVATION. 
 
 20. Reverting now to the expression at the end of 
 Article 18, and making the proper changes of notation, we 
 find for the value of the integral at the end of Article 1G, 
 
 e 4»(n+l)« t 
 
 2yV' ij[n(n+ l)s\ 
 
 This expression for the probability that the error, pro- 
 duced by the combination of numerous errors (see Article 
 14), will be I, is based on the supposition that the changes 
 of magnitude of I proceed by a unit at a time. If now we 
 pass from Errors of Integers to Graduated Errors, we may 
 consider that we have thus obtained all the probabilities 
 that the error will lie between I and ? + l. In order to 
 obtain all the probabilities that the error will lie between 
 I and I + 81, Ave derive the following expression from that 
 above, 
 
 e in(n+lj7a , g£ 
 
 fjir' \/{-in (n + 1) . s] 
 
 Here Hsa very large number, expressing the magni- 
 tude x of an error which is not strikingly large, by a large 
 multiple of small units. 
 
 Let I = mx, where m is large ; 81= m8x ; and the pro- 
 bability that the error falls between x and x + 8x is 
 
 1 \/G . m , , "' *" » 
 
 . jl . . e 4n.(»+l).s $ x 
 
 v%' »/{4m . (n + l).s}' 
 
 T 4n(n + l) .8 - , . . 
 
 .Let — ^ — ,/ — = c, wliere c may be a quantity of 
 
laplace's investigation of their law. 15 
 
 magnitude comparable to the magnitudes which we shall 
 use in applications of the symbol x; then we have finally 
 for the probability that the error will fall between x and 
 x + Bx, 
 
 1 -t 
 
 j- . 6 c- . Bx. 
 
 C\fir 
 
 This function, it will be remarked, possesses the cha- 
 racters which in Article 11 we have indicated as necessary. 
 We shall hereafter call c the modulus. 
 
 21. Laplace afterwards proceeds to consider the effect 
 of supposing- that the probabilities of individual errors, in 
 the different series mentioned in Article 14, are not uniform 
 through each series, as is supposed in Article 14, but vary 
 according to an algebraical law, giving equal probabilities 
 for + or — errors of the same magnitude. And in this case 
 also he finds a result of the same form. For this, however, 
 we refer to the Theorie Analytique des Probabilites. 
 
 22. Whatever may be thought of the process by 
 which this formula has been obtained, it will scarcely be 
 doubted by any one that the result is entirely in accord- 
 ance with our general ideas of the frequency of errors. In 
 order to exhibit the numerical law of frequency (that is, 
 
 the variable factor e c 2 , which, when multiplied by Bx, 
 gives a number proportioned to the probability of errors 
 falling between x and x + Bx), the following table is com- 
 puted ; 
 
16 
 
 SIMPLE ERRORS OF OBSERVATION. 
 
 Table of Values of e~c\ 
 
 X 
 
 X 2 
 
 X 
 
 .r 2 
 
 c 
 
 e c s 
 
 c 
 
 e~S 
 
 0-0 
 
 1-0000 
 
 2-6 
 
 0-001159 
 
 01 
 
 0-9901 
 
 2-7 
 
 0-0006823 
 
 0-2 
 
 0-9608 
 
 2-8 
 
 0-0003937 
 
 0-3 
 
 0-9139 
 
 2-9 
 
 0-0002226 
 
 0-4 
 
 0-8521 
 
 30 
 
 0-0001234 
 
 0-5 
 
 0-7788 
 
 3-1 
 
 0-00006706 
 
 0-6 
 
 0-6977 
 
 3-2 
 
 0-00003571 
 
 0-7 
 
 0-6126 
 
 6 6 
 
 0-00001864 
 
 0-8 
 
 0-5273 
 
 3-4 
 
 0-000009540 
 
 0-9 
 
 0-4449 
 
 3-5 
 
 0-000004785 
 
 1-0 
 
 0-3679 
 
 3-6 
 
 0-000002353 
 
 1-1 
 
 0-2982 
 
 3-7 
 
 0-000001134 
 
 1-2 
 
 0-2369 
 
 3-8 
 
 0-0000005355 
 
 1-3- 
 
 0-1845 
 
 3-9 
 
 0-0000002480 
 
 1-4 
 
 0-1409 
 
 4-0 
 
 0-0000001125 
 
 1-5 
 
 0-1054 
 
 4-1 
 
 0-00000005006 
 
 1-6 
 
 0-07731 
 
 4-2 
 
 0-00000002183 
 
 1-7 
 
 0-05558 
 
 4-3 
 
 0-000000009330 
 
 1-8 
 
 0-03916 
 
 4-4 
 
 0-000000003909 
 
 1-9 
 
 0-02705 
 
 4-5 
 
 0-00000000-1605 
 
 2-0 
 
 0-01832 
 
 4-6 
 
 0-0000000006461 
 
 2-1 
 
 0-01216 
 
 4-7 
 
 0-0000000002549 
 
 2-2 
 
 0-007907 
 
 4-8 
 
 0-00000000009860 
 
 2-3 
 
 0-005042 
 
 4-9 
 
 0-00000000003738 
 
 2-4 
 
 0-003151 
 
 5-0 
 
 0-00000000001389 
 
 2-5 
 
 0-001930 
 
 
 
 23. And to present more clearly to the eye the import 
 of these numbers, the following curve is constructed, in 
 
LAW OF PROBABILITY OF ERRORS. 
 
 17 
 
 which the abscissa represents - , or the proportion of the 
 
 c 
 
 magnitude of an error to the modulus, and the ordinate 
 
 represents the corresponding frequency of errors of that 
 
 magnitude. 
 
 H 
 
 Here it will be remarked that the curve approaches 
 the abscissa by an almost uniform descent from Magnitude 
 of Error = to Magnitude of Error = 1*7 x Modulus ; and 
 that after the Magnitude of Error amounts to 20 x Modulus, 
 the Frequency of Error becomes practically insensible. This 
 
 A. B 
 
18 SIMPLE ERRORS OF OBSERVATION. 
 
 is precisely the kind of law which we should a priori have 
 expected the Frequency of Error to follow; and which, with- 
 out such an investigation as Laplace's, we might have 
 assumed generally ; and for which, having assumed a 
 general form, we might have searched an algebraical law. 
 For these reasons, we shall, through the rest of this treatise, 
 assume the law of frequency 
 
 1 _ J ' 2 
 ■ — 7— e c- . Bx, 
 c y7r 
 
 as expressing the probability of errors occurring with mag- 
 nitude included between x and x + Bx. 
 
 § 3. Consequences of the Law of Probability or Frequency 
 of Errors, as applied to One System of Measures of 
 One Element. 
 
 24. The Law of Probability of Errors or Frequency of 
 Errors, which we have found, amounts practically to this. 
 Suppose the total number of Measures to be A, A being a 
 very large number ; then we may expect the number of 
 errors, whose magnitudes fall between x and x + Bx, to be 
 
 A -~ 
 
 ■ — i— e & . Bx, 
 
 C V7T 
 
 where c is a modulus, constant for One System of Measures, 
 but different for Different Systems of Measures. It is partly 
 the object of the following investigations to give the means 
 of determining either the modulus c, or other constants 
 related to it, in any given system of practical errors. ' 
 
 25. This may be a convenient opportunity for remark- 
 ing expressly that the fundamental suppositions of La- 
 
MEAN ERROR. 19 
 
 place's investigation, Article 14, assume that the law of 
 Probability of Errors applies equally to positive and to 
 negative errors. It follows therefore that the formula in 
 Article 24 must be received as applying equally to positive 
 and to negative errors. The number A includes the whole 
 of the measures, whether their errors may happen to be 
 positive or negative. 
 
 26. Conceive now that the true value of the Element 
 which is to be measured is known (we shall hereafter con- 
 sider the more usual case when it is not known), and 
 that the error of every individual measm^e can therefore 
 be found. The readiest method of inferring from these a 
 number which is closely related to the Modulus is, to take 
 the mean of all the positive errors without sign, and to 
 take the mean of all the negative errors without sign 
 (which two means, when the number of observations is 
 very great, ought not to differ sensibly), and to take the 
 numerical mean of the two. This may be called the 
 JVTean Error. It is to be regarded as a mere numerical 
 quantity, without sign. Its relation to the Modulus is 
 thus found. Since the number of errors whose magnitude 
 
 A - x ~ 
 is included between x and x + Sx is — — e c 2 . Bx, and the 
 
 c sjTT 
 
 magnitude of each error does not differ sensibly from x, 
 
 A _*! 
 the sum of these errors will be sensibly — r— e c ' 2 . xSx ; and 
 
 J c sjir 
 
 the sum of all the errors of positive sign will be 
 
 A r, * cA 
 
 a x . e fi2 . x 
 
 B 2 
 
20 SIMPLE ERRORS OF OBSERVATION. 
 
 The number of errors of positive sign is 
 
 CtJlTjQ 2 
 
 Dividing the preceding expression by this, 
 
 Mean positive error = —— . 
 
 1 V 73 " 
 
 Similarly, 
 
 Mean negative error = —.— . 
 
 And therefore, 
 
 Mean Error = -=cx 0o64189. 
 
 V7T 
 
 And conversely, 
 
 c = Mean Error x 1 772454. 
 By this formula, c can be found with ease when the series 
 of errors is exhibited. 
 
 27. It is however sometimes convenient (as will ap- 
 pear hereafter, Article 61) to use a method of deduction 
 derived from the Squares of Errors. The positive and 
 negative errors are then included under the same formula. 
 If we form the mean of the squares, and extract the square 
 root of that mean, we may appropriately call it the Error 
 of Mean Square. This, like the Mean Error, is a numerical 
 quantity, without sign. To investigate it in terms of c, 
 we remark that the sum of the squares of errors between 
 x and x + Bx (formed as in the last Article) w r ill be 
 
 A -** 
 
 —— € «* . Bx X X 2 
 
ERROR OF MEAN SQUARE. 21 
 
 and the sum of all the squares of errors will be 
 
 A f +0 ° T -- 9 + co (-Ac _*! 
 dx.e <? .x — i~ — i- x.e c2 
 
 The first term vanishes between the limits — oo and + x , 
 
 Ac 2 
 and the second term = + -~- . The whole number of errors 
 
 c 2 
 is .4. Hence the Mean Square is — , and the Error of Mean 
 
 Square is 
 
 '1 
 
 cj^ =cxO-707107; 
 
 or c = Error of Mean Square x T414214. 
 
 28. It has however been customary to make use of a 
 different number, called the Probable Error. It is not 
 meant by this term that the number used is a more pro- 
 bable value of error than any other value, but that, when 
 the positive sign is attached to it, the number of positive 
 errors larger than that value is about as great as the num- 
 ber of positive errors smaller than that value : and that, 
 when the negative sign is attached to it, the same remark 
 applies to the negative errors. The Probable Error itself 
 is a numerical quantity, without sign. To ascertain the 
 algebraical condition which this requires, we have only to 
 remark that, as the number of positive errors up to the 
 
 value x is- — —I dx.e ° 2 , and as the whole number of 
 
oo 
 
 SIMPLE ERRORS OF OBSERVATION. 
 
 positive errors is -_- , and half the whole number of positive 
 
 errors is — , we must find the value of x which makes 
 4 
 
 1 p -f' 1 [ w 1 
 
 — — dx.e.c 2 , or—— dw.e w , equal to j. 
 
 C \/7T J o V 7T Jo ■* 
 
 2!). For this purpose, we must be prepared with a 
 
 table of the numerical values of 
 
 dw . e u ' 2 . It is not 
 
 V 7r J o 
 
 our business to describe here the process by which the 
 numerical values are obtained (and which is common to 
 the integrals of all expressible functions) ; we shall merely 
 give the following table, which is abstracted from tables 
 in Kramp's Refractions and in the Encyclopaedia Metro- 
 politan, Article Theory of Probabilities. 
 
 Table of the Values of -— dw. e" 
 
 VtJo 
 
 w 
 
 o-o 
 
 Integral. 
 
 w 
 
 Integral. 
 
 w 
 2-2 
 
 Integral. 
 
 0-000000 
 
 11 
 
 0-440103 
 
 0-499068 
 
 o-i 
 
 0-036232 
 
 1-2 
 
 0-455157 
 
 2-3 
 
 0-499428 
 
 0-2 
 
 0-111351 
 
 1-3 
 
 0-467004 
 
 2-4 
 
 0-499655 
 
 0-3 
 
 0-164313 
 
 1-4 
 
 0-476143 
 
 2-5 
 
 0-499796 
 
 0-4 
 
 0-214196 
 
 1-5 
 
 0-483053 
 
 2-6 
 
 0-499881 
 
 0-5 
 
 0-260250 
 
 1-6 
 
 0-488174 
 
 2-7 
 
 0-499932 
 
 0-6 
 
 0-301928 
 
 1-7 
 
 0-491895 
 
 2-8 
 
 0-499!" 12 
 
 0-7 
 
 0-338901 
 
 1-8 
 
 0-494545 
 
 2-9 
 
 0-499979 
 
 0-8 
 
 0-371051 
 
 1-9 
 
 0-496395 
 
 30 
 
 0-499988 
 
 0-9 
 
 0-398454 
 
 2-0 
 
 0-497661 
 
 
 
 10 
 
 0-421350 
 
 2-1 
 
 0-498510 
 
 00 
 
 0-500000 
 
PROBABLE ERROR. 23 
 
 By interpolation among these, we find that the value 
 of w which gives for the value of the integral 025, is 
 0*476948 ; or the Probable Error, which is the corresponding 
 value of x, is c x 0476948. And, conversely, c = Probable 
 Error x 2096665. 
 
 30. The reader will advantageously remark in this 
 table how nearly all the errors are included within a small 
 
 value of w or - . For it will be remembered that the Inte- 
 c 
 
 gral when multiplied by A (the entire number of positive 
 
 and negative errors) expresses the number of errors up to 
 
 error 
 that value of w or . Thus it appears that from w = 
 
 error 
 up to w = 1*65 or = 1*65, we have already obtained 
 
 49 
 
 — - of the whole number of errors of the same sign ; and 
 
 49999 
 from w = up to w = 30, we have obtained K 0nno °f the 
 
 whole number of errors of the same sign. 
 
 31. Returning now to the results of the investigations 
 in Articles 26, 27, 28, 29 ; we may conveniently exhibit 
 the relations between the values of the different constants 
 therein found, by the following table : — 
 
24 
 
 SIMPLE ERRORS OF OBSERVATION. 
 
 PROPORTIONS OF THE DIFFERENT CONSTANTS. 
 
 
 Modulus. 
 
 Mean 
 Error. 
 
 Error of 
 
 Mean 
 Square. 
 
 Probable 
 Error. 
 
 In terms of Modulus . . . 
 
 1-000000 [ 0-564189 
 
 0-707107 
 
 0-476948 
 
 In terms of Mean Error 
 
 1-772454 
 
 1-000000 
 
 1-253314 
 
 0-845369 
 
 In terms of Error of") 
 Mean Square J 
 
 1-414214 
 
 0-797885 
 
 1-000000 0-674506 
 
 In terms of Probable") 
 Error J 
 
 2-096665 
 
 1-182916 
 
 1-482567 
 
 1-000000 
 
 32. To distinguish each of the errors, really occurring 
 in observations, from the " Mean Error," " Error of Mean 
 Square," "Probable Error," which are mere numerical 
 deductions made according to laws framed for convenience 
 only, we shall usually designate an error really occurring 
 (whether its magnitude be known or not) by the term 
 " Actual Error." 
 
 § 4. Remarks on the application of these jwocesses in 
 particular cases. 
 
 33. It must always be borne in mind that the law of 
 frequency of errors does not exactly hold except the num- 
 ber of errors is indefinitely great. With a limited number 
 of errors, the law will be imperfectly followed ; and the 
 deductions, made on the supposition that the law is strictly 
 followed, will be or may be inaccurate or inconsistent. 
 
UNUSUALLY LARGE ERRORS. 25 
 
 Thus, if we investigate the value of the modulus, first 
 by means of the Mean Error, secondly by the Error of 
 Mean Square, we shall probably obtain discordant results. 
 We cannot assert a priori which of these is the better. 
 
 34. There is one case which occurs in practice so fre- 
 quently that it deserves especial notice. In collecting 
 the results of a number of observations, it will frequently 
 be found that, while the results of the greater number 
 of observations are very accordant, the result of some 
 one single observation gives a discordance of large mag- 
 nitude. There is, under these circumstances, a strong 
 temptation to erase the discordant observation, as having 
 been manifestly affected by some extraordinary cause of 
 error. Yet a consideration of the law of Frequency of Error, 
 as exhibited in the last Section (which recognizes the pos- 
 sible existence of large errors), or a consideration of the 
 formation of a complex error by the addition of numerous 
 simple errors, as in Article 14 (which permits a great num- 
 ber of simple errors bearing the same sign to be aggregated 
 by addition of magnitude, and thereby to produce a large 
 complex error), will shew that such large errors may fairly 
 occur ; and if so, they must be retained. We may perhaps 
 think that where a cause of unfair error may exist (as in 
 omission of clamping a zenith-distance-circle), and where 
 we know by certain evidence that in some instances that 
 unfair cause has actually come into play, there is sufficient 
 reason to presume that it has come into play in an in- 
 stance before us. Such an explanation, however, can only 
 be admitted with the utmost caution. 
 
2G COMBINATION OF ERRORS. 
 
 PART II. 
 
 ERRORS IN THE COMBINATION OF FALLIBLE 
 MEASURES. 
 
 § 5. Law of Frequency of Error, and values of Mean 
 Error and Probable Error, of a symbolical or numeri- 
 cal Multiple of one Fallible Measure. 
 
 35. This case is exceedingly simple ; but it is so im- 
 portant that we shall make it the object of distinct treat- 
 ment. Suppose that, in different measures of a quantity X, 
 the actual errors x v x 2 , x 3 , &c. have been committed. 
 Then it is evident that our acceptations of the value of the 
 quantity Y=nX (an algebraical or numerical multiple of 
 X), derived from these different measures, are affected by 
 the Actual Errors y 1 =nx 1 , y 2 =nx 2 , y 3 =nx s , &c; and that, 
 generally speaking, where X is liable to any number of 
 errors of the magnitudes x, x 4- Bx, or any thing between 
 them, Y is liable to exactly the same number of errors of 
 the magnitudes nx = y, nx + nSx =y + By, or of magnitudes 
 between them. Therefore the expression for the Frequenc}^ 
 of Errors in Article 24 becomes this : 
 
 The number of errors of Y or nX, whose magnitudes 
 fall between y and y + 8y, may be expected to be 
 
 A _** 
 
 . e c% . Sx, 
 
 C *JlT 
 
ERROR OF MULTIPLE OF A MEASURE. 27 
 
 which is the same as 
 
 — . e »v- . 071. 
 
 nc V7r 
 
 From this we at once derive these conclusions: (1) The 
 
 law of Frequency of Errors for nX is exactly similar to that 
 
 for the errors of an original measure X; and therefore, in 
 
 all future combinations, nX may be used as if it had been 
 
 an original measure. (2) The modulus for the errors of 
 
 nX is, in the formula, nc. (3) Referring to the constant 
 
 proportions in Articles 26, 27, 29, 31; the Mean Error 
 
 of nX will therefore = n x mean error of X; the Error of 
 
 Mean Square of nX= n x error of mean square of X ; the 
 
 Probable Error of nX= n x probable error of X. 
 
 36. It may be useful to guard the reader against one 
 misinterpretation of the meaning of nX. We do not mean 
 the measure of a simple quantity Y which is equal to n X. 
 The errors (whether actual, mean, or probable) of the 
 quantity X cannot in any way be made subservient to the 
 determination of the error of another simple quantity Y. 
 Thus, reverting to our instances in Article 5, &c, a judg- 
 ment of the possible error in estimating the length of a 
 road about 100 yards long will in no degree aid the judg- 
 ment of the possible error in estimating the length of a 
 road about 10000 yards long. The quantity nX is in fact 
 merely an algebraical multiple or a numerical multiple of 
 X, introduced into some algebraical formula, and is not 
 exhibited as a material quantity. 
 
 37. Another caution to be observed is this ; that we 
 must most carefully distinguish between nX the multiple 
 
28 COMBINATION OF ERRORS. 
 
 of X (on the one hand), and the sum of a series of n inde- 
 pendent quantities X, + X 2 + &c. ... + X n (on the other 
 hand) ; even though the mean error or probable error of 
 each of the quantities X v X 2 , &c. is equal to the mean 
 error or probable error of X. The value of mean error or 
 probable error of such a sum will be found hereafter 
 (Article 53). 
 
 § 6. Law of Frequency of Error, and values of Mean 
 Error and Probable Error, of a quantity formed by 
 the algebraical sum or difference of two independent 
 Fallible Measures. 
 
 38. Suppose that we have the number C of measures 
 of a quantity X, in which the law of frequency of errors 
 is this (see Article 24), that we may expect the number 
 of errors whose magnitudes fall between x and x + h, to be 
 
 —r- . e c~.li- 
 
 C\/7T 
 
 c being the modulus of these errors, and the number C 
 being very large. 
 
 And suppose that we have the number F of measures 
 of a quantity Y, absolutely independent of the measures 
 of the quantity X, in which we may expect the number 
 of errors whose magnitudes fall between y and y + h, to be 
 
 F -t . 
 e r-.h, 
 
 / being the modulus of these errors, and the number F 
 being very large. 
 
ERROR OF SUM OF TWO MEASURES. 29 
 
 And suppose that a new quantity Z is formed by adding 
 X and Y together. 
 
 It is required to ascertain the law of frequency of errors 
 of ^or X+ Y. 
 
 39. Instead of supposing the errors to be graduated, 
 
 we will suppose that all the errors whose number is 
 
 C — 
 — — . e & . h have the uniform magnitude x : and so for 
 
 other equal intervals h in the value of x. Thus, putting 
 
 x n , x t , x, x", for x — 2h, x — h, x + h, x+ 2h, and putting 
 
 C 
 C for — 7- , the numbers of errors of X will be 
 
 C\Jtt 
 
 of magnitude x — 2h, C . e <? . h; 
 
 of magnitude x — h, C'.e a . /i ; 
 
 a" 2 
 
 of magnitude x, C'.e~c 3 .h; 
 
 of magnitude x + h, C'.e e-.h; 
 
 x"- 
 
 of magnitude x + 2h, Ce'^.h; 
 
 and it is plain that, by making h small enough, this state 
 of things will approach as near as we please to that of 
 graduated errors. 
 
 In like manner, the numbers of errors of Y will be 
 
 of magnitude y+2h, F'.e f .h; 
 
 - v - 
 of magnitude y + h, F'.e f 2 .h; 
 
30 COMBINATION OF ERRORS. 
 
 of magnitude y, F'.e f-.h; 
 
 of magnitude y — h, F'.e f-.h; 
 
 v" 
 of magnitude y — 2h, F' . e~ 7- . h. 
 
 40. Let x + y = z. Now in order to form all the 
 possible values of Z, we must combine every possible 
 value of X (0 in number) with every possible value of Y 
 (F in number), forming a total number CF of combina- 
 tions. (This process implies that the errors of A' are ab- 
 solutely independent of those of Y.) The number of com- 
 binations of errors will be the same. If we examine the 
 result of combining the two series of errors in the last 
 Article, we find that the magnitudes of the errors formed 
 will be 
 
 z - 4/<, z-Sh, z- 2h, z- h, z,z + h,z + 2h, z + 3h, z + -ih. 
 
 We shall therefore have a series of magnitudes of error 
 of Z in our result, varying by a step of magnitude h every 
 time, and therefore similar to those which we have adopted 
 for the errors of X and Y. 
 
 41. Now if we examine the combinations of errors 
 that will produce z, and the numbers of these combina- 
 tions (which apply to a step of magnitude h), we find the 
 following: 
 
 combining x — 2h with y + 2h, the number is 
 
 C'.e~*.hxF'.e S'.h or C'F'.hxe'* P.h; 
 
ERROR OF SUM OF TWO MEASURES. 31 
 
 combining x — li with y 4- h, the number is 
 
 G'F'.hxtT + 'P.h; 
 
 combining x with y, the number is 
 
 C'F' . h x e~*~f- .h; 
 combining x + h with y — h, the number is 
 
 C'F'.hx e~^~%.h; 
 combining x + 2h with y — 2h, the number is 
 
 C'F' .h x e~~^~f r -.h; 
 and so on, continued indefinitely both ways. If we put 
 z — x for y, and remark that 
 
 y" = y + %h = z — x + 2h — z— x u , 
 and so for the others, we see that all the last factors in 
 the series just exhibited are the values of 
 
 XV ?/2 ,3-2 (Z-X)- 
 
 e~ c 2 ~f- . h, or e~&~ "T* - . h, 
 when for x we put successively the values 
 
 x — 2h, x — h, x, x + h, x + 2h, 
 continued indefinitely both ways, without altering the 
 value of z. The sum of these, supposing h made indefi- 
 nitely small, is the same as 
 
 + oo X? (Z-X)% 
 
 - 00 
 
 where z is considered constant. Introducing the factors, 
 and remarking that 
 
 C F OF 
 
 C'F' = ~-x/ T = ^, 
 c\>7r yv 77 " c 7 7r 
 
32 COMBINATION OF ERRORS. 
 
 the whole number of errors of magnitude z when a step of 
 magnitude h is made each time, or, as in Article 40, the 
 whole number of errors of Z whose magnitudes are included 
 between z and z + h, will be 
 
 CFh r+°° 7 _* 2 _^r*l 2 
 dx . e & / a : 
 
 cfrr 'J- 
 where z is to be regarded as constant. 
 
 42. The index of the exponential is easily changed 
 into this form; 
 
 (- 
 
 c z +f ef V c 2 +/V 
 
 Let c 2 +/ 2 = /, C c ~^=- 2 , a--q~^=£ 
 
 z* £ 2 
 
 Then the index is :, — K . 
 
 9' 7 
 
 And, (as dx = d%, and z is constant for this investiga- 
 tion), the whole number of errors of Z, whose magnitudes 
 are included between z and z + h, will be 
 
 CFh * [ +c ° Jy -? 
 cpr J _oc 
 
 But (see Article 17, and remark that in this case 
 
 - - X 
 
 + 00 _|5 c f qf 
 
ERROR OF SUM OF TWO MEASURES. 33 
 
 Therefore, finally, the whole number of errors of Z whose 
 magnitudes are included between z and z + h, will be 
 
 CF _*? , 
 
 where the whole number of combinations which can form 
 errors is CF. 
 
 43. Comparing this expression with that in Article 
 24, it appears that the' law of frequency of error for Z is 
 precisely the same as that for X or for Y; the modulus 
 being g or 
 
 V(c 2 +/ 2 ). 
 
 Hence we have this very remarkable result. When 
 two fallible determinations X and Y are added algebrai- 
 cally to form a result Z, the law of frequency of error 
 for Z will be the same as for X or Y, but the modulus 
 will be formed by the theorem, 
 
 square of modulus for Z= square of modulus for X+ square 
 of modulus for Y. 
 
 44. And as (see Articles 26, 27, 28, 29, 31) the Mean 
 Error, the Error of Mean Square, and the Probable Error, 
 are in all cases expressed by constant numerical multiples 
 of the Modulus, we have 
 
 (m.e. ofZ) 2 = (m.e. ofX) 2 + (m. e. of Y) 2 . 
 (e. m. s. of Zf = (e. m. s. of Xf + (e. m. s. of Y)\ 
 (p. e. of Zf = (p. e. of X) 2 + (p. e. of Y)\ 
 
 A. C 
 
34 COMBINATION OF ERRORS. 
 
 These are the fundamental theorems for the Error of 
 the Result of the Addition of Fallible Measures. They 
 constitute, in fact, but one theorem ; inasmuch as, using 
 one, the others follow as matter of course. "We shall 
 commonly make use of Probable Errors (as most exten- 
 sively adopted), unless any difference is expressly noted ; 
 "but the reader, who prefers Mean Errors, may form the 
 theorems in the corresponding shape, by merely substi- 
 tuting "m. e." for "p.e." throughout. 
 
 45. It cannot be too strongly enforced on the student 
 that the measures which determine X must be absolutely 
 and entirely independent of those which determine Y. If 
 any one of the observations, which contributes to give a 
 measure of X, does also contribute to give a measure of Y; 
 then the single measure of X founded on that observation 
 must be combined with the corresponding single measure 
 of Y to form its value of Z, and with no other ; and the 
 freedom to combine any possible error of X with any possi- 
 ble error of Y, on which the whole investigation in Articles 
 40 and 41 depends, is to that extent lost. As an illustra- 
 tion : suppose that differences of astronomical latitude upon 
 the earth, or ' amplitudes,' are determined by observations 
 of the same stars at the two extremities of a meridian arc : 
 and suppose that X, the amplitude from a station in the 
 Isle of Wight to a station in Yorkshire, is determined by 
 observing stars in the Isle of Wight and the same stars in 
 Yorkshire; and suppose that Y, the amplitude from the 
 Yorkshire station to a Shetland station, is determined by 
 
ERROR OF SUM OF TWO MEASURES. 35 
 
 observing stars in Yorkshire and the same stars in Shet- 
 land. First suppose that the observations of stars used 
 in the measure of X are not the same which are used in 
 the measure of Y. Then the errors in the determination 
 of X are totally independent of the errors in the deter- 
 mination of Y; any one determination of Xmay be com- 
 bined with any one determination of Y; and if Z= X 
 + Y= amplitude from Isle of Wight to Shetland, the 
 theorem 
 
 (p. e. of Z)' = (p. e. of Xf + (p. e. of Y)* 
 
 applies strictly. But suppose now that one and the same 
 set of star-observations made in Yorkshire are used to 
 determine X (by comparison with Isle of Wight observa- 
 tions) and Y (by comparison with Shetland observations). 
 Then the determination of X, based upon a star-observa- 
 tion in Yorkshire, will be combined only with a deter- 
 mination of Y, based upon the same star-observation in 
 Yorkshire (as will be seen on taking the means of zenith 
 distances at the stations, and forming the amplitudes). 
 The Yorkshire observations are of no use at all for deter- 
 mining Z, and may be completely omitted. Their errors 
 have no influence on the result ; for if the observations of 
 any star in Yorkshire make X too small, the same observa- 
 tions make Y equally too large, and in forming Z=X+ Y 
 these errors disappear. In fact, the determination of Z 
 here is totally independent of those of X and Y; and the 
 investigation of its mean error or probable error will not 
 depend on those of X and Y. It will depend on the ob- 
 servations at the Isle of Wight and Shetland only: whereas 
 
 C2 
 
3G COMBINATION OF ERRORS. 
 
 the probable error of X will depend on observations at the 
 Isle of Wight and Yorkshire only, and the probable error 
 of Y will depend on the observations at Yorkshire and 
 Shetland only. Thus it may happen that, although 
 Z = X + Y, the probable error of Z is less than either the 
 probable error of X or the probable error of Y. 
 
 The investigation of the probable error of Z, when a 
 portion of the stars observed are common to two or three 
 stations, will be explained hereafter (Article 80). 
 
 4G. Suppose that we have determinations of X and 
 Y, as in Article 38, and W=X— Y; it is required to 
 ascertain the law of frequency of errors and the mean error 
 or probable error of W. 
 
 The fundamental supposition, upon which we have 
 
 gone throughout the investigation, is, that the law of 
 
 frequency is the same for positive and for negative errors 
 
 of the same magnitude. And this is implied in our final 
 
 formula for the number of errors between x and x + Bx, 
 
 A jp 
 namely, — j— e c 2 . Bx, which gives equal values for x = + s 
 
 and for x = — s. Inasmuch therefore as Y is liable to 
 positive and negative errors of the same magnitude in 
 equal numbers, it follows that — l r is liable to the same 
 errors as + Y; and therefore the probable error of — I 7 " is 
 the same as the probable error of + Y. 
 
 47. Now W= X+ (— Y), and therefore 
 (p. e. of Wf = (p. e. of A') 2 + (p. e. of - Y)\ 
 
ERROR OF DIFFERENCE OF TWO MEASURES. 37 
 
 Substituting in the last term from Article 4G, 
 
 (p. e. of Wf = (p. e. of X) 2 + (p. e. of F)\ 
 
 48. The theorems of Article 44 may therefore be ex- 
 tended, in the following form ; 
 
 [m. e. of (X ± Y)Y = (m. e. of X) 2 + (m. e. of Y)\ 
 
 {e. m. s. of (X ± Y)}' = (e. m. s. of X) 2 + (e. m. s. of F) s , 
 
 {p. e. of (X ± Y)Y = (p. e. of X) 2 + (p. e. of Y)\ 
 
 and. the law of frequency of errors for X+ Y will be similar 
 to that for a simple fallible measure. 
 
 49. The reader's attention is particularly invited to 
 the following remark. We have found in Article 35 that 
 when the errors of a fallible measure are subject to our 
 general law of Frequency of Errors, the errors of any con- 
 stant multiple of that measure are subject to the same 
 laws ; and we have found in Articles 44, 47, 48, that when 
 the errors of each of two fallible measures are subject to 
 that law, the errors of their sums and differences are sub- 
 ject to the same law. Now all our subsequent combina- 
 tions of fallible quantities will consist of sums, differences, 
 and multiples. Consequently, for every fallible quantity 
 of which we shall treat hereafter, the General Law of 
 Frequency of Errors will apply. Regarding this as suf- 
 ficient notice, we shall not again allude to the Law of 
 Frequency of Errors. 
 
38 COMBINATION OF ERRORS. 
 
 § 7. Values of Mean Error and Probable Error, in 
 combinations which occur most frequently. 
 
 50. To find the Probable Error of kX+ IY, k and I 
 
 being constant multipliers. 
 
 By Article 35, the probable error of kX=k x probable 
 error of AT; and the probable error of IY=1 x probable 
 error of Y. Now, considering kX and I Y as two indepen- 
 dent fallible quantities, 
 
 {p. e. of (kX + 1 Y)}* = (p. e. of kX) 2 + (p. e. of I Y)\ 
 
 Substituting the values just found, 
 {p.e. of (7cX+lY)Y = k\(p.e. of X) 2 + 1 2 . (p. e. of Y)\ 
 
 In like manner, 
 {m. e. of (kX+ I Y)f = k 2 . (m. e. of X) 2 + I 2 . (m. e. of F) 2 . 
 
 51. To find the Probable Error of the sum of any 
 number of independent fallible results, 
 
 B + S+T+ U+&c. 
 
 This is easily obtained by repeated applications of the 
 theorem of Article 44, thus : 
 
 [p. e. of (R + S)} 2 = (p. e. of X) 2 + (p. e. of £) 2 ; 
 
 [p.-e.of{(i2 + ^) + r}] a 
 
 = {p. e. of (It + S)Y + (p. e. of T) % 
 
 = (p. e. of R) 2 + (p. e. of S) 2 + (p. e. of T) s ; 
 
ERROR OF AGGREGATE OF MEASURES. 3,9 
 
 [ip.e.o{{(R + S+T)+U}Y 
 
 = {p. e. of (R + S+ Tff + (p. e. of Uf 
 
 = (p. e. of Rf+ (p. e. of £) 2 + (p. e. of T) 2 +(p. e. of U) 2 ; 
 
 and so on to any number of terms. 
 
 A similar theorem applies to the Error of Mean Square, 
 and the Mean Error, substituting e. m. s. or m. e. for p. e. 
 throughout. 
 
 52. In like manner, using the theorem of Article 50, 
 the probable error of rR + s8+ tT + uU+&c, where r, s, 
 t, u, are constant multipliers, is given by the formula, 
 
 {p. e. of (rR + sS + tT + u U) } * 
 
 =r 2 .(p.e. of R)*+s\(-p.e. of S) z +t\ (p.e.of T) 2 +w 2 .(p.e. f U)\ 
 
 And a similar theorem for Error of Mean Square and 
 Mean Error, substituting e. m. s. and m. e. for p. e. 
 
 Measures thus combined may be called " Cumulative 
 Measures." 
 
 53. To find the Probable Error of X x + X 2 + . . . + X n , 
 where X lt X v X v ...X nf are n different and independent 
 measures of the same physical quantity, or of equal phy- 
 sical quantities, in every one of which the probable error 
 is the same, and equal to the probable error of X x . 
 
 By the theorem of Article 51, 
 
40 COMBINATION OF ERRORS. 
 
 {p.e.of (T 1 + X+... + Z„)} 2 =(p.e. of A^+fc.e. of A',) 2 ... 
 
 + (p.e.ofX„) 2 
 = (p. e. of A^) 2 + (p. e. of XJ 2 + . . . + (p. e. of A^) 2 to n terms 
 = n . (p. e. of X X Y; 
 and therefore, 
 
 p. e. of (X, + X 8 + ... + XJ = *Jn x p. e. of X v 
 
 54. In Article 35, we found that 
 
 p. e. of nX 1 = n x p. e. of X x ; 
 
 but here we find that 
 
 p. e. of (X, 4- X 2 ... + X n ) = 01 x p. e. of X v 
 
 although the probable error of each of the quantities X 2 , 
 X 3 , &c. is equal to that of X v A little consideration will 
 explain this apparent discordance. When we add together 
 the identical quantities X 1} X v X v &c. to n terms; if there 
 is a large actual error of the first X v there is, necessarily, 
 the same large actual error of each of the other X x , X l5 &c: 
 and the aggregate has the very large actual error n x large 
 error of X v But when we add together the independent 
 quantities X t , X 2 , &c, if the actual error of X x is large, it 
 is very improbable that the simultaneous actual error of 
 each of the others X 2 , A" 3 , &c, has a value equally large 
 and the same sign, and therefore it is very improbable 
 that the aggregate of all will produce an actual error equal 
 or approaching to n x large error of A^. The magnitude 
 of the probable error (which is proportional to the mean 
 error, see Article 31) depends on the probability or fre- 
 
ERROR OF AGGREGATE OF MEASURES. 41 
 
 quency of large actual errors, (for in Article 26, to make the 
 mean error large, we must have many large actual errors) ; 
 and therefore the probable error of X x + X 2 + . .. + X n will 
 be smaller than that of X t + X ± + . ... to n terms, although. 
 p. e. of Xj = p. e. of X 2 = . . . = p. e. of X n . 
 
 55. To find the probable error of the mean of X v 
 
 X 2 , X n , where X v X 2 , ... X n , are n different and 
 
 independent measures of the same physical quantity, in 
 every one of which the probable error = p. e. of X x . 
 
 The mean of X, I, ... X = X i + X 2+~- + X » 
 
 n 
 
 n ' n a n 
 
 and the square of its probable error, by Article 52, 
 
 = J (p. e. of XJ 2 + * (p. e. of X 2 Y + . . . + i (p. e. of X n )\ 
 — —-i (P- e - °f X iY + "2 (P- e - of X iY + ••• to w terms, 
 
 = J( P .e.ofX 1 ) 2 = ^(p.e.ofX i r; 
 
 and therefore, 
 
 p. e. of mean of X v X 2 , . . . X n = — x p. e. of X r 
 
 SRStTY 
 
42 COMBINATION OF ERRORS. 
 
 § 8. Instances of the Application of these Theorems. 
 
 56. Instance (1). The colatitude of a geographical 
 station is determined by observing, on times, the zenith- 
 distance of a star at its upper culmination ; and by 
 observing, n times, the zenith-distance of the same star at 
 its lower culmination ; all proper astronomical corrections 
 being applied. The probable error of each of the upper 
 observations is p. e. u. and that of each of the lower is 
 p. e. 1. To find the probable error of the determination 
 of colatitude. 
 
 The probable error of the upper zenith-distance, which 
 
 is derived from the mean of to observations, is , ' — ; 
 
 ywi 
 
 and the probable error of the lower zenith-distance, which 
 
 tj e 1 
 is derived from the mean of n observations, is ' * • * . 
 
 1 1 
 
 Now the colatitude = 9 upper zenith-distance + ^ lower ze- 
 nith-distance ; and the determinations of these zenith- 
 distances, as facts of observation, are strictly independent. 
 Therefore, by Article 52, 
 
 (p. e. of colatitude) 2 
 
 = - (p. e. of u. zen. dist.) 2 + v (P- e - °f !• zen - dist.) 2 
 
 _l (p.e.Ti,)' | 1 (p.e.l.) 2 
 4 ' to 4 ' n 
 
INSTANCES OF AGGREGATES OF MEASURES. 43 
 
 If the observations at upper and lower culmination are 
 equally good, so that 
 
 p. e. u. = p. e. 1. = p. e., 
 then (p. e. of colatitude) 2 = ^' '' . (— + -) ; 
 
 or p. e. of colatitude = ^— . I - 
 
 mn 
 
 57. Instance (2). In the operation of determining 
 geographical longitude by transits of the moon, the moon's 
 right-ascension is determined by comparing a transit of the 
 moon with the mean of the transits of several stars ; to 
 find the probable error of the right-ascension thus deter- 
 mined. 
 
 If p. e. m. be the probable error of moon-observation, 
 and p. e. s. the probable error of a star-observation, and 
 if the number of star-observations be n, then we have 
 
 p. e. of mean of star- transits = "' ,*- ' , 
 p. e. of moon-transit = p. e. m. 
 
 Hence, by Article 48, 
 p. e. of (moon-transit — mean of star-transits) 
 
44 COMBINATION OF ERRORS. 
 
 If p. e. s. = p. e. m. = p. e., 
 p. e. of (moon-transit — mean of star-transits) 
 
 =p-°VS +1 )- 
 
 It will be remarked here that, when the number of stars 
 amounts to three or four, the probable error of result is very- 
 little diminished by increasing the number of stars. 
 
 § 9. Methods of determining Mean Error and Probable 
 Error in a given series of observations. 
 
 58. In Articles 26, 27, 28, we have given methods 
 of determining the Mean Error, Error of Mean Square, and 
 Probable Error, when the value of every Actual Error in 
 a series of measures or observations is certainly known. 
 But it is evident that this can rarely or perhaps never 
 apply in practice, because the real value of the quantity 
 measured is not certainly known, and therefore the value 
 of each Actual Error is not certainly known. We shall 
 now undertake the solution of this problem. Given a 
 series of n measures of a physical element (all the mea- 
 sures being, so far as is known to the observer, equally 
 good) ; to find (from the measures only) the Mean Error, 
 Error of Mean Square, and Probable Error, of one measure, 
 and of the mean of the n measures. 
 
 59. We shall suppose that (in conformity with a re- 
 sult to be found hereafter, Article GS,) the mean of the 
 
CORRECTED DETERMINATION OF MEAN ERROR. 4.) 
 
 n measures is adopted as the true result. Yet this mean 
 is not necessarily the true result ; and our investigation 
 will naturally take the shape of ascertaining how much 
 the formulEe of Articles 26, 27, 28, are altered by recog- 
 nizing its chance of error. And first, for Mean Error. In 
 the process of Article 26, suppose that, in consequence of 
 our taking an erroneous value for the true result, all the 
 + errors are increased by a small quantity, and all the 
 — errors are diminished (numerically) by the same quantity. 
 Then the mean + error and the mean — error will be, one 
 increased and the other diminished, by the same quantity, 
 and their mean, which forms the mean error, will not be 
 affected. And if, from the same cause, one or more of 
 the — errors become apparently + errors, the mean + error 
 and the mean — error are very nearly equally affected in 
 magnitude but in different ways (numerically), and their 
 mean is sensibly unaffected. Thus the determination of 
 Mean Error is not affected ; and the process of Article 2G 
 is to be used without alteration. A result may follow from 
 this which is slightly inconsistent with that to be found in 
 Article 60, as has been remarked in Article 33. 
 
 60. Secondly, for Error of Mean Square. Suppose 
 that the Actual Errors of the n measures are a, b, c, d, &c. 
 to n terms ; then the Actual Error of the mean is 
 
 a + h + c + d + &c. 
 n ~ ; 
 
 and therefore if, for the process of Article 27, we form 
 the sum of the squares of the Apparent Error of each mea- 
 
46 COMBINATION OF ERRORS. 
 
 sure, that is of the difference of each measure from the 
 mean ; we do not form the squares of a, b, c, d, &c, but of 
 
 a+b+c+d+ &c. 
 a , 
 
 7 a+b+c+d+ &c. 
 o 
 
 a + b + c + d J r Sec 
 c 
 
 The sum of their squares (that is, the sum of 
 cf apparent errors) is 
 
 a 2 + l 2 + c" + &c 
 -~(a + b + c + &c.) x(a + b + c + d + &c.) 
 
 + nx —x(a + b + c + d + &c.Y 
 
 = a 2 + b 2 + c- + &c - x (a + b + c + d + &e. '. 
 
 n K ' 
 
 Now, in the long run of observations, -we may consider 
 
 each of the squares in the first part of this formula as 
 being equal to the Mean Square of Error ; so that for a", 
 or o 2 , or c 2 , &c, we may put (Error of Mean Square)* using 
 the definition of Article 27. But for a + b + c + d - 
 which enters as an aggregate quantity, we must remark 
 that, by Article 51, 
 
CORRECTED DETERMINATION OF PROBABLE ERROR. 47 
 
 Mean Square of -Error of (a + b + c + d + &c.) 
 = (m. s. e. of a) + (m. s. e. of b) + &c. 
 = n x (Error of Mean Square) 2 . 
 
 Thus the sum of squares which we form is truly 
 
 n x (e. m. s. of a measure) 2 — x n x (e. m. s. of a measure) 2 , 
 v ' n 
 
 = (n — 1) x (e. m. s. of a measure) 2 . 
 
 And from this, 
 
 , /sum of squares of apparent errors 
 e. m. s. of a measure = A / —^ , 
 
 V 71—1 
 
 „ , /sum of squares of apparent errors 
 e. m. s. ot the mean = . / - — -. ~ . 
 
 V 7i (n- 1) 
 
 And by the table of Article 31, 
 p. e. of a measure 
 
 'sum of squares of apparent errors 
 
 = 06745 x A 
 
 n — 1 
 
 p. e. of the mean 
 
 4 
 
 , sum of squares of apparent errors 
 — O'o/4o x A / . =-r 
 
 7i [n — 1) 
 
 61. The quantities which enter into the formation of 
 the mean error, error of mean square, and probable error, 
 will be most conveniently computed thus. It is supposed 
 that the different measures are A, B, C, &c, and that 
 their mean is M. 
 
48 COMBINATION OF ERRORS. 
 
 First, for the mean error. Select all the measures 
 A, B, C, &c. which are larger than M: supposing their 
 number to be I, form the quantity 
 
 A + B+G+&C. , r 
 1 M ' 
 
 which gives one value of mean error. Select all the 
 measures P, Q, B, &c, which are smaller than M\ sup- 
 posing their number to be s, form the quantity 
 
 M P+Q + B + &C. 
 
 s 
 
 which gives the other value of mean error. The mean of 
 these two values of mean error is to be adopted. 
 
 Second, for the error of mean square and probable error. 
 
 We wish to form (A - Mf +(B- Mf +(C- M) a + &c. 
 This = A 2 + £ 2 +6' 2 + &c.-2J/. (A + B+C + &c.)+n.3r. 
 
 But A + B+C + &c. = n.M; 
 
 so that the expression 
 
 = A 2 +B*+ C" + &c. - n . M\ 
 
 This is the "Sum of squares of apparent errors," to be 
 used in the formula) of Article 60. 
 
USE OF COMBINATION-WEIGHTS. 49 
 
 PART III. 
 
 PRINCIPLES OF FORMING THE MOST ADVANTAGEOUS 
 COMBINATION OF FALLIBLE MEASURES. 
 
 § 10. Method of combining measures; meaning of "com- 
 bination-weight ;" principle of most advantageous 
 combination : caution in its application to " entangled 
 measures." 
 
 62. The determinations of physical elements from 
 numerous observations, to which this treatise relates, are 
 of two kinds. 
 
 The First is, the determination of some one physical 
 element, which does not vary or which varies only by 
 a certainly calculable quantity during the period of 
 observations, by means of numerous direct and immediate 
 measures. Thus, in the measure of the apparent angular 
 distance between the components of a double star, we are 
 making direct and immediate measures of a quantity 
 sensibly invariable; in measuring the difference of moon's 
 right ascension from the right ascension of known stars at 
 two or more known stations, in order to render similar 
 observations at an unknown station available for determin- 
 ing its longitude, we are making direct and immediate 
 measures of quantities which are different at the two or 
 more stations, but whose difference can be accurately com- 
 puted. 
 
 A. D 
 
50 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 63. The measures thus obtained are all fallible, and 
 the problem before us is, How they shall be combined ? It 
 is not inconceivable that different rules might be adopted 
 for this purpose, depending (for instance) upon the products 
 of different powers of the various measures, and ultimate 
 extraction of the root corresponding to the sum of the 
 indices of powers: or upon other imaginable operations. 
 But the one method (to which all others will approximate 
 in effect) which has universally recommended itself, not 
 only by its simplicity, but also by the circumstance that it 
 permits all the measures to be increased or diminished by 
 the same quantity (which is sometimes convenient), is, to 
 multiply each measure by a number (either different for eacli 
 different measure, or the same for any or all) which number 
 is here called the "combination-weight;" to add together 
 these products of measures by combination-weights; and 
 to divide the sum by the sum of combination-weights. 
 
 64. The problem of advantageous combination now 
 becomes this, What combination-weights will be most 
 advantageous ? Arid to answer this, we must decide on 
 the criterion of advantage. The criterion on which we 
 shall fix is: — That combination is best which gives a 
 result whose probable error, or mean error, or error of 
 mean square, is the smallest possible. This is all that 
 we can do. We cannot assert that our result shall be 
 correct ; or that, in the case before us, its actual error shall 
 be small, or smaller than might be given by many other 
 combinations; but Ave can assert that it is probable that its 
 actual error will be the smallest, and that it is certain that, 
 
PROBABLE ERROR OF RESULT TO BE MINIMUM. 51 
 
 by adopting this rule in a very great number of instances, 
 we shall on the whole obtain results which are liable to 
 smaller errors than can be obtained in any other way. 
 
 65. Now if we know the probable errors, or the pro- 
 portion of probable errors, of the individual observations, 
 (an indispensable condition,) we can put known symbols 
 for them, and we can put undetermined symbols for the 
 combination-weights; and, by the precepts of Part II, we 
 can form the symbolical expression for the probable error 
 of the result. This probable error is to be made mini- 
 mum, the undetermined quantities being the combination- 
 weights. Thus we fall upon the theory of complex maxima 
 and minima. Its application is in every case very easy, 
 because the quantities required enter only to the second 
 order. Instances will be found in Articles 68 to 72. 
 
 66. It sometimes happens that, even in the measures 
 of an invariable quantity, combinations of a complicated 
 character occur. Different complex measures are some- 
 times formed, leading to the same result; in which some 
 of the observations are different in each measure, but 
 other observations are used in all or in several of the 
 measures; and thus the measures are not strictly inde- 
 pendent. We shall call these "entangled measures." 
 The only caution to be impressed on the reader is, to be 
 very careful, in forming the separate results, to delay the 
 exhibition of their probable errors to the last possible 
 stage; expressing first the actual error of each separate 
 result of the form ultimately required, by the actual error 
 
 D2 
 
52 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 of each observation. It will often be found that, in this 
 way, the results of observations will be totally or partially 
 eliminated (and justly so), which, if the probable errors 
 had been formed at an early stage, would have vitiated 
 the result. Instances of this will be given below 
 (Articles 74 to 85). 
 
 G7. The Second class is, the simultaneous deter- 
 mination of several physical elements. It may be illus- 
 trated by one of its most frequent applications, that of 
 determining the corrections to be applied to the orbital 
 elements of a planet's orbit. The quantities measured are 
 right ascensions and north polar distances, observed when 
 the planet is at different points in its orbit, and in 
 different positions with respect to the observer. If ap- 
 proximate orbital elements are adopted, each having an 
 indeterminate symbol attached to it for the small correction 
 which it may require; it will be possible to express, by 
 orbital calculation, every right ascension and north polar 
 distance by numerical quantities, to which are attached 
 definite multiples of the several indeterminate symbols. 
 Equating these to the observed right ascensions and ninth 
 polar distances, a long series of numerous equations is 
 obtained, with different multiples of the indeterminate 
 symbols; each equation being subject to its own actual 
 error of observation. And the question before us is now, 
 How r shall these numerous equations be combined so as to 
 form exactly as many equations as the number of indeter- 
 minate symbols, securing at the same time the condition 
 that the probable error of every one of the values thus ob- 
 
COMBINATION OF SIMPLE MEASURES. 53 
 
 tained shall be the smallest possible? This is also a case 
 of complex maxima and minima. Numerous problems in 
 astronomy, geodesy, and other applied sciences, require this 
 treatment. It will be fully explained in Articles 87 to 122. 
 
 § 11. Combination of simple measures; meaning of "the- 
 oretical weight;" simplicity of results for theoretical 
 weight; allowable departure from the strict rules. 
 
 68. Supj)ose that we have n independent measures of 
 some element of observation [e.g. the angular distance be- 
 tween two stars), all equally good, so far as we can judge 
 a priori; to find the proper method of combining them. 
 
 Let E v E 2 ,...E M be the actual errors of the individual 
 measures, which are not known, but which will affect the 
 result. Let their probable errors be e v e 2 , ... e n , each of 
 which = e. And let the combination-weights required be 
 w v w.„ ... w n . Then the actual error of the result, formed 
 as is described in Article 63, will be 
 
 w 1 E 1 + w 2 E 2 ... + w n E n 
 w t + w 2 + ... +w n 
 
 E t -\ —r— -&2+&C 
 
 w 1 + w 2 ... + w n w 1 + w 2 ...+ tu n 
 
 The (p. e. of result) 2 , by Article 52, is 
 
 w i + w z ...+ W n J ' \w t + w a ... + W 
 
 (w t + w 2 . . . + w n y 
 
 which in this instance becomes 
 
 £■ ^ l € _ 
 
54 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 Making the fraction minimum with respect to w v we obtain 
 2w, 2 
 
 W* + W* ... + W n ~ W^ + M? 2 . . . + w n 
 
 Similarly, by w# 
 
 2w„ 2 
 
 = 0. 
 
 = 0, 
 
 w* + w 2 ... + w* w l + iu 2 . . . + w n 
 and so for the other weights. 
 
 It follows that m/j = w 2 = w 3 , &c, but that all are inde- 
 terminate. That is, the measures are to be combined by 
 equal combination-weights; or, in other words, the arith- 
 metical mean is to be taken. The (probable error of result) 2 
 e 2 
 
 =V or ' 
 
 probable error of result = —r- ; 
 as was found in Article 55. 
 
 69. Suppose that we have n independent measures or 
 results which are not equally good. (For instance: the 
 atmospheric or other circumstances of individual observa- 
 tions may be different : or, if individual observations are 
 equally good, the results of different days, formed by the 
 means of different numbers of observations on the dif- 
 ferent days, would have different values. In determina- 
 tions of colatitude by means of different stars, the values 
 of results from different stars will be affected by their north 
 polar distances, as well as by the other circumstances.) 
 
 The notations of Article G8 may be retained, rejecting 
 only the simple letter e. Thus we have for (p. e. of result) 8 , 
 w*e* + vr*e* ...+ w n V t 
 
 (w x + w., . . . -f w n y * 
 
THEORETICAL WEIGHT. 55 
 
 and w v w 2 , &c, are to be so determined as to make this 
 minimum. 
 
 Differentiating with respect to w v 
 
 2"A' _1 =0 . 
 
 «*i \ + w 2 e a ■ ■ • + w n V ^, + lU 2 ...+W n 
 
 Differentiating with respect to tv 2 , 
 
 2w 2 e 2 * J^ =a 
 
 w{e* + iv *e 2 ' ... -I- w n \" w l + w s ... + w n 
 And so for the others. 
 
 It is evident that w^e* = w 2 e* = &c. = w n e^ = G some 
 indeterminate constant. Hence 
 
 CO c 
 
 and (p.e. of result) 2 
 
 _ C(w t + w a ...+ w tl ) _ C_ 
 
 1 111 
 
 Or = — I H...H . 
 
 (p. e. of result)' 1 ' e* e* e n 
 
 70. We shall now introduce a new term. Let 
 1 
 (probable error)*' 
 
 be called the "theoretical weight," or t. w. Then we have 
 these two remarkable results: — 
 
 When independent fallible measures are collateral, that 
 is, when each of them gives a measure of the same un- 
 known quantity, which measures are to be combined by 
 combination-weights in order to obtain a final result ; — 
 
5G ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 First. The combination-weight, for cacli measure ought 
 to be proportional to its theoretical weight. 
 
 Second. When the combination -weight for each mea- 
 sure is proportional to its theoretical weight, the theoretical 
 weight of the final result is equal to the sum of the theo- 
 retical weights of the several collateral measures 1 . 
 
 When the theoretical weights of the original fallible 
 measures are equal, and they are combined with equal 
 combination-weights, the theoretical weight of the result 
 is proportional to the number of the original measures. 
 
 71. These rules apply in every case of combination of 
 measures leading to the value of the same simple quantity, 
 provided that the observations on which those measures are 
 founded are absolutely independent. Thus, we may com- 
 bine by these rules the measures of distance or position of 
 double stars made on different days; the zenith distances of 
 the same star (for geographical latitude) on different days ; 
 the results (for geographical latitude) of the observations of 
 different stars ; the results (for geodetic amplitude) of the 
 observations of different stars ; the results (for terrestrial 
 longitudes) of transits of the moon on different days, &c. 
 
 72. Instance. In Article 56 we have found for the 
 probable error of colatitude determined by m observations 
 
 1 The reader is cautioned, while remembering these important theo- 
 rems, also to bear in mind the following (Articles 44 to 52) : — 
 
 When independent fallible measures or quantities are cumulative, that 
 is, when they are to be combined by addition or subtraction to form a new 
 fallible quantity; then the square of probable error of the new fallible 
 quantity is equal to the sum of the squares of probable errors of the several 
 cumulative measures or quantities. 
 
INSTANCE, DETERMINATION OF COLATITUDE. 57 
 
 of a star at its upper culmination, and n observations at 
 its lower culmination, 
 
 e /m + n 
 
 ~2\ mn ' 
 
 where e is the probable error of an observation, all being- 
 supposed equally good. Another star, whose observations 
 are equally good, observed m 1 times at upper and n l times 
 at lower culmination, gives a result with probable error 
 
 e /»?, + Wj 
 2 V lujT^ ' 
 
 a third gives a result with probable error 
 
 a A / — > &c - 
 
 ■L \ in s n 2 
 
 Their theoretical weights are 
 
 4 mn 4 m,n, 4 
 
 -*-, &c. 
 
 2 2 
 
 e' J ' m + n ' e 2 ' m y + n t ' e" ' m 2 + n s 
 The different results ought to be combined (to form a 
 mean) with combination-weights proportional respec- 
 tively to 
 
 mn m,n, m„n a 
 
 &c .; 
 
 m + n m 1 + n t m a + n 2 
 
 and the theoretical weight of the mean so formed will be 
 
 4 / mn m l n l »yi a „ \ 
 
 e~ \m + n vn, x + n 1 m % + n 2 ^ 7 ' 
 
 and its probable error will be the square root of the re- 
 ciprocal of this quantity. 
 
 It is supposed here that the zenith-point is free from 
 error. If it is not, the case becomes one of " entangled 
 observations," similar to that of Article 75. 
 
58 ADVANTAGEOUS COMBINATION OF MEASUEES. 
 
 73. We may however depart somewhat from the 
 precise rule of combination laid down in Article 70, with- 
 out materially vitiating our results. We have in Article Gi) 
 determined the conditions which make p. e. of result mini- 
 mum ; and it is well known that, in all cases of algebraical 
 minimum, the primary variable may be altered through a 
 considerable range, without giving a value of the derived 
 function much differing from the minimum. Thus, sup- 
 pose that we had two independent measures, for the same 
 physical element, whose probable errors were e and e' = 2e. 
 We ought, by the rule of Article 70, to combine them by 
 combination-weights in the proportion of 4 : 1. But sup- 
 pose that we use combination-weights in the proportion of 
 n : 1. Put E and E' for the actual errors ; the actual 
 error of result will be 
 
 n + 1 n + 1 n+1 
 
 the p. e. will be (by Article 52) 
 
 n V - / 1 V J VOr + 4) 
 
 n + l)- e+ [n + V- e \= e --nr + r 
 
 /f 3-2m 
 
 Using special numbers, we find 
 
 With combination-weights as 2 : 1, the p. e. of result 
 
 = cf = «x 0943. 
 
 o 
 
RELAXATION OF RULE. 59 
 
 With combination-weights as 4:1, the p. e. of result 
 
 
 
 5 
 
 . e x 0-894. 
 
 8 : 
 
 1, 
 
 the p. e. of result 
 
 
 
 V68 
 
 ex 0-916. 
 
 1G 
 
 :1, 
 
 the p. e. of 
 
 result 
 
 
 
 V2G0 
 = e ~T7 = 
 
 e x 0-947. 
 
 Thus it appears that we may use combination-weights 
 in any proportion between those of 2 : 1 and 1G : 1 without 
 
 increasing the p. e. of result by more than — part. 
 
 But if we used a proportion of combination-weights 
 less than 3 : 2, the probable error of the result would be 
 greater, and the value of the result less, than if we used 
 the principal measure alone. 
 
 The values of the result obtained by these combina- 
 tions will be different, but we have no means of knowing 
 with certainty whether one approaches nearer to the 
 truth than another. All that we know is that, in repeat- 
 ing combinations of these kinds in an infinite number of 
 instances, that which we have indicated as best will on the 
 whole produce rather smaller errors than the others. 
 
 When, however, we depart from the strictness of the 
 First rule in Article 70, the Second theorem of that Article 
 no longer holds. 
 
60 ADVANTAGEOUS COMBINATION , OF MEASURES. 
 
 § 12. Treatment of Entangled Measures. 
 
 74. The nature and treatment of entangled measures 
 will be best understood from instances. 
 
 Instance (1). Suppose that the longitude of an unknown 
 station is to be determined by the right ascension of the 
 moon at transit (as found by ascertaining the difference be- 
 tween the moon's time of transit arid the mean of the times 
 of transit of n stars) compared with the right ascension at 
 transit determined in the same manner at a known station 
 (where the number of stars observed is a); and suppose the 
 probable error of transit of the moon or of any star to be e. 
 Then, as has been found in Article 57, the probable error of 
 
 right ascension at the unknown station is e » /(-+ 1 
 that at the known station is e A / ( - + 1 ) ; and therefore 
 
 a/CH- 
 
 by Articles 47 and 48, as these two determinations are in 
 every respect independent, the probable error of the differ- 
 ence of right ascensions at transit (on which the longitude 
 
 a 
 
 depends) iseW^- 
 
 Supposc that a second comparison is made, of the same 
 transits at the unknown station, with transits of the moon 
 and b stars at a second known station. The probable 
 error of the quantity on which the longitude depends is 
 
 found in like manner to be e A / ( + j + 2 
 
ENTANGLED MEASURES. Gl 
 
 Now if we combined these two results, (leading to the 
 same physical determination, and both correct,) by the rules 
 of Article 70, we should obtain an erroneous conclusion. 
 
 OS. V 
 
 For, the two results are not independent, inasmuch as the 
 observations at the unknown station enter into both. 
 
 75. To obtain a correct result, we must refer to the -* 
 actual errors. In strictness, we ought to refer to the actual 
 error of each individual observation ; but, inasmuch as it is 
 perfectly certain that all the observations at each of the 
 stations, separately considered, are entirely independent of 
 all the observations atjihe other stations, we may put a sym- 
 bol for the aatiim error of excess of moon's Ii.A. above mean 
 of stars' R.A. at each of the stations. Let these symbols 
 be N, A, B, respectively. Then the aefcfc*£=eErors of tin- 
 quantities on which longitude depends, as found by com- 
 paring the unknown station with each of the known stations, 
 are respectively N — A, N — B. Let the quantities be 
 combined with the combination-weights a, /3. Then the 
 hnal &e&ral error will be 
 
 a + /3 a + j3 a + /S 
 
 And the square of probable error of final result 
 
 - K e - of ^ + dw (p - e - 0{Ay + (=. w? (p - e - of *>'}• 
 
 To make this minimum, w r e must make 
 
 a 2 (p. e. of A)' + /3 2 (p. e. of Bf 
 
G2 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 minimum. This algebraical problem is exactly the same 
 as that of Article 69, and the result is 
 
 G R- ° 
 
 "-fae.ofAy /3 "~(p.e.of£) S! ' 
 
 where G is an indeterminate constant. And this gives for 
 (p. e. of final result) 2 , 
 
 {(p.e.ofAT + ^1 
 
 ( . , „, t (p.e. of Af x(p.e. of£ ) 2 ) 
 
 = |(p. e. of Nf + ( ; e . of ^ + ^e.of^)4 
 
 (1 (l + q)(l + 6) 
 
 70. If we put r for the "theoretical weight" of final 
 result (see the definition in Article 70) ; n, a, b, for those 
 of the observations N, A, B, respectively; then the last 
 formula but one becomes 
 
 11 1 
 
 r n a + b ' 
 
 _ (a + b) n 
 ° r 1_ n+(a-fb)* 
 
 Let n be divided into two parts n a and n b , such that 
 
 a b 
 
 n„ = . n, n, = ,- n. 
 
 a a + b ' b a-t-b 
 
 Now if the theoretical weight n a at the station N had 
 
 * Instances of a more complicated character may be seen in the Me- 
 moirs of the E. Astronomical Society, Vol. xix. p. 213. 
 
PARTITION OF THEORETICAL WEIGHT. 63 
 
 been combined with the theoretical weight a at the station 
 A, they would have given for theoretical weight of their 
 result 
 
 a'n 
 a . n. a + b an 
 
 r. = 
 
 D a + a an n + (a + b) ' 
 
 a + b 
 
 And if the theoretical weight n b at the station N had 
 
 been combined with the theoretical weight b at the station 
 
 B, they would have given for theoretical weight of their 
 
 result 
 
 b'n 
 
 b . n,, a + b bn 
 
 r, = 
 
 n b + b _bn . n + (a + b) ' 
 a + b + 
 
 And consequently, 
 
 r a + r b = r. 
 And it is easy to see that, as there are two conditions 
 to be satisfied by the two quantities n a , n b , no other 
 quantities will produce the same aggregates n and r. 
 
 77. Hence we may conceive that the theoretical 
 weight n is divided into two parts proportional to a and b, 
 and that those parts are combined separately with a and b 
 respectively, and that they produce in the result the sepa- 
 rate parts r a and r b , which united make up the entire 
 theoretical weight of result r. The same, it would be 
 found, applies if there are any number of stations A, B, 
 C, D, &c. 
 
G4< ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 78. The partition of theoretical weight of final result 
 thus obtained, producing separate theoretical weights of 
 result depending on the combination of N with A and 
 N with B respectively, does in fact produce separate 
 theoretical weights for comparison of A T with A, and com- 
 parison of N with B, without necessarily distinguishing 
 whether the element (as moon's place) to which JSf relates 
 is inferred from that to which A relates, or whether the 
 element to which A relates is inferred from that to which 
 N relates. Hence it is applicable to such cases as the 
 following. 
 
 79. Instance (2). A geodetic theodolite being con- 
 sidered immoveable, observations (whose actual error is 
 M) are made with it for the direction of the north meri- 
 dian, and observations (subject to actual errors A, B, C, 
 &c.) are made on different triangulation-signals : to find 
 the weight to be given to the determination of the true 
 azimuth of each signal. 
 
 Using analogous notation, the theoretical weight m is 
 to be divided into parts m a , m b , m c , &c. ; and then the 
 weights of the determinations for separate signals are 
 those produced by combining m a with a, m b with b, &c, 
 or are 
 
 am bm 
 
 m + (a + b + c &c.) ' m + (a + b + c &c.) 
 
 , &c. 
 
 80. Instance (3). In the observation of zenith-dis- 
 tances of stars for the amplitude of a meridian arc divided 
 
 f^S. 
 
ENTANGLED MEASURES. 65 
 
 into two sections by an intermediate station : suppose that 
 a stars are observed at all the stations, the means of actual 
 errors being respectively A t , A 2 , A 3 : suppose that b stars 
 are observed at the first and second stations only, the 
 means of the actual errors being respectively B t , B 2 : 
 that c stars are observed at the second and third only, 
 the means of actual errors being C 2 , C 3 : and that d stars 
 are observed at the first and third only, the means of 
 actual errors being J) 1 , D 3 . They may be represented 
 thus : 
 
 «S oj . o3 
 
 Ti A ■— ; 3 r s a 
 
 a> S « -S ?o 
 
 fc» .-< ►" "S r "1 
 
 u ■*» h o !h -w 
 
 a> o3 <K -£ id c3 
 
 O _^ O g Or^ 
 
 w £ m © aj.jj 
 
 J** |I I- 
 
 OQ U2 02 
 
 Stars a -4^ ^4 2 , -4 3 . 
 
 Stars 6 B v B 2 . 
 
 Stars c C 2 , C 3 . 
 
 Stars cZ D x , D 3 . 
 
 Suppose the probable error of every individual obser- 
 vation to be e. It is now required to find the combina- 
 tion proper for determining the amplitude of the first 
 section of the arc. 
 
 81. Besides the direct measures of the first section, 
 there are indirect measures produced by subtracting the 
 measures of the second section from the measures of the 
 
OG ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 whole arc. All the possible measures of the first section 
 are therefore the following : 
 
 I. A 2 -A x \ 
 II. Bt -BJ BireCt 
 
 in. 4, -^-(4, -4,)] 
 iv. A-^-(c 5 -a y[ 
 
 V. D 3 - Dl -(A 3 -A 2 )\ 
 VI. D t -D t -{C 9 -Cj] 
 
 But of these, III is a mere reproduction of I : and of 
 the four measures I, IV, V, VI, it is easily seen that one 
 may be formed from the three others ; and the retention of 
 all would introduce indeterminate solutions. The following 
 may be retained, as substantially different ; 
 
 A 2 —A t with combination-weight v, 
 
 B 2 -B x w, 
 
 A 3 -A-C 3 +C 2 x, 
 
 D z -D x -C z +C 2 y. 
 
 These are entangled measures, inasmuch as A lt C 2 , C s , 
 appear in different measures. 
 
 82. The actual error of their mean will be 
 
 v(A t -A l )+to{B t -B l ) + x (A z -A,-C a +C t )+y (B-D- C 3 + < [) 
 
 v + w + x + y 
 
 = -{v+x)A 1 +vA 2 +xA-wB 1 +wB 2 +(x+ y) C 2 -(x+y) C.-yD.+ yD, 
 
 v+w+x+y 
 
ENTANGLED MEASURES. 67 
 
 The independent fallible quantities are now separated ; 
 and, by Article 52, remarking that (p. e. of A l f= -, and 
 so for the others, we find 
 
 p. e. of resultN 2 
 e ) 
 
 (y + w + x+yf 
 
 Making this minimum with regard to v, w, x, y, as in 
 Article 69, 
 
 (v + x)-+v- = C, 
 
 v 'a a 
 
 11 n 
 
 W r +W T = (J, 
 
 
 
 (y + x) - + x - + {x+y) - + (x +y) - = C, 
 'a a c c 
 
 {x+y)\+ {*+y)]+y\+y\ = g 
 
 From which 
 
 _ 4a 2 + 2ac + 4<ad „ 
 ~8a + 6c + 6^T ' 
 
 b n 4a& + She + Sbd n 
 W ~2 C - 8a + 6c + 6d °' 
 
 2ac — 2nd „ 
 X ~8a + Gc + 6d ' 
 
 2ad + Scd „ 
 y ~ 8a + 6c + 6d 
 
 E2 
 
68 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 It is remarkable here that in some cases x may be 
 negative, indicating that advantage will be gained by sub- 
 tracting that multiple of measure from the others. 
 
 If a = b = c = d, and D = aC, the combination- weights 
 become 
 
 D ' D ' D 
 
 83. If we thought fit to reject the combination 
 A a -A 1 -G a +G a> 
 
 there would be no entanglement ; and it would easily be 
 found, by Article 70, that the combination-weights ought 
 
 cd 
 
 to be proportional to a, b, ; and the theoretical weight 
 
 of the result 
 
 1 fa b cd 
 
 In like manner, for the second section of the arc, the 
 measures to be used are 
 
 a 3 -a 2 , c 3 -a 2 , zv-A-'B.+tfi; 
 
 and the theoretical weight of result 
 1 (a c bd 
 
 ' e 2 U + 2 + 2b + 2d. 
 
 84. Now if Ave combined these two sections to form 
 the whole arc, and inferred the probable error of the whole 
 from the probable errors of the sections by the rule of 
 
ENTANGLED MEASURES. 69 
 
 Article 44, we should obtain an erroneous result. For, the 
 observations on which the determinations of value of the 
 two sections are founded are not independent ; both contain 
 the observations A 2 , B x , B^, C„, C z , D t , D 3 ; and they are 
 therefore entangled results. 
 
 The correct result for the whole will be obtained by 
 an investigation exactly similar to that for each part. 
 There is the direct measure by the a stars, with error 
 A 3 — A x ; the direct measure by the d stars, with error 
 L\ — Dj ; and the indirect measure obtained by adding 
 the result of the b stars to the result of the c stars, with 
 error B 2 — B x + C 3 — C r The theoretical weight of the 
 result will be found to be 
 
 1 fa d be 
 
 ? \2 + 2 + 26 + 2c, 
 
 If the number of observations at the intermediate sta- 
 tion is very small, (as if a is small, b and c = 0, d large,) 
 the theoretical weight of the value of each section will be 
 small, while that of the entire arc may be great. 
 
 This instance is well adapted to give the reader a clear 
 idea of the characteristic difference between actual error 
 and probable error. So far as actual error is concerned, 
 if we add the measure of one section with its actual error, 
 to the measure of the other section with its actual error, 
 we entirely (and correctly) destroy so much of the actual 
 error as depends on the observations at the intermediate 
 station. But the probable error (see Article 8) is a mea- 
 
70 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 sure of uncertainty; and if, without looking carefully in 
 each case to the origin of the uncertainty, we simply add 
 together the two separate measures charged with their 
 respective uncertainties, we obtain for the whole arc a 
 sum with an apparently large imc&rtainty which is very 
 incorrect. 
 
 85. If the observations at the three stations are to be 
 combined in one connected system ; it will be best to use 
 each batch of stars separately, giving to each resulting 
 amplitude its proper weight as deduced from that batch 
 only. For the batches B, C, D, the operation is perfectly 
 clear; for the batch A, the principles of Articles 75, 
 7G, 79, must be used, which here give a very simple 
 result. 
 
 86. It is scarcely necessary to delay longer on the 
 subject of entangled measures. The caution required, and 
 which in all cases suffices, is : — to commence the investi- 
 gations by the use not of probable but of actual errors ; to 
 collect all the coefficients of each actual error, and to 
 separate them from the coefficients of every other error ; 
 and when the formulae are in a state fit for the introduc- 
 tion of probable errors, to investigate, by a process special 
 to the case under consideration, the magnitudes of the 
 combination-weights which will produce the minimum 
 probable error in the result. 
 
DETERMINATION OF SEVERAL ELEMENTS. 71 
 
 § 13. Treatment of numerous equations applying to 
 several unknown quantities : introduction of the term 
 " minimum squares." 
 
 87. The origin of equations of this class has been 
 explained in Article G7. It has there been seen that, 
 putting x, y, &c, for the corrections to orbital elements, 
 &c. which it is the object of the problem to discover, 
 (the number of which elements we shall for clearness 
 suppose to be three, though the investigation will evi- 
 dently apply in the same form to any number of such 
 corrections,) every equation will have the form 
 
 ax + by + cz = f 
 
 where f is the difference between a quantity computed 
 theoretically from assumed elements and a quantity ob- 
 served, and is therefore subject to the casual error of 
 observation. If the last terms of the equations, as given 
 immediately by observation, have not the same probable 
 error, we shall suppose that the equations are multiplied 
 by proper factors (see Article 35), so that in every case 
 the probable error of the last term f is made = e ; e being 
 an arbitrary number, for which sometimes it is very con- 
 venient to substitute the abstract value 1. We shall use 
 the letters a, b, c,f and others which are to be introduced, 
 without subscripts, in their general sense ; but for the 
 separate equations we shall affix the subscripts 1, 2, &c. 
 
/2 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 88. The number of equations being greater than 
 three, and it being requisite to reduce the final equations 
 to three in number ; the only method which suggests itself, 
 for giving every one of the fundamental equations a proper 
 share in the formation of those three equations, is : — first 
 to multiply the equations by a series of factors h 1} k 2 , &c, 
 and to adopt their sum as one fundamental equation ; 
 secondly, to multiply them by another series l 1} l 2 , &c. ; 
 thirdly, to multiply them by another series m,, m„, &c- 
 Thus having the series of fundamental equations 
 
 a l x + b 1 y + c l z=f l , 
 
 a i x+b a y + c^=f s , 
 
 &c. 
 
 Ave form the three series 
 
 h^x + k&y + \c x z = \f y , 
 
 h 2 a 2 x + hp 2 y + k,c,z = k.,f 2 , 
 
 &c. 
 
 l/i.jc+l 2 b 2 y+1 2 c 2 z = lj], 
 &c. 
 
 m^x + vrijbjy + m^z = m^, 
 m 2 a 2 x + m 2 b 2 y + m 2 c 2 z = m % f 2 , 
 &c. 
 
DETERMINATION OF SEVERAL ELEMENTS. 73 
 
 of which the sums are 
 
 x . t (Jca) + y . 2 (kh) + z.t (Ice) = 2 (A/), 
 x . X (la) + y.S (lb) +z.Z (Ic) = X (If), 
 x . 2 (ma) + y . S (mb)+z . 2 (mc) = S (mf). 
 
 These are our three final equations for determining 
 x, y, and z : and our problem now is, to ascertain the law 
 of formation of the factors h, I, m, which will give values 
 of x, y, z, for each of which the probable error may be 
 minimum. 
 
 89. Let us confine our attention, for a short time, 
 to the investigation of the value of x. The process of 
 solving the last three equations will consist, in fact, in 
 finding different factors wherewith the equations may be 
 multiplied, such that, when the multiplied equations are 
 added together, y and z may be eliminated, and the terms 
 depending on x and /may alone remain. But, remarking 
 how the three equations are composed from the original 
 equations, this multiplication of equations formed by sums 
 of multiples of the original equations is in fact a collection 
 of sums of other multiples 'of the original equations. Let 
 n be the general letter for the multipliers (formed by 
 this double process) of the original equations ; then the 
 final process for solution of the equations is thus ex- 
 hibited; 
 
 x xX (na) = 2 (nf) ; 
 
 t{nb)=0; 
 
 $(nc)=0; 
 
74- ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 which can be solved with an infinity of different values 
 of n. 
 
 90. From these, 
 
 S(n/) = tt 1 / 1 +»j p ,+&c.. 
 
 S (na) ?? 1 a 1 + n 2 a. z + &c. ' 
 
 2 2 
 
 from which the actual error of x 
 
 — x actual error of f t 
 
 » 1 a 1 + >i 2 a 2 + &c 
 
 H — — w— x actual error of f a 
 
 « 1 a 1 + n 2 a 2 + &c. y 2 
 
 + &c, 
 
 and, as the probable error of each of the quantities/^, f 2 , 
 &c. = e, the square of probable error of x 
 
 •i n i + " 2 2 + & c - 
 = e~ x 
 
 = e" x 
 
 (7* 1 a 1 +?? 2 a 2 + &c.) 5 
 1 (if) 
 
 The numbers n v v. 2 , &c. are so to be chosen that the 
 square of probable error of x shall be minimum ; and 
 therefore its variation produced by simultaneous small 
 variations in each of them shall be 0. 
 
 If we put Bn t , 8n 2 , &c. for such small variations, we 
 must have, by the formulae of ordinary differentiation, 
 
DETERMINATION OF SEVERAL ELEMENTS. ~0 
 
 _ n^Sw, + nj>n 2 + &c. _ a 1 Sn l + a 2 Sn 2 + &c. t 
 Wj" + n* + &c. w^j + n 2 a 2 + &c. ' 
 
 ~ n,Sn, +n a 8n a + &c. a,$n, + a„8i> + &c. rin 
 or 0= l — l ^ , 2 , 2 — v ; — \ .... L1J. 
 
 But the variations 6^, S« 2 , &c, are not independent 
 here, as were the corresponding variations in Articles 68 
 and 69 ; for they are affected by the antecedent conditions 
 2 (nb) = 0, 2 (mc) = ; from which we derive 
 
 = b l 8n 1 + b 2 Sn 2 + &c [2], 
 
 = 0^ + c 2 8n 2 + &c [3]. 
 
 These three equations must hold simultaneously for 
 the values of n v n 2 , &c., which we require. 
 
 91. It would perhaps be a troublesome matter to ex- 
 tract analytically from these equations the values of n v n 2 , 
 &c. We are however able to shew synthetically that a 
 certain form given to the numbers n v n 2 , &c. satisfies 
 the conditions. Let h 1 = a v k 2 = a 2> &c. ; l l = b l , l 2 — b 2 , &c; 
 m 1 = b 1 , m 2 = b 2 , &c.; so that the final equations of Article 
 88 take the form 
 
 x . t (a 2 ) + y.S (ab) + z.X (ac) =2 (a/) [4], 
 
 x.S(ab) + y.Z(b"~)+z.Z(bc)=S(bf) [•>], 
 
 x.Z(ac)+y.Z(bc)+z.Z(c*) =S(c/) [6]. 
 
 Then the values of x, y, z, which are deduced from 
 these equations, possess the properties required. 
 
7f> ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 92. For, suppose that we obtain the value of x by 
 multiplying the first of these by p, the second by q, the 
 third by r, and taking their sum, the coefficients of y and .3 
 being made to vanish. Then 
 
 x x [p . 2 (a 2 ) + q . 2 (a b) + r . 2 (ac)} 
 
 -p.S(a/) + 2 .S(J/) + r.X(o/}; 
 p.S.(a6) + 2.X(6 s ) + r.S(&c) = 0; 
 l> . 2 («c) + ^. 2 (Jc) + r.2 (c 2 ) =0; 
 which are the same as 
 
 x x 2 (a ( pa+ qb + re)} = 2 [f(pa + qb+ re)} .... [7], 
 
 ${b(pa+qb+rc)}=0 [8], 
 
 2{c(pa + qb + rc)} = [9]. 
 
 Comparing these equations with those of Article 89, n 
 is now replaced by pa + qb + re. Therefore 
 
 2 (n~) = 2 {(pa + qb + re) {pa + qb + re)} 
 
 = p . 2 (an) + q . 2 [b [pa + qb + re) } + r . 2 (c (_/;« + qb + ?*c)j . 
 
 The last two quantities vanish, by virtue of equations [8] 
 and [9]; and therefore 2 (n 2 ) =p. 2 (an). Substituting 
 this in the first denominator of equation [1], the equation 
 becomes 
 
 ("i~2 Kl i) & n i + ( n a ~~P a t) ^ n 2 + ^ c - = 0> 
 or (#&, + 7'oJ Bn 1 + (qb„ + re„) Sn a 4- &c. = ; 
 
 or 
 
 r q (bSn. + b 9 Sn 9 + &C.)) 
 
 J iV ' l ' 2 , = [10]; 
 
 1+ r (cjw, + c„8» s + &c.)J 
 
DETERMINATION OF SEVERAL ELEMENTS. 7/ 
 
 which is, under the new assumptions, the equivalent of 
 equation [1], and on the truth of which will depend the 
 validity of the new assumptions. Now the equation [10] 
 is true ; for its left hand consists of two parts, of which 
 one =0 by equation [2], and the other = by equation 
 [3]. Consequently, the equations [1], [2], [3], are simul- 
 taneously satisfied : and therefore the assumption of 
 Article 91 gives the values of x, whose probable error is 
 minimum. 
 
 93. If we investigate, by a similar method, the as- 
 sumption which will give for y the value whose probable 
 error is minimum, we have only to remark that the equa- 
 tions [4], [5], [6], are symmetrical with respect to x, y, 
 and z, and therefore when treated for y in the same 
 manner as for x, they will exhibit the same result for y 
 as for a;; that is, the probable error of y, as determined 
 from their solution, is minimum. In the same manner, 
 the probable error of z, as determined from the solution 
 of the same equations, treated in the same manner, is 
 minimum. 
 
 The problem, therefore, of determining values of x, 
 y, z, to satisfy, with the smallest probable error of x, y, 
 and z, the numerous equations 
 
 a x x + l x y + c x z =f v 
 
 a a x + b 2 y + c z z =/ s , 
 
 &c. 
 
78 ADVANTAGEOUS COMBINATION OF MEASUEES. 
 
 is completely solved by solution of the equations 
 x . 2 (a 2 ) + y . 2 {ah) + z . 2 iac) = 2 (a/), 
 a? . 2 (aft) + y . 2 (//) +2.2 {bo) = 2 (*/), 
 x . 2 (ac) + y . 2 (Jc) + ^ . 2 ( c 2 ) = 2 (</). 
 
 94. Suppose that, instead of proposing to ourselves 
 the condition that the probable errors of the deduced 
 values of x, y, z, shall be minimum, we had proposed this 
 condition ; that the sum of the squares of the errors re- 
 maining after correction for the deduced values of x, y, 
 and z, or 
 
 ^.{ax + hj + cz-fy, 
 
 shall be minimum. On differentiating each equation 
 with respect to x, and taking their sum, we should have 
 obtained 
 
 2 [a . {ax +by + cz -/)] = ; 
 
 and similarly for y and z 
 
 Z{b.{ax + by + cz-f)}=0, 
 
 %{c.{ax+hj + cz-f)}=0; 
 
 the very same equations as those found above. In conse- 
 quence of this property of the equations, of giving such 
 values of x, y, and z, that the sum of squares of errors 
 remaining after their application shall be minimum, the 
 method is very frequently called " the method of minimum 
 squares." This term is very unfortunate ; it has fre- 
 quently led investigators to suppose that the subject of 
 
MINIMUM SQUARES OF RESIDUAL ERRORS. 79 
 
 the minimum is the sum of squares of discordances as first 
 presented ; whereas it ought to be the sum of squares of 
 discordances, when so multiplied as to have the same 
 probable error. 
 
 95. It is easy to see that the same principles apply, 
 the same remarks hold, and the same result is obtained, 
 when the number of unknown elements, instead of being 
 restricted to three, is any whatever. The rule is universal; 
 multiply every equation by such a factor that the pro- 
 bable error of the right-hand term will be the same for 
 all ; multiply every altered equation by its coefficient 
 of one unknown quantity, and take the sum for a new 
 equation ; the same for the second unknown quantity ; 
 and so on for every unknown quantity ; and thus a num- 
 ber of equations will be found equal to the number of 
 unknown quantities. 
 
 96. In order to exhibit the probable error of x thus 
 determined, we may proceed by a purely algebraical pro- 
 cess. It will however soon be found that it leads to 
 results of intolerable complexity. We would recommend 
 the reader to introduce numbers as soon as possible for 
 every symbol except f (that quantity from whose error 
 all errors spring). In the following explanation, however, 
 of the succession of steps, the reader will easily see to 
 what extent he can advantageously retain the symbols. 
 
 It is first necessary to find the factors of the equations 
 [4], [5], [6], of Article 91, or the last equations of Article 
 
SO ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 93, which will eliminate y and z. They are easily found 
 to be, 
 
 For equation [4], S.^xS.c 2 -^. he)'. 
 
 For equation [5], 2 . ac x S . be— 2 . ab x "Z . c\ 
 
 For equation [6], S . «& x 2 . &c — 2 . ac x2. b". 
 
 There is no difficulty in finding the factors when the 
 number of unknown quantities exceeds three ; but the 
 trouble is so great that it will always be best to use 
 numbers. 
 
 Applying these, we obtain 
 
 x = P.%(af) + Q.^(bf) + R.t(cf) } 
 
 where P, Q, B, are numbers, but af, bf, cf, are for the 
 present retained in the symbolical form. 
 
 Now if we examine the form in which the individual 
 quantities f lt f s , &c. enter into this expression, and if we 
 collect together all the multiples of each individual quan- 
 tity, we shall find 
 
 x = (p fli + Qh t + Bc^f x + (Pa, + Qb, + Bc & ) f % + &c. 
 
 We have here a number of independent fallible quan- 
 tities, to which the formula of Article 52 will properly 
 apply. Remarking that the probable error of each of the 
 quantities f^f^, &c. is supposed to =e, we obtain 
 
 /p. r.oixj = (Pa, + Q\ + Re,) 3 + (Pa, + QK + PcJ + &c. 
 
 = z.(r a + Qb + iicy; 
 
SMALL VARIATION OF THE FACTORS IS PERMITTED. 81 
 
 which may be exhibited in symbols of great complexity, 
 but which it will be very far easier to evaluate in numbers 
 by an entirely numerical process. 
 
 The operation for finding the probable errors of y and 
 s would be exactly similar. 
 
 97. The relaxation of the rules for determining the 
 most advantageous values of the factors of the equations, 
 which in reference to the treatment of simple measures is 
 explained in Article 73, is admissible also in the treatment 
 of equations applying to several unknown quantities, and 
 for the same theoretical reason. By taking advantage of 
 this relaxation, the labour may sometimes be materially 
 diminished. In actual applications, the numbers a t , « 2 , &c. 
 b 1} &c, usually consist of troublesome decimals. In prac- 
 tice, all desirable accuracy will be secured for the result, 
 by striking off, in the factors only, all the latter decimals, 
 leaving only one or two significant figures. The use of 
 different factors will produce different results, but not 
 necessarily more inaccurate results ; we have no means of 
 certainly knowing which are the best ; we only know that, 
 if we repeat the process in an infinity of instances, the fac- 
 tors corresponding accurately to minimum will furnish us 
 with results whose errors are, on the whole, a little smaller 
 than those originating from other factors. 
 
82 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 § 14. Instances of the formation of equations applying to 
 several nnknoivn quantities. 
 
 98. It will perhaps bo instructive to shew how equa- 
 tions, such as those treated above, arise. For this purpose, 
 we will take two instances ; one of very simple and one of 
 very complicated character. 
 
 99. Instance 1. It is required to determine the most 
 probable values of the personal equations between a 
 number of transit observers A, B, C, D, &c; where the 
 observers have been brought into comparison in many 
 combinations, or perhaps in every possible combination ; 
 but never more than two at a time. 
 
 Use the symbol (ab) to denote the number of compari- 
 sons between A and B, and A — B for the symbolical 
 value of the personal equation between A and B, (AB) 
 for its numerical value deduced from the mean of com- 
 parisons. And suppose that the probable error of each 
 single comparison is e. Then the probable error of (AB) 
 
 is , ; . . Therefore when we have formed the equation 
 
 A-B = (AB), 
 srm is liable tc 
 , we must, in conformity with the recommendation 
 
 in which the last term is liable to the probable error 
 
 e 
 /(ab) 
 in Article 87, multiply the equation by *J(ab), and then 
 
INSTANCE: PERSONAL EQUATIONS. 83 
 
 its probable error will be e. Thus we find, for the dif- 
 ferent comparisons, the following equations, all liable to the 
 same probable error e : } 
 
 V(aj) . A - s/(ab) . B = sjiab) . (AB), 
 V(«c) . A - *J(ac) . G = V(«c) . (A C), 
 
 &c. 
 Vt&c) . B-*J(bc) . C= V(6c) . (J5C), 
 
 &c., 
 
 and these equations are exactly such as those in Article* 
 88, though in an imperfect form. The determining equa- 
 tions must therefore be formed by the rule of Article 93.: 
 Thus we find ; — • 
 
 The first equation is to be formed by the sum of the 
 following, 
 
 (ah) . A - (ab) .B = + (ab) . (AB), 
 
 (ac) . A - (ac) . C = + (ac) . (A C), 
 
 &c. 
 
 The second equation is to be formed by the sum of 
 
 the following, 
 
 (ab) . B - (ab) .A = -(ab). (AB), 
 (bc).B-(bc). C= + (bc).(BC), 
 &c. 
 
 F 2 
 
84 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 Thus we obtain the simple rule : — 
 
 Form each equation for comparison of two observers by 
 taking the mean of all their comparisons. 
 
 Multiply each such equation by its number of compari- 
 sons. This is, in fact, the same as if the sum of all the 
 individual comparisons of those two observers had been 
 taken. 
 
 In the various multiplied equations which contain A ; 
 make the coefficient of A in every equation positive (by 
 changing all the signs of the equation where necessary), 
 and then add all together to form a determining equation. 
 
 In the various multiplied equations which contain B, 
 including, if necessary, one from the last-mentioned series ; 
 make the coefficient of B in every equation positive (by 
 changing signs if necessary), and add all together to form 
 a determining equation. 
 
 In the various multiplied equations which contain C, 
 including, if necessary, one from each of the last two series ; 
 make the coefficient of C in every equation positive (by 
 changing signs if necessary), and add all together to form 
 a determining equation. 
 
 And so through all the observers. 
 
 It will be found that one of the determining equations 
 may be produced by a combination of all the other de- 
 termining equations ; and therefore it is necessary to 
 assume a value for one of the quantities, A, or B, or 
 C, &c. 
 
INSTANCE: GEODETIC TRIANGULATION. 85 
 
 100. Instance 2. In a net (not necessarily a simple 
 chain) of geodetic triangles ; one or more sides have been 
 actually measured, or so determined by immediate refer- 
 ence to measured bases that they may be considered as 
 measured; in some of the triangles, three angles have 
 been measured, in others only two ; at some stations, all 
 the angles round the circle have been observed, at others 
 not all ; at some stations, astronomical azimuths have been 
 observed : it is required to lay down the rules for de- 
 termination of the positions of the different stations. 
 
 101. It is first necessary to determine the value of 
 probable error in each of the observations. And this is 
 not to be done by a simple rule, because the observa- 
 tions are not all alike. For instance, the horizontal angle 
 between two signals is liable to error from (1) error of 
 instrument, (2) error of pointing to one signal, (3) error 
 of pointing to the other signal ; and when each probable 
 error is ascertained, the probable error of horizontal angle 
 between signals is easily formed. But for the azimuth 
 of a given signal, the sources of error are, (1) error of 
 instrument, (2) error of pointing to the signal, (3) error 
 of pointing in the direction of the meridian ; and the pro- 
 bable error of this last may be very different from the 
 others. The linear measures will require a peculiar esti- 
 mate of probable error. 
 
 All must, however, antecedent to all other treatment, 
 be so found that the probable error of every measure, of 
 whatever kind, can be specified. 
 
86 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 102. The next step in this (and in all other compli- 
 cated cases) will be, to assume that every co-ordinate of 
 station which we are seeking, is approximately known, 
 and numerically expressed. Thus, if the triangulation is 
 so small in scale that its area may be supposed a plane, 
 we may assume, for the two rectangular co-ordinates of 
 every point, numerical values, each subject to a small 
 correction (which corrections it is the object of the whole 
 investigation to ascertain). If the triangulation is so large 
 in scale that the spheroidal form of the earth must be 
 regarded, we may assume, for the astronomical latitude 
 and longitude of every point, numerical values, each sub- 
 ject to a small correction. 
 
 103. With these numerical values and symbolical cor- 
 rections, every fact which has been the subject of measure 
 must be computed ; and the computation-result must con- 
 sist of two parts, one numerical, and the other multiply- 
 ing the symbolical corrections. Thus; suppose that the 
 area is plane, and that the rectangular co-ordinates are 
 
 For 1st station, o^ + Ba^ b^&b^ 
 
 For 2d station, or 2 +Sa.,, b,_, + 8b^, 
 
 For 3d station, o 3 + Ba 8 , b s + Bb & , 
 &c. 
 
 (where a x , « 2 , &c, l\, \, &c, are numerical, and Ba t , 
 Ba„, &c, Bb lf Bb„, &c. are symbols only). Suppose that the 
 direction of a is parallel to the meridian. Then the azimuth 
 of the second station as viewed from the first is 
 
instance: geodetic triangulation. 87 
 
 ■ (« 2 -«x) +(h-K) * K - ' 
 
 fa-ad +(K-h) a lJ 
 
 where tan (7„ = * , and where all quantities are nu- 
 
 merical, except Ba t , 8l\, Ba 2 , Bb, y For convenience we will 
 write this, 
 
 True azimuth of 2d station as seen from 1st station 
 
 = a + a, 2 • *h -A*- Ba > + ^ • S K - A, 2 • % 
 
 And iu like manner, 
 
 True azimuth of 3d station as seen from 1st station 
 
 = C 3 + A lfi . 8a 3 -A lfi . Ba, + B h3 . cb 3 - B l3 . Bb t . 
 
 Now if the azimuth of the 2d station had been observed 
 at the 1st, and found —<y 2 (subject to error of observation), 
 then the comparison of the first formula with this would 
 give the equation 
 
 A* • K - A* • Sf 'i + B * • Bh * ~ B i,* • hh i = v* - G *' 
 
 If, however, no azimuth from the meridian had been 
 observed, but only the horizontal angle # 23 between the 
 2d and 3d stations, which is the same as <y 3 — %, and is 
 subject to errors of observation ; then we should have 
 
 A*M - VV ( J '.s- A J -^ + *»&* ~ B ^h 
 
88 ADVANTAGEOUS COMBINATION OF MEASURES. 
 
 The distance of the 2d station from the first 
 
 = V[K-« 1 ) 2 + (^ 2 -W 
 
 a„ — a 
 
 vKa,-a>+(K-w (K ~ Sai) 
 
 which may be written 
 
 A + M 1A (Sa a - SaJ + N, t (BK - Bb,). 
 
 Now if the distance from the first station to the second 
 had been measured, and found =X 2 , subject to error of 
 observation; then the comparison with this formula would 
 give 
 
 M lfi . ca 2 - M lfi . Ba x + N Ui . 8b a - X 12 . S& x = \ - L 2 . 
 
 104. Each of these equations contains, on the right 
 hand, a fallible, quantity ; the first contains <y a , the second 
 contains # 23 , the third contains X.,. The probable error 
 of each of these, as we have said in Article 101, must be 
 supposed known ; and then, the equation must be divided 
 by a divisor proportional to that probable error. This 
 operation being effected, we shall have a scries of equa- 
 tions whose probable errors are all equal ; and the rule 
 of Article 93 can be applied ; and we shall have a series 
 of determining equations equal in number to the number 
 of unknown quantities ; that is, to double the number of 
 stations. 
 
GENERALITY OF THE THEORY. .89 
 
 105. The reader who will well examine such an in- 
 stance as this will be struck with the perfect generality 
 and great beauty of the theory. He will see that, what- 
 ever has been observed, and in whatever shape, as a fact 
 of observation, will enter with its proper weight into the 
 formation of the final determining equations. He may 
 exercise his fancy in introducing different circumstances. 
 Suppose, for instance, that a referring-signal has been 
 used. The correction of assumed azimuth of that refer- 
 ring-signal v/ill be a new unknown quantity. Sometimes 
 it is combined only with observations of signals, in which 
 case it produces one form of equation ; sometimes it is 
 combined also with meridian-observations, in which case 
 it produces another form of equation. In some batches 
 of observations, it may be necessary to use the theory of 
 "entangled measures" (see Articles 74 to 80) before the 
 probable errors can be properly found. Whatever mea- 
 sure is made, a proper corresponding equation can be 
 formed, and the proper cautions accompanying it will soon 
 present themselves. 
 
 10G. It may occur to the reader as a difficulty, that 
 the quantities concerned are not homogeneous ; the mea- 
 sure of 0, for instance, being in angle, and that of X in 
 units of length. This, however, is only apparent. Any 
 measure is expressed by means of certain units, and its 
 probable error is expressed by the same units ; and, when 
 we divide each equation by a divisor proportional to its 
 probable error, we do in fact produce an equation of abs- 
 
00 ERRORS OF OBSERVATIONS REQUIRED 
 
 tract numbers, whose probable error also is an abstract 
 number. The whole are then strictly comparable, in 
 whatever kinds of measure they have originated. 
 
 107. The solution of so numerous a series of equa- 
 tions is of course troublesome. It is, however, no more 
 troublesome than the nature of the subject strictly re- 
 quires. And it is to be considered that it gives to every 
 observation of every kind exactly the weight that is due 
 to it : that it leads to one distinct system of results, with- 
 out leaving any opening for uncertainty ; and that that 
 system of results is the most probable. 
 
 § 15. Treatment of Observations in which it is required 
 that the Errors of Observations rigorously satisfy 
 some assigned conditions. 
 
 108. In the equations considered in Article 87, each 
 of which gave the effects of combining, with different fac- 
 tors, the unknown corrections of certain elements, in order 
 that observations might thereby be better represented, — it 
 is to be remarked that there was no expectation that the 
 result of combination of these corrections of elements 
 would exactly represent the observations, or that any exact 
 relation could be assigned as existing among the cor- 
 rections which were to be found; or between "the result of 
 applying those corrections," and "observations". 
 
 109. But there are instances in which the nature of 
 the problem requires that, among the corrections which 
 
/ 
 
 TO SATISFY ASSIGNED CONDITIONS. 91 
 
 are to be found, a prescribed equation shall be rigorously 
 maintained. The nature and treatment of these will best 
 be understood from examples. 
 
 110. Instance (1). In a geodetic triangle, observa- 
 tions of the three angles are made. On comparing their 
 sum with the quantity 180° + spherical excess, to which 
 that sum ought to be equal, it is found that the sum 
 requires the correction A. How ought that correction to 
 be divided among the three angles, their probable errors 
 of observation being known ? 
 
 111. Let E v E, v E z be the three corrections required, 
 the probable errors of observation being e t , e 2 , e s . Then 
 the first consideration is, 
 
 E l is to be as small as possible, 
 
 E 2 
 
 e[ 
 
 The quantity E 1 here corresponds to the quantity 
 ax-\-by-\- cz —f 
 in Article 87; the latter being the residual difference 
 between an observed quantity and a quantity computed 
 from corrected elements, which difference is to be made 
 "as small as possible." The equations are therefore to 
 be exhibited in the same form. 
 
 Hence we may state the equations thus : 
 E x = 0, with probable error e v 
 E 2 = 0, e 2 , 
 
 tf.>fc V 
 
92 ERROKS OF OBSERVATIONS REQUIRED 
 
 If wo stopped at this form, we could not obtain a valid 
 solution : the number of equations being tlie same as the 
 number of unknown quantities, in which case no solution 
 depending on probabilities can be obtained. 
 
 112. Now we introduce the condition 
 
 and use it to eliminate one of the quantities, as E s . Then 
 the equations become, 
 
 i^=0, with probable error e x , 
 
 E. z = 0, e,, 
 
 A-E^E^O, e s . 
 
 Here we have three equations to determine two quan- 
 tities, and the process of Article 93 may be followed., 
 
 113. Dividing by the probable errors, we have these 
 equations, in each of which the probable error = 1 ; 
 
 !-»■ 
 
 A E E 
 
 e 3 e 3 e 3 
 
 and therefore, by the process of Article 93, forming a 
 final equation principally for E v by multiplying each 
 
TO SATISFY ASSIGNED CONDITIONS. 93 
 
 equation here by its. coefficient of E l and adding the pro- 
 ducts, 
 
 {ey W (?/ W 
 
 Similarly, forming a final equation principally for E,, 
 by multiplying each equation by its coefficient of E z , and 
 adding the products, 
 
 & WWW * 
 
 E E , 
 
 Comparing these two equations, -r— ^ = — y 2 . W e might 
 
 at once infer from this that — h has the same value, 
 
 W 
 
 but *it may be more satisfactory to solve the equations 
 completely. Eliminating E 2 from the first equation, by 
 the relation just fouud, 
 
 Therefore ^-^.jgr^q^ii 
 
 whence, by the relation found, 
 
 w+w+w 
 
1)4* ERRORS OF OBSERVATIONS REQUIRED 
 
 and by subtracting their sum from A, 
 
 \2 
 
 E-A -^ 
 ' W+W' + to" 
 
 Hence, the corrections to be assigned to the different 
 angles ought to be proportional to the squares of their 
 respective probable errors. 
 
 114. Instance (2). From a theodolite station, n sig- 
 nals can be seen ; the angles, between each signal and the 
 next in azimuth, are independently observed ; their sum, 
 which ought to be 360°, is 360° - B : how ought the cor- 
 rection B to be divided ? 
 
 115. The equations in this instance will be 
 
 E x = 0, with probable error e 1 , 
 E,= 0, e 2 , 
 
 K = 0, e n . 
 
 Then, by the equation 
 
 E^E.^&c. + E^B, 
 
 the last of the equations is changed into 
 
 B — E 1 —E 2 — &c — E n _ x = 0, with probable error e n ; 
 
 and the equations are to be treated in the same manner as 
 in Instance (1) ; and a similar result is obtained ; namely, 
 
TO SATISFY ASSIGNED CONDITIONS. 
 
 95 
 
 that the corrections to be assigned to the different 
 angles must be proportional to the squares, of their re- 
 spective probable errors. 
 
 The next instance will be more complicated. 
 
 116. Instance (3). In the survey of a chain of tri- 
 angles, a hexagonal combination of the following kind 
 occurs, in which every angle is observed independently; 
 all are liable to error ; to find the correction which ought 
 to be made to each. 
 
 Let the angles be denoted by the simple numbers ; 
 let their corrections sought be [1], [2], &c, and their 
 probable errors of observation (1), (2), &c. 
 
06 ERRORS OF OBSERVATIONS REQUIRED 
 
 Then we have the equations, 
 
 [1] = 0, with probable error (1), 
 [2] = 5 (2); 
 
 [18] = 0, (18). 
 
 And now Ave have to consider how many of these 
 unknown quantities can be eliminated by virtue of the 
 geometrical relations. 
 
 117. Adding the angles at the central station, and 
 comparing the sum with 300°, 
 
 [1] + [2] + [3] + [4] + [5] + [G] = a known quantity A: 
 
 Then in the six triangles, 
 
 [1] + [ 7 ] + [ 8 ] = a known quantity B, 
 
 [2] + [9] + [10] C, 
 
 [3] + [H] + [12] D, 
 
 [4] + [13] + [14] E, 
 
 [5] + [15] + [10] F, 
 
 [6] + [l7] + [18] G. 
 
 When corrections satisfying these equations are ap- 
 plied, we shall have a set of six triangles, with angles 
 consistent in each triangle ; and which so adhere together 
 that they fill up 360° at the central station ; nevertheless 
 
TO SATISFY ASSIGNED CONDITIONS. 97 
 
 it might happen that, in calculating b from a, c from b, 
 d from c, e from d, f from e, and a from f, we should 
 find a value a' differing from a. But it is necessary that 
 a' be found rigorously equal to a. Tracing the calcu- 
 lations through the several triangles, it is found that this 
 equation gives (with corrected angles), 
 
 sin 7 sin 9 sin 11 sin 13 sin 15 sin 17 
 sin 8 sin 10 sin 12 sin 14 sin 16 sin 18 
 
 and, taking the logarithms, with the addition of symbols 
 for the corrections, 
 
 log . sin 7 — log . sin 8 + log . sin 9 — log . sin 10 + log . sin 11 
 
 — log.sinl2 + log.sinl3 — log.sinl4 + log.sinl5— log.sinlG 
 
 -f log . sin 17 — log . sin 18 
 
 + cot7 x[7] -cotSx [8] + cot9x [9]-&c 
 
 + cotl7x [l7]-cotl8 x [18] 
 
 = 0. 
 
 We shall use the symbol L to denote the first part of 
 this expression, which is a known quantity. 
 
 Thus we have eight equations to be rigorously satis- 
 fied. By means of these, we are to eliminate eight of the 
 quantities [1], [2], &c, and there will remain ten quan- 
 tities to be determined by eighteen equations. 
 
 118. Suppose for instance we decide to eliminate the 
 corrections [1], [2], &c, as far as [8]. We have 
 
 [2]= C- [9] -[10], 
 [3]=D-[11]-[12], 
 
 A. G 
 
98 ERRORS OF OBSERVATIONS REQUIRED 
 
 [4] =£'-[13] -[14], 
 
 [5] = ^-[15] -[16], 
 
 [6] =£-[17] -[18]. 
 
 Substituting these iu the first equation, 
 
 [\]=A-C-D-E-F-G 
 
 + [9] + [10] + [H]+[12]+[13]+[14]+[15]+[16]+[l7]+[18]. 
 
 Then 
 
 [7] = £-[l]-[8] 
 
 =-A+B+C+D+E+F+G 
 -[8]-[9]-[10]-[ll]-[12]-[13]-[14]-[15] 
 _[1G]_[17]-[18]. 
 Substituting this in the last equation of Article 117, 
 = 
 L + cot 7 x (- A + B + C+ D + E+ F+ G) 
 
 -cot7x{[8]+[9]+[10]+[ll]+[12] + [13]+[14]+[15] 
 
 + [16] + [17] +[18]} 
 
 - cot 8 x [8] + cot 9 x [9] - cot 10 x [10] + cot 11 x [11] 
 -cot 12 x [12] + cot 13 x [13] -cot 14 x [14] 
 + cot 15 x [15] - cot 1G x [16] + cot 17 x [17] 
 -cot 18 x [18]. 
 
 From this equation, [8] is found in terms of [9], [10], 
 &c. as far as [18]. And substituting it in the preceding 
 expression, [7] is found in terms of [9], [10], &c. as far 
 as [18]. Thus all the corrections [1], [2], ...[8], are ex- 
 pressed in terms of [9], [10]... [18], 
 
TO SATISFY ASSIGNED CONDITIONS. 99 
 
 119. The primary equations of probabilities are, 
 [1] = with probable error (1), 
 [2]=0 (2), 
 
 [18]=0 (18), 
 
 Of these, the first eight will now be changed into the 
 following : 
 
 [9] + [10]+... + [18] = -vl+ C+D + E + F+G, 
 
 with probable error (1) 
 
 [9] + [10] = C, (2) 
 
 [11] + [12]=2>, (3 ) 
 
 [13] +[14]=^, (4) 
 
 [15] + [16]=^ (5) 
 
 [17] + [18] = G, (6) 
 
 ("series of multiples] r series of ] 
 
 |of[9],[10]...[18],U known (7), 
 
 [ expressing [7] J (quantities! 
 
 i series of multiples') r series of ] 
 of[9],[10]...[18],l = i known | (8). 
 expressing [8] J [quantities' 
 
 The remaining equations will retain their simple form, 
 
 [9] = 0, with probable error (9), 
 [10] = 0, (10), 
 
 [18] = 0, (18). 
 
 G 2 
 
100 ERRORS OF OBSERVATIONS REQUIRED 
 
 120. Each of these eighteen equations is then to 
 be divided by its probable error, and we thus obtain the 
 following equations, in each of which the probable error 
 
 = i; 
 
 [9] [10] [18] _ -A + C + D+E+F+G 
 
 (i) + w + W " (1) 
 
 with probable error 1. 
 
 W (2) (2)' ' 
 
 and so through all the equations. 
 
 The equations so divided, having the same probable 
 error, are in a fit state for application of the method of 
 Article 93. The first of the final equations (principally 
 for [9]) will be formed by multiplying each equation by 
 the coefficient of [9] in that equation, and adding the pro- 
 ducts ; the second of the final equations (principally for 
 [10]) will be formed by multiplying each equation by the 
 coefficient of [10] in that equation, and adding the pro- 
 ducts; and so on to [18], From the equations thus formed, 
 the values of [9], [10].. .[18], are found; and by substi- 
 tuting these in the formula? of Article 118, the values of 
 [1], [2]... [8] are -found. 
 
 121. It is particularly to be observed that, although 
 in the changed equations of probabilities we eliminate 
 such quantities as [1], [2], &c, we do not eliminate their 
 corresponding probable errors (1), (2), &c, each of which 
 must be left in its place. This retention of the probable 
 error will be remarked in Instances (1) and (2). 
 
TO SATISFY ASSIGNED CONDITIONS. 101 
 
 122. The complete solution is so troublesome that it 
 would scarcely ever be used in practice. Probably some 
 process like the following would be employed, with suf- 
 ficient accuracy : 
 
 Divide the error A by the process of Instance (2), and 
 use the corrected angles in the process that follows. 
 
 Divide the errors B, C,...G, by the process of Instance 
 (1), and use the corrected angles in the process that fol- 
 lows. 
 
 Apply the last equation of Article 117, by a process 
 nearly similar to that for A. 
 
 Repeat the process for dividing A' (the discordance at 
 the center produced by the angles as last corrected). 
 
 Repeat the process for dividing B', C ... G'. And 
 continue this operation as often as may be necessary. 
 
PART IV. 
 
 ON 3IIXED EKKOES OF DIFFERENT CLASSES, AND 
 CONSTANT ERRORS. 
 
 § 16. Consideration of the circumstances under which the 
 existence of Mixed Errors of Different Classes may 
 he recognized, and investigation of their separate 
 values. 
 
 123. When successive series of observations are 
 made, day after day, of the same measurable quantity, 
 which is either invariable (as the angular distance be- 
 tween two components of a double star) or admits of being 
 reduced by calculation to an invariable quantity (as the 
 apparent angular diameter of a planet) ; and when every 
 known instrumental correction has been applied (as for 
 zero, for effect of temperature upon the scale, &c.) ; still 
 it will sometimes be found that the result obtained on 
 one day differs from the result obtained on another day 
 by a larger quantity than could have been anticipated. 
 The idea then presents itself, that possibly there has been 
 on some one day, or on every day, some cause, special to 
 the day, which has produced a Constant Error in the 
 measures of that day. It is our business now to consider 
 the evidence for, and the treatment of, such constant 
 error. 
 
ADMISSIBILITY OF CONSTANT ERROR. 103 
 
 124. The existence of a daily constant error, that is, 
 of an additional error which follows a different law from 
 the ordinary error, ought not to be lightly assumed. 
 When observations are made on only two or three days, 
 and the number of observations on each day is not ex- 
 tremely great, the mere fact, of accordance on each day and 
 discordance from day to day, is not sufficient to prove a 
 constant error. The existence of an accordance analogous 
 to a "run of luck" in ordinary chances is sufficiently pro- 
 bable. If this be accepted, as applying to each day, the 
 whole of the observations on the different days must be 
 aggregated as one series, subject to the usual law of error. 
 More extensive experience, however, may give greater con- 
 fidence to the assumption of constant errors ; and then the 
 treatment of which we proceed to speak will properly 
 apply. 
 
 125. First, it ought, in general, to be established that 
 there is possibility of error, constant on one day but vary- 
 ing from day to day. Suppose, for instance, that the 
 distance of two near stars is observed with some double- 
 image instrument by the method of three equal distances, 
 alternately right and left. It does not appear that any 
 atmospherical or personal circumstance can produce a con- 
 stant error; and, unless we are driven to it by considerations 
 like those to be mentioned in Article 129, we must not 
 entertain it. But suppose, on the other hand, that we have 
 measured the apparent diameter of Jupiter. It is evident 
 that both atmospheric and personal circumstances may 
 
104 MIXED ERRORS, AND CONSTANT ERRORS. 
 
 sensibly alter the measure; and here we may admit the 
 possibility of the error. 
 
 126. Now let us take the observations of each day 
 separately, and, by the rules of Articles GO and Gl, investi- 
 gate from each separate day the probable error of a single 
 measure. We may expect to find different values (the 
 mere paucity of observations will sufficiently explain the 
 difference); but as the individual observations on the dif- 
 ferent days either are equally good, or (as well as we can 
 judge) have such a difference of merit that we can approxi- 
 mately assign the proportion of their probable errors, we 
 can define the value of probable error for observations of 
 standard quality as determined from the observations of 
 each day; we must then combine these, with greater 
 weight for the deductions from the more numerous obser- 
 vations, and we shall have a final value of probable error 
 of each individual observation, not containing the effects 
 of Constant Error. From this we can, by the rule of 
 Article 55, infer the "Probable Error of Each Day's 
 Eesult ;" still not containing the effects of Constant Error. 
 The " Result of Each Day," also not containing any cor- 
 rection for Constant Error, is given by the mean of deter- 
 minations for each day. 
 
 127. We must now attach to the numerical value of 
 "Result of Each Day" a symbol for "Actual Error of 
 Result of Each Day;" and take the mean of all these 
 compound quantities, numerical and symbolical, for all the 
 days ; (the combination-weights being proportional to the 
 
PROCESS FOR DISCOVERY OF CONSTANT ERROR. 105 
 
 number of observations on each day, unless any modifying 
 circumstance require a different proportion). This mean 
 may be regarded as "Final Result." The " Final Result" 
 is to be subtracted from the "Result of Each Day;" the 
 remainder is the "Discordance of Each Day's Result." 
 For each day it consists of two parts; a number, and a 
 series of multiples of all the symbols for " Actual Error of 
 Result of Each Day." 
 
 128. Now treat the Discordance (consisting of the 
 number accompanied with multiples of symbols) as being 
 itself an Error, and investigate the " Mean Discordance " 
 by the rule of Article 26 or 59; a value of "Mean Dis- 
 cordance" will thus be obtained, consisting of a number 
 accompanied with a series of multiples of symbols of 
 "Actual Error." Consider each day's "Actual Error" as 
 an independent fallible quantity whose Probable Error is 
 that obtained in Article 126, and form the "Probable 
 Error of Mean Discordance" by the rule of Article 52. 
 Thus we have, for Mean Discordance, a formula consisting 
 of two parts, namely, 
 
 (1) A numerical value. 
 
 (2) A number expressing the probable error in the 
 determination of that numerical value. 
 
 129. And now it will rest entirely in the judgment of 
 the computer to determine whether the simple numerical 
 value (1) just found, is to be adopted for Mean Discordance 
 or not. It is quite clear that, if (2) exceeds (1), there is no 
 
106 MIXED ERRORS, AND CONSTANT ERRORS. 
 
 sufficient justification for the assumption of a Discordance, 
 that is, of a Constant Error. If (2) is much less than (1), 
 it appears equally clear that a Constant Error must be 
 assumed to exist, and (1) or any value near it may he 
 adapted for Mean Discordance. The Probable Discordance, 
 or Probable Constant Error, will be found by multiplying 
 this by 08153, as in Article 31. 
 
 130. The reader must not be startled at our referring 
 these decisions to his judgment, without material assist- 
 ance from the Calculus. The Calculus is, after all, a 
 mere tool by which the decisions of the mind are worked 
 out with accuracy, but which must be directed by the 
 mind. In deciding on the admissibility of Constant Error, 
 after giving full weight to the considerations of Article 129, 
 it will still be impossible, and would be wrong, to exclude 
 the considerations of Article 125, and these cannot be 
 brought under algebraical or numerical rule. 
 
 131. These investigations suppose that the "Dis- 
 cordance of Each Day's Result" cannot, so far as we know 
 antecedently, be referred to any distinct assignable cause. 
 But if there should appear to be any such cause, as, 
 for instance, if we conceive that the observations of one 
 person always give a greater measure than the observations 
 of another person, it will be easy to apply an investigation, 
 analogous to that just given. The observations of each 
 person should be separated from those of other persons and 
 collected together; from the collected group of each per- 
 son's observations, a Mean Result and Probable Error of 
 
DISTINCT CAUSE OF CONSTANT ERROR. 107 
 
 Mean Result for each person must be found ; and then the 
 reader must judge whether, in view of the amount of Pro- 
 bable Errors, a Personal Difference of Results is admissible 
 or required. The investigation is simpler than the preced- 
 ing, in this respect, that it arrives at a Simple Personal 
 Difference of Results, and not at a Mean Discordance. 
 And the result is simpler than the last, because it is a Con- 
 stant Correction to the results of one person, instead of an 
 uncertain correction liable to the laws of chance. 
 
 § 17. Treatment of observations when the values of Pro- 
 bable Constant Error for different groups, and probable 
 error of observation of individual measures within each 
 group, are assumed as known. 
 
 132. When numerous and extensive series of observa- 
 tions have been made, as in Articles 126, &c, sufficient to 
 determine the Probable Value of the so-called Constant 
 Error (which is in fact an Error varying from group to 
 group) and the ordinary probable error of an individual 
 observation in each group ; suppose that there are made 
 occasional observations, in limited groups, for which it is 
 desirable to define the rides of combination. We are not 
 justified, for each of these limited groups, in assuming a 
 value for the Constant Error, or Variable Error of the 
 Second Class, applicable to that group; we must treat it 
 as an uncertain quantity, and ascertain the combination- 
 weights, and the probable error and theoretical weight of 
 final result, under the effects of the errors of the two classes, 
 
108 MIXED ERRORS AND CONSTANT ERRORS. 
 
 by an operation analogous to those which are applied when 
 the errors are only of one class. 
 
 133. In the first group of observations, let the actual 
 value of the error of second class be ^G) in the second 
 group 2 C ; in the third group Z C, &c. ; the probable value 
 of each being c. And in the first group, let the actual 
 values of the errors of fir^st class (or ordinary errors) for the 
 successive observations be l E v r E 2 , X E 3> &c. ; for those in 
 the second group 2 E X , 2 E 2 , &c. ; the probable value of each 
 being e. And let the number of observations in the suc- 
 cessive groups be jn, 2 n, &c. Let the combination factors 
 be t 1} x z % , t z 3 , &c; 2 z v 2 z 2 , 2 z 3 , &c. ; # xt z z 2 , z z 3 , &c. ; and 
 so for successive groups. 
 
 Then the actual errors of the separate measures will be 
 &c. 
 &c. 
 
INVESTIGATION OF COMBINATION- WEIGHTS. 109 
 
 And the actual error of the final result, obtained by 
 combining the separate measures with the combination- 
 weights above given, will be the fraction, whose nume- 
 rator is 
 
 (a + A+ A + &c -) i#+ (a + a + &c ^ 
 
 + (A. 1 ^ + A-A + A-A + &c.) 
 
 + <a-A+a-A+&°.) 
 
 + (A>*^i + «■»:«*■ + **) + *«■ 
 
 and whose denominator is 
 
 (A + A + A + &C + (A + 2* 2 + &c + (A + 3 ? 2 + &c) +&c. 
 
 134. The square of the probable error of the final 
 result, found in exactly the same way as in all preced- 
 ing cases, will be the fraction whose numerator is 
 
 + { (A) 2 + (A) 2 + & c.} e 2 + {(^Y + (, /2 ) 2 + &c.} e 2 
 
 and whose denominator is 
 {(A + A + &c + (A + 2 * 2 + &c -) + (A + s* 2 + &c -) + &c.} 2 . 
 
 This is to be made minimum with respect to the varia- 
 tion of each of the quantities x s x , x z# &c, „z v „z 2 , &c, 
 9 Z i> s z -2> & c - & c - Differentiating with respect to each, 
 making each differential coefficient = 0, and treating as 
 in former instances, we find successively, (putting A for 
 an indeterminate constant), 
 
110 MIXED ERRORS AND CONSTANT ERRORS. 
 
 First, x z x = x z % = x ~3 = &c. 
 
 therefore, for each of these we may use the symbol x z. 
 Second, x n. x z.c 2 + x z.e 2 = A, 
 
 2 n. 2 z.c 2 + 2 z.e 2 = A, 
 s n. 3 z.c 2 + ii z.e 2 = A, 
 &c. 
 
 from which we obtain 
 
 _A_ 
 
 xZ x n . c 2 + e 2 ' 
 
 which is applicable to every observation in the first group ; 
 
 A 
 
 which is applicable to every observation in the second 
 group ; and so on through all the groups. 
 
 135. In the numerator of expression for the square of 
 probable error of result, if for x z x , x z 2 , &c, we insert x z, and 
 so for other groups, it becomes 
 
 y . / . c 2 + 2 n 2 . / . c~ + &c. + x n . / . e 2 + ji . / . e 2 + &c. 
 
 = A ( x n . x z + 2 n . „z + &c), 
 
 and the same substitution converts the denominator to 
 
 d u • i z + 2 n • 2 Z + ^ c -) 2 i 
 and the square of probable error of result 
 
 A 
 
 x n . x z + ji . 2 z + &c. ' 
 
DISTINCT CAUSE OF CONSTANT ERROR. Ill 
 
 which with the values of x z, 2 z, &c. found above, becomes 
 1 
 
 77 + Vt-5 + &C. 
 
 Or 
 
 1 1 
 
 (probable error of result) 2 
 
 c -\ — c + 
 
 136. If, as in Article 131, we conceive that we can 
 fix upon some distinct cause of Constant Error for one 
 group, all the others being assumed free from Constant 
 Error, aud can ascertain with confidence the amount of 
 the Constant ; that group of results may then be reduced 
 by application of the Constant. For the determination 
 of the probable error of the result of the group so cor- 
 rected, it must be borne in mind that the determination 
 of the Constant is liable to error. Let A, B, C, D, &c. 
 to n terms, be the actual errors, and a, b, c, d, &c. the 
 probable errors of the means of various groups, A cor- 
 responding to that in which we suspect sufficient reason 
 for assuming a Constant Error. The actual error of de- 
 termination of Constant Error will be 
 
 . J3+C+D + &C. 
 
 A-- — , 
 
 ii — 1 
 
 and the probable error of determination of Constant Error 
 will be 
 
 b~ + <y + d~ + &c. 
 
 V? 
 
 + 
 
 (n-lf 
 
112 CONCLUSION. 
 
 But the corrected result of that group will be liable to the 
 actual error 
 
 _B+ C+D + &C.) B+G+D + &C. 
 
 n — 1 J n - 
 
 and its probable error will be 
 
 ft* + c 2 + d* + &c. \ 
 
 1 (n-iy r 
 
 In fact, by referring that result to the mean of other re- 
 sults, and so determining its correction, we entirely deprive 
 that result of any original value in the application of these 
 groups. 
 
 But if, on another occasion, there were observations 
 made by the same person or under the same circumstances 
 as the observations A, then the determination of Constant 
 Error and of its probable error just found would be pro- 
 perly applicable. 
 
 These conclusions will be varied according to the 
 various assumptions made ; the reader will have little dif- 
 ficulty in applying the theory of preceding Articles to any 
 of them. 
 
 CONCLUSION. 
 
 137. In the practical applications of the Theory of 
 Errors of Observations and of the Combination of Obser- 
 vations which have fallen under our notice, the following 
 are the principal sources of error and inconvenience. 
 
ERRORS TO BE AVOIDED. 113 
 
 (1) In some instances, measures have been combined 
 by a method of " minimum squares " without reference to 
 the value of probable error of each of the separate obser- 
 vations ; and an erroneous result has been deduced. The 
 computer, apparently, has had his attention engrossed by 
 " minimum squares " as the important result to be ob- 
 tained ; whereas, in reality, the satisfying of the equations 
 for minimum squares produces a merely accidental coinci- 
 dence of results in certain cases (not in all) with those 
 leading to the " minimum probable error of final result," 
 which is the legitimate object of search. 
 
 (2) In some instances, entangled observations have 
 been treated as if they were independent, and an erroneous 
 result has been inferred. 
 
 (3) In some instances, the labour of application of the 
 theory has been greatly and unnecessarily increased by the 
 use of numerical coefficients proceeding to several places 
 of decimals ; when simple factors would have given results 
 possessing all desirable accuracy. 
 
 We believe that, by avoiding these errors, and by 
 otherwise conforming to the principles of this Treatise, 
 the Theory of the Combination of Observations may, 
 without great labour, be made a valuable aid in the com- 
 putation of Physical Measures. 
 
APPENDIX. 
 
 PRACTICAL VERIFICATION OF THE THEORETICAL LAW 
 FOR THE FREQUENCY OF ERRORS. 
 
 With the view of examining the practical accuracy of the 
 
 formula Axe~c 2 .8x (see Article 24 and preceding Articles) 
 for the frequency of the occurrence of Errors of Measure &c. 
 between the error-magnitudes x and x + &x, I collected the 
 results (636 in number) of all the observations of the N.P.B. 
 of Polaris made at the Royal Observatory in the years 1869 
 to 1873, as reduced in each year to exhibit the mean N.P.D. 
 at the beginning of the year from every observation in the 
 year. In every separate year the difference between each of 
 these mean N.P.D. and the annual mean of all was taken. 
 Prom the large number of observations in each year, and from 
 the perfect certainty as to the elements of reduction from one 
 year to another, it was evident that there would be no appre- 
 ciable error in considering these " differences from the mean " 
 in each year as identical with the " diffei'ences from the mean " 
 of all which would have been obtained if all the results had 
 been referred to one epoch of time and treated in one group. 
 I therefore extracted from the various groups all the " differences 
 from the mean," and arranged them in order of magnitude, 
 from the largest negative "difference" — 2"*35 to the largest 
 positive "difference" + 3"*51. These may be considered as 
 veritable errors of observation. The sum of negative errors 
 was-215" - 88, and the sum of positive errors +213"*73 (the 
 
APPENDIX. 115 
 
 small discordance between them arising from the loss of figures 
 
 in the succeeding decimal places). The mean error or 
 
 215"-88 + 213"-73 ™ „* eK * ,., , , 
 
 -^rp; was = -0755 : from which, by the table 
 
 ooo 
 
 in Article 31, the Modulus was found to be 1" , 1973, and the 
 
 Probable Error 0"'57 11. 
 
 The errors were then divided into small groups, each group 
 extending over an error-range of 0" # 05 ; from 0" - 03 to 0" - 07, 
 0"*08 to 0" - 12, 0"-13 to 0""17, and so on, both in the positive 
 and in the negative direction. But, as the frequency of errors 
 for the large values was very small, one group was extended 
 from - 2"-38 to - 2"-13, one from - 2"-12 to - l"-98, one from 
 - l"-97 to l"-73, one from - 1"'72 to - l"-58, one from + l"-58 
 to + l"-72, one from + 1"73 to + l"-97, one from + l"-98 to 
 + 2"-12, one from + 2"\L3to + 2"-38, one from +2"-39 to +3"-58. 
 
 The only result extracted from these groups was, the number 
 of observations in each group : and this was considered as 
 representing the frequency through an error-range of 0"'05, 
 corresponding in formula to the magnitude of the central error 
 of the group as the independent variable : thus the number of 
 errors between 0""63 and 0"*67 was taken to represent the 
 frequency through an error-range of 0"-05, which must cor- 
 respond in any mathematical formula to the independent 
 variable 0""65. For those cases in which longer groups were 
 employed, the actual number of observations was reduced so 
 as to make it justly comparable with the number of obser- 
 vations in other parts of the series of groups : thus in the group 
 extending from — 2"'38 to 2"-13, which extends over an error- 
 range of 0""25, or five times the ordinary error-range, the 
 actual number of observations was divided by 5 to make it 
 comparable with the others, and the resulting quotient was 
 held to correspond to the error or independent variable — 2" - 30. 
 
116 APPENDIX. 
 
 It soon became evident that there was no marked dis- 
 cordance between the laws of distribution for positive and for 
 negative values : and therefore the corresponding numbers 
 were added together. Then, in order to remove small irregu- 
 larities, the first number was added to the second, the second 
 to the third, &c, and then the first sum was added to the 
 second, the second to the third, &c, the first sum of the second 
 order being held to correspond to the second original number ; 
 and so throughout. The extreme first and last were adopted 
 without change. The numbex^s thus formed are evidently, on 
 the whole, eight times as large as the original numbers. The 
 numbers thus produced were laid down in a graphical repre- 
 sentation, in which the abscissa was the magnitude of the 
 "difference," or error of observation, and the ordinate was 
 the corresponding number for eight times the frequency. Then 
 a free curve was drawn by hand passing through the points 
 representing these numbers. And this terminated the re- 
 ference to the facts of observation. It appeared that, in the 
 hand-drawn curve, the ordinate for error — might be taken 
 as 124. 
 
 For a similar exhibition of the results of theory, or of the 
 
 numbers given by the formula A x e <? 3 , where c = Modulus 
 = 1-1973, and where A evidently =124, it was only necessary 
 to calculate the expression, 
 
 -** 
 frequency = 124 xe (1 ' 1973) \ 
 or 
 
 log. frequency =2-0934217- ^^r^ x (Error) 2 . 
 
 This calculation was made for every value of Error 05, 
 0-10, 0-15, <fcc. to 1-G5, and then for 1-75, 2-10, 2-30, 2-50. 
 
APPENDIX. 
 
 117 
 
 The comparison of the observed and the theoretical results 
 is given in the followin» table. 
 
 Error of Observation. 
 
 Observed Frequency of Errors 
 multiplied by the factor 8, 
 corresponding to the error- 
 range = 0"'05. 
 
 Ordinate of the Frequency- 
 curve drawn by hand. 
 
 o 
 a . 
 
 .2 5" 
 
 H 
 
 o-oo 
 
 112 
 
 124 
 
 124-00 
 
 0-05 
 
 127 
 
 124 
 
 123-78 
 
 o-io 
 
 135 
 
 123-5 
 
 123-14 
 
 0-15 
 
 129 
 
 122 
 
 122-07 
 
 0-20 
 
 125 
 
 120 
 
 120-59 
 
 0-25 
 
 116 
 
 118 
 
 118-71 
 
 0-30 
 
 114 
 
 116 
 
 116-45 
 
 0-35 
 
 106 
 
 1135 
 
 113-84 
 
 0-40 
 
 95 
 
 110 
 
 110-90 
 
 0-45 
 
 90 
 
 105-5 
 
 107-67 
 
 0-50 
 
 88 
 
 102 
 
 104-16 
 
 0-55 
 
 97 
 
 98 
 
 100-41 
 
 0-60 
 
 105 
 
 93 
 
 96-46 
 
 0-65 
 
 104 
 
 88-5 
 
 92-35 
 
 0-70 
 
 91 
 
 84 
 
 88-10 
 
 0-75 
 
 75 
 
 80 
 
 83-76 
 
 0-80 
 
 72 
 
 76 
 
 79-35 
 
 0-85 
 
 69 
 
 71-5 
 
 74-91 
 
 0-90 
 
 55 
 
 66-5 
 
 70-47 
 
 0-95 
 
 52 
 
 62-5 
 
 66-07 
 
 1-00 
 
 68 
 
 58 
 
 61-73 
 
 1-05 
 
 67 
 
 54 
 
 57.47 
 
 1-10 
 
 53 
 
 50 
 
 53-32 
 
 115 
 
 48 
 
 46 
 
 49-29 
 
 1-20 
 
 39 
 
 42 
 
 . 45-41 
 
 1-25 
 
 31 
 
 38-5 
 
 41-69 
 
 
 ''" i 
 
 
 
 
 2°°"S 
 
 b 
 
 
 d 
 
 A £s h 
 
 £t3 
 
 »_ 
 
 
 of 
 
 fact 
 le e 
 5. 
 
 y 
 
 a> 
 
 0) 
 
 gS-So 
 
 £j? 
 
 
 1 
 
 g, S»M II 
 
 2 P 
 
 "ca S 
 
 
 
 
 
 o 
 
 fa s a a 
 
 Oro 
 
 2-£ 
 
 U 
 
 o 
 s 
 
 ■a a g 2 
 
 IE 
 
 v 
 
 w 
 
 £ll 
 
 •S3 
 
 H 
 
 
 .238 
 o 
 
 o 
 
 
 1-30 
 
 29 
 
 35-5 
 
 38-14 
 
 1-35 
 
 33 
 
 325 
 
 34-78 
 
 1-40 
 
 38 
 
 30 
 
 31-60 
 
 1-45 
 
 35 
 
 27 
 
 28-61 
 
 1-50 
 
 30 
 
 24 
 
 25-81 
 
 1-55 
 
 22-3 
 
 22 
 
 23-21 
 
 1-65 
 
 132 
 
 18 
 
 18-56 
 
 1-75 
 
 8-4 
 
 14-5 
 
 14-64 
 
 2-10 
 
 6-4 
 
 6 
 
 573 
 
 2-30 
 
 4-1 
 
 2-5 
 
 310 
 
 2-60 
 
 0-4 
 
 o-o 
 
 1-11 
 
 To exhibit more clearly to the eye the result of this com- 
 parison, the following diagram is prepared. 
 
Ordinate representing the number of errors in each group ranging 
 through 0-"05 of magnitude, multiplied by the factor 8. 
 
APPENDIX. 119 
 
 It is evident that the formula represents with all practicable 
 accuracy the observed Frequency of Errors, upon which all the 
 applications of the Theory of Probabilities are founded : and 
 the validity of every investigation in this Treatise is thereby 
 established. 
 
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