Z7S 
 3e 
 
 UC-NRLF 
 
 $C IbS 112 
 
v^ 
 
 '^..^-^ 
 
 ON 
 
 THE USE OF 
 
 EQUIVALENT NUMBERS 
 
 IN THE 
 
 METHOD OF LEAST SQUARES. 
 
 BY 
 
 GEORGE P. BOND, A.M., 
 
 ASSISTANT AT THE OBSERVATOBY OF HARVAKD COLLEGE. 
 
 [From the Memoirs of the American Academy of Arts and Sciences, New Series, Vol. YI.] 
 
 CAMBRIDGE: 
 METCALF AND COMPANY, 
 
 PEINTEBS TO THE UNIVERSITT. 
 
 1856. 
 
ON 
 
 ^^/y :/^^ /' 
 
 THE USE OF 
 
 EQUIVALENT NUMBERS 
 
 IN THE 
 
 METHOD OF LEAST SQUARES. 
 
 BT 
 
 GEORGE P. BOND, A.M., 
 
 ASSISTANT AT THE OBSEEVATORY OF HAKVAED COLLEGE. 
 
 [From the Memoirs of the American Academy of Arts and Sciences, New Series, Vol. VI.] 
 
 CAMBRIDGE: 
 METCALF AND COMPANY, 
 
 PBINTEBS TO THE UNIVEESITT. 
 
 1856. 
 

 
ON THE 
 
 USE OF EQUIYALENT NUMBERS 
 
 METHOD OF LEAST SQUAHES. 
 
 One of the most important applications which has been made of mathematics to 
 investigations in physical science has for its object to ascertain the best manner of 
 combining data affected by unknown errors of observation, so that the probable effect 
 of these errors shall be the least possible. The method of least squares proposes 
 to accomplish this, by reducing to a minimum value the sum of the squares of the out- 
 standing errors, and, by conforming to this single criterion, to fulfil the condition, so 
 desirable in the prosecution of thorough and exact research, of reducing to its least 
 possible amount the influence of errors in the data employed. 
 
 The investigations here presented have been entered upon with the design of deter- 
 mining the degree of numerical exactness proper to be observed in making use of the 
 method of least squares, in order to secure its peculiar advantages with the least out- 
 lay of labor. 
 
 Some detail in the discussion seems to be called for from the prevalence of a practice, 
 almost universal among computers, of adhering to the letter of the method of least 
 squares with a strictness which implies a misapprehension of its true spirit. It is 
 impossible to adduce any valid reasons to justify such a course when it must be fol- 
 lowed at a serious expense of time and labor in the computations. 
 
 It has not escaped the observation of Gauss, in his original exposition of the method, 
 that some freedom of interpretation may be allowed when its theoretical results are 
 applied in practice, as the following passage, referring to the solution of equations by 
 least squares, will show : — 
 
 I/) 9 1.5 9 
 
4 ON THE USE OF EQUIVALENT NUMBERS 
 
 " When the number of functions or equations proposed for solution is considerable, 
 the computations become laborious, the more so from the circumstance that the co- 
 efficients by which the primitive equations are to be multiplied are almost always com- 
 plicated decimal fractions. If it is not thought worth the trouble in such a case to 
 calculate the products with exactness by means of logarithms, it will generally be suf- 
 ficient to substitute for them (i. e. for the multiplying factors) more simple numbers 
 differing but slightly from them." * 
 
 In his subsequent researches, it does not appear that Gauss has given any further 
 development to the suggestion here put forth. Indeed, the introduction of modifications 
 of a like nature, however desirable in a practical point of view, would have deprived 
 a purely theoretical discussion of much of its elegance and symmetry. Yet the passage 
 above quoted lends the support of the highest authority to the leading proposition 
 which we shall have occasion most frequently to insist upon ; namely, the propriety of 
 allowing some relaxation of theory in applying the calculus of probabilities to the dis- 
 cussion of data affected by ordinary errors of observation, whenever the modification 
 conduces to convenience and the saving of labor at the sacrifice of no appreciable ad- 
 vantages. 
 
 Even an unqualified admission of the superior probability of results which exactly 
 fulfil the criterion proposed in the method of least squares, does not relieve us from the 
 necessity of restricting it to examples which never actually occur, that is, if the ques- 
 tion be made a rigorous one ; f — to such, for instance, as involve the discussion of obser- 
 vations which are entirely free from unknown constant errors, or errors foUomng any 
 law of facility which does not imply the assumption that the mean error is proportional 
 to the square root of the mean of the squares of the individual errors. But we know 
 that this proposition, which lies at the foundation of the whole subject, is not suscepti- 
 ble of absolute demonstration by any process of mathematical reasoning. Further than 
 this, we know from constant experience that the law of distribution of errors recognized 
 in the method of least squares practically faUs, in extreme cases, both for very large and 
 for very small errors. If any illustration of the failure of the assumed law be needed, 
 it will be found in the familiar instance of computing by it the probable error of the 
 arithmetical mean of a very large number of observations, where common sense assures 
 us that the theoretical probable errors of the result are invariably smaller than they 
 should be. 
 
 Why, then, should an implicit adherence to its minutest details be required as essen- 
 
 * Theoria Motus, § 185. t Theor. Comb., % 17. 
 
IN THE METHOD OF LEAST SQUARES. 9 
 
 tial to its successful application, or to the attainment of all the advantages which its 
 employment may confer upon the discussion of any practical problem ? 
 
 It is true that no other system can be proposed which is free from similar objections, 
 or which can be mathematically demonstrated to be exclusively the best, without quali- 
 fication, and therefore the arguments above stated are of no force whatever, if employed 
 as reasons for the rejection of the method of least squares. They nevertheless greatly 
 weaken the position of those who Avould insist upon a strict compliance with its pre- 
 cepts, and effectually preclude all alignments of a purely theoretical character in support 
 of such a course. Still it is desirable that the force of any objections which may be 
 made to an attempt to modify the theoretical conditions for eiFecting the most favorable 
 combination of equations should be appreciated at their true value. We therefore pro- ^ 
 pose to show that the spirit of the method of least squares, rightly apprehended, in 
 reality rather invites than discountenances a liberal construction of its rules. 
 
 Admitting that the best possible solution is attained when the sum of the squares 
 of the outstanding errors, represented by S2, is a minimum, it is evident that /2 is a 
 minimum relatively to the manner in which the original equations have been treated. 
 And since the peculiarity of the solution consists in the employment of a system of 
 factors, a, a, &c., by which the original equations are multiplied before combination, 
 the first differential coefficient of fl relatively to either of these factors, in the case of 
 the least-square solution, must have the value for each factor, 
 
 dJl 
 a a 
 
 When, therefore, the factors are varied by small amounts, Ba, B a, &c., the conse- 
 quent variations of fl developed in a series, will contain only terms multiplied by the 
 second and higher powers of 8 a ; or, in general terms, if we deviate from the exact 
 (1.) precepts of the method of least squares hy small variations of the first order, we shall fail 
 to satisfy its fundamental criterion hy small terms of the second order only. 
 
 Looking thus at the most elementary principle of the method, we find a warrant 
 for some degree of liberty in applying it, — a liberty which we can scarcely hesitate to 
 avail ourselves of, if we further consider the peculiar circumstances attending its actual 
 employment in the discussion of data furnished directly by observation. 
 
 Among its first requirements is the assignment of weights to the original observations ; 
 but it is one which it is not possible to fulfil correctly, for we are provided neither with 
 a theory nor with data for the purpose. All that can be done is to accept, as indices of 
 the relative value of the different observations, certain numbers depending either proxi- 
 mately or remotely upon no other authority than the mere exercise of the judgment 
 
 2 
 
b ON THE USE OF EQUIVALENT NUMBERS 
 
 alone. No one can pretend that this is a process susceptible of strict accuracy ; yet an 
 error here is as fatal as if we had disregarded any other of the precepts of the method. 
 
 This step being an arbitrary one, although one of fundamental importance, we may 
 properly appeal to it as a precedent for the modification of others suggested by con- 
 siderations of convenience, though they may not, like this, be justified on the plea of 
 actual necessity. In this view of the subject, we find support for the modification 
 suggested by Gauss, in the passage we have quoted above. Each of the complicated 
 factors which it is there proposed to simplify is itself a product of two other factors, 
 one of which is the weight of the equation under treatment ; if one of these, that 
 is, the number representing the weight, is erroneous, the product is of course errone- 
 ous, with whatever accuracy the other is expressed. 
 
 Again, as a matter of convenience, it is usual to express the conditional equations 
 proposed for solution in a linear form, by reducing the indeterminates to small quanti- 
 ties and neglecting the terms multiplied by their second and higher powers, and to con- 
 struct, from them normal equations, as they are called, previously to applying the 
 method of least squares. Both of these may be practices perfectly allowable under the 
 circumstances, but since they are almost always theoretically incorrect, their admis- 
 sion is a virtual relinquishment of all pretensions to a rigorous course of computation, 
 and cannot be compensated for by any subsequent refinements. 
 
 We will now proceed to examine the limits of accuracy appropriate to the arithmeti- 
 cal operations required in the combination of conditional equations by the method of 
 least squares, and afterwards to develop in detail some proposed modifications of that 
 method, having for their object the reduction to its minimum value of the amount of 
 labor requisite for its successful application. 
 
 It is scarcely necessary to remark, that the subject .is plainly one which is in its 
 nature somewhat vague and insusceptible of rigorous treatment, though it is at the 
 same time interesting from its practical bearings. If no very precise or definite rules 
 for regulating the degree of numerical exactness suited to the discussion of any given 
 problem can be arrived at, it may still be of service to point out the principles which 
 ought to guide the computer in the choice of such limits as shall perfectly meet all 
 reasonable requirements of accuracy, without imposing upon him the unprofitable labor 
 of multiplying the extent and difficulties of calculation, to no useful purpose, and 
 without the remotest prospect of sensibly improving the real value of the results. 
 
 Let us suppose a series of equations, 
 
 a X -\- b y-j- -\- m =e, 
 
 a' X -|- J' y -j- -\- m' = e'l 
 
IN THE METHOD OF LEAST SQUARES. 7 
 
 in which m is the element demed from observation, and e the unknown error of the 
 equation, to be solved by the method of least squares, giving for x the value x, with its 
 probable error, £, obtained from a compai-ison between the observed and the computed 
 values of m, after substituting x, y, &c. in the primitive equations. 
 
 If Xq be the true value of x, we may represent by x a quantity such that it is 
 an even chance whether Xq — x is comprised between the limits s ~\- x and f — x. 
 The magnitude of the limit defined by x has an evident relation to the question how 
 far the simplification of the arithmetical processes may be carried without detriment 
 to the results. 
 
 For instance, the solution of the above equations may be repeated with small varia- 
 tions from the process at first applied, giving for x a new value Xi with a probable 
 error E^ , differing but little from a. If we were in entire ignorance of the relative 
 amount of the probable errors f, and s, there would be no reason at all for giving 
 the preference to x rather than to Xi . If only the single circumstance were known 
 that fi exceeded £ by a given small amount, we should be equally at a loss, while 
 the value of s remained unknown, to state the relative weight of Xi compared with 
 X, and should, in fact, be again obliged to resort to the hypothesis that £ and fj were 
 sensibly equal. And in general, the greater the uncertainty of f, or, in other words, 
 the larger the value of x, the less reason would there be for excluding from competition 
 with X any other determination of x, such as x^, of which the probable error e^ dif- 
 fered but little from s. 
 
 In order to employ the limit x as here proposed, its value must be known before the 
 computations have reached an advanced stage. That this is not ordinarily practicable 
 will readily appear. On the other hand, it must be left entirely to the judgment of the 
 computer to decide as to the precise manner in which x is to be applied in limiting the 
 allowable amount of difference fj — £. 
 
 Objections of a similar character apply equally to other standards which might be 
 proposed for the same object. As has before been remarked, the question must be 
 treated, if at all, upon a somewhat arbitrary basis, and we must be content with sug- 
 gestions addressed to the judgment or common sense of the computer, in cases where 
 no fixed rule is admissible. 
 
 Viewed in this light, there will ordinarily be no difficulty in recognizing the point 
 at which there will be danger of compromising accuracy in the attempt to simplify the 
 computations, nearly enough at least for practical purposes, if we are prepared to 
 admit, at least in its general spirit and tendency, the truth of the following propo- 
 sition : — 
 
O ON THE USE OF EQUIVALENT NUMBERS 
 
 The application of the method of least squares to the discussion of observations of physi- 
 cal phenomena, with the exception of a few special cases of rare occurrence, requires the 
 (2.) use of such numbers only, in the arithmetical processes peculiar to it and characteristic of 
 
 the method, as may be designated by one of the numerals 0, 1, 2 9, or of the fractions 
 
 — , — —, or by a product of one of these numbers by an integral power of 10. 
 
 An idea may be formed of the amount of the intentional errors occasioned by these 
 substitutions, by noticing that if by N is represented any number whatsoever, and by 
 N' a number chosen from the proposed series which most nearly coincides with N, we 
 shall have 
 
 N— isr 1 
 
 /„ , The maximum value of — — := - nearly. 
 
 \o.) iV " 
 
 JV — iV' 1 
 The probable value of — zr^ — <^ — . 
 
 Before proceeding to a detailed investigation of the consequences of the changes pro- 
 posed, it will be useful to point out the degree of insecurity attaching to the values 
 which must ordinarily be adopted to represent the probable error of x ; the different 
 sources which may be supposed to contribute to the increase of s ; and their relative 
 importance in connection with the question of the comparative accuracy of the two 
 results X and Xi. 
 
 s may be referred to the combined influence of two mutually independent errors 
 r) and rj, T) being the probable value of Xg — x which would result from the errors of 
 observation alone, supposing the theory of the method of least squares and its applica- 
 tion to the data to be rigorously exact, and V the probable amount of error in x having 
 its origin in errors necessarily committed in the discussion of the observed data, sup- 
 posing the mode of discussion, although the best practicable, to fall short of strict 
 conformity with the theory. tjI represents the value of rf when the same data have 
 been reduced, by a process made intentionally still less exact, to a small extent, both 
 in its theory and in its arithmetic, than that which gives the error •??'. tjI will bear 
 to Xi a relation similar to that which ■>/ bears to x. j; cannot be completely eliminated, 
 so long as the errors of observation remain unknoAvn, by any treatment, and the same 
 may be said of rf ; but 7)1 can always be reduced to its least limit, •»?', by suitable refine- 
 ments of theory and of computation. In view of the fact that ij and rj' must have 
 always sensible, but very uncertain values, it will be of but little consequence that 
 ij'i should be reduced to its utmost limit without regard to the labor and inconvenience 
 which it may cost. At all events, the attempt will be ineffectual as a means of im- 
 proving the substantial accuracy of the results, as we shall presently see. 
 
IN THE METHOD OF LEAST SQUARES. 9 
 
 Since ■;; is independent of ij and Vi » we have, assuming ?;' to be the least attainable 
 value of 9;'i, 
 
 If rl' be used to designate the probable value of x — x^ vphich would result from 
 small intentional deviations from that treatment of the data which is recognized to be 
 the best, we have 
 
 (4.) ,;a = ,'^ + ,"2, ,2 ^,,2 + ,'3^ ,,2 = ,2 ^ ,'2 _|_ ,"2. 
 
 As regards the uncertainty of £, some estimate of its extent may be obtained in the 
 following manner. 
 
 If it is an even chance that the error of which the probable value is 17 is comprised" 
 somewhere between the limits ?? -f- \ and 17 — X, «? having been derived from compari- 
 sons of a given system of equations with observation, the number of individual equa- 
 tions thus compared being represented by n, and the number of unknown quantities 
 entering into them by «', \ may be found from the expression* 
 
 (5.) X = 0.477 
 
 ^/n — nf 
 
 Any value of « — w' less than 100 gives 
 
 ^ 21 ' 
 
 The scale of substituted numbers (2) admits, as we have before stated, of represent- 
 ing 17 within the probable amount of — ??; hence, for any value of ?i — «' less than 100, 
 the series will afford numbers representing t? with a probable error less than \. A 
 slight examination M'ill show that a similar remark applies still more decisively to £. 
 
 The considerations which oblige us to attribute a sensible value to ij are too many 
 and too obvious to require to be specified in detail. It will be sufficient to cite one or 
 two which have already been alluded to. The existence of unknown constant errors in 
 the data will render the application of the method of least squares, strictly speaking, 
 inexact. From this source »?' will inevitably acquire some influence. Again, the un- 
 certainty incident to any attempt to assign to the original data their proper relative 
 weights, will have a similar effect. No process more loose and arbitrary can well be 
 conceived, than that by which the relative precision of the elements afforded directly by 
 observation is graduated. Yet, imperfect as it is, improvement in this particular is 
 scarcely to be hoped for. Exact conformity with a theory which requires a previous 
 knowledge of the relative weight of observations is quite impossible. 
 
 • Gauss, Zeitschrift fiir Astr., B. I. Theor. Comb.,§ 40. 
 
10 ON THE USE OF EQUIVALENT NUMBERS 
 
 At the same time, theij, that the existence and influence of i?' are admitted, its amount 
 is altogether uncertain, to an extent suiRcient at least to make the uncertainty of e 
 which is dependent on that of 17 and 7?' not less in proportional amount than that of ij ; 
 consequently we shall obtain from (5) the expression 
 
 (5 a.) x> 0.477 
 
 ■v/n — n' 
 
 by which to measure the uncertainty of e. If n — n' < 100, 
 (55.) K>^.. 
 
 When £ is represented by a number chosen from the series (2), the probable error of 
 the representation is, by (3), less than — £ ; in other words, it is more than an even 
 chance that this number will fall within the limits s -\- —£ and £ — — £ ; and since 
 
 I 25 25 
 
 the inherent uncertainty of £ makes it more than an even chance that its actual value is 
 outside of the limits £ -f- ^ « and £ — — f , in accordance with the above determination 
 of X, we conclude that £ can be represented by one of the series of numbers 0, 1, 2 
 
 9, or of the fractions -, - -, or by a product of one of these numbers by 
 
 an integral power of 10, with more accuracy than we can determine its amount by 
 one hundred comparisons between the observed and the computed values of m. It 
 would be easy to show, from the probable existence of constant errors alone, that an 
 indefinite increase of the number of comparisons with observation would not sensibly 
 diminish the uncertainty of £ below the amount stated. The proposition (2) would thus 
 be sustained, as far as relates to all expressions for probable errors and weights, since 
 they must depend upon conditions similar to those limiting the accuracy of f . 
 
 An immediate consequence of this admission will be the extension of the proposition 
 in question, in the qualified sense, at least, in which alone it is to be understood, to all 
 other arithmetical expressions required in the application of the method of least squares, 
 since the peculiar province of the latter is restricted entirely to the solution of equations 
 of the form 
 
 (5 c.) a (a; — ar,) -f J (y — y,) 4- -f (m — m,) = e, 
 
 in which each separate term and factor may be defined as proposed in (2). 
 
 To illustrate this, let us suppose for the moment that <r, has been derived from the 
 same primitive equations, but by an essentially difiierent process from that by which 
 X has been obtained ; Xi would still be precisely equal to x, if it were not for the errors 
 e, e', &c. Any such process, not intentionally bad, must evidently lead to a determina^ 
 tion of .Vi differing from x by an amount of an order not higher than that of f, while 
 
IN THE METHOD OF LEAST SQUARES. 11 
 
 one adopted expressly on account of its good qualities, though without bearing any in- 
 tended resemblance to the method of least squares in its characteristic features, will 
 diminish the difference x — Xi to a value either less in absolute amount than f , or very 
 nearly equal to it; so that (cZ? — Xi), {y — ^J, &c. may be sufficiently well expressed 
 by the series (2) in the same sense that s may be. The equations 
 
 ax -\-hy -\- -\- m ^ e, 
 
 axi + hyi-]- -f m, = 0, 
 
 in which e is the value of e obtained by substituting x, y, &c. in the primitive equa- 
 tions, give at once 
 
 (5 c.) a (a: — a;,) + J (y — y,) + + (m — m,) = e, 
 
 The solution of which by the method of least squares gives {x — ^i), {y — ^i) , 
 
 and thence 
 
 X ^= Xi -\- {x — a;, ) , 
 
 y = yi + {y — yd,, 
 
 Therefore the application of that method may be confined to the equations (5 c) alone. 
 
 But by what has already been said, (x — x^), (y — yi) , and e, are quantities 
 
 sufficiently well expressed by the series (2), from which we readily infer that there can 
 be rib appreciable advantage in giving to (m — m,) or to the products a(x — Xi), 
 b(y — yi), &c., or to the factors a, b, &c., any higher exactness of expression. Hence 
 the proposition (2) may be extended to every term and factor of the equations (5 c), 
 and therefore to all the niunerical processes peculiar to, arid characteristic of, the meth- 
 od of least squares. 
 
 This extreme application cannot, however, be recommended even on the ground of 
 convenience or simplicity; on the contrary, the indiscriminate use of the fractional 
 terms of the series would often be highly inconvenient ; and a form of solution like 
 that just indicated would not always be deskable. 
 
 In considering the different sources of error from which s and Si acquire their value, 
 it will be convenient to compare the increase given to s by the introduction of small 
 intentional inaccuracies, in consequence of which s becomes Si, either with f itself or 
 with £ — 17, by means of the following relations derived from (4) : — 
 
 (6.) 
 
 «i — 
 
 <K:-)'' ^:<(^y 
 
12 ON THE USE OF EQUIVALENT NUMBERS 
 
 Used in connection with (3) and the limit defining the uncertainty of s, 
 
 X ■> 0.477   , or usually x > — £. 
 
 -^ sfn — n' 21 
 
 «, will thus be comprised between the limits £ -f- ^ x and t — ^ x, when 17" has such a 
 value that 
 
 \ £ J 10 n — n'' 
 
 The relative accuracy of x and x^ will now be investigated for some special exam- 
 ples of deviation from a strict compliance with the method of least squares. 
 Let the equations proposed for solution be the following : — 
 
 ax-j-iy-j- 4-m =e, weight = w, 
 
 (7.) a' x' + J' y + + ?n' = e', " = w', 
 
 where e is the difference between the observed and computed value oi m ; m being the 
 element derived from observation. 
 
 In solving these by least squares, the final equation for x is formed by taking the 
 sum of all the equations after multiplying the first by a w, the second by a w', and 
 so on, and then making 
 
 a w e -\- a' w' e' -\- =0, 
 
 and for y 
 
 h w e -\- I' w' e' -\- =0, • 
 
 continuing in succession to form new equations until a final equation is obtained for 
 each unknown quantity. 
 
 We shall compare the results of two solutions of the above equations (7), in one of 
 
 which (I.) the factors aw, a w' conform strictly to the method of least squares. In 
 
 the other (II.}, these factors are replaced respectively hy a, al ; a being that one of 
 
 the numbers 0, 1, 2 9, or of the fractions j, y ^1 or of their products by an in- 
 tegral power of 10, which approaches most nearly to a given ratio with aw, and a that 
 which approaches most nearly to the same ratio with a w, 8cc. In a similar manner, 
 )8, y are used in the place of bw, cw 
 
 The true values of x and y we will indicate by Xq, y^ Those deduced by 
 
 (I.) will be denoted hy x,y , and those deduced by (II.) will be denoted by Xi, 
 
 yi For the final equation for x, we make 
 
 a w e + o! w' e' + := 0. 
 
IN THE METHOD OF LEAST SQUARES. 18 
 
 For the final equation for Xi 
 
 « ei + «' ej' + = 0. 
 
 For the corresponding final equations for c^o, which must be rigorous, we make either 
 a w ec, -{- a' ra' e'o -\- ^ a w e,, -\- a' w' e'o -\- 
 
 or 
 
 o Co + «' e'o + = a Co + ce' e'o + 
 
 according as the first (I.) or the second (II.) form of combination is adopted. 
 
 Co, e, and Si are the values of e when the indeterminates Xq, y^ , x, y 
 
 Xi,yi , &c., replace x, y in (7). 
 
 The final equations for the combination (I.) are : — 
 
 P x-\- P' y-{- P" Z+ + I, = 0, 
 
 . P'x+Q y-j-Q'z-\- + M=0, 
 
 ^ '' P"a;4- Q'2/+ Q"«4- + iV = 0, 
 
 P = waa -{- w' a' a' -{- , Q = to i b -{- w' b' h' -\- , R = w c c -{- w' c' c' -\- 
 
 (9.) P' = wab-\-w' a'b' -}- , Q — ivb c -\- w' b' c' -\- , R' = w c d -\- w'c'd' -\- 
 
 From the conditions I. and II. applied to the original equations (7), if we make 
 
 Aa =^ a W -{- da , Bj3=&«)-f-8i3, Cy — c w -\- 8y , 
 
 Aa' = a'w' -\-8a' , B/3' = 5'w' + 8/3' , Cy' = c'^mj' + Sy' , 
 
 may be obtained 
 
 P{x-x,)-\-P{y-y,)-\- -^L+Px,-\-Py, + = 
 
 L + Pa;, + P'3/', + — — e^da — e\da' —   
 
 Hence, 
 
 P (x — xi) + P' (y — y,) + = 8« e, + 8a' e'. -(- 
 
 (10.) P'{x — x,) + Q {y — y,) + = 8^e. + 8^' e'. + 
 
 And in a similar manner, 
 
 P (xo — x) -\- P' {yo — y) -\- = a w e, + a' w' e'o + 
 
 (11.) P' (a;„ _ a;) + Q (y„ _ y) + = bwe,-\-b'to'e'o-{- 
 
 Since e is the probable value of Xq — x, and 17" the probable value of (x — Xi), to ob- 
 
 3 
 
14 ON THE USE OP EQUIVALENT NUMBERS 
 
 n 
 
 tain the ratio — , we will compare the probable values of Xq — x and x — Xi, having, as 
 above, 
 
 /■in\ n" Probable value of {x — a;,) 
 
 f Probable value of {xg — x) 
 
 Xq — OS and x — w^ must be derived from a solution of equations (10) and (11), but 
 since (II.) differs from (I.) by small variations only, we have, very nearly, 
 
 (13.) w e,= _|_ «,/ e',2 _(- = ^e'' + w'e" + 
 
 For the second member of (13) is a minimum relatively to the mode of solution, and, 
 as has already been shown, (1), it differs from the first member by small terms of the 
 second order only, those of the first order vanishing with the first differential coefficient 
 
 oi SI = w e^ -\- vf e'^ -\- 
 
 If, then, /^o, /i, and ^i represent the probable values of Bq, e, and Ci corresponding 
 to the unit of weight of the equations (7), we may assume, for the purpose of deter- 
 mining X — ,2?!, that /i — /lii is a small quantity compared with /i, since we have 
 
 a w e'' -I- tti' e" -I- 
 
 — = ri 7-^-7 = 1, very nearly. 
 
 /ti w e, -|- w' e'c -\- 
 
 Moreover, in the absence of exact knowledge of the magnitude of the errors of e^, 
 
 e'o, , it is necessary to admit that they are best represented by the errors e, e, ; 
 
 hence we have — = 1, and consequently - = 1, very nearly. 
 
 The conditions of the solution (II.) give for the probable value of either of the ratios 
 
 0' w' 
 
 aw ^ bw ^ 
 
 8 a' 8 fi' 
       ' a'w'' b'w'' 
 
 
 
 
 
 (14.) 
 
 8a 
 
 — = g^ 
 aw 
 
 I'-g..... 
 ow 
 
 8 a' 
 
 a'w' 
 
 = g> 
 
 Because a being by (II.) nearly proportional to a w, the probable value of S a will also 
 be proportional to a iv ; and a similar remark applies equally to S fl, S y, &c. 
 
 The probable values of the second numbers of (10) and (11) are then, respectively, 
 
 8aei-\-8a' e'i-\- = g Mi V? ' a w e^ -\- a' w' e' ^ -\- = ,j^ ^p^ 
 
 8 ^ e, + S ^' e'l + = g' Fi \/Q ' J to 60 + i' M!' e'o + = ^0 \/Q, 
 
 Hence, in consequence of the identity of the coefficients P,!^ , Q, Q' , &c. in 
 
 the two systems (10) and (11), we obtain 
 
IN THE METHOD OF LEAST SQUARES. 15 
 
 Probable value of (a; — a;,) _ /i, ^ _ Probable value of {y — y,) _ ^i 
 
 Probable value of (a?o — a;) ims 
 
 And from (12) the general expression 
 
 '' Probable value of (a?o — a;) im, Probable value of (^o — y) p-a 
 
 (16.) 
 
 e 
 
 Giving to g the value (3) 
 
 ^ 1 
 
 we shall have in the present case 
 and by (6) 
 
 V 1 
 
 
 ^ 1250 
 
 In other words, by using the foi-m of solution (II.) in the place of a rigorous appli- 
 cation of the method of least squares, the probable errors of the concluded results will 
 not be increased by one one-thousandth part, — a difference entirely too small to be 
 sensible. The two processes, as far as regards accuracy, therefore, may be consid- 
 ered as perfectly identical. On the other hand, the advantages of simplicity and 
 convenience are altogether in favor of the second, in which all the operations of mul- 
 tiplication and division required in the construction of the final equations are reduced 
 to their simplest arithmetical forms. 
 
 The necessity of distinguishing between the probable error of x^, that is, the proba- 
 ble value of (a?o — x^, and the difference between x^ and x, or (<ri — x), deserves par- 
 ticular attention here. While A'l will often differ very much from x^ this fact taken by 
 itself by no means indicates that the chances that a?! is the true value are not sensibly 
 as good as that x is. The discrepancy really proves that the original observations upon 
 which the discussions have been based are so faulty, that very little confidence can be 
 placed in either result, or in any other that can be deduced from' the same data. 
 
 To give completeness to the investigation, we will compare the processes by which 
 the probable errors of the values of the indetermiuates in the two solutions (I.) and (II.) 
 are obtained. 
 
 For the system (I.), let x, y, z, &c. be eliminated in succession from the equations 
 (8), in the following manner. 
 
 Multiply the final equation for ^ by — and subtract it from the final equation for i/, 
 
 P" 
 and again by — and subtract it from the final equation for s, forming the new equa- 
 tions, in which x does not enter : — 
 
16 ON THE USE OF EQUIVALENT NUMBERS 
 
 (Q - 1^ P')2/ + (Q' - jP") z+ + ilf _ I' L = 0, 
 
 /iw\ pii pii pi I 
 
 ^ ^^ {q-yP')y + {R -^P")z+ + jv_£-L = o, 
 
 to whicli we shall give the following notation : 
 
 Q, 2/ + Q', z + + M. = 0, 
 
 (18.) Q'.2/+E, 2 + + JV, =0, 
 
 Let y be eliminated from these new equations (18) in the same manner that x was 
 from the equations (8), multiplying the first by -~ and subtracting it from the second, 
 
 (J?,-2-'Qg.+ 4-iV,-|-'M. = 0, 
 
 with the corresponding notation, 
 
 (19.) K,,z + + iV,, = 0. 
 
 For a more complete illustration of the notation, we will collect in one view the 
 equations obtained by the successive eliminations : 
 
 (a) P a; + P' y + P" 2 + + i = ^ 
 
 (5) F' ^J^qyj^q ^J^ + M = 
 
 (e) P" a; + Q' jr + R z + + iV = 
 
 (20.) (J.) Q, y + Q'. 2 + + m; = 
 
 (c.) Q'^y + R, 2 + + iV, = 
 
 Final 
 Equations. 
 
 Equations formed by 
 eliminating x. 
 
 (c,y) ^xy2 + + -^iy^ I Equations formed by 
 
 . ) eliminating x and y. 
 
 Let /t4 be the probable error of one of the original equations of the unit of weight. 
 The probable errors of the second members of the equations (20) will be, 
 
 Probable error of the equation (a) = /i /^p 
 
 (I) =a*VQ 
 
 (c) =,.vr 
 
 (21.) " « " (J,) =m\/q: 
 
IN THE METHOD OF LEAST SQUARES. 17 
 
 When the final equations obtained by (II.) are solved in an analogous manner, and 
 
 the notation is changed so as to indicate the coefficients of x^, y^, &c., Pj , P'l , 
 
 which replace P, P, &c., we have 
 
 (a) F. a;. + P', y, + P", 2, + + L, =0-| 
 
 (6) 'Q.-i;i+ Qi yi+Q'i zi + + M. =o[ Final 
 
 (c) "-Ri a-i + 'P, 2/i + Ki z, + + JV, = I Equations. 
 
 (22.) (5.) Qi,y,+ Q',. 2, + + ilf., =0] 
 
 , ., , r> I T> I 1 AT « I Equations formed by 
 
 I eliminating Xi. 
 
 (c,y) Pi »j, Zi -|- + -^i^j, = •» Equations formed by 
 
 . , . . j eliminating Si and y^. 
 
 1 IF 
 
 Probable error of the equation (22) (a) = (1 ± - ^) M ^ — i 
 
 (( u 
 
 t( (t 
 
 (23.) 
 
 
 (c,,) = (1 ± 
 
 i^^'^Jf"^' 
 
 To demonstrate these results, (23), it is to be observed that the probable errors of the 
 
 second members of the equations (22), (a), (6^), are the probable sums of the 
 
 second members of the equations 
 
 i'l (^0 — a;,) + P', (yo — 2/,) + = « eo + a' e'o + 
 
 Q,.(y«-!/,) + = (i3-^ «)fo+(^'-^«')e'„ + 
 
 The probable value of (j?e\ is, by (13) and (14), . , . 
 
 «''ej = — - a ej := w ej 1 -^ — = /* (1 iff) — , 
 
18 ON THE USE OF EQUIVALENT NUMBERS 
 
 and the probable value of the sum of the terms aco -\- a' e'^ -\- is 
 
 «e„+«'e'„ + = _^ ^(1 ± g) V« a + «' a' + = ^ {1 ± I g) \^ , 
 
 which is the probable error of the equation (22), (a). 
 Again, since 
 
 Au = aw(l + —'), Bp^bw(l + ^-^\, 
 
 \ a id) \ b loj 
 
 and 
 
 Pi = a a -{- a' a' -\- , Q, = J + 0' J' + 
 
 P\ = a b -\- a' b' -\- , 'Q,=pa-]-^'a' -\- 
 
 or, substituting the probable values 
 
 \ aw J \ bwj 
 
 we have 
 
 Moreover, 
 
 The sum of all the terms ('^ „a—'-^^a\ = ('-^ p, _ !^ /Q \ = o, 
 
 Therefore the probable sum of the terms 
 
 0-^„)e„ + 0'-^a')e'„ + 
 
 will be 
 
 which is the probable error of the equation (22), (b^). 
 
 The other probable errors in (23) are readily supplied by analogy. 
 
 If we neglect - g, of which the probable value is less than — , the probable errors of 
 (22), (a), (b^) become 
 
 Probable error of equation (22) (a) = /* — ,• Probable error of equation (22) (b^) = fi -^ • 
 We shall now proceed to explain a third form of solution, (III.)- 
 
IN THE METHOD OF LEAST SQUARES. 19 
 
 Returning to the equations (10), 
 
 P (x — ic,) + -P' (y — yi) + = e,8« +e'.8«' + 
 
 (24.) p/(^_^,)_^Q (y-y.) + = e,d^+e',8^'4-, 
 
 we find, for the probable values of their second numbers, 
 
 g li \/w a a -\- w' a' a' -\- = g' F V^" > 
 
 (25.) g /* Vwb J 4- w' J' J' + = g- ^ V Q , 
 
 It is evident that the probable sum CiS a-j-e\S a,' -^ , being proportional to the 
 
 square root of the sum of the squares of the individual terms, depends mainly upon the 
 large terms ; or, since e^Sa = ei -v/mT— ^ and e^ \/w'= fi, this sum will be 
 
 
 +^'+ 
 
 w' 
 
 S a J. , . Sa ^ i a' 
 
 If any two or more of the coefiicients -7=, as, for instance, -7=. and -7==, were equal, 
 any small change increasing the former and diminishing the latter by equal amounts 
 would not alter the coefficient of u: but if -7-^ were much smaller than —7=^, we 
 shoidd have, very nearly, 
 
 www' 
 
 and a small change in -7^ would affect the coefficient of /x by an amount insensible 
 compared with the effect of an equal change in —r^. 
 
 Let (P) represent the sum of a certain number of the largest of the terms composing 
 
 the series 
 
 P =^ w aa-\- w' a' a' -\- 
 
 and (p) the sum of a number of the smallest of the terms of the same series. Let also 
 (S P) be the sum of the terms — ^ corresponding to the series (P), and (Bp) the sum of 
 
 the terms — , corresponding to the series (p). 
 Then we have the probable values 
 
 For the large terms, (8 P) = g' (P), 
 For the small terms, (^p) =^ g' (p), 
 
 g representing the general probable value of — for all the terms, whether of large, 
 small, or medium value. 
 
 ^ Or irti 
 
 if UNIVERSITY 
 
 OF J 
 
20 ON THE USE OF EQUIVALENT NUMBERS 
 
 Let US suppose the mode of solution (II.) to be itself varied by changing the factors 
 «, a , &c., corresponding to the large and small terms, so that for the large tenns, g, or 
 the probable value of — , -7^, , for these particular terms, becomes 
 
 ■^ a 10 a IB 
 
 g=H, 
 and for the small terms, 
 
 We shall then have the probable values, 
 
 For the large terms (8 P,) = IP (P), 
 ^ '' For the small terms (S;?,) = A' {p). 
 
 (8 P) and (pp) becoming (S Pj) and {^Pi) when g becomes H and h. 
 In order that the probable sum of the second member of the equation. 
 
 P{x — x,) + P'{y — y,)^ = e, 8 « + e'l S «' + 
 
 should not be increased by the proposed changes of S a, we must have 
 
 (8P) +{hp) >(8P,)+(Si.,)> 
 
 or, by (26) and (27), 
 
 W (P) + h- (p)< g= (P) + g^ (p). 
 
 We shall assume, for the terms corresponding to {p), that the probable value of A is 
 
 A = —1. 
 
 This condition involves only small changes in the factors a, a' , because, for the 
 
 terms corresponding to {p), a w being small, B a ^ aw h = — aw, will also be small ; 
 
 we then have 
 
 H^(P)<^(P) + (^-l)(p), 
 
 or, since we can put ^^ — 1 = — 1 very nearly, g being small compared with imity, 
 we obtain 
 
 (28.) g^ _ H^ > — , H^ < g^ — — , 
 
 s ^ (P) ^ ® (P) 
 
 representing the condition to be observed in order that the second members of (24) 
 should not be increased by the changes made in the large and small values of a. 
 
 This, it will be remembered, can be applied only when the condition h ^ — 1 in- 
 volves only small changes in the factors a, a of the order of the mean value of 
 
 S a for all the factors. ^^ being necessarily a positive quantity, H must always be 
 less than g. 
 
IN THE METHOD OF LEAST SQUARES. 21 
 
 (28) may easily be extended to the analogous cases of the second members of the 
 
 equations 
 
 P' {x — X,) -\- Q {y - Vr) + = e.«j3 + e',8^' 4- 
 
 P" (x — a;.) + Q' (y — 2/i) + = ei8y + e'l 8 y + 
 
 so that 
 
 (?) (r) 
 
 w «'-^>^- ''-•''>wy 
 
 give the limits within which the proposed changes of the factors ^, ^ 7, 7' 
 
 will not increase the probable sums eiB^-{-e\B^ -\- and ejBy-\-e\Sy-\- 
 
 For the factors a, a' , /3, /3' corresponding to the equations most important 
 
 in their influence upon the final determination of a?, y respectively, if we use num- 
 
 JV N' . 1 
 
 hers chosen from a series for which — — — is only — as large as it is for the series (2), 
 
 we shall have 
 
 And if at the same time we omit altogether a certain number of the unfavorable 
 
 equations by making in these instances a = 0, /S = , that is, S a = — aw, 
 
 S /3 ^ — b IV, ov h = — 1, we find 
 
 ^631 
 
 We shall therefore keep within the limits (28) and (29) as long as the coefiicients in 
 the omitted equations satisfy the conditions 
 
 (P)^68l' (Q)^68l' 
 
 The probable values of a? — a?!, j/ — i/^, will not have been increased, and conse- 
 quently the solution may be accepted as equivalent to II. 
 
 A general method, III., of adjusting the degree of numerical accuracy which should 
 
 be observed in the expression of the factors a, a' , P, ^' > inay be derived 
 
 from the following considerations. 
 
 In II. the adjustment is evidently not so favorable as it might be, since the limit of 
 the intentional inaccuracies S a, S a' ,B^,B0 has been fixed by the relations 
 
 ha — aw g, 6 a' =^ a' w' g', 8^ = hwg, 8^' = b' w' g', 
 
 g having the same average value whether aw, bw be large or small ; thus the 
 
 4 
 
22 ON THE USE OF EQUIVALENT NUMBERS 
 
 largest iaaccuracies are committed when aw,bw are largest, that is, when the 
 
 equation has most influence upon the final result for any particular indeterminate. 
 
 In order to secure a more advantageous distribution, it will be necessary to recur 
 again to the equations (10). It appears that, for a given limit of inaccuracy in the 
 
 expression of the factors, the probable values of 07 — <3?i, 3/ — 2/i • ^'^^^ ^^ least 
 
 when the separate terms of the second members of these equations, irrespective of their 
 signs, are equal to each other, or 
 
 e,8a = e',fi«'= Ci 8 /3 = e'l 8 /3' = 
 
 S a, S /3 ought therefore to be inversely proportional to the probable errors of the 
 
 primitive equations, or directly as the square roots of their weights. 
 We shall, then, define III. hy the relations 
 
 ba—kgs/w, 8fi=BgVw, 
 
 (30.) da' = A.gVw', 8^'=BgV 
 
 w 
 
 A, B and g heing constant quantities. 
 
 To secure the equivalency of II. and III., the values of A, B must depend on 
 
 the condition that the probable values of the second members of (10) should remain 
 unchanged, or 
 
 g/i Va? + A^ + = gM Vp, g/. Vb' + b" + = ^^ Vq, 
 
 lL.\/n= VP, BV'»= V'Q, 
 
 Hence it is easy to conclude, that, if we make in (30) 
 
 A = mean value of a \^w, B = mean value of b Vw, 
 
 the means being in every instance taken without regard to signs, the probable values of 
 
 X — cVi, t/ — y, will be smaller in III. than in II., while III. in point of facility 
 
 has a decided advantage over II. ; since by making a = 0, ;S = in all cases in 
 
 which a Vw <. Ag, b \/w <Bg, , a considerable amount of unnecessary computa- 
 tion may often be avoided. 
 
 The following will then be the limits of intentional numerical inaccuracies allowed 
 in the expression of the factors a, a' , /S, /3' in the three methods. 
 
 I. 11. III. 
 
 8a=0, 8^=0 8a=awg,8^=hwg 8a = Ag Vw , 8 ^ = B ^ l/w 
 
 8 a' = 0, 8 ^' = , 8 a' = a'w'g,8^' = h'v}' g 8 «' = A g Vw', 8 ^' = B g Vw' 
 
IN THE METHOD OF LEAST SQUARES. 23 
 
 The following tests of tlie correctness of all the numbers entering into the final equa- 
 tions will be found useful. They apply equally to the three methods. 
 Let the sums S, S' , 2, 2 be formed as follows : — 
 
 «+'^+ +'«=S, «+^+y+ =2, 
 
 (31-) a' + 5'+ + m' = S', „' + /3' + y + =^, 
 
 We shall then have the following test controlling at once all the computed quan- 
 tities, Pi, P'l , 'Qi, Q, , ii, Jfi 
 
 2:{Pi,Qi,Ri L„M„N, ) = 2" (S 2). 
 
 For we have 
 
 P,-\- P\-i- -^L, =aa-\-ah-\- 
 
 + a' a' + «' J' + J> = S « + S' a' + , 
 
 + 
 
 'Qi + Q. + + M, = ^ a + ^ J + 
 
 (32.) _|- ^' a' + iS' 5' + } = S^ + S'/3' + 
 
 "Ri + 'Ri-\- + JV. = y a + y J + 
 
 + ya' + yj' + } =Sy + S'y' + , 
 
 The sum of all the equations (32) is 
 
 (A + P', + -{- i,) + CQ. + Q. 4. + ilf.) + = S-?+S'-S'+ 
 
 or 
 
 (33.) Jr(Pi,Qt,Ki Ly,M„N, ) = ^(S^). 
 
 When this is not fulfilled, the particular equation at fault may be detected by using 
 the partial tests, 
 
 Pi + P'. + X. = Sa + S'«' + 
 
 (^^•^ 'Q1+Q1+ Jii;=s/3 + s'j3' + •.. 
 
 If all the numbers are not carried out to the same number of figures beyond the 
 decimal point, it will be advisable, in order to apply the test to the best effect, to alter 
 arbitrarily the decimal point, making, for the time being, the number of figures to the 
 right hand of it the same in all the equations. 
 
 The successive stages in the computations by (II.) and (III.) will be as follows : — 
 
 a) To assign the values of the equivalent factors a, /3 , which can readily be 
 
 done by simple inspection. The factors once adopted must be used rigorously in mul- 
 tiplying every term of the equations to which they belong. 
 
24 ON THE USE OP EQUIVALENT NUMBERS 
 
 b) After the multiplications have been performed, and the sums taken, the numbers 
 adopted in the final equations are to be tested by (33). 
 
 c) The solution of the final equations. 
 
 d) The determination of weights. 
 
 If changes have been made in the decimal pointing, or otherwise, by introducing 
 
 the constants A, B , it must be remembered that, although the final equations 
 
 thus formed will give the same values of o^i , j/, , &c. that Avould have been obtained 
 if no such alteration had been made, the determination of the weights and probable 
 errors of Xi, ^i , &c. requires that the correct pointing be restored in the coefficients, 
 or else that the probable errors be computed in conformity with the formulse (23). 
 
 When the number of indeterminates is considerable, it will be advisable, in solving 
 the final equations, to eliminate x, y, z, &c., in succession, and then to repeat the 
 operation, commencing the elimination in the reverse order, s, y, x, &c. One of the 
 advantages of so doing is a complete check upon the work by the comparison of the 
 value of that indeterminate which is obtained last by both eliminations. It is, however, 
 mostly recommended from its facilitating the computation of the weights. In this case, 
 the following formulae may be used, if the number of indeterminates does not exceed 
 six. Let these be x, y, z, f, -n, ^, and their weights, W^^y , W^y , &c. The ordinary 
 formulae for computing the weights give 
 
 (35.) TF(0 = U^ y z i,, TFw = P< , 5 J y . 
 
 W^■Q is the coefficient of f in the equation resulting from the elimination of x, y, z, f , 
 and r}, by the process indicated in (18), (19), and (20), and W^^^ the coefficient of x 
 after ^, r}, f, z, and y have been eliminated. We have, also, 
 
 (36.) W(,, = ^J-JTF(o, W^, = ^^^W,:.,. 
 
 '-'X y z i ■'■ t, " i * 
 
 The factors and divisors required in (36) will have been already computed during 
 the eliminations which have preceded. 
 
 From the equations containing only f , 17, ?, the latter is to be eliminated ; and from 
 the equations containing 2;, y, and x, x is to be eliminated. We then have 
 
 Trco = |^J^pr„, w,z, = ^;^^w,,,. 
 
 ix y z Z Hi H i X 
 
 For the weight of Xi,yi, &c., when (II.) or (III.) is used, we shall have, from (23), 
 
 Weight o{xi= A T^d) , 
 (37.) « yi = B W(y) , 
 
IN THE METHOD OF LEAST SQTJAKES. 25 
 
 The limits of effective accuracy appropriate to the numerical representation of the 
 
 coefficients a, 6 may be investigated in the following manner : — 
 
 If we determine cr, , j/, by the method of least squares from the equations, 
 
 (38.) (a — da) Xi-{- {b — dh) i/i-\- -\- m — d m = ei, weight = w, 
 
 we shall have from (8) and (38), i£ da, db , which may be employed to represent 
 
 the errors of the adopted coefficients, are small, 
 
 (39.) 
 
 P(xi — x) -\- P' (yi — y) -\- = aw( — e -\- xda-{- y dh -{- -\- dm) -\- 
 
 If fi is the probable value of dm \/w, the most suitable values of a; da, y dh . 
 evidently fulfil the conditions 
 
 (40.) xda\w =ydb\^w = dm\^w = ii', xda'Vw'^^ydb'Vw' := dm' \ w' =^ (>.' . 
 
 Observing that we may substitute in the second members of (39) the probable values 
 
 e <^ a X, e' <. a' X , e Khy, e' <. V y , we may conclude, by comparing (39) 
 
 with (11), (12), and (6), that, if;*' is less than the limit 
 
 (41.) ^ = -4^=, 
 
 'd denoting the number of indeterminates in (38), the difference Cj — i of the probable 
 errors of x^,yi derived from (38) and of x,y derived from (8) will be 
 
 (42.) e.-e<ig'^.. 
 
 Consequently, if ^ < — , i^ — f will be less than — — f or less than in 11. 
 
 When /a' is known, the limits for admissible values of da, db in (38) will be 
 
 (43.) da<C — ^-^ , da'<C — ^-^r db^-da, db'=:-da' dm = xda, dm'^xda' 
 
 xVw xVw' y y 
 
 If the mean values of ax, by , irrespective of their signs, are all of the same 
 
 order of magnitude, we may substitute in (43) the a priori probable values 
 
 (44.) a;<^ I " 
 
 A-^Jn + n'— 1 
 
 a; _ B 
 
 y~A. 
 
 where n is the number of equations (38), n' the number of indeterminates, and, taking 
 the means without regard to signs, 
 
 (45.) A = mean value of a Vw , B = mean value of b \/w M = mean value of m 's/w . 
 
26 ON THE USE OF EQUIVALENT NUMBERS 
 
 We will now proceed to compare the three methods of solution (I.), (II.), (HI-)' ^7 
 applying them to the following series of equations of six indeterminates, taken from a 
 memoir, by Gauss, on the elliptic elements of the orbit of the planet Pallas.* 
 
 Original Equations.^ 
 
 1 = — 183"93 + 0.79363(21,+ 143.66dy + 0.39493d jr + 0.95920 tZ 9 — 0.18856 dS2 + 0.17387 df 
 
 2 = — 6.81 — 0.02658(21.+ 46.71 (Zy + 0.02658 (Ztt — 0.20858(2 9 + 0.15946(2 0+ 1.25782 di 
 
 3 = — 0.06 + 0.58880(21,+ 358. 12 (2 y + 0.26208 (2 n- — 0.85234 (2 <p + 0.149 12 (2 + 0.17775(2 i 
 
 4 = — 3.09 + 0.01318(21,+ 28.39(2y — 0.01318(2 ,r — 0.07861 (2q) + 0.91704(2Q + 0.54365<2i 
 
 5 = — 0.02 + 1.73436 (2 L + 1846. 17 dy — 0.54603 dn — 2.05662 dcp — 0. 18833 (2 a — 0. 17445 d i 
 
 6 0=— 8.98 — 0.12606(21,— 227.42(2y + 0.12606(27r — 0.38939 (29 + 0.17176(20— 1.35441 (2t 
 
 7 = — 2.31 + 0.99584 (21,+ 1579.03 dy-\- 0.06456 (2 tt + 1.99545 d 9 — 0.06040 da. — 0.33750 d i 
 
 8 0=+ 2.47 — 0.08089 (2 L— 67.22(2y + 0.08089<2,r — 0.09970(2 9 — 0.46359(2 0+ 1.22803(2* 
 
 9 = + O.01 + O.65311(2L+1329.09(2y + O.38994(2jr — 0.08439(2 9 — 0.04305 (2 + 0.34268(2 i 
 9, 0=+ 38.12 — 0.00218 (2 L+ 38.47 (2y + 0.00218(27r — 0.18710(29 + 0.47301 (2o— 1.14371 (2i 
 
 10 = — 317.73 + 0.69957 (2 L+ 1719.32 (2y + 0.12913 (2 TT- 1.38787(2 9 + 0.17130(2 — 0.08360 (2i 
 
 11 = + 117.97 — 0.01315(2i— 43.84(2y + 0.01315(2,r + 0.02929 (29+ 1.02138(2 0- 0.27187(2i 
 
 The probable error of one of these equations is /4 = + 90", the weights being 
 equal, excepting for 9^, which has been excluded from each of the solutions. 
 
 As an illustration of the mode of applying the limits (43), we will make in (41) and 
 (42) y = — , we shall then have 
 
 V -/ ^ 25 ' 
 
 p. — ± 90", n — 11, 
 
 A = ± 0.5 , 
 
 M = ± 60" , 
 
 {x = dL)<C± 100" , 
 
 (2a < ± 0.013, 
 
 g'= i ' «' = 6, 
 
 B = ± 700, 
 
 m' = ± 1".3, 
 
 (2/= (2y)<±0".07, 
 
 d J < ± 18. 
 
 0, D, E, and F, being of the same order of magnitude with A, we may conclude from 
 (42), (43), and the above values of d a and d b, that, if we reject in all the equations the 
 two right-hand figures from the values of m, and the three right-hand figures from all 
 the other numbers, writing, for instance, the first equation 
 
 = _ 184" + 0.79 (2 1, + 140 (2y + 0.39 (2 tt + 0.96 dcp — 0.19 (2 Q + 0.17 d i, 
 and the second 
 
 = — 7" — 0.03 (2 L+ 50 (2 y + 0.03 (2 TT- 0.21 (2 9 + 0.16 (2 + 1.26 (2 i, 
 
 the probable error of the value of either of the quantities of dL, dj , when the 
 
 equations thus written are solved by the method of least squares, will exceed the prob- 
 
 * Disquisitio de Elementis Ellipticis Palladis. Comment. Soc. Reg. Gottingensis Recent. Vol. I. 
 t The computations necessary for the solution of these equations have been executed by Messrs. J. F. 
 Flagg and T. H. Safford, jr. 
 
IN THE METHOD OF LEAST SQUARES. 27 
 
 able error of the same quantity obtained with the employment of the exact coefficients 
 by less than — of that error. 
 
 ^ 1250 
 
 We shall, however, for the present confine our attention to a direct comparison be- 
 tween the results of the solutions I., II., and III., and retain in each the exact coeffi- 
 cients of the original equations, adopting the constants A, C, D, E, F = 1.0, and 
 
 B = 1000.0, and for the factors a/3 a somewhat ruder system of representation, 
 
 especially in II., than that upon Avhich the previous discussions have been based. 
 
 If we employ the limits (28), (29), Ave find that the most favorable equations are (1), 
 (5), (7), and (10), and the least favorable (2), (4), (8), (11). Applying (III.) in the 
 particular form to which it is limited in (28) and (29), we shall omit (2), (4), (8), and 
 (11), as ineffective in the final equations for d L. (1), (5), (7), and (10) give 
 
 (P) = 1 X 0.8' + 1 X 1-7'' + 1 X 1.0' + 1 X 0.7' = 5.0000, 
 
 and (2), (4), (8), and (11), 
 
 (p) = 1 X 0.03' + 1 X 0.01' + 1 X 0.08' + 1 X 0.01' = 0.0075. 
 
 (P) 667 ' 
 
 According to (28), we must have, in this case, 
 
 ^100 
 
 in order to preserve the equivalency of (II.) and (III.), if the original equations (2), 
 (4), (8), and (11) are omitted in forming the final equation for d L. We may, then, 
 adopt the following system of factors. 
 
 d L. 
 
 No. of 
 
 Factors in the Method 
 
 Original Equation. 
 
 of Least Squares. 
 I. 
 
 1 
 
 + 0.79363 
 
 2 
 
 — 0.02658 
 
 3 
 
 + 0.58880 
 
 4 
 
 + 0.01318 
 
 5 
 
 + 1.73436 
 
 6 
 
 — 0.12606 
 
 7 
 
 + 0.99584 
 
 8 
 
 — 0.08089 
 
 9 
 
 + 0.65311 
 
 10 
 
 + 0.69957 
 
 11 
 
 — 0.01315 
 
 Equivalent 
 Factors. 
 
 n. 
 
 + 0.8 
 
 — 0.03 
 
 + 0.6 
 
 + 0.01 
 
 + 1.7 
 
 — 0.1 
 
 + 1.0 
 
 — 0.08 
 
 + 0.7 
 
 + 0.7 
 
 — 0.01 
 
 Equivalent 
 Factors. 
 
 
 ni. 
 
 
 + 0.8 
 
 
 0.0 
 
 
 + 0.6 
 
 
 0.0 
 
 
 + (.- 
 
 — 0.1 
 
 ■0 
 
 + 1.0 
 
 
 0.0 
 
 
 + 0.7 
 
 
 + 0.7 
 
 
 0.0 
 
 
28 ON THE USE OF EQUIVALENT NUMBERS 
 
 The mean value of H'= — for the equations (1), (5), (7), and (10) is 
 
 IT 1 ^1 
 
 if = — or <[ — , 
 
 200 ^ 100 
 
 as required by (28). 
 
 For the second indeterminate dy, the favorable equations are (5), (7), (9), (10). 
 The unfavorable ones are (2), (4), (8), and (11). 
 
 (Q) = 10600000, (?) = 9400, 
 = , H must be < — 
 
 (Q) 1130' ^37 
 
 dy. 
 
 Equivalent Equivalent 
 
 Factors. Factors. 
 
 n. III. 
 
 + 0.1 +0.1 
 + 0.05 0.0 
 
 + 0.4 + 0.4 
 + 0.03 0.0 
 
 + 2.0 +(^-0 
 
 — 0.2 — 0.2 
 + 2.0 + 1.6 
 
 — 0.07 0.0 
 
 +1.0 +(i+-;) 
 
 + 2.0 + 1.7 
 
 — 0.04 0.0 
 
 The mean value of H= — ^ fpr the equations (5), (7), (9), (10), is 
 
 H =: — or < — , 
 125 87 
 
 the limit given by (29). 
 
 For dir the favorable equations are (1), (3), (5), (9) ; the unfavorable, (2), (4), (11). 
 
 (»■) 1 rr , ^ 1 
 
 y~ = — , H must be <; — . 
 
 (B) 675 ' ^ 92 
 
 No. of 
 
 Factors in the Method 
 
 Original Equation. 
 
 of Least Squares. 
 L 
 
 1 
 
 + 143.66 
 
 2 
 
 + 46.71 
 
 3 
 
 + 358.12 
 
 4 
 
 + 28.39 
 
 5 
 
 + 1846.17 
 
 6 
 
 — 227.42 
 
 7 
 
 + 1579.03 
 
 8 
 
 — 67.22 
 
 • 9 
 
 + 1329.09 
 
 10 
 
 + 1719.32 
 
 11 
 
 — 43.84 
 
IN THE METHOD OP LEAST SQUARES. 29 
 
 dir. 
 
 Equivalent 
 
 Factors. 
 
 m. 
 + 4.0 
 
 0.0 
 + 2.6 
 
 0.0 
 — 5.5 
 + 1.0 
 + 0.6 
 + 1.0 
 + 4.0 
 + 1.0 
 
 0.0 
 
 For dq) the favorable equations are (1), (5), 7), (10); the unfavorable, (4), (8), 
 (9), (11). 
 
 (S) 465   
 
 The value of H is imaginary, or < I . 
 
 The rejection of the unfavorable equations cannot in this instance be compensated 
 for by increasing the accuracy of treatment of the favorable ones. 
 
 dq). 
 
 Equivalent Equivalent 
 
 Factors. Factors. 
 
 II. ffl. 
 
 + 1.0 
 
 No. of 
 
 Factors in the Method Equivalent 
 
 Original Equation. 
 
 of Least Squares. Facters. 
 I. 11. 
 
 1 
 
 + 0.39493 + 4.0 
 
 2 
 
 + 0.02658 + 0.3 
 
 3 
 
 + 0.26208 + 3.0 
 
 4 
 
 — 0.01318 —0.1 
 
 5 
 
 — 0.54603 — 5.0 
 
 6 
 
 + 0.12606 + 1.0 
 
 7 
 
 + 0.06456 + 0.6 
 
 8 
 
 + 0.08089 + 0.8 
 
 9 
 
 + 0.38994 + 4.0 
 
 10 
 
 + 0.12913 + 1.0 
 
 11 
 
 + 0.01315 +0.1 
 
 The mean value of H 
 
 is — , which exceeds the limit H 
 
 on ' 
 
 No. of 
 Original Equation. 
 
 Factors in the Method 
 
 of Least Squares. 
 
 L 
 
 1. 
 
 + 0.95920 
 
 2 
 
 — 0.20858 
 
 3 
 
 — 0.85234 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 — 0.07861 
 
 — 2.05662 
 
 — 0.38939 
 + 1.99545 
 
 — 0.09970 
 
 — 0.08439 
 
 10 
 
 — 1.38787 
 
 11 
 
 + 0.02929 
 
 — 0.2 — 0.2 
 
 — 0.9 
 
 ' V 100/ 
 
 — 0.2 
 
 -('-0 
 
 — 0.08 0.0 
 
 — 2.0 — 2.0 
 
 — 0.4 — 0.4 
 + 2.0 + 2.0 
 
 — 0.1 0.0 
 
 — 0.08 0.0 
 
 — 1.0 
 
 + 0.03 0.0 
 
 -('+0 
 
 The mean value of H is — : it should be as above < 0. 
 
 56 ^ 
 
30 
 
 ON THE USE OF EQUIVALENT NUMBERS 
 
 For dil the favorable equations are (4) and (11); the unfavorable ones, (7) and (9). 
 
 (T) 344- 
 
 H is again imaginary. In rejecting (7) and (9), we have passed the prescribed limits. 
 
 dn. 
 
 No. of 
 Original Equation. 
 
 Factors in the Method 
 
 of Least Squares. 
 
 I. 
 
 Equivalent 
 Factors. 
 
 n. 
 
 Equivalent 
 
 Factors. 
 
 ffl. 
 
 a 
 
 — 0.18856 
 
 — 0.2 
 
 — 0.2 
 
 2 
 
 + 0.15946 
 
 + 0.2 
 
 + 5 
 
 3 
 
 + 0.14912 
 
 + 0.1 
 
 + \ 
 
 4 
 
 + 0.91704 
 
 + 0.9 
 
 + (1—0.08) 
 
 5 
 
 — 0.18833 
 
 — 0.2 
 
 — 0.2 
 
 6 
 
 + 0.17176 
 
 + 0.2 
 
 + 5 
 
 7 
 
 — 0.06040 
 
 — 0.1 
 
 0.0 
 
 8 
 
 — 0.46359 
 
 — 0.5 
 
 1 
 
 2 
 
 9 
 
 — 0.04305 
 
 — 0.04 
 
 0.0 
 
 10 
 
 4- 0.17130 
 
 + 0.2 
 
 + 5 
 
 11 
 
 -f 1.02138 
 
 + 1.0 
 
 + (1+0.(B) 
 
 ii the favorable equations are (2), (6), (8), and the rmfavorable (10). 
 
 
 ([/) 769' 
 
 ^68 
 
 
 
 d 
 
 i. 
 
 
 No. of 
 Original Equation. 
 
 Factors in the Method 
 
 of Least Squares. 
 
 L 
 
 Equivalent 
 Factors.   
 U. 
 
 Equivalent 
 Factors. 
 
 m. 
 
 1 
 
 + 0.17387 
 
 + 0.2 
 
 + 5 
 
 2 
 
 + 1.25782 
 
 + 1.0 • 
 
 + (' + 
 
 3 
 
 + 0.17775 . 
 
 + 0.2 
 
 + 5 
 
 4 
 
 + 0.54365 
 
 + 0.5 
 
   +\ 
 
 5 
 
 — 0.17445 
 
 — 0.2 
 
 1 
 6 
 
 6 
 
 — 1.35441 
 
 — 1.0 
 
 -('+0 
 
 7 
 
 — 0.33750 
 
 — 0.3 
 
 1 
 
 3 
 
 8 
 
 + 1.22803 
 
 + 1.0 
 
 +('+0 
 
 9 
 
 + 0.34268 
 
 + 0.3 
 
 +i 
 
 10 
 
 — 0.08360 
 
 — 0.08 
 
 
 
 11 
 
 — 0.27187 
 
 — 0.3 
 
 1 
 
 ~4 
 
IN THE METHOD OF LEAST SQUARES. 31 
 
 The mean value of iT is — , the computed limit being i? < — . 
 Applying these factors to the original equations, we obtain, — 
 
 By the Method of Least Squares. I. 
 
 Oocf. of Coef. of Coef. of Coef. of Coef. of Coef. of 
 
 dL. dy. dn, da. da. di. 
 
 Coef. of 
 
 Coef. of 
 
 Coef. of 
 
 Coef. of 
 
 dy. 
 
 dn. 
 
 d<f. 
 
 da. 
 
 7203.91 - 
 
 - 0.09344 — 
 
 2.28516 — 
 
 0.34664 — 
 
 = — 371.09 4- 5.91569 + 7203.91 — 0.09344 — 2.28516 — 0.34664 — 0.18194 
 = — 580104.00 -f 7203.91 + 10834225.00 — 49.06 — 3229.77 — 198.64 — 143.05 
 = — 113.45— 0.09344— 49.06+ 0.71917+ 1.13382+ 0.06400+ 0.26341 
 
 = + 268.53— 2.28516— 3229.77+ 1.13382+ 12.00340— 0.37137— 0.11762 
 = + 94.26— 0.34664— 198.64+ 0.06400— 0.37137+ 2.28215— 0.36136 
 
 = — 31.81— 0.18194— 143.05+ 0.26341— 0.11762— 0.36136+ 5.62456 
 
 By the Equivalent Factors. II. 
 
 Coef. of Coef. of Coef. of Coef. of Coef. of Coef. of 
 
 dL. dy. dit. d(f. da. di. 
 
 = — 372.23 + 5.89945 + 7208.64 — 0.04729 — 2.22332 — 0.33807 — 0.19932 
 = — 662.04 + 7.85800 + 11830.88 — 0.20089 — 3.15658 — 0.16440 — 0.48494 
 = — 1051.81 — 0.02274 + 126.84 + 7.02511 + 10.51379 + 0.45680 + 3.10941 
 = + 137.76 — 1.90276 — 2453.82 + 1.15752 + 11.43969 — 0.33172 — 0.14026 
 ^ + 84.26 — 0.42389 — 250.17 + 0.05158 — 0.45069 + 2.27706 — 0.39471 
 = — 42.98 — 0.20005— 247.16 + 0.29878— 0.04721 — 0.30249 + 4.50959 
 
 By the Equivalent Factors. III. 
 
 Coef. of Coef. of Coef. of Coef. of Coef. of Coef. of 
 
 dL. dy. dn. dcp. da. di. 
 
 = — 37i'.03 + 5.97864 + 7296.25 — 0.06706 — 2.33932 — 0.37876 — 0.08024 
 = — 560.48 + 7.21446+10853.11—0.05221— 3.26575 — 0.20615 — 0.18982 
 = — 1061.36 — 1.13102 — 959.72 + 7.19887 + 11.91488 + 0.34031 + 3.07535 
 = + 251.17 — 2.11274— 2917.00+1.13986 + 11.78434 — 0.37820 + 0.08665 
 = + 97.46 — 0.29118— 75.36 + 0.07546— 0.22584 + 2.29772 — 0.39567 
 • = — 54.37 — 0.12928 — 4.62 + 0.26533 — 0.24548 — 0.37541 + 5.56738 
 
 In order to test the numerical values in these final equations by (33), the decimal 
 points for L, M, N, &c., and for the coefficients of dy, should be changed three 
 places to the left. This being done, we have for 11. 
 
 2 (Pi, Qi, L„M„ ) = + 57.87362, 
 
 JS(S:s) = + 57.87354. 
 
 And for III. 
 
 i: (Pi, Qi, L„M„ ) = + 57.56917, 
 
 2:(S:s) = + 57.56916. 
 
32 
 
 ON THE USE OF EQUIVALENT NUMBERS 
 
 In both cases, the agreement is as near as could be desired. 
 
 We have, then, the following equations by successive eliminations : — 
 
 
 
 Coef. of 
 
 Coef. of Coef. of 
 
 
 Coef. of 
 
 Coef. of Coef. of 
 
 
 II 
 
 dL. 
 
 dy. dn. 
 
 
 dif. 
 
 dL>. di. 
 
 = — 
 
 371.09 4- 
 
 5.91569 + 
 
 7203.91— 0.09344 
 
 — 
 
 2.28516 — 
 
 0.34664— 0.18194 
 
 = —138534.00 
 
 + 
 
 2458225.00 + 62.13 
 
 — 
 
 510.58 4- 
 
 213.84 + 73.45 
 
 = — 
 
 115.81 
 
 
 + 0.71612 
 
 + 
 
 1.11063 — 
 
 0.06392+ 0.25868 
 
 = + 
 
 25.66 
 
 
 
 + 
 
 9.29213 — 
 
 0.36175 — 0.57384 
 
 = + 
 
 75.23 
 
 
 
 
 + 
 
 2.22346— 0.37766 
 
 = + 
 
 17.11 
 
 
 II. 
 
 
 
 + 5.42383 
 
 
 
 Coef. of 
 
 Coef. of Coef. of 
 
 
 Coef. of 
 
 Coef. of Coef. of 
 
 
 II 
 
 dL. 
 
 dy. dn. 
 
 
 dtf. 
 
 da. di. 
 
 = — 
 
 372.23 -f 
 
 5.89945 + 
 
 7208.64— 0.04729 
 
 — 
 
 2.22332 — 
 
 0.33807— 0.19932 
 
 = — 
 
 166.23 
 
 + 
 
 2229.06— 0.13790 
 
 — 
 
 0.19504 + 
 
 0.28590— 0.21945 
 
 = — 
 
 1041.71 
 
 
 + 7.03449 
 
 + 
 
 10.51875 + 
 
 0.43566 + 3.12386 
 
 = + 
 
 176.07 
 
 
 
 + 
 
 9.01520 — 
 
 0.49449— 0.72094 
 
 0:= + 
 
 100.43 
 
 
 
 
 + 
 
 2.17690— 0.46611 
 
 = + 
 
 16.00 
 
 
 III. 
 
 
 
 + 4.24758 
 
 
 
 Coef. of 
 
 Coef. of Coef. of 
 
 
 Coef. of 
 
 Coef. of Coef. of 
 
 
 II 
 
 dL. 
 
 dt. dn. 
 
 
 dip. 
 
 da. di. 
 
 = — 
 
 371.03 + 
 
 5.97864 + 
 
 7296.25— 0.06706 
 
 — 
 
 2.33932 — 
 
 0.37876— 0.08024 
 
 = — 
 
 112.76 
 
 + 
 
 2048.68+ 0.02871 
 
 — 
 
 0.44288 + 
 
 0.25090 — 0.09299 
 
 = — 
 
 1108.40 
 
 
 + 7.18029 
 
 + 
 
 11.56325 + 
 
 0.21715+ 3.07926 
 
 = + 
 
 274.45 
 
 
 
 + 
 
 9.07933 — 
 
 0.50448 — 0.43778 
 
 = + 
 
 117.10 
 
 
 
 
 + 
 
 2.22130— 0.43492 
 
 = + 
 
 30.76 
 
 
 
 
 
 + 5.33957 
 
 From which are finally deduced the values of the six unknown quantities : 
 
 II. 
 
 III. 
 
 di =— 3.15 
 
 — 3.77 
 
 — 5.76 
 
 da =— 34.37 
 
 — 46.94 
 
 — 53.85 
 
 dq, =— 4.29 
 
 — 22.41 
 
 — 33.50 
 
 dTT =+ 166.44 
 
 + 186.17 
 
 + 212.41 
 
 d y = + 0.054335 
 
 + 0.08963 
 
 + 0.05115 
 
 dL = — 3.06 
 
 — 56.32 
 
 — 14.58 
 
 The following are the outstanding errors of the original equations : — 
 
IN THE METHOD OF LEAST SQUARES. 
 
 33 
 
 I. 
 
 II. 
 
 III. 
 
 1 — 111.00 
 
 — 155.51 
 
 — 127.23 
 
 2 — 8.31 
 
 — 3.70 
 
 — 7.29 
 
 3 + 59.18 
 
 + 59.08 
 
 + 84.73 
 
 4 — 36.67 
 
 — 47.02 
 
 — 54.55 
 
 5 + 19.92 
 
 + 21.77 
 
 + 33.12 
 
 6 + 0.07 
 
 -1- 6.92 
 
 + 19.63 
 
 7 4- 85.77 
 
 + 54.52 
 
 + 16.00 
 
 . 8 + 25.01 
 
 + 35.37 
 
 + 38.66 
 
 9 -f- 135.88 
 
 + 157.61 
 
 + 144.46 
 
 10 — 204.64 
 
 — 155.64 
 
 — 174.86 
 
 11 + 83.44 
 
 + 69.63 
 
 + 64.28 
 
 ms of the squares of these errors 
 
 are 
 
 
 I. 
 
 II. 
 
 III. 
 
 Q = 92919 
 
 88556 
 
 85205 
 
 It is obvious that the solution I., as given by Gauss in the memoir above quoted, 
 is incorrect, since the sum of the squares of the errors should be less by the method 
 of least squares than by any other mode of combination.* A revised solution gives 
 the following equations for I. Those for II. and III. are repeated for the sake of 
 comparison. 
 
 I. (Revised solution.) 
 
 Coef. of 
 dL. 
 
 Coef. of 
 dt. 
 
 7203.91 — 0.09344 
 
 Coef. of 
 dTt. 
 
 Coef. of 
 df. 
 
   2.28516 — 
 446.986 
 
 Coef. of 
 di. 
 
 0.34664— 0.18194 
 
 Coef. of 
 dii. 
 
 = — 371.09 + 5.91569 + 
 
 = — 128204.00 + 2061567.00 + 64.72780 — 446.986 + 223.4856 + 78.50985 
 
 = — 115.28 + 0.71566+ 1.11176+ 0.05150+ 0.25807 
 
 = + 276.46 + 9.29666 — 0.53681 — 0.57179 
 
 0=+ 110.68 + 2.20290— 0.43212 
 
 = + 41.94 + 5.40298 
 
 II. 
 
 Coef. of 
 dL. 
 
 = — 372.23 + 5.89945 + 
 
 = — 166.23 + 
 
 0=z— 1041.71 
 
 = + 176.07 
 
 = + 100.43 
 
 = + 16.00 
 
 Coef. of 
 
 Coef. of 
 
 Coef. of 
 
 Coef. of 
 
 Coef. of 
 
 dy. dn. d<p. 
 
 7208.64— 0.04729— 2.22332 — 
 
 2229.06— 0.13790— 0.19504 + 
 
 + 7.03449+ 10.51875 + 
 
 + 9.01520 — 
 
 + 
 
 dn. di. 
 
 0.33807— 0.19932 
 
 0.28590— 0.21945 
 
 0.43566+ 3.12386 
 
 0.49449 — 0.72094 
 
 2.17690— 0.46611 
 
 + 4.24758 
 
 * The errors of equations (10) and (11) given in the Disq. de Elementis Palladis are 
 + 83".01 ; the joint effect of these would increase still further the discrepancy in O. 
 
 ■216".54 and 
 
34 
 
 ON THE USE OF EQUIVALENT NUMBERS 
 
 III. 
 
 Coef. of 
 
 Coef. of 
 
 Coef. of 
 
 Coef. of 
 
 Coef. of 
 
 Coef. of 
 
 
 ilL. 
 
 dy. 
 
 dn. 
 
 d(f. 
 
 da. 
 
 di. 
 
 = — 
 
 371.03 + 5.97864 + 
 
 7296.25 — 
 
 0.06706 — 
 
 2.33932 — 
 
 0.37876 — 
 
 0.08024 
 
 = — 
 
 112.76 + 
 
 2048.68 + 
 
 0.02871 — 
 
 0.44288 + 
 
 0.25090 — 
 
 0.09299 
 
 = _ 
 
 1108.40 
 
 + 
 
 7.18029 + 
 
 11.56325 + 
 
 0.21715 + 
 
 3.07926 
 
 = + 
 
 274.45 
 
 
 + 
 
 9.07933 — 
 
 0.50448 — 
 
 0.43778 
 
 = + 
 
 117.10 
 
 
 
 + 
 
 2.22130 — 
 
 0.43492 
 
 = + 
 
 30.76 
 
 
 
 
 + 
 
 5.33957 
 
 Giving, for the unknown quantities, 
 
 (46.) 
 
 I. (Revised 
 
 solution.) 
 
 11. 
 
 III. 
 
 di =— 7.76 
 
 
 — 3.77 
 
 // 
 — 5.76 
 
 da =— 51.76 
 
 
 — 46.94 
 
 — 53.85 
 
 dq, =— 33.20 
 
 
 — 22.41 
 
 — 33.50 
 
 diT =4-219.19 
 
 
 + 186.17 
 
 + 212.41 
 
 d y = + 0.05401 
 
 
 + 0.08963 
 
 — 0.05115 
 
 dL = — 15.68 
 
 
 — 56.32 
 
 — 14.58 
 
 These results will form the basis for a comparison of the relative probability of the 
 three systems. 
 
 In view of the apparently gross discrepancies of the solutions II. and III. from I., 
 and with the admitted fact before us that the latter reduces perfectly to a minimum the 
 sum of the squares of the residual errors, we should at first sight scarcely be disposed 
 to admit that there is in reality no sensible difference in the accuracy of the three 
 systems. This unfavorable impression must have its origin in the erroneous infer- 
 ence, that, inasmuch as I. is theoretically the best of all solutions, the errors of the 
 results of other systems must be proportional to their deviation from I. 
 
 For instance, in I. we have rfi = — 16", but in II. dL = — 56"; from which 
 is inferred that in II. the error of <^i is proportional, or nearly so, to — 16" -|- 56" 
 = -|-40"; and in a similar way, dI2, dtp, &c. will be condemned or approved by 
 the standard of their disagreement or accordance with the corresponding values 
 in I. 
 
 Such a mistake may, we repeat, be easily committed, in a cursory glance at the num- 
 bers in question. That it is a violation of the fundamental principles of the method 
 of least squares will appear on a very little consideration. 
 
 To apply the criterion proposed in that method for determining the relative proba- 
 
IN THE METHOD OP LEAST SQUARES. 35 
 
 bility of the three systems, we obtain first the residual errors of the original equa- 
 tions as follows : — 
 
 Residual Errors of Original Equations. 
 
 I. (Revised solution.) II. III. 
 
 // // " 
 
 1 — 125.45 
 
 2 — 9.14 
 
 3 + 86.62 
 . 4 — 53.78 
 
 5 + 32.12 
 
 6 + 22.92 
 
 7 + 21.00 
 
 8 + 35.51 
 
 9 + 149.37 
 
 10 — 169.77 
 
 11 + 66.95 
 
 The sums of the squares of these errors are 
 
 I. 
 
 Q = 85091 
 
 The probable value of a residual error for one of the original equations — since there 
 are eleven equations* and six indeterminates — will be obtained from the expression 
 
 u= 0.67459 J — ^— , 
 
 giving the values 
 
 I. II. III. 
 
 (47.) f= ± 88".00 ± 89".77 ± 88".06. 
 
 The agreement of fi in I. and III. is a conclusive proof of the equivalency of the two 
 solutions, notwithstanding the freedom which has been exercised in the choice of fac- 
 tors for the latter. In the case of II., the discrepancy amounts to - of /x ; but it will 
 be noticed that the factors actually employed were based upon a system of representa- 
 tion considerably less exact than the series (2). It might easily be shown that this 
 difference would have been reduced to less than one fifth of its present amount if the 
 numbers for the factors had been selected from (2), agreeably to the conditions by 
 which II. has been defined. 
 
 • Equation 9o having been excluded from each of the solutions. 
 
 — 155.51 
 
 
 — 127.23 
 
 — 3.70 
 
 
 — 7.29 
 
 -f 59.08 
 
 
 + 84.73 
 
 — 47.02 
 
 
 — 54.55 
 
 -f 21.77 
 
 
 + 33.12 
 
 + 6.92 
 
 
 + 19.63 
 
 + 54.52 
 
 
 4- 16.00 
 
 + 35.37 
 
 
 + 38.66 
 
 + 157.61 
 
 
 -f 144.46 
 
 — 155.64 
 
 
 — 174.86 
 
 + 69.63 
 
 1 VO 
 
 
 + 64.28 
 
 II. 
 
 III. 
 
 
 88556 
 
 85205 
 
 
36 ON THE USE OF EQUIVALENT NUMBERS. 
 
 If we compute the limits of these values of /*, we find that it is only an even 
 chance that /*, for the best solution, is comprised within the limits 
 
 ^= ± 69".3, and M = ± 106".7. 
 
 Such is the extreme uncertainty of the only element by which the question of pref- 
 erence between I., II., and III. can be decided. Compared with it, the inconsiderable 
 differences which we find between the values of fi in I., II., and III. will admit of 
 but the single inference, that either of the systems of values of di, dfl, dtp, &c., 
 presented in (46), notwithstanding their great disagreements with each other, actu- 
 ally fulfils the criterion of accuracy proposed in the method of least squares so 
 nearly, that it is impossible to give a decisive reason for adopting one rather than 
 another as the most probable solution. It is worthy of notice, moreover, that the 
 arithmetical mean of the above residual errors, irrespective of their signs, is less 
 in II. and III. than in I. Thus in this instance the latter would rank lowest of 
 the three, if we were to compute the relative probabilities according to a process 
 recommended by the highest authorities* as the most suitable for ordinary use, in 
 which the probable error is directly proportional to the arithmetical mean of the errors 
 irrespective of their signs. 
 
 * Laplace, Theorie Analytique des Probahilitis ; Gauss, Zeitschrift fur Astronomic, B. I. ; Peters, Astr. 
 NacJi., No. 1034. 
 
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