Reducing Observations relating to Several Quantities. 185 



The point thus designated must be on each of two, or more, 

 loci analogous to the Normal equations of the ordinary method. 

 Accordingly the intersection of the " Median loci " was at first 

 proposed by me as the solution. But Mr. Turner has shown 

 that these loci are apt to have in common, not only several 

 points, but even lines and spaces*. Instructed by his mas- 

 terly criticism, I now restate the rule as follows ; confining 

 myself, for convenience of enunciation, to the case of two 

 unknown quantities. 



Trace the observation-lines in f he neighbourhood of the true 

 point approximately knownf. Beginning at a point on one 

 of those lines, move continaallyj along one or other of the 



lines in the direction for which - 1 - is least ; R being the sum 



dll 

 of the residuals, each taken positively§ . The value of -=- for 



any path ax + by — v — at any assigned point is thus to be 

 ascertained. Let the sum of the a's (coefficients of x) be- 

 longing to the lines on the right of that point be A ; and, to 



the squares of the residuals may be the least possible. This rule is deri- 

 vable from, and specially correlated with, the hypothesis that the law of 

 facility is the Probability-curve. But it is thought legitimate by Laplace 

 and other eminent authorities to employ the rule even where the hypo- 

 thesis is not assumed. No doubt the use of either method divorced from 

 the law of facility appropriate to it is open to logical objections. But 

 the difficulties are not greater for one method than for the other. The 

 present writer's explanation of the philosophical difficulty common to 

 both methods is stated in the Appendix to a little treatise on the Art of 

 Measurement, entitled 'Metretike' (London: Temple Co., 1887). It is 

 briefly summarized in the ' Cambridge Philosophical Transactions ' for 

 1887. 



* "■ On Mr. Edgeworth's Method of Reducing Observations relating to 

 Several Quantities," by H. H. Turner, M.A., B.Sc, Fellow of Trinity 

 College, Cambridge. Philosophical Magazine, December 1887. 



t We may, as Mr. Turner says, " leave out of consideration those with 

 large residuals." Or, more exactly, those for which the residual, divided by 

 the precision, is large. For the residual ax+by — v represents the perpen- 

 dicular let fall from the points?/, multiplied by the factor \/d 2 + b' 2 ; which 

 factor may be called the precision. That perpendicular, then, might be 

 short, and the line might run into the little field which we have to 

 explore, if, though the residual is large, it is matched by a large coefficient 

 of precision. 



| The true point is in general on an observation-hue. For, as Mr. 

 Turner has pointed out, the Median loci are in general made up of obser- 

 vation-lines. The exceptions to this statement will be noticed presently. 



§ That is, supposing all the v's to be observations of equal worth, 

 ranging under one and the same facility-curve. Otherwise it is proper to 

 multiply each of the residuals entering into R by a factor proportional to 

 the Greatest Ordinate of the corresponding facility-curve (supposed 

 symmetrical). See Laplace, Theor. Analyt. Suppl. 2, subjinem. 



