Reducing Observations relating to several Quantities. 467 



Mr. Edgeworth thus describes his method in the case of 

 two variables x and y : — " Find an approximate solution by 

 some rough process (such as simply adding together several 

 of the equations so as to form two independent simultaneous 

 equations). Take the point thus determined as a new origin, 

 and substitute in the n (transformed) equations for one of the 

 variables x a series of values ±8, ±28, &c. Corresponding 

 to each of these substitutions we have n equations for y. For 

 each of these systems determine the Median according to 

 Laplace's Method of Situation. This series of Medians forms 

 one locus for the sought point. A second locus is found by 

 transposing x and y in the directions just given. The inter- 

 section of these loci is the required point." 



Some of the labour of this process, and sometimes the pre- 

 liminary search for an approximate solution, may be avoided 

 by the use of a graphical method, which will be best described 

 by considering first a simple case. Suppose we have five 

 equations, 



x + '01 y = a ly 



x + '01?/ = a 2 , 



x + y = b ly 



X + y = b. 2 , 



x + y = b s . 



Geometrically these represent two lines nearly parallel to the 

 axis of y, and three inclined at 45° to it. 



Now in forming the first normal equation according to the 

 Method of Least Squares, we should multiply each of the 

 equations by the coefficient of x, which is unity in each case. 

 In Mr. Edgeworth's method we are to find the median line, 

 weighting all the equations according to the coefficient of x, 

 i. e. equally. This is evidently the broken line A B C D E F; 

 for an ordinate drawn through any point of it cuts the system 

 of five lines in five points, one of which is on the locus 

 A B C D E F and two others are on either side. 



It is obviously very easy to' draw this locus when once we 

 have any portion of it ; for we simply traverse the network, 

 changing our line at every corner. 



The second median locus is obtained by weighting the 

 equations, or lines according to the coefficients of y. The 

 first two count for very little, and the last three again count 

 equally. The median is thus the line K L throughout, for the 

 crossing of the slightly weighted lines does not disturb the 

 balance of weights. 



We are now to take the point of intersection of these loci 

 2 12 



