Reducing Observations relating to several Quantities. 469 



ease. It is necessary, however, to select the scale of the dia- 

 gram with some care, so as to prevent confusion when there 

 are many equations. 



If the final solution is roughly known, the origin should be 

 taken in the neighbourhood ; and substitution of this rough 

 solution in the equations will enable us to leave out of con- 

 sideration those with large residuals, the lines representing 

 which will often not fall within the limited area of paper at 

 command. [If it should happen, when the line m crosses I, 

 that 



A + m = B — m-f I, 



the locus does not follow either line, but consists of the whole 

 space between them up to the next crossing. This is the 

 special case referred to above.] 



(2) In the method of least squares the normal equations 

 give a unique solution ; but the intersection of two broken 

 lines may be a series of points, and the two median loci may 

 also have a common portion. The solution then becomes to 

 some extent indeterminate. It is difficult to decide whether 

 this is generally the case ; but in all the simple examples 

 which I have tried, and in two real examples of 93 and 67 

 equations respectively, the intersection of the loci consisted of 

 a finite line and one or more points : and there are also cases 

 where the loci approach very closely, and where they would 

 have again met had one only of the observations been almost 

 infinitesimally different. For instance, in the longest example 

 tried, I took ninety-three equations of the form «£ + Ay = B. 

 B is the tabular error in the semidiameter of Venus, as ob- 

 tained by a certain observer at Greenwich. This is supposed 

 divisible into two parts ; one constant, a:, and the other, y, 

 varying as the diameter A. The values of x and y obtained 

 by the method of least squares were 



^=l // -22 ±0"-222, 



y = 0"-024±0"-011. 



By the method of median loci, I obtain 



^=0^93, y = 0"-037, 



or #=1"-13, y = 0"'032; 



or any value on the line joining these two points. There is 

 also a point of meeting of the loci at 



z=l // '15, y = 0"'029; - 



and a point of near approach at 



a? = l"-37, y=0"-017. 



