Reducing Observations relating to Several Quantities. 189 



is therefore the required path. Proceeding to B we find 



-5-, for the paths Bs and B5 1; positive ; and for the path BC, 



zero. Also at the point C, all the paths except CB present a 

 positive increment. Hence the most probable solution is any 

 point on the line BO ; and the best* solution is the middle of 

 that line. 



Thus the weighty objection to the new method on account 

 of its "failure to give a unique solution" f is completely 

 removed. It remains to consider the objection that it is not 

 less laborious than the ordinary method. 



Of course the issue cannot be decided on the score of con- 

 venience alone. We must take accuracy also into account. 

 For this purpose we may distinguish two species of cases : 

 (a) those in which the Method of Medians has an advantage 

 in respect of accuracy, and (ft) those where the Probability- 

 curve is presumed to prevail, (a) Suppose that the law of 

 facility is " discordant/ ' made up of two Probability-curves, 

 thus : 



1 _iE? 1 _£? 



V 7rO V 7TC 



By a formula of LaplaceJ, the probable error of the Median 

 is the reciprocal-of-the-Grreatest-ordinate divided by \/2n ; 

 that is, in the case before us, 



2n 2 C + c 



The corresponding error for the Arithmetic Mean prescribed 

 by the Method of Least Squares is the square root of twice 

 the Mean- Square- of-Error, divided by y/n; that is, 



s/2n 

 It is evident that, if C and c be very unequal, the former 

 solution may be ever so much better than the latter. 



(/3) In the ordinary case of laws of facility which are 

 Probability-curves, I should like to express myself with a 

 cautious deference to the opinion of practical astronomers. 

 On the one hand, the probable error is increased by about 

 twenty per cent, when we substitute the Median for the 

 Arithmetic or Linear Mean§. On the other hand, the labour 

 of extracting the former is rather less : especially, I should 



* See note 3 to p. 186. t Mr. H. H. Turner, he. cit. 



% Supplement 2, Theor. Anal. § Ibid. 



