1885.] Mr Edgeworth, On observations and statistics. 311 



In the case of facility-curves which may be regarded as pro- 

 bability-curves whose modulus is not given, nor known to be 

 identical (cp. Glaisher, loc. cit), the writer recommends a limiting 

 (derived) function of the equation of {n — 1) degree, 



1 + -^ + -L_=o, 



on __ nr* //» ^_ rp 



where a?,, x 2 , &c. are the observations. 



In the general case of facility-curves not probability-curves, 

 it is argued that, though a perfect solution of the problem : What 

 is the best Mean is unattainable (the writer retracting his hasty 

 statement to the contrary in Phil. Mag., Feb. 1884), yet an 

 approximate solution in the form of a weighted arithmetical mean 

 is very generally afforded, if the facility-curves can be regarded 

 as of the form 



a -Mx°+Px i ...+Rx r 



by taking the (r — l) th differential of 



f-{M l (x-x 1 f + M 2 (x-x 2 Y+&c.} 

 1 + P t (x- xj + P 2 (x - x.y + &c. 

 [+ R x {x- xj + B 2 (x- xj -f &c. 

 a 'limiting' function of the equation whose solution is required. 



This method, imperfect as it is, preferred to the method of 

 least squares, which is criticised at length. It is shewn that its 

 fundamental principle is identical with what Mr Todbunter calls 

 "assumed inversion" (Todbunter, Prob., p. 566): namely, that we 

 can test a Mean, or method of reduction, say 6{x v x 2 ...), by putting 

 ourselves as it were at the source of error, taking every set of 

 values such as 



(t) x t , x 2 , x 3 

 (z) x 1 , x 2 , x s 



which as it were emanate from the source according to the (sup- 

 posed known) facility-curves of the observations, forming for each 

 set the mean 6, e.g. 



(1) 6{x v x 2 , cb 3 ....)) 



(2) 6(x;, <, <...)' 



and then observing the divergence of the facility-curve presented 

 by the values of 6. This principle, it is argued, is theoretically 

 incorrect and practically leads to wrong conclusions, e.g. if it were 

 proposed to find the best value of n in the function 



1 l I 



s + x,; + &c. + x' i ) n 



— ^r — J' 



considered as a method of reduction. 



