Reduction of Observations, 139 



reasoning may be extended to facility-curves not identical. It 

 will be found that the absolutely most advantageous coincides 

 with the absolutely most probable values. This is the weighted 

 arithmetic mean, where the weight of each observation is the 

 inverse mean-square-of-error ; which is also the relatively 

 both most advantageous and most probable value. Thus our 

 four guides, though theoretically and in general divergent, do 

 in a particular but doubtless very extensive case all lead to 

 the same point. 



In general, where neither of the conditions above stated is 

 known to be fulfilled, the absolutely most advantageous, ac- 

 cording to us the supreme end, becomes lost to view. Some 

 indications as to its whereabouts may be suggested by the 

 secondary end, the most probable value, and confirmed by the 

 relative method. Unfortunately not much is likely to be 

 known about the facility-functions on which the calculation 

 of the absolutely most probable value depends ; and the rela- 

 tive method has not been extended, as far as I know, to (the 

 determination of the constants in) other means beside the 

 arithmetical. A few general remarks may here be offered. 



(u) Suppose it is known that the facility-curve for each 

 observation is of the form <j>(w-, c), where c may have any 

 value between -indefinitely wide limits ; then the most probable 

 value is obtained by differentiating Laplace's y f , not only 

 with regard to x f , but also with regard to each constant. 

 (]8) Another general remark is, that the most probable value 

 is not necessarily a continuous function of the observations. 

 For instance, let there be two observations, x± and w 2 , each 



generated by the curve - -. 2 . The most probable value is 



-~r — -, provided that 1 9 2 < 1. But for higher values of 



the distance between the observations, the mean is a position 

 of minimum probability. The (absolutely) most probable 

 value is 



0i+* 2 ± / {x^x 2 y L 



This incident is illustrated by the theorem in the relative 

 method, that in the case supposed* there is no advantage in 

 taking in, including in our arithmetic mean, an additional 

 observation. (7; The preceding example shows that the 

 mean of two observations is not, as has been saidf, the (abso- 



* Phil. Mag. 1883, p. 308. 



t ' Memoirs of the Astronomical Society of London/ xl. p. 92. 



