140 Mr. F. Y. Edgeworth on the 



lutely) most probable value, in the complete sense of that term. 

 It need not be in any sense. It may be a position of minimum 

 probability for all values of the distances between the obser- 

 vations. The most probable value afforded by two or more 

 observations may be, not any intermediate point, but any one 

 of the observations. This incident is illustrated by the relative 

 theorem that for some species of facility-curves we obtain a 

 worse result by taking in additional observations. For in- 

 stance, if the facility-curve* is of the form 



h C™-J l 7 



-I e cos ethos doc 



7T J 



(a not changing sign), the multiplication of observations di- 

 minishes probability and advantage if t is less than unity ; the 

 weightedf arithmetic mean ^ 1 ^ 1 +#2^2 + & c -~ H S# is the posi- 

 tion of minimum advantage and probability. (8) Generally 

 speaking, the absolutely most advantageous value is not out- 

 side %, though it may be coincident with, the outmost observa- 

 tions ; and is most likely to be in the neighbourhood of those 

 observations which have most weight — which flow from sources 

 known by previous experience or by deduction to yield obser- 

 vations least divergent from the real§ value. 



I propose now, in the light of this additional theory, to 

 review the practical rules which were stated in a former essay. 

 Adopting the classification there given ||, and beginning with 

 symmetrical curves, we see that the solution of species (1) in 

 Class I. is not only, as universally admitted, the absolutely 

 most probable value, but also, according to Laplace's reasoning, 

 the absolutely most advantageous. Species (1) of Class II. 

 is now raised to the same level of certainty. Where Laplace's 

 method of " observations non faites encore" is applicable to 

 this case, there his method of " observations deja faites " is 

 also applicable. The necessity of an initial step in the dark 

 under which the Method of Least Squares has hitherto laboured 

 is done away with. Species (3) of both classes is, I think, 

 raised to the same level of certainty. The solution of species 

 (4) which I have suggested^ has not perhaps the highest 

 certainty of the absolutely most advantageous value, but such 

 certainty as the most probable value affords**; that is as much 



* ( Memoirs of the Astronomical Society of London,' xl. p. 307. 



t The weights are the inverse mean powers t~t—l. 



X Supposing the function of detriment symmetrical. 



§ Or, more generally, what is believed to be the most advantageous 

 value — to cover the hyperphysical problems alluded to at the end of this 

 paper. 



|| Phil. Mag. 1883, p. 365. If Ibid. p. 370. 



** Above remark (a). 



