504 Mr. F. Y. Edge worth on the Determination 



contribution of the remaining contents is a maximum, when 

 they are all massed at a point as remote as possible from 0. 

 that is at one of the corners. The maximum contribution of 

 the residue is then 



i 2 x area of residue = 2i 3 (h —h 1 ). 



Hence, as a superior limit for the whole compartment, we 

 have 2^(Ao"-f^i)- And as a superior limit of the sum of 

 squares of error for the whole curve, 



SOP 2 x h+ Sf i 3 h (inclusive of h ) + fi s (h -J h ). 



This quantity must be divided by Sn, and multiplied by 2, in 

 order to obtain a superior limit for the square of modulus- 

 constant. 



But what we require is not merely a superior limit, but one 

 which is not very much superior. The value which we have 

 found will possess this additional property, provided that the 

 highest of the given compartments (h ) is in the neighbour- 

 hood of the Greatest Ordinate of the real facility- curve with 

 which we are concerned, and that the real Greatest Ordinate 

 is in the neighbourhood of the real Centre of Gravity. For 

 then the assumed centre of gravity will be in the neighbour- 

 hood of the real one. The first condition may be taken for 

 granted. And there is reason to think that the second is very 

 generally fulfilled. It is true of facility-curves which are of 

 the Probability form, and that form is always tending to arise 

 in rerum naturd. It is true* of symmetrical Binomials, 

 which, according to Mr. Galton, are very prevalent. It is 

 true * also of unsymmetrical Binomials — a form which there 

 is some reason for attributing to many of the unsymmetrical 

 facility-curves which occur. For the hypotheses made by 

 Mr. Galton with regard to symmetrical Binomials, mutatis 

 mutandis, may, I think, be applied to the unsymmetrical 

 species. Again, the simple hypothesis by which Quetelet and 

 others have explained the rise of the Probability-curve, or 

 infinite symmetrical Binomial, slightly f modified, will ex- 

 plain the rise of unsymmetrical Binomials. 



So far, then, as Binomials, infinite or finite, symmetrical or 

 unsymmetrical, prevail, our method is serviceable. For ex- 

 ample, it is applicable to anthropometrical statistics ; I have 

 applied it to Dr. Baxter's statistics of human stature, above 



* See note to p. 320 of my paper " On the Law of Error " &c. in 

 the last No. of this Journal. 



f A form of the modified hypothesis ^will be found at p. 317 of my 

 paper on the Law of Error. 



