108 Prof. F. Y. Edgeworth's Exercises 



same as that for the differences between the entries in 

 columns 1 and 2, namely '04. 



To these estimates an addition is to be made in virtue of 

 the error of the species (7) . It will be found that the errors of 

 this species are nearly cancelled in both 1 and 2, but not so 

 in 3 or 4 ; a circumstance which perhaps accounts for the 

 considerable divergence between 3 and 4 as compared with 

 that between 1 and 2. 



The differences actually occurring are 0, *01, "05, '06, 

 •07, -08. 



II. We have next to consider the errors incident to the 

 proportional values of the coefficients a, b, c, f, g, h — the case 

 of three variables being taken as an example. It will be 

 recollected that these proportions are thus obtained. The 



ratios — , -r-, -r are respectively equal to the principal minors, 



■P 1 



and the ratios 4* a j t to the other minors, of the de- 

 terminant 



/ 9 



A' A ? 



— to the other 



A 



1 



P12 P31 



P12 



1 P23 



Pz\ 



?23 1 



Accordingly the first set of results, the proportional coeffi- 

 cients of the squares of the variables, have the least possible 

 relative error when the p's are all zero (the deviations of the 

 organs independent) ; and the greatest possible relative error 

 when the p's are each unity (the correlation a case of simple 

 law unmixed with chance). Contrariwise, the relative error 



of 'jr, -^-, — , the proportional coefficients of the products yz, 



zoe, xy, is greater, the less the coefficients p are. 



The relative error of jr , ~ , &c. is apt to be greater than 



that of the p's from which they are calculated. For, put e 12 , 

 e 2B , &c. as the absolute errors of p 12 , p 2 ^ &c. Then the 

 relative errors of p 12 , p2& &c. are e 12 -i-p 12 , e 23 +p 2d , &c. ; while 



f 

 the relative error of, for example, jr { = pupu — p 2 s) is 



(Pn^nplus p ls e l2 minus e n ) -^ {pi 2 p n ~-p 2z ) ; 



which is made up of three terms, each of which seems as 

 likely as not to be of the same order as e 2 ^-p 2% (it being 

 recollected that all the p's are proper fractions). 



III. We pass to the absolute values of the coefficients by 



