in the Calculation of Errors. 103 



at the required point. This calculation being not disturbed 

 by the leverage of a limb stretching to infinity, affords an 

 inferior limit to the true value : say u', determined from the 

 equation 



( io ° i r°° 1 



„ u , V7T J 2 u> V7T 



The root of this equation is found from the Tables to lie 

 between "35 and *4. Concluding, then, that the limit is in 

 the neighbourhood of *4, we ought to reject about two fifths of 

 the observations (that being the proportion of the deviations 

 which fall within a distance of *38 from the centre). 



The gain in accuracy is seen by comparing the modulus of 

 error before correction, viz. 



s/ir ^l—p 2 -r- \/n, 



with the modulus after correction, that of the expression 

 S^-r-S^ integrated between limits co and (say) *4. Each 

 y, as before, being liable to an error whose modulus is 

 Vl—p 2 l2 , we have for the modulus of error incident to /o 12 , 



= Vl-p? 2 x \/n V-572-T-- 



n -'16 



= \/l-P? 2 \/^x-756x^ + ' 16 = Vl^xV^ '89 nearly. 



Thus the modulus of the purified observations is about ten per 

 cent, smaller than the modulus of the whole set. To this slight 

 gain in accuracy is to be added a considerable saving of trouble. 

 As compared with the best possible method, the corrected 

 second-best is very much less troublesome and very little less 



accurate — the modulus of the former being 

 the modulus of the latter V/ - 



(ii.) We have next to consider the method of treating the 

 statistics to which Mr. Galton's tables in the paper already 

 cited lend themselves. For each degree or small dif- 

 ference, e. g. a tenth (of the unit modulus), on the axis x take 

 the mean of the corresponding ?/'s ; and put the latter, divided 

 by the former, as a value of the qucesitum p 12 : e.g.. 



