102 Prof. F. Y. Edgewortfrs Exercises 



weighted mean ceases to become smaller by the addition of a 

 new observation ; that is, when 



begins to be greater than 



Expanding, and neglecting terms of an inferior order, we 

 have the equation above written. 



To apply this principle to the case before us, we may, 

 without loss of generality, regard the given values of x 

 as ranged on one branch of a probability-curve from zero 

 to infinity. Each x divided into the associated y gives a 

 value for p 12 of which the true weight is x 2 and the weight 

 used in the method under consideration is x. Putting p 2 = x 2 

 and q 2 = x in the general formula, we have for the limiting 

 condition : 



x m+ i x Sftf begins to be greater than 2# m+ i x Sfl, 

 or 



x 



»i+i 



Dins to be less than hS?w-i-n. 



'O 



Now, as we move outwards from the centre towards 

 infinity, the right-hand member of the above inequality 



converges to a constant — -=, while the left-hand member 



V7T 



increases indefinitely. There is therefore no upper limit. 



To determine the lower limit, putting u for x m+ i, find a 

 point on the #-axis of a probability-curve (of unit modulus), 

 distant u from the origin such that w = half the distance from 

 the origin of the centre-of-gravity of the area which is inter- 

 cepted by the axis of abscissse, the ordinate at the point u, 

 and the curve. In symbols, 



u — \ I xe~ x2 dx-r- j e~ x2 dx, 



ux4:xe +u2 x) e~ x2 dx = l. 



Taking logarithms, and using the second of the tables 

 appended to Be Morgan's " Calculus of Probabilities " (Encye. 

 Metrop. vol. ii.), I find a value for u between *4 and '45 ; 

 which is presumably the solution. 



But, as it may be suspected that this result is exaggerated 

 by the prolongation of the curve to infinity in theory, though 

 not in fact, I have verified the solution by substituting for the 

 centre of gravity the Median of the tract outside the ordinate 



