500 ON THE ESTIMATION OF 



which arises from the element of luck mixing itself in the 

 competition. This it does the more, the fewer the shots 

 allowed to each, nor can it be eliminated, so as to make 

 skill the sole determining power, but on the average of 

 very enormous numbers, such as, for instance, ten or 

 twenty thousand arrows discharged by each marksman. 

 Every shooter, of course, aims to the best of his ability, 

 exclusively with a view to hit the centre of the gold ; 

 nor is it conceivable that, having that intention, there 

 should exist in any individual such specialty of aiming 

 as should disperse his shots, failing the gold, so as to 

 strike preferentially (say) the blue, rather than the red 

 ring on one side of it and the black on the other. 



(5.) The following table shows the respective num- 

 bers of "hits" per 1000 shots, which may be expected 

 to occur on a calculation from our formulae, within the 

 several coloured areas of the five equidistant rings (con- 

 sidering the central gold as the first ring) into which an 

 ordinary target is divided. Considering its diameter as 

 divided into ten equal parts, the outside diameters of 

 those rings will be respectively 2, 4, 6, 8, 10; their radii, 

 i, 2, 3, 4, 5 ; and the areas of their containing circles, 

 in the proportion of the squares of these numbers, i, 

 4, 9, 16, 25, so that the areas of the several coloured 

 spaces form the progression i, 3, 5, 7, 9. The usual 

 rule of valuation, then, which accords to hits in any of 

 the rings (from the white inwards), values in the propor- 

 tion of these numbers ; assumes the probability of hitting 

 to be in the 'simple proportion of the area struck (as 

 would be the case were the shooting entirely at random), 



