SKILL IN TARGET-SHOOTING. 497 



progression as the square of the error increases in arith- 

 metical." Now, it is perfectly true that the deviation 

 of the point of incidence from the mark is error. But 

 it is something more special. It is error in that one 

 particnlar direction vn which the point of incidence lies 

 from the mark aimed at. In estimating, therefore, the 

 probability of striking a target at a certain definite dis- 

 tance from the centre aimed at, we must multiply the prob- 

 ability of striking a determinate point at that distance 

 from the centre, by the number of points within the 

 extent of the target which actually do lie at that distance 

 from it, without regard to the directions in which they 

 lie : i.e.^ we have to multiply the fractional number ex- 

 pressing the abstract probability of committing a given 

 error out of an indefinite xivcixCo^x o{ eqiuilly possible oxit's,^ 

 by a number proportional to the degree of opportunity 

 which the circumstances of the special case afford for 

 its commission. In this case that degree of opportunity 

 is evidently measured by, and proportional to, the length 

 of the circumference of the circle about whose centre, 

 at the distance specified, an arrow may strike, or a ball 

 drop from a height. 



(2.) Reasoning on this (the correct principle in the 

 case of target-shooting), any one conversant with mathe- 

 matical analysis will find no difficulty in arriving at tlie 

 following singularly neat and simple formula for the p7-ob- 

 ability of missi?ig, in any one single shot, a circular area 

 of given radius (r), at whose centre the shooter aims.* 



* The demonstration of this formula is annexed in the form of a 

 note at the end of this essay. 



2 I 



