5o6 ON THE ESTIMATION OF 



(lo.) The same principles apply of course equally to 

 rifle-shooting as to archery, provided the target aimed at 

 be circular. If rectangular, and especially if an elon- 

 gated rectangle, the same formulae will not apply ; and 

 the appropriate formulae would be necessarily much more 

 complex and their results proportionably more difficult 

 of calculation. This is a strong argument for the use 

 of circular targets : for, though for the mere decision of 

 the order of merit in a distribution of prizes almost any 

 impartial rule, rough and readily applicable, may suffice, 

 the same cannot be said when the object is to obtain a 

 true numerical measure of the national skill in the use 

 of that great weapon for which purpose it is highly 

 desirable that the data afforded by our rifle prize meet- 

 ings should be preserved, collected, and reduced syste- 

 matically. 



NOTE. 



Demonstration of the formula in (2.) and {^^.^ 



The probability of committing the specific error r [all errors pre- 

 senting equal facility for their commission^ is proportional to 

 E ( kr"^)-, the characteristic sign E being used to denote the expo- 

 nential or anti-logarithmic function ; and k being some certain con- 

 stant to be determined or eliminated. And in the case of aiming at 

 the central point of a circular target, the degree of facility afforded 

 for the commission of a lineal error r, no matter in what direction, 

 is proportional to 27rr, the circumference of a circle of that radius, 

 or, simply to r : so that the probability of planting a shot some- 

 where on the circtimference of that circle is measured by r. E (kr^), 

 and therefore the probability of making a hit anywhere within its 

 area is proportional iofrdr. E {kr'-) taken between the limits 

 o and r. Representing certainty therefore by i ; this probability 

 (which we have denoted by H in the foregoing pages) will be ex- 



