MATHEMATICAL SECTION. 145 
which may be called the average shot, has been recently introduced 
by the United States Ordnance Department, under the name 
“radius of the circle of shots,” in place of the extremely defective 
quantity, the mean absolute deviation, the insufficiency of which 
was pointed out by Henry Metcalfe, Captain of Ordnance, in the 
Report of the Chief of Ordnance of 1882. Thus the adopted 
method of discussion of the precision of firearms, as used by that 
department, is in agreement with Liagre’s theory, only the shots 
are not referred to the true center, but to the “center of shots,” 
viz.: their center of gravity. 
We have, now, the following three quantities, each of which may 
be used as a measure of precision, sighting and leveling being 
equally good. 
1, the even chance shot, p. 
2, the most probable shot, <, (or mean error of sighting and lev- 
eling). 
3, the average shot, 7,, also called radius of the circle of shots; 
and they are related to each other as follows: 
p 2 
2a > ONG pe 
The preceding formule I regard as complete, for practical discus- 
sion of target records, provided there is no evidence for a constant 
vitiating cause. If, for example, during a shooting match the wind 
is blowing constantly in the same direction, the effect of this might 
be partially revealed by computing for the whole match the quan- 
tity : 
= 8 
If the sign of this quantity is consistent with the observed direc- 
_ tion of the wind, it might, perhaps, be proper to refer the shots to 
a new center, to the right or left of the true center, by this quan- 
tity. In that case we have, however, 
he | ne? 
‘ - fa" : —1 i 
In leveling there may be a somewhat constant individual habit 
of holding too high or too low, which, however, ought not to be 
eliminated in a fair discussion of a match, although it would be of 
interest to compute the quantity 
Ly] 
Oo 
n 
for each marksman and for a whole team. 
