2 E. L. De Forest—Law of Error in Target-Shooting. 
the simplest, it is often adopted in discussing the errors of target- 
shooting. Even when the form (1) is retained, it is robbed of a 
portion of its generality by assuming that the axes of X and Y are 
respectively horizontal and vertical, instead of being coincident with 
the free axes. Although this assumption seems to have been uni- 
versally made, it has appeared to me to be of doubtful propriety. 
The only reasons I have seen stated for its adoption are, that errors 
caused by the wind are horizontal, while those which depend on the 
range and the force of gravity are vertical. This takes no account 
of errors produced by other causes, such as defects or peculiarities in 
the weapon, imperfect sighting, and fatigue or nervousness in the 
marksman. ‘These may act obliquely, and so far as we know, are as 
likely to occur in one direction as another. 
As a result of accident, it will happen in general, that the centre 
of gravity of the shot-marks does not exactly coincide with the true 
point aimed at, namely, the centre of the target. Accidental devia- 
tions from the centre of gravity, or from the free axes drawn through 
it, are thus of the nature of residual errors, while such deviations 
from the centre of the target, or from axes drawn through it parallel 
to the former ones, are of the nature of true errors. In any given 
case, we can compute the amount of probable deviation of the centre 
of gravity from the centre of the target. If the actual deviation 
falls within this amount, or does not much exceed it, we may pre- 
sume that it is purely accidental, and shifting the position of the 
computed probability surface (1) so as to make its origin coincide 
with the centre of the target while its coordinate axes remain par- 
allel to their former positions, we shall have the law of probability 
of error for future shots. But if the actual deviation is far beyond 
the probable amount, it indicates the probable existence of some 
constant causes of error, likely to affect future shots in the same 
way, and the probability surface must not be shifted, unless we also 
correct the aim of future shots to correspond with it. 
It will in general happen also, as the result of accident, that an 
actual group of shot-marks will be more elongated in one direction 
than in the direction at right angles to it, so that one of the squared 
g.m. errors p,” 9,” will be greater than the other, even when the 
probability of error is really the same in all directions. The con- 
stants , and A, computed by (2) will thus appear to be different, 
and the law of error will seem to be as in (1), when it is really of 
the simpler form (5). But here too, in any given case, we can com- 
pute the amount of probable difference between p,’ and p,’, suppos- 
