56 
ARTILLERY FIRE. 
so that the centre of the group of points obtained by constantly firing 
at that elevation may be, as near as possible, at the object. 
The case in which the range is not found by a range-finder does not 
differ materially, The artilleryman, in fact, constitutes himself the 
range-finder, and estimates the range to the best of his ability. The 
difference is merely that between a good and an indifferent range¬ 
finder, and the principles explained below apply with equal force to 
both. 
The range having been estimated, by the best means at disposal, 
at r yards, the gun is laid for the range. The point of impact obtained 
may be any one of the scattered group of points that would be ob¬ 
tained if the gun were fired a larger number of times at this range. 
This point, however, is more likely to be that point which will ulti¬ 
mately be the centre of impact than any other point of the group. The 
reason is that in the immediate neighbourhood of the centre of impact 
the points of impact are denser than they are anywhere else. The 
probability that the first shot strikes at the ultimate centre of impact 
is not great, but only greater than the probability of its striking at 
any other point. 
Suppose this first shot to strike yq yards over the object (where ob¬ 
serve that yq may be negative and that the meaning then is that the 
shot strikes so many yards short of the object) so that, the gun being 
laid for a range r, the actual range obtained is x + yq. The centre of 
impact for the gun as laid is more likely to be x + yq yards from the 
gun than at any other point that can be assigned. 
The wisest course, in view of the subsequent shooting, is to lay the 
gun for the second shot for a range of r—yq yards, since, under this 
circumstance, the centre of impact of the gun, as laid for the second 
shot, is more likely to be at the object (x yards from the gun) than at 
any other point that can be assigned along the range. In other words, 
the best chance of obtaining a range of x yards is to lay the gun for a 
range of r — yq yards. 
The second shot is fired, with the above correction, and the result is, 
suppose, that the shot strikes p 2 yards over (observe again that p 2 may 
be negative). A range of x + y> 2 yards has been obtained by laying 
the gun for a range of r — yq yards. Had this second shot been fired 
for a range of r yards we may assume, without sensible error, that the 
range obtained would have been 
x + p { + y> 3 yards. 
Virtually we have now the result of two shots fired for a range of r 
yards and the mean ranges obtained has been 
i [x + yq + x + yq + p 2 ) 
= x + yq + \ y> 2 yards. 
As a result of the firing of the two shots we may say that the centre of 
impact of projectiles fired for a range of r yards is more likely to be 
at a point 
x + yq + i y> 2 yards 
from the gun than at any other point that can be assigned along the 
range. 
