ARTILLERY FIRE. 
57 
The third shot should, therefore, be fired for a range of 
r - Pi “ 2 Pz yards, 
since, then, the centre of impact of shots so fired 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. 
Thus the correction for the third shot is 
- 2 Pi yards, 
which is, numerically, \ of the actual error of the second shot. 
The third shot is fired with this correction (that is for a range of 
r ~~ Pi ~ 2 P% yards) and, suppose, it strikes p 3 yards over. We may 
assume, as before, that if it had been fired for a range of r yards the 
shot would have struck 
Pi + hp* + Ps yards over, 
or, the same thing, that the range obtained would have been 
x + Pi + hpz + yards. 
Thus, virtually, we have before us the results of three shots fired for 
the range r; the mean of these is 
i {x + pi + & -1- jPi + jp 2 + x + £>i + \ y> 2 + p 3 ) yards 
= X +Pi + hp*+ hp 3 yards. 
The point which is at this distance from the gun is now more likely 
to be the centre of impact of shots fired for the range r than is any 
other point that can be assigned along the range. 
Accordingly the fourth shot should be fired for a range 
r - Pi - iPi ~ iPa yards, 
or, the same thing, the correction for the fourth shot should be 
— iPs yards, 
which is, numerically, -J- of the actual error of the third shot. 
Proceeding in this way it is not difficult to see that the correction 
for the ti th shot should be 
-- ^—rPn-i yards, 
n — 1 
which is, 
numerically, 
1 
n — 1 
of the actual error of the (n — l) th shot. 
To establish the law, assume it to hold for the first n — 1 shots. 
Virtually we have before us the results of n — 1 shots fired for a 
range r. 
The mean of these has been 
Vj | (* + px) + (* + Pi + p 2 ) + (* + Pi + £ Pt + j 
+ (« + Pi + ipz + ip 3 + Pi) +. 
. . . . + (® + Pi + \p 2 + . . + ^~Pn-i + 14-i) | yards. 
8 
